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[ "First application of combined isochronous and Schottky mass spectrometry: Half-lives of fully-ionized 49 Cr 24+ and 53 Fe 26+ atoms", "First application of combined isochronous and Schottky mass spectrometry: Half-lives of fully-ionized 49 Cr 24+ and 53 Fe 26+ atoms" ]	[ "X L Tu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n\nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n", "X C Chen \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "J T Zhang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n\nSchool of Nuclear Science and Technology\nLanzhou University\n730000LanzhouPeople's Republic of China\n", "P Shuai \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "K Yue \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "X Xu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "C Y Fu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "Q Zeng \nSchool of Nuclear Science and Engineering\nEast China University of Technology\nNanChang 330013People's Republic of China\n", "X Zhou \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "Y M Xing \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "J X Wu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "R S Mao \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "L J Mao ", "K H Fang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n\nSchool of Nuclear Science and Technology\nLanzhou University\n730000LanzhouPeople's Republic of China\n", "Z Y Sun \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "M Wang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "J C Yang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "Yu A Litvinov \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n\nGSI Helmholtzzentrum für Schwerionenforschung\nPlanckstraße 164291DarmstadtGermany\n", "K Blaum \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany\n", "Y H Zhang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "Y J Yuan \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "X W Ma \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "X H Zhou \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n", "H S Xu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China\n" ]	[ "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "School of Nuclear Science and Technology\nLanzhou University\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "School of Nuclear Science and Engineering\nEast China University of Technology\nNanChang 330013People's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "School of Nuclear Science and Technology\nLanzhou University\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "GSI Helmholtzzentrum für Schwerionenforschung\nPlanckstraße 164291DarmstadtGermany", "Max-Planck-Institut für Kernphysik\nSaupfercheckweg 169117HeidelbergGermany", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China", "Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouPeople's Republic of China" ]	[]	Lifetime measurements of β -decaying highly charged ions have been performed in the storage ring CSRe by applying the isochronous Schottky mass spectrometry. The fully ionized 49 Cr and 53 Fe ions were produced in projectile fragmentation of 58 Ni primary beam and were stored in the CSRe tuned into the isochronous ion-optical mode. The new resonant Schottky detector was applied to monitor the intensities of stored uncooled 49 Cr 24+ and 53 Fe 26+ ions. The extracted half-lives T 1/2 ( 49 Cr 24+ ) = 44.0(27) min and T 1/2 ( 53 Fe 26+ ) = 8.47(19) min are in excellent agreement with the literature half-life values corrected for the disabled electron capture branchings. This is an important proof-of-principle step towards realizing the simultaneous mass and lifetime measurements on exotic nuclei at the future storage ring facilities. 29.20.db	10.1103/physrevc.97.014321	[ "https://arxiv.org/pdf/1804.02653v1.pdf" ]	119,472,654	1804.02653	6a167881fac1687668d1dd4a9cd85bf4b841465a	First application of combined isochronous and Schottky mass spectrometry: Half-lives of fully-ionized 49 Cr 24+ and 53 Fe 26+ atoms 8 Apr 2018 X L Tu Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany X C Chen Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China J T Zhang Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China School of Nuclear Science and Technology Lanzhou University 730000LanzhouPeople's Republic of China P Shuai Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China K Yue Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China X Xu Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China C Y Fu Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China Q Zeng School of Nuclear Science and Engineering East China University of Technology NanChang 330013People's Republic of China X Zhou Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China Y M Xing Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China J X Wu Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China R S Mao Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China L J Mao K H Fang Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China School of Nuclear Science and Technology Lanzhou University 730000LanzhouPeople's Republic of China Z Y Sun Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China M Wang Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China J C Yang Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China Yu A Litvinov Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China GSI Helmholtzzentrum für Schwerionenforschung Planckstraße 164291DarmstadtGermany K Blaum Max-Planck-Institut für Kernphysik Saupfercheckweg 169117HeidelbergGermany Y H Zhang Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China Y J Yuan Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China X W Ma Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China X H Zhou Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China H S Xu Institute of Modern Physics Chinese Academy of Sciences 730000LanzhouPeople's Republic of China First application of combined isochronous and Schottky mass spectrometry: Half-lives of fully-ionized 49 Cr 24+ and 53 Fe 26+ atoms 8 Apr 2018(Dated: April 10, 2018)PACS numbers: 2340-s, 2920db Lifetime measurements of β -decaying highly charged ions have been performed in the storage ring CSRe by applying the isochronous Schottky mass spectrometry. The fully ionized 49 Cr and 53 Fe ions were produced in projectile fragmentation of 58 Ni primary beam and were stored in the CSRe tuned into the isochronous ion-optical mode. The new resonant Schottky detector was applied to monitor the intensities of stored uncooled 49 Cr 24+ and 53 Fe 26+ ions. The extracted half-lives T 1/2 ( 49 Cr 24+ ) = 44.0(27) min and T 1/2 ( 53 Fe 26+ ) = 8.47(19) min are in excellent agreement with the literature half-life values corrected for the disabled electron capture branchings. This is an important proof-of-principle step towards realizing the simultaneous mass and lifetime measurements on exotic nuclei at the future storage ring facilities. 29.20.db I. INTRODUCTION The half-life is a fundamental property of an atomic nucleus. The knowledge of nuclear half-lives is important for understanding nuclear structure and in nuclear astrophysics [1]. Even though the half-life of a neutral atom is almost independent of external physical and chemical conditions [2], the modification of the number of orbital electrons in the atom has a great effect on its half-life [3,4]. Furthermore, since atoms are highly ionized in hot stellar environments, the β decay rates of highly charged ions (HCI) are worth being investigated in order to accurately determine the time scales and pathways of stellar nucleosynthesis processes [5]. To experimentally address half-lives of HCIs, one needs to be able to produce exotic nuclides in the required high atomic charge states and to preserve the ions in these states for sufficiently long time. The development of heavy-ion cooler storage rings coupled to in-flight fragmentation facilities offers such a possibility. Highly-charged radioactive ions are produced at relativistic energies in a production target, which simultaneously acts as an electron stripping target. Owing to the ultra-high vacuum conditions of a storage ring, the produced ions of interest can be stored in the defined atomic state. The storage time is defined by the radioactive decay on the one side and by the unavoidable beam losses in the ring on the other side. Apart from first half-life measurements of HCIs in Electron-Beam Ion Traps [6], extensive investigations of decays of HCIs were done only in the heavy-ion storage ring ESR at GSI in Darmstadt [1,7]. There, a non-destructive monitoring of the number of stored ions in the ring is done * Electronic address: [email protected] by employing the technique of time-resolved Schottky Mass Spectrometry (SMS) [8]. Numerous highlight results were obtained and reported over the 25 years of running the ESR such as the investigations of bound-state beta decay [9][10][11] or "modulated" orbital electron capture decays [12,13]. All these experiments were performed with electron cooled ion beams. Besides wellknown advantages of electron cooling there are several disadvantages if half-life measurements are considered. Firstly, the cooling takes time, ranging from a few seconds to a few minutes, depending on the momentum distribution of particles and its offset to the velocity of electrons in the cooler. To some extend, this can be improved by implementing the stochastic pre-cooling of the particles before applying the electron cooling [14,15]. Secondly, the recombination of stored ions with cooler electrons is the main atomic beam loss mechanism in the storage ring [1]. Last but not least, the electron cooling damps the amplitudes of betatron oscillations, which prevents two different, electron-cooled ions with very close mass-overcharge ratios to pass each other in the ring. This effect can cause systematic errors in the determination of the number of stored particles [16]. Isochronous mass spectrometry (IMS) was developed at the ESR to address masses of nuclei with half-lives shorter than the cooling time [17][18][19]. No electron cooling is applied in the IMS thus avoiding the disadvantages listed above. However, only time-of-flight (TOF) detectors [20,21], based on the recording of secondary electrons from thin foils penetrated by the stored ions, could be used in the rings. Such detectors destroy the stored beam within a few hundred revolutions, corresponding to a storage time of about a millisecond. Still, the half-lives of short-lived isomeric states could be addressed with IMS with TOF detectors [22,23]. Owing to the development of high-sensitivity non-destructive Schottky detectors [24][25][26], which are capable to detect single stored heavy highly-charged ions within a few ten milliseconds, it became attractive to develop an isochronous Schottky mass spectrometry with time-resolved Schottky detection technique for halflife measurements of HCIs. Furthermore, this is a basis for half-life measurements of exotic nuclides foreseen at the future storage ring projects at FAIR in Darmstadt and HIAF in Huizhou [27,28]. We note, that within the later projects further developments of Schottky detectors to include positionsensitivity has been proposed [29,30]. Schottky mass spectrometry in the isochronously tuned storage ring has been shown to work in the ESR [19,31]. In this work we present the first application of IMS+SMS technique to half-life measurements of highly-charged radionuclides. II. EXPERIMENT The operation of the Cooler-Storage Ring at the Heavy Ion Research Facility in Lanzhou (HIRFL-CSR) enables a new opportunity for β decay lifetime measurements of HCIs. The HIRFL-CSR is quite similar to the high-energy part of the GSI facility, consisting of the heavy-ion synchrotron SIS, the inflight fragment separator FRS, and the experimental storage ring ESR. At HIRFL-CSR, the main storage ring (CSRm), used as a heavy-ion synchrotron, is connected to an experimental storage ring (CSRe) via a projectile fragment separator (RIBLL2) [32]. With these facilities, HCIs like bare, hydrogen-like (H-like) and helium-like (He-like) ions can be produced by fragmenting primary beams extracted from CSRm at a sufficiently high energy. The produced fragments are separated by the RI-BLL2 and transferred and injected into the CSRe for experiments. The CSRe has already been running for about ten years [33]. Many masses of short-lived nuclei have been precisely measured by applying the IMS technique at ESR and CSRe [18,19,[34][35][36][37][38]. The experiment reported here was performed in the context of isochronous mass measurement on 52 Co at CSRe, where RIBLL2 and CSRe were set to a fixed magnetic rigidity of Bρ = 5.8574 Tm [38]. The details of the standard IMS measurements at CSRe can be found in Refs. [36][37][38]. The 58 Ni 19+ beam accumulated and accelerated in CSRm was fast extracted and focused upon a 15 mm-thick beryllium production target placed at the entrance of RIBLL2. The beam intensity was about 10 8 particles per spill. According to the CHARGE calculations [39], at this kinetic energy more than 99.9% of all fragments emerged the target as fully-stripped atoms. These HCIs were separated by RIBLL2, which was operated as a pure magnetic rigidity analyzer, and then injected into the CSRe. The fragments produced in the projectile fragmentation reaction show a quite broad velocity distribution. The nuclides with different mass-to-charge ratios (m/q), e.g. nuclides of interest and calibration nuclides having different velocities, can simultaneously be stored in the CSRe. This allows for an access to the properties of a series of nuclides simultaneously. Compared to the SMS with electron cooling the mass resolving power is reduced from about 10 6 to 10 5 [40]. An energy of 430.8 MeV/u for the primary 58 Ni beam was chosen to optimize the transmission and storage of the A = 2Z + 1 and A = 2Z + 2 nuclides, where A and Z are the atomic mass and the proton numbers, respectively. A new, highly-sensitive Schottky resonator was manufactured by GSI [24]. It has been installed at one of the straight section of the CSRe [41], as indicated in figure 1. The principle of the resonator is similar to the one of an antenna. The stored ions were circulating in the CSRe with revolution frequencies of about 1.6 MHz and at each revolution they excited the resonator. The signal from the Schottky resonator was analyzed with a commercial real-time spectrum analyzer Tektronix RSA5100A. Typically the frequencies around the 150 th harmonic of the revolution frequency were analyzed. The IQ (in-phase and quadrature phase) data in the time domain were acquired. The sampling frequency was set to 3.125 MS/s for IQ. The RSA5100A was triggered with a period of 5 s by a logical signal. Each acquired data file contains in total 409600 samples, corresponding to about 131 milliseconds of recording time. In order to obtain Schottky frequency spectra for the A = 2Z + 1 and A = 2Z + 2 nuclides simultaneously, a span of 2.5 MHz centered at 243.85 MHz was chosen in the data acquisition. Duration of each measurement between two adjacent injections into the CSRe was about 30 min. III. DATA ANALYSIS Schottky frequency spectra were obtained by Fast Fourier Transform (FFT) of the IQ data. 4096 IQ data were used to produce one frequency frame. In order to improve the signalto-noise characteristics, 600 subsequent frequency frames obtained from 6 subsequently acquired data files were averaged to produce a final frequency spectrum. One part of the final frequency spectrum is shown in figure 2. The resonance response of the Schottky detector was obtained from data where no ions were stored and was subtracted from the spectrum. In the isochronous ion-optical mode of the CSRe, the energy of the stored ions of interest is chosen such that γ ≈ γ t = 1.4, where γ is the relativistic Lorentz factor and γ t denotes the transition point of the storage ring [17]. The velocity spreads of ions, which are due to the nuclear reaction process, are in first order compensated by the lengths of orbits in the CSRe and can be neglected. Hence, the revolution frequencies ( f ) reflect directly the m/q ratios of the stored ions, see Ref. [17] for more details: ∆ f f = − 1 γ 2 t ∆(m/q) (m/q) .(1) Particle identification of the frequency peaks in the Schottky frequency spectrum can be done by comparing it with a reference spectrum [37]. We note, that without cooling, the mean velocities for different ion species are different. This leads to the situation, see figure 2, that two different series of nuclides from two different harmonics (154 th for A = 2Z + 1, 156 th for A = 2Z + 2) are present in the same observation frequency window. The basic property of Schottky signals is that the area under a frequency peak is proportional to the number of stored ions [1], S ∝ 2N[k( f )Qe f ] 2 ,(2) where S is the area, N and Q are the number and charge state of stored ions, respectively. The functionk( f ) represents the sensitivity of the Schottky resonator at different frequencies, f . In our analysis, this factork( f ) is treated as almost constant for a narrow frequency span. Once β decay happens, the m/q ratio changes and the area under the frequency peak will decrease, which is the cornerstone of the lifetime measurements [4,9,42,43]. The RIBLL2 was operated purely as a magnetic rigidity analyzer. The advantage is that the nuclides of interest with A = 2Z + 1 and the calibration nuclides with A = 2Z + 2 could be stored simultaneously, as it is the case for in figure 2. The range of mass-to-charge ratios is about 2%. Taking into account that averaging of 600 frequency frames was done for each frequency spectrum, the statistical uncer-tainty of the amplitude at a given frequency point is about 4% [4,44]. The areas under the frequency peaks were extracted by means of Gaussian fitting. The normalized peak areas of 49 Cr-50 Cr and 53 Fe-54 Fe as a function of the time elapsed since injection are plotted in figure 3. The β decay constant (λ β ) was extracted by fitting the ratios (R=S aim /S cali. ) of the peak areas with a single exponential function: R = S aim 0 exp(−λ t T ) S cali. 0 exp(−λ s T ) = S aim 0 exp[−(λ s + λ β )T ] S cali. 0 exp(−λ s T ) ,(3) where S aim and S cali. are the peak areas of aimed nuclide and calibration nuclide, respectively. T is the time elapsed since injection. The total decay constant, λ t , is the sum of the beam loss constant, λ s , and β decay constant, λ β . The bare 50 Cr and 54 Fe ions are stable, which means that their β decay constants are equal to zero. The beam loss constants depend typically only on Z and follow a quadratic form for different nuclei [42]. Please note, that the main loss mechanism in Ref. [42] was the recombination in the electron cooler, which is disabled in our case. (19) The λ β of 49 Cr determined in this work is about 0.0116(7) min −1 in the laboratory frame. Taking into account the Lorentz factor γ = 1.362, deduced from the magnetic rigidity of CSRe, T 1/2 ( 49 Cr 24+ ) = 44.0(27) min in the rest frame. The half-life for neutral 49 Cr is 42.3(1) min and the β + branching ratio I β = 92.8% [45]. Thus, the estimated half-life for bare 49 Cr is 45.6(1) min, assuming that the rest 7.2% entirely proceed via orbital electron capture decay. The peak of 53 Fe in the ground state contains feeding from the isomer due to isomeric transition. The half-life for the bare 53 Fe isomer is 2.48(5) min [42]. Therefore, we fit the data for 53 Fe only starting from time T > 9 min, when about 85% of the isomers have already decayed. The λ β of 53 Fe determined in this work is 0.0601(14) min −1 in the laboratory frame. The Lorentz factor γ is 1.363 leading to T 1/2 ( 53 Fe 26+ ) = 8.47 (19) min in the rest frame. The half-life for neutral 53 Fe is 8.51(2) min and the β + branching ratio I β = 97.04% [46]. Thus, the half-life of 8.77(2) min is expected for bare 53 Fe. The experimental half-life for bare 53 Fe measured at GSI is 8.5(3) min [42]. The differences of β decay Fermi functions between neutral and bare ions are not considered in the estimations of half-lives [42]. The half-lives obtained in this work are in excellent agreement with both theoretical and available experimental values. The latter indicates that the IMS+SMS technique can be used to study β decay lifetimes of HCIs. IV. SUMMARY AND OUTLOOK Beta decay half-lives of highly charged ions were measured in the CSRe storage ring tuned into the isochronous ionoptical mode by applying the non-destructive time-resolved Schottky mass spectrometry technique. The half-lives of bare 49 Cr and 53 Fe nuclei were determined. The results are in excellent agreement with the literature values corrected for the disabled decay channels involving electrons. The results demonstrate the capability of such isochronous Schottky mass spectrometry to lifetime studies of HCIs, which is an important proof-of-principle step towards realizing the simultaneous mass and lifetime measurements on exotic nuclei at the future storage ring facilities FAIR in Germany and HIAF in China. Concerning the near-term future, half-life measurements at CSRe will be continued with the investigation of the orbital electron capture decay of bare, H-like, and He-like 111 Sn ions [47]. The half-life for neutral 111 Sn is 35.3(6) min and the β branching ratio I β = 30.2% [48]. It is expected that, due to the conservation of the total angular momentum of the nucleus-lepton system, the hyperfine ground state of H-like 111 Sn ions can not decay via electron capture. The highly charged 111 Sn ions have already been produced and stored at the CSRe. Compared to the results for 49 Cr and 53 Fe from this work, the charge state of 111 Sn is much higher which will cause a better signal-to-noise characteristics of the Schottky signal. The stable primary beam 112 Sn in relevant atomic charge states will be used to calibrate the beam loss constants. FIG. 1 : 1The heavy ion storage ring CSRe. The Schottky resonator (see photo in the insert) is installed in a straight section of CSRe. FIG. 3 : 3Peak areas normalized with the initial area (S 0 ) as a function of the time elapsed since the injection of 49 Cr-50 Cr and 53 Fe-54 Fe ions into the CSRe. The error bars represent statistical uncertainties of the peak areas extracted from the corresponding Gaussian fits. The solid lines represent fits with single exponential function. The data for 53 Fe is fitted only starting from time T > 9 min, when the isomers have mostly decayed. For details see text. TABLE I : IMeasured half-lives of bare 49 Cr and 53 Fe nuclides (T CSRe 1/2 = ln(2)/λ β /γ). Listed are the half-lives of neutral atoms (T 1/2 ), theoretical half-lives of bare ions (T cal 1/2 ), and the experimental half-lives of bare ions (T ESR 1/2 ) from previous ESR measurements [42]. Nucl. T 1/2 /min T cal 1/2 /min T ESR 1/2 /min T CSRe 1/2 /min Neutral Bare Bare Bare 49 Cr 42.3(1) 45.6(1) 44.0(27) 53 Fe 8.51(2) 8.77(2) 8.5(3) 8.47 The authors thank the IMP accelerator team for the excellent support. 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[ "Glass Transition and Lack of Equipartition in a Statistical Mechanics model for Random Lasers", "Glass Transition and Lack of Equipartition in a Statistical Mechanics model for Random Lasers" ]	[ "G Gradenigo \nNANOTEC CNR, Soft and Living Matter Lab\nPiazzale A. Moro 2I-00185Roma, RomaItaly\n\nDipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly\n", "F Antenucci \nDipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly\n\nInstitut de Physique Théorique\nCEA\nUniversité Paris-Saclay\nF-91191Gif-sur-YvetteFrance\n", "L Leuzzi \nNANOTEC CNR, Soft and Living Matter Lab\nPiazzale A. Moro 2I-00185Roma, RomaItaly\n\nDipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly\n" ]	[ "NANOTEC CNR, Soft and Living Matter Lab\nPiazzale A. Moro 2I-00185Roma, RomaItaly", "Dipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly", "Dipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly", "Institut de Physique Théorique\nCEA\nUniversité Paris-Saclay\nF-91191Gif-sur-YvetteFrance", "NANOTEC CNR, Soft and Living Matter Lab\nPiazzale A. Moro 2I-00185Roma, RomaItaly", "Dipartimento di Fisica\nUniversità di Roma \"Sapienza,\" Piazzale A\nMoro 2I-00185RomaItaly" ]	[]	Recent experiments suggest that the onset of lasing in optically active disordered media is related to an ergodicity-breaking transition for the degrees of freedom of the electromagnetic field. We test this hypothesis in numerical simulations of the dynamics of nonlinearly coupled light modes under external pumping. The collective behavior of light mode amplitudes appears to be akin to the one displayed in glass formers around the ergodicity breaking glass transition: a critical pumping exists, beyond which the thermodynamic phase is fragmented into a multitude of states. The probability distribution of the overlap between such states, i.e., the glass order parameter, turns out to be well described by the replica symmetry breaking scheme. The unprecedented observation is that such symmetry breaking occurs at the same pumping power values at which a lack of equipartition among light modes arises. Finally, we show that the mean-field scenario for the glass transition is quite robust for the description of the physics of random lasers. * [email protected] 1 This is the dominant term in absence of non-centrometric, non-time translational invariant phenomena like double harmonic generation.That is, when χ (2) contributions to non-linear optical susceptibility are negligible with respect to χ (3) . arXiv:1902.00111v1 [cond-mat.stat-mech]	null	[ "https://arxiv.org/pdf/1902.00111v1.pdf" ]	91,184,381	1902.00111	dac5dc308a7e98ce78e6fdf2ce127342b5fa303e	Glass Transition and Lack of Equipartition in a Statistical Mechanics model for Random Lasers 31 Jan 2019 G Gradenigo NANOTEC CNR, Soft and Living Matter Lab Piazzale A. Moro 2I-00185Roma, RomaItaly Dipartimento di Fisica Università di Roma "Sapienza," Piazzale A Moro 2I-00185RomaItaly F Antenucci Dipartimento di Fisica Università di Roma "Sapienza," Piazzale A Moro 2I-00185RomaItaly Institut de Physique Théorique CEA Université Paris-Saclay F-91191Gif-sur-YvetteFrance L Leuzzi NANOTEC CNR, Soft and Living Matter Lab Piazzale A. Moro 2I-00185Roma, RomaItaly Dipartimento di Fisica Università di Roma "Sapienza," Piazzale A Moro 2I-00185RomaItaly Glass Transition and Lack of Equipartition in a Statistical Mechanics model for Random Lasers 31 Jan 2019 Recent experiments suggest that the onset of lasing in optically active disordered media is related to an ergodicity-breaking transition for the degrees of freedom of the electromagnetic field. We test this hypothesis in numerical simulations of the dynamics of nonlinearly coupled light modes under external pumping. The collective behavior of light mode amplitudes appears to be akin to the one displayed in glass formers around the ergodicity breaking glass transition: a critical pumping exists, beyond which the thermodynamic phase is fragmented into a multitude of states. The probability distribution of the overlap between such states, i.e., the glass order parameter, turns out to be well described by the replica symmetry breaking scheme. The unprecedented observation is that such symmetry breaking occurs at the same pumping power values at which a lack of equipartition among light modes arises. Finally, we show that the mean-field scenario for the glass transition is quite robust for the description of the physics of random lasers. * [email protected] 1 This is the dominant term in absence of non-centrometric, non-time translational invariant phenomena like double harmonic generation.That is, when χ (2) contributions to non-linear optical susceptibility are negligible with respect to χ (3) . arXiv:1902.00111v1 [cond-mat.stat-mech] Recent experiments on optically-active disordered media, displaying random lasing above a given pumping threshold, have provided the evidence of particularly non-trivial correlations between shot-to-shot fluctuations [1][2][3][4][5][6][7]. A statistical analysis of the distribution of such fluctuations shows that, crossing the lasing threshold, their variance has a fast accelerating increase with the power. A further probe of the distribution of the similarities of pairs of fluctuations in different shots, termed intensity fluctuation overlaps (IFO's), has shown that in such compounds strong correlations among fluctuations are compatible with an organization of emission mode configurations in clusters of states, akin to the one occurring for the multitude of thermodynamic states composing the glassy phase in glass formers. Such a correspondence has been theoretically explained proving the equivalence between the distribution of IFO's and the distribution of the overlap between states, the so-called Parisi overlap, that is the order parameter of the phase transition in glassy systems. The analytical proof has been, though, derived assuming very narrow-band spectra, such that all modes can be considered at the same frequency. This is not the case, however, for many realistic multimode lasers, both ordered and random. In particular, a non-linear feature of standard multimode lasers -providing short pulses -is mode-locking. A necessary condition for mode-locking to occur is that a matching condition is satisfied among the frequencies of sets of coupled modes. E.g., in the four-wave mixing case 1 theory predicts that given four waves of indices i 1 , i 2 , i 3 , i 4 they can be actually coupled if and only if their frequencies satisfy \|ω i1 −ω i2 +ω i3 −ω i4 \| < γ; γ being the typical line-width of a mode. It has been recently experimentally demonstrated that mode-locking occurs in the GaAs powder random laser [8]. The above observations make the investigation of multimode models of random lasers a necessary step to understand the fundamental mechanisms at the ground of the fascinating phenomenon of lasing in random media. The first goal of this work is to undertake such investigation by means of numerical simulations and bridge the (random) laser phenomenology to the one of critical phenomena in glassy systems. In particular, the question we want to answer is whether the narrowing of the emission spectrum experimentally observed at the lasing threshold and the related crossover to coherent light emission are due to an underlying phase transition. In panel a) of Fig. 1 we show an instance of spectra of ZnO nanoparticles in a rhodamine 640 dye solution, taken from [9]. We will show that the phase transition scenario is correct: a glass transition takes place for the degrees for freedom of the electromagnetic field in the optically active disordered medium. Another goal of the present work is to bridge the phenomenology of ergodicity breaking in systems with coupling frustration and in non-linear systems with ordered interactions, as for instance the famous Fermi-Pasta-Ulam (FPU) model [10], a one dimensional chain of classical particles coupled by non-linear springs. For the FPU model the first numerical simulation ever performed, in Los Alamos in 1955, showed that, taking an initial condition far from equilibrium, there is a non-ergodic regime persisting on very long time-scales. Such a regime is typically revealed as a persistent lack of equipartition between the fundamental degrees of freedom of the system [11][12][13][14][15][16]. This kind of ergodicity breaking phenomenology, which is also typical of non-linear systems displaying breathers or solitonic excitations [17][18][19], was never observed, to our knowledge, in systems with disordered interactions in their glassy phase. The present work is the first one focused on a model where the lack of equipartition between the fundamental degrees of freedom and the non-trivial features of the Parisi overlap distribution appear at the same time, strongly suggesting that these are just two complementary way to detect the same underlying phenomenon: the breaking of ergodicity. This observation is supported by our numerical data on a statistical mechanics model for non-linear waves interaction in random media, the mode-locked 4-phasor model. These kind of models with random couplings, pertaining to the set of p-spin models, are not usually studied in numerical simulations, because of their high computational cost, but are used to carry out analytical computation. In that context they are never studied in inhomogeneous interaction networks, such as the one produced by the frequency matching condition among light mode frequencies, which on the contrary characterizes the mode-locked 4-phasor model studied here. In the random laser case the 4-phasor model emerges naturally to describe mode dynamics in the stationary regime [20][21][22][23][24]. Since the model variables -complex continuous -can change their magnitude, unlike Ising, XY or Heisenberg spins, one can contemporarily probe energy equipartition among degrees of freedom and the occurrence of a phase transition. This is, thus, the first unprecedented example of a model where lack of equipartition and state fragmentation at the critical point appear as concomitant effects. RESULTS Model We study a statistical mechanics model for the non-linear interactions of the electromagnetic field in a disordered optically active medium: the mode-locked 4-phasor model. The phasors, dynamic variables of the system, are the complex amplitudes a k (t) = A k (t) e iφ k (t) of the electromagnetic field expansion in normal modes E(r, t) = N k=1 a k (t) e iω k t E k (r) + c.c.. Being A k = \|a k \| ∈ R + and φ k = arg a k ∈ [0, 2π], the dynamics of the stationary regime can be shown to be a stochastic potential dynamics whose Hamiltonian reads [21,25]: H[a] = − ı J ı A i1 A i2 A i3 A i4 cos(φ i1 − φ i2 + φ i3 − φ i4 ),(1) where the coefficients J ı={i1,i2,i3,i4} are quenched random variables (we will take them Gaussian) and the ı specifies a particular light mode network. We also implement gain saturation into the model, that can be formally rephrased into a constraint on the total intensity: N = N k=1 A 2 k [20,24]. Despite the fact that energy is continuously injected and dissipated within a random laser, according to [20,21,24,25] one can assume an effective equilibrium distribution for the amplitudes: P (a 1 , . . . , a k ) = e −βH[a] δ N − N k=1 A 2 k ,(2) where β is some effective inverse temperature and measures the optical power per mode available to the system. Rescaling A k → A k / √ in Eq. (2), the new variables are constrained on the same hypersphere at the cost of a rescaling of the effective temperature as β → β 2 = P 2(3) where P is the so-called pumping rate parameter. In these rescaled variables the Boltzmann weight reads ρ[{a}] = 1 Z exp{−P 2 H[a]} , making explicit the role of the pumping as effective heat bath for the stationary regimes of the lasing random media. We have investigated how the system behaves varying the pumping rate P. According to Eq. (2) and Eq. (3), in numerical simulations it is identical to fix the constraint and change the temperature T or work at fixed temperature varying the value of . We have done our simulations varying the temperature T , in order to leave a clear term of comparison with the literature on glassy systems (see Methods for the numerical algorithm) but we will often discuss our results in terms of pumping rate P. The reader has just to bear always in mind that P ∼ 1/ √ T : high pumping rates correspond to low temperatures and vice versa. Our goal is to study what happens beyond the narrow-band mean-field approximation of [21,[25][26][27]. In particular, we consider the situation where the non-linear interactions in Eq. (1) are chosen according to a selection rule which depends on mode frequencies, the so-called Frequency Matching Condition (FMC) [28]: \|ω i1 − ω i2 + ω i3 − ω i4 \| γ,(4) where γ is the typical line-width. The FMC constraint introduce in the topology of the interaction network inhomogeneities such that standard mean-field approximations used solve the thermodynamics of disordered systems [29,30] are not exact. Let us consider for instance the simple case of a linear dispersion relation with equispaced angular frequencies, ω j = ω 0 + j δ, with δ γ and i = 1, . . . , N . Eq. (4) very simply reads as the constraint \|i 1 − i 2 + i 3 − i 4 \| = 0 on the summation indices in Eq. (1), diluting by a factor N the number of interactions from the fully connected case, [31]. We call the resulting interaction network the mode-locked graph, whose detailed discussion can be found in the corresponding section in Methods. N (N − 1)(N − 2)(N − 3)/24 Observables The assumption that the stationary probability distribution of light mode amplitudes is an effective equilibrium one [see Eq. 2] allows us to use an equilibrim Monte Carlo dymamics to sample phase space. To speed up the equilibration we use the Parallel Tempering algorithm [32]. To show that the crossover from fluorescence to random lasing is an ergodicity breaking transition, we have measured standard order parameters to detect ergodicity breaking. The first, and simplest conceptually, is the spectral entropy: S sp = − N i=1Î k log(Î k )(5)I k = A 2 k N k=1 A 2 k ,(6) measuring the energy degree of equipartition in the spectrum,Î k being the normalized thermal average of the intensity for given wave number k. The angular parentheses denote the thermodynamic average (see Methods). The information content of the spectral entropy can be conveniently expressed by the effective number of degrees of freedom, introduced in the framework of nonlinear systems [11,13]: n eff = exp (S sp ) N ,(7) where n eff = 1 for perfect equipartition and n eff = 0 when the total energy is concentrated on a finite fraction of modes. Values n eff < 1 signal the breaking of equipartition [11,13]. We also measured the order parameter introduced to detect ergodicity breaking in the context of disordered systems thermodynamics, the Parisi's overlap [21,22,33,34]: q αβ = 1 N N k=1ā α k a β k = 1 N N k=1 A α k A β k cos(φ α k − φ β k ).(8) The greek indices in Eq. (8) denote different replicas, i.e., independent configurations at equilibrium at the same temperature T . The Parisi's overlap is characterized by a low temperature non-trivial distribution in presence of a glass phase [29,33]. We have, further, introduced the overlap between the energy stored on different interactions, i.e., the plaquette overlap: Q αβ = 1 N 4 N4 i=1 P α i P β i ,(9) where P i is the plaquette degree of freedom, Intensity spectrum I λ = A 2 λ as a function of the wavelength λ averaged over many istances of the quenched randomness, numerical simulations with N = 64 degrees of freedom. The wavelength λ = 2π/k is expressed here in arbitrary units ([a.u.]). Notice the narrowing of the spectrum at larger values of the pumping rate P. Panel c): Intensity spectrum I λ as a function of the wavelength λ for a single istance of the quenched randomness, numerical simulations with N = 64 degrees of freedom. Color code of the curves: P increases from bright (yellow) to dark (purple). Notice the crossover from a smooth, almost equipartited, spectrum at low P to a disordered pattern of isolated peaks at high P. and where N 4 is the number of four-body couplings (plaquettes) appearing in the definition of energy in Eq. (1). We have studied the two probability distributions P (q) and P(Q). P i = A i1 A i2 A i3 A i4 cos(φ i1 − φ i2 + φ i3 − φ i4 ),(10) As is clear from their definitions in Eqns. (9) and (10), in order to measure P (q) and P(Q) one needs to know the phases of the electromagnetic modes [35], which are experimentally not retrieved so far in random lasers [8,[36][37][38]. For this reason it was introduced and measured in [1,27] breaking order parameter for complex waves amplitudes which do not depend on phases: C αβ = 1 N N k=1 ∆ α k ∆ β k (11) ∆ α k ≡ A α k − A α k 2 √ 2(12) where the indices α and β label independent equilibrium configurations. From the point of view of an experiment the idea is to consider, for a given experimental setup (quenched randomness) and a fixed value P of the pumping rate, the average value A α k for each line of the spectrum and then study the shot-to-shot, i.e., the replica-to-replica, correlation between fluctuations around this average value. We have, thus, numerically measured the distribution P (C αβ ). All the above mentioned distributions have been characterized by measuring their Binder parameter B = (3 − κ)/2 and their bimodality parameter b = (γ 2 + 1)/κ, where κ and γ are respectively kurtosis and skewness (more details in Methods). Breaking of equipartition We first characterize the behaviour of the intensity spectrum vs the pumping rate P. As reported in panel a) of Fig. 1 for one instance of experimental measurements [39], a narrowing of the spectrum is commonly observed as P increases. The spectra computed in our numerical simulations, shown in panel b) of Fig. 1, display the same behaviour. Besides the possibility of tuning at our will the control parameters of the system, one of the great advantages of numerical simulations is the possibility to look at the spectrum dependence on the temperature for a single instance of disordered couplings. In panel c) of Fig. 1 we see that increasing P a crossover occurs from an almost equipartited regime to a regime where energy is concentrated on some peaks arranged in a disordered pattern. Data from a single instances of disorder, thus, suggest that the spectral narrowing accompanying the onset of random lasing in experiments [9,37,39] has to be related to a breaking of equipartition. The degree of equipartition is characterized by the effective number of degrees of freedom n eff [see Eq. (7)], a parameter introduced in the FPU literature to detect detect non-ergodicity through the lack of equipartition [11,13,15]. In panel a) of Fig. 2 we show the behaviour of n eff , the average over different istances of the disordered couplings, as a function of the pumping rate P. We plot data as function of the squared inverse pumping rate, that is, the temperature T = P −2 . At low pumping rate (high T ) we find a good degree of equipartition between electromagnetic field modes, n eff ≈ 1, for all the sizes N studied. By increasing the pumping rate (decreasing T ) n eff drops to lower values at a breaking point which slightly depends on N . We can roughly identify this breaking point as the lasing threshold P c (N ) = T c (N ) for the pumping rate. By looking at the curves in panel a) of Fig. 2 it is clear that the decrease of n eff is the steeper the larger is N : in the framework of critical phenomena this is the first indication that the crossover to non-equipartition is a first-order transition. A stronger evidence of the first-order nature of the transition comes from the study of the Binder parameter B and of the bimodality parameter b of the distribution P (n eff ) varying P, shown respectively in panel b) and panel c) of Fig. 2. The value P c (N ) estimated as the point where B signals a maximal deviation from Gaussianity corresponds to the breaking point of n eff , as well as to the peak of the bimodality indicator b. The last observation confirms that the deviation from Gaussianity is due to a bimodal nature of the n eff distribution, thus confirming the first-order transition scenario [40]. We believe that to reproduce such an analysis in experiments would be of primary interest. Finally, in order to emphasize the difference in the degree of equipartion at different values of P, in panel d) of fig. 2 spectra obtained for different istances of the random coupligs are compared at low pumping rate, panel d1), and high pumping rate, panel d2). As we already mentioned, in the context of non-linear systems the lack of equipartition is usually regarded as a landmark of ergodicity breaking [11][12][13][14][15][16], often also related to the apparence of "localized objects" as breathers or solitons [17][18][19]. But for that class of systems there is no example showing that the lack of equipartition is also related to a non trivial distribution of the overlap order parameter introduced by Parisi 40 years ago to characterize ergodicity breaking in disordered systems [29]. The goal of the next section is to show that, on the contrary, for the mode-locked 4-phasors model this is precisely what happens: the lack of equipartition takes place together with, following the terminology of [29], the breaking of the symmetry between replicas. Glass transition As promised, we now investigate to which extent the narrowing of the emission spectrum, see Fig. 1, is related to a glass transition. More precisely, to the so-called Random First-Order Transtion (RFOT) [41][42][43][44]: a mixed-order transition with no latent heat but with a discontinuous order parameter. This is the overlap distribution P (q) changing from a single peaked distribution to a bimodal one at the critical point, a behaviour typical of first-order transitions. In the high T ergodic phase the distribution P (q) is peaked at the origin q 0 = 0. Below the glass transition temperature P (q) develops a secondary peak at a finite distance q 1 > q 0 from the origin [45]. This behaviour is typical when phase space splits into disjoint ergodic components. The overlap of configurations inside the same ergodic component is q 1 while the mutual overlap of configurations belonging to different ergodic components is q 0 . Let us show that the numerical results on the mode-locked 4-phasor model are in complete agreement with this scenario. In panels a) and b) of Fig. 3 we display the specific-heat per mode C V N = E 2 − E 2 N T 2 vs T = 1/P 2 for different system sizes. The overbar denotes the average over disordered couplings realizations, whereas the angular parentheses denote the thermodynamic average (see Methods). Besides the numerical data for the model on the mode-locked graph we also report, as a comparison to standard RFOT systems, numerical data obtained for another similar system, the random diluted 4-phasor model. The latter is defined by the same variables and interactions of the mode-locked 4-phasor model, but lives on a different graph, characterized by the same number of non-zero interactions and the same scaling with N as the mode-locked one, but obtained by randomly diluting a fully connected graph in which each mode interacts in every possible set of four modes (see Methods for details). At all sizes C V /N has a characteristic non-monotonic behaviour, with a peak at a T that depends on N . We identify this point as a finite size "critical temperature" T c (N ) = 1/P 2 c . The good collapse of the curves at different Fig. 3, demonstrates critical scaling. In panel a) the scaling form for the mode-locked p-phasor model turns out to be C V = N 3/2 h(τ N 3/2 ), where h(y) is a universal (size-independent) function and τ ≡ T /T c − 1 is the distance from the critical point in adimensional units. The scaling regime shrinks with size as τ ∼ N −3/2 . The scaling exponent 1/ν = 3/2 deviates remarkably from mean-field [46], which would predict 1/ν = 1/2 for a theory of second-order transition with quartic non-linearity. As a comparison, in panel b) of Fig. 3 the curves of the randomly diluted p-phasor model (with the same amount of dilution of the mode-locked one) are displayed. The scaling regime of the specific heat is, here, τ ∼ N −1/2 , in agreement with a simple mean-field model of the glass transition [41]. The strong deviation from mean-field scaling in the mode-locking case is due to the correlated way in which the interaction network is diluted and mainly depends on the fact that, because of FMC, Eq. (4), modes whose frequencies are at the center of the spectrum tend to interact more than modes whose frequencies are at the boundaries. Though this effect is expected to vanish in the thermodynamic limit N → ∞, this very strong -pre-asymptotic -finite size effects reflect typical features of random laser compounds, such as the spectral narrowing with P. In panel c) and d) of Fig. 3 we display the distribution of the modes overlap P (q), respectively for the randomly diluted 4-phasor model [panel c)] and the mode-locked model [panel d)], across the critical scaling region. For both models the probability distribution P (q) is a Gaussian centered on the origin at low pumping rate P. In the case of random dilution, above the critical value P c (N ) one finds secondary peaks at a finite distance from the origin: they are the signature of the broken ergodicity phase. In the mode-locked graph, on the contrary, at high pumping rates we just find two pronounced shoulders at identical system size, i.e., with the same number N of modes. Although shoulders at small N might develop in peaks at larger N , it would hard to demonstrate the presence of an ergodicity-broken phase solely on the basis of the data in panel d) of Fig. 3. In mean-field models the information content of variable and interaction overlaps is theoretically the same, though finite size corrections may depend on the observble [47]. Therefore, we consider, as well, the distribution P(Q) of plaquette overlap, Eq. (9). This, indeed, turns out to display a much clearer signal of the broken ergodicity phase: data are shown in panel a) of Fig. 4. At low P the distribution is symmetric around the origin whereas in proximity of the transition a secondary peak arises at q 1 > 0. In the inset of panel a) of Fig. 4 the corresponding multimodality parameter b is shown (see also Methods). The region where the parameter b signals a bimodal distribution of the overlap is precisely the interval of pumping rates around P c . The Intensity Fluctuation Overlap We have presented the numerical evidence that in a statistical mechanics model for random lasers there is a RFOT glass transition concomitant with the lack of equipartition. One problem is to which extent this picture can be assessed even in experiments. The available technology for the measurements of light mode phases [35,36] applies only to high-power directional impulses: unfortunately this is not the operating regime of standard random lasers [37,38]. We cannot therefore rely on observables which require the measurement of phases. The modes and plaquette overlaps defined Eqns. (8) and (9) depend on phases, so they are not useful for current experiments. A further intensity fluctuation overlap C αβ , cf. Eq. (11), can be introduced, that -at the mean-field level -is proved to be in a one-to-one correspondence with the standard overlap, C αβ ∝ q 2 αβ , ∀ α, β [27] and can be measured in real random lasers [1][2][3][4][5][6][7]. In panel b) of Fig. 4 is shown the distribution P(C) for N = 64 at four different values of the pumping rate P, two above and two below the critical P c , as determined from the caloric curve (Fig. 2). At first sight there is no clear evidence of secondary peaks at high pumping rates for this system size, although non-Gaussian tails appear in the vicinity of the transition. It is the study of the Binder parameter B dependence on P which reveals how P(C) brings the signature of a first-order transition: data are shown in the inset of panel b) of Fig. 4. The behaviour of B as a function of P (plotted as B vs P −2 = T to have a clearer term of comparison with the literature) is the one characteristic of first-order transitions [40]. This behaviour of the IFO order parameter is, therefore, also consistent with the RFOT scenario and with the study of the breaking of equipartition in the spectrum: in all the three cases the ergodicity-breaking parameter behaves as the order parameter of a first-order transition. CONCLUSIONS By means of Monte Carlo numerical simulations of a statistical mechanical model of the non-linear interactions of light in a random medium, we have shown that the onset of random lasing has the properties of a glassy phase transition. More precisely it is an ergodicity-breaking transition characterized by a diverging second order susceptibility (the specific-heat) and an order parameter, the overlap, with first-order features at the transition. As weird as it can sound, light multiply scattering in random media really looks like, among physical systems, a very good benchmark to test the existence and the hidden nature of a glass transition. Further on, thanks to this feature and to the properties of the model adopted, we were able to show unprecedented evidence of a deep connection between lack of equipartition -typical of ergodicity breaking in non-disordered systems with non-linear interactions [10,11,15] -and the breaking of ergodicity as described within the paradigm of replicasymmetry-breaking [30]. The mode-locked 4-phasor model is the first example of a system where ergodicity breaking manifest itself at the same time as a breaking of the symmetries between replicas and as a lack of equipartition. This important result suggest a possible way to overcome the intrinsic difficulty usually encountered in the measure of q, which is not a single-experiment observable. The standard protocol is that one needs to compare the results of several experiments done on the same sample or of several numerical simulations with the same realization of quenched disorder to obtain a measure P (q). Using the jergon of disordered systems one needs more replicas of the same system. As we showed, when the occurrence of a non-trivial P (q) is simultaneous with the loss of spectral equipartition, one can simply take advantage of latter to detect the ergodicity breaking glass transition. METHODS The Mode-Locked graph The first step of the numerical study is the generation of the mode-locked graph. It is easier to describe the structure of the interaction network as a bipartite graph where interaction nodes J µ labeled by greek letters are connected to variables nodes A k labeled with latin letters. Since we have a four-body interaction each interaction node is always attached to 4 variable nodes and is defined by the ordered list of their indices J µ (i, j, k, l). The order is relevant because from the point of view of the energy stored in the interaction (see Eq. 1) there are non-equivalent permutations. The steps to generate the mode-locked graph are as follows: 1. a virtual complete graph with N !/(N − 4)!4! interaction nodes is generated; 2. for each interaction node the three non equivalent permutations are considered: P µ (i 1 , i 2 , i 3 , i 4 ), P µ (i 2 , i 1 , i 3 , i 4 ) and P µ (i 1 , i 2 , i 4 , i 3 ); each time a non-equivalent permutation satisfies the FMC, the corresponding interaction of the virtual graph is added to the real graph. 3. The procedure at point 2 is repeated until a preassigned number of interactions in the complete graph is reached, for computational reasons this number is always a power of 2. For large N , the above procedure tends to cut O(N ) of all interacting quadruplets [31]. Operatively, for a system with N complex variables we have drawn a bipartite graph with a number of interactions scaling as O(N 3 ) and equal to the power of 2 soon smaller than the number of all possible interactions fulfilling the FMC constraint. The number of interaction nodes N 4 corresponding to each N is listed in Tab. I. Concerning the structure of the topology of the interaction network it is important to stress that the dimensionality of the disordered optical medium in real space is scarcely important for the thermodynamics of the problem: the interaction among modes remains in any case highly non-local in the basis of normal modes. The only quantity which depends on the real-space dimensionality is the spatial overlap between the normals modes, the information about which is stored in the disordered coefficients J ı of the Hamiltonian in Eq. (1) as [21,26] J i1i2i3i4 = i 2 ω i1 ω i2 ω i3 ω i4 V dr χ (3) α (ω i1 , ω i2 , ω i3 , ω i4 ; r) E α1 i1 (r) E α2 i2 (r) E α3 i3 (r) E α4 i4 (r).(13) where χ (3) α is the non-linear susceptibility of the system. 2 Without any loss of generality we assumed such couplings to be independent Gaussian random variables with zero mean and variance J 2 ∼ N −2 , which guarantees energy extensivity. In random lasers couplings as expressed in Eq. (13) are, in general, disordered because modes display different spatial shape and extension [48,49]. The constituents of the integrals in Eq. (13) are very difficult to calculate from first principles. The only specific form of the non-linear susceptibility has been computed by Lamb [50,51] for few-modes ordered lasers and no analogue study for RLs has been performed so far, to our knowledge. Integrals like Eq. (13) in a random medium can be regarded as a sum over many random variables. Different couplings involving a given mode might, in general, be correlated [52]. Since, however, we are interested in the critical behavior, thus in the large size limit of our simulated systems and since correlations decay with the size of the system, we adopt as working hypothesis a Gaussian distribution for each J ı : P (J ı ) = N 2 2π exp − N 2 J 2 ı 2(14) We further stress that, from the perspective of probing RFOT, considering correlated J's leads to qualitatively analogue phase diagram as it is well known in spin-glass systems such us, e.g., the Random Orthogonal Model [53,54]. If we look at interactions in the space of normal modes the system is infinite-dimensional: any degree of freedom participates to O(N 2 ) interactions. That is why we expect the mean-field glass transition scenario drawn in [21,27] to be quite robust for the mode-locked p-phasor, even if the hypothesis of narrow-band is removed. Last but not least, the non-locality of interactions between light modes also guarantees that phenomena like energy localization, a pathology of sparse networks [28,55], are avoided. Numerical Algorithm We have studied systems with N complex variables a k = \|A k \|e iφ k interacting with Hamiltonian in Eq. (1). The sampling of the probability distribution in Eq. (2) was done by means of a Parallel Tempering Monte Carlo algorithm (PT). In the PT algorithm one runs M simulations with local Metropolis dynamics for identical replicas of the same system, i.e. with the same quenched disorder J ı . For each of the M copies the equilibrium distribution e −βiE is sampled with a local Metropolis algorithm. We have runned K independent simulations at temperatures T i = β −1 i , from T 0 = T min to T N P T = T max . Each 50 steps of the local Metropolis algorithm an exchange of configurations among simulations running at neighbouring temperatures is proposed. That is, if the Metropolis algorith for a i runs with β i and that of a j with β j one tries the following attempt: {(a i , β i ); (a j , β j )} =⇒ (a j , β i ); (a i , β j ), which is accepted with probability p swap : p swap = min 1, e − The replica exchange update is proposed sequentially for all pairs of neighbouring temperatures β i and β i+1 . For all simulations the N P T temperatures where taken with a linear spacing in β, i.e. β i+1 = β i + ∆β. In the local Metropolis algorithm the configuration of complex "spins" (a 1 , . . . , a N ) is updated with attention to keep k \|a k \| 2 = const. In order to fulfill this constraint each update is realized chosing at random two spins a i = A i e iφi and a j = A j e iφj and proposing un update to a i and a j such that: \|a i \| 2 + \|a j \| 2 = \|a i \| 2 + \|a j \| 2(16) This is simply achieved by extracting three random numbers A i , A j and θ with uniform probability in the interval [0, 2π] and proposing the four simultaneous updates A i → A i , A j → A j , A i → A cos(θ) and A j → A sin(θ), with A 2 = A 2 i + A 2 j . For the parallel tempering we used 32 temperatures for all sizes. We have simulated systems at four different sizes, N = 32, 48, 64, 102. The update of the the variables A i must be done sequentially, since the interaction network is dense and there is no way to partition the variables in subsets which can be updated independently. We have implemented parallelization on GPU graphic cards at two levels. In order to accept or reject the update of two spins a i and a j one has to compute the energy update on O(N 2 ) quadruplettes, i.e. ∆E = Ñ k=1 ∆E k , with N = O(N 2 ): the calculation of all the ∆E i is realized exploiting the parallelism of GPU. The execution of the M simulations at different temperatures is implemented in parallel on the GPU. We have runned simulations on two type of graphic cards: GT X − 680, K − 20. The best overall speedup achieved with respect to the sequential code on CPU is of a factor 8. Binder and Bimodality parameters The quantitative indicators used to characterize the order parameter distributions are the Binder parameter B and the multimodality parameter b. Both these indicators are buildt from cumulants of the distribution. In particolar one needs the definition of curtosis κ, κ = (∆q) 4 (∆q) 2 2 ,(17) and skewness γ, γ = (∆q) 3 (∆q) 2 3/2 ,(18) where ∆q = q − q . The angle parantheses denote thermal average while the overline denotes average over disorder realizations. The Binder parameter is B = (3 − κ)/2 and the multimodality parameter reads as b = (γ 2 + 1)/κ. In the case of the distribution obtained from a sample with n data, which is the case considered here, the definition of the multimodality parameter is b = γ 2 + 1 κ + 3(n−1) 2 (n−2)(n−3) . (19) Replicas and Overlap In the analytic calculation of disordered systems free energy one needs to consider the analytic continuation to non-integer values of replicas number, n → 0 [29]. Nevertheless, to make a sense of replicas, one has to think to integer values of n. Replicas are independent equilibrium configurations sampled with the same quenched disorder. This definition corresponds to the protocol used in numerical simulation. One "replica" of the system is represented by the swarm of N P T configurations used for a given instance of the Parallel Tempering MC dynamics. Different istances of the PT dynamics characterized by the same set of quenched couplings J ı and the same interaction network between modes are different replicas. For N = 32, N = 48 and N = 64 for each disorder instance we simulated 4 replicas, which gave us the availability of 6 independent values of q αβ : q 12 , q 13 , q 14 , q 23 , q 24 , q 34 . For N = 102 we simulated two replicas for each instance of the disorder. To accumulate statistics for P (q) we measured values of q αβ comparing replicas at the same iteration of the PT dynamics, each 500 iterations. Since the distribution P (q) is not self averaging [30], for each size of the system we have sampled the equilibrium measure for N sample ≈ 100 istances of the disorder. It is useful to clarify also how we measured in practice all thermal averages indicated with angular brackets, O[A] . Thermal averages were measured as "time" averages along the PT dynamics, along the second half of each run: O[A] = 2 N sweep Nsweep i=Nsweep/2 O[A i ].(20) Figure 9 .FIG. 1 . 91Spectra of emission from the rhodamine 640 dye solution containing ZnO nanoparticles. The ZnO particle density is ∼6 × 10 11 cm −3 . The incident pump pulse energy is (from bottom to top) 0.74, 1.35, 1.7, 2.25 and 3.4 µJ. mode. A further increase of optical gain leads to lasing in more low-loss modes. Laser ion from these modes gives discrete peaks in the emission spectrum (figure 9). hen the scattering strength increases further, the decay rates of the eigenmodes and the ing among them continue decreasing. There are a small number of eigenmodes with ely long lifetime and nearly decoupled from other modes. The threshold gain for lasing Panel a): Emission spectrum of ZnO nanoparticles in a rhodamine 640 dye solution. From bottom to top: increasing injected optical power P. The emission spectrum of rhodamine shown here is a reproduction of the original Fig. 9 appearing in [H. Cao, Waves Random Media 13, R1 (2003)]. Panel b): FIG. 2 . 2the so-called Intensity Fluctuation Overlap (IFO), an ergodicity Panel a): Effective number of degrees of freedom n eff (disorder average) as a function of the inverse (squared) pumping rate P −2 = T , corresponding to the temperature in the numerical algorithm. n eff ≈ 1 signals equipartion, n eff < 1 lack of equipartition. Panel b): Binder parameter B for the probability distribution of n eff , measured as a function of P −2 . Panel c): Bimodality parameter b for P (n eff ) measured as a function of P −2 . Panel d1) & d2): Spectra obtained from different instances of the disorder are shown for the equipartited phase, (Pc/P) 2 ≈ 2, in d1, and the non-equipartited phase, (Pc/P) 2 ≈ 0.5, in d2. FIG. 3 . 3Panel a) and Panel b): Specific heat CV (T ) = E 2 − E 2 /T 2 as a function of T ; different curves represent different sizes of the system. Panel a): 4-phasor model on the mode-locked graph. Inset: Specific heat as a function of τ N 3/2 , where τ = T /Tc(N ) − 1, curve collapse in the scaling region. The four sizes are N = 32, 48, 64, 102. Panel b): 4-phasor model on the randomly diluted graph. Inset: Specific heat as a function of τ N 1/2 , curve collapse in the scaling region. The four sizes are N = 32, 48, 64, 96. Panel c): Random Dilution. Normal modes overlap probability distribution for system size N = 64 at temperatures T /Tc(N ) = 1.71, 1.15, 0.98, 0.7. Panel d): mode-locked graph. Normal modes overlap probability distribution for N = 64 at temperatures T /Tc = 1.71, 1.15, 0.97, 0.64. N , shown in the inset of both panel a) and b) of FIG. 4 . 4Panel a) Plaquettes overlap distribution P(Q) for N = 64 and N4 = 2 14 . Inset: Multimodality parameter b measured for P(Q), values above the threshold b * = 5/9 (full black line) indicate a bimodal distribution. Panel b): Intensity Fluctuation Overlap (IFO) probability distribution P (C) for system size N = 64 (Np = 2 14 ), four different values of the pumping rates: (Pc/P) 2 = 1.71, 1.15, 0.97, 0.64. Inset: Binder parameter B measured for P (C) as a function of P −2 = T ; the behaviour is the typical one of first-order transitions, with the transition at the minimum of B. Acknowledgements. The authors thank D. Ancora, G. Benettin, L. Biferale, A. Crisanti, G. Parisi, A. Ponno and A. Vulpiani for useful discussions. The research leading to these results has received funding from the Italian Ministry of Education, University and Research under the PRIN2015 program, grant code 2015K7KK8L-005 and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program, project LoTGlasSy, Grant Agreement No. 694925. 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[ "Charge transfer across transition metal oxide interfaces: emergent conductance and new electronic structure", "Charge transfer across transition metal oxide interfaces: emergent conductance and new electronic structure" ]	[ "Hanghui Chen \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n\nDepartment of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA\n", "Hyowon Park \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n\nDepartment of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA\n", "Andrew J Millis \nDepartment of Physics\nColumbia University\n10027New YorkNYUSA\n", "Chris A Marianetti \nDepartment of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA\n" ]	[ "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA", "Department of Physics\nColumbia University\n10027New YorkNYUSA", "Department of Applied Physics and Applied Mathematics\nColumbia University\n10027New YorkNYUSA" ]	[]	We perform density functional theory plus dynamical mean field theory calculations to investigate internal charge transfer in an artificial superlattice composed of alternating layers of vanadate and manganite perovskite and Ruddlesden-Popper structure materials. We show that the electronegativity difference between vanadium and manganese causes moderate charge transfer from VO 2 to MnO 2 layers in both perovskite and Ruddlesden-Popper based superlattices, leading to hole doping of the VO 2 layer and electron doping of the MnO 2 layer. Comparison of the perovskite and Ruddlesden-Popper based heterostructures provides insights into the role of the apical oxygen. Our first principles simulations demonstrate that the combination of internal charge transfer and quantum confinement provided by heterostructuring is a powerful approach to engineering electronic structure and tailoring correlation effects in transition metal oxides. FIG. 1: Simulation cells of A) bulk SrVO 3 , B) bulk SrMnO 3 and C) SrVO 3 /SrMnO 3 superlattice; D) bulk Sr 2 VO 4 , E) bulk Sr 2 MnO 4 and F) Sr 2 VO 4 /Sr 2 MnO 4 superlattice. The green atoms are Sr. The blue and purple cages are VO 6 and MnO 6 octahedra, respectively. The stacking direction of the superlattice is the [001] axis. itative support and quantitative corrections to the schematic. The conclusions are in Section VI. Four Appendices present technical details relating to the insulating gaps of Sr 2 VO 4 and Sr 2 MnO 4 , alternative forms of the double counting correction, and the possibility of two consecutive repeating layers (i.e. 2/2 superlattices instead of 1/1 superlattices).II. COMPUTATIONAL DETAILS The DFT [30, 31] component of our DFT+DMFT [32, 33] calculations is performed using a plane-wave basis [34], as implemented in the Vienna Ab-initio Simulation Package (VASP) [35-38] using the Projector Augmented Wave (PAW) approach [39, 40]. The correlated subspace and the orbitals with which it mixes are constructed using maximally localized Wannier functions [41] defined over the full 10 eV range spanned by the p-d band the Appendix A: Metal-insulator transition of Sr 2 VO 4In this appendix, we show that within single-site DMFT and with the p-d separation fixed by the experimental photoemission data, there is a metal-insulator transition in Sr 2 VO 4 with an increasing Hubbard U V (U ′ V is determined by the p-d separation for each given U V ).	10.1103/physrevb.90.245138	[ "https://arxiv.org/pdf/1408.0217v2.pdf" ]	52,214,428	1408.0217	46936813eac6d2eaab5867682aa085be49cc906e	Charge transfer across transition metal oxide interfaces: emergent conductance and new electronic structure 1 Aug 2014 Hanghui Chen Department of Physics Columbia University 10027New YorkNYUSA Department of Applied Physics and Applied Mathematics Columbia University 10027New YorkNYUSA Hyowon Park Department of Physics Columbia University 10027New YorkNYUSA Department of Applied Physics and Applied Mathematics Columbia University 10027New YorkNYUSA Andrew J Millis Department of Physics Columbia University 10027New YorkNYUSA Chris A Marianetti Department of Applied Physics and Applied Mathematics Columbia University 10027New YorkNYUSA Charge transfer across transition metal oxide interfaces: emergent conductance and new electronic structure 1 Aug 2014(Dated: August 4, 2014) We perform density functional theory plus dynamical mean field theory calculations to investigate internal charge transfer in an artificial superlattice composed of alternating layers of vanadate and manganite perovskite and Ruddlesden-Popper structure materials. We show that the electronegativity difference between vanadium and manganese causes moderate charge transfer from VO 2 to MnO 2 layers in both perovskite and Ruddlesden-Popper based superlattices, leading to hole doping of the VO 2 layer and electron doping of the MnO 2 layer. Comparison of the perovskite and Ruddlesden-Popper based heterostructures provides insights into the role of the apical oxygen. Our first principles simulations demonstrate that the combination of internal charge transfer and quantum confinement provided by heterostructuring is a powerful approach to engineering electronic structure and tailoring correlation effects in transition metal oxides. FIG. 1: Simulation cells of A) bulk SrVO 3 , B) bulk SrMnO 3 and C) SrVO 3 /SrMnO 3 superlattice; D) bulk Sr 2 VO 4 , E) bulk Sr 2 MnO 4 and F) Sr 2 VO 4 /Sr 2 MnO 4 superlattice. The green atoms are Sr. The blue and purple cages are VO 6 and MnO 6 octahedra, respectively. The stacking direction of the superlattice is the [001] axis. itative support and quantitative corrections to the schematic. The conclusions are in Section VI. Four Appendices present technical details relating to the insulating gaps of Sr 2 VO 4 and Sr 2 MnO 4 , alternative forms of the double counting correction, and the possibility of two consecutive repeating layers (i.e. 2/2 superlattices instead of 1/1 superlattices).II. COMPUTATIONAL DETAILS The DFT [30, 31] component of our DFT+DMFT [32, 33] calculations is performed using a plane-wave basis [34], as implemented in the Vienna Ab-initio Simulation Package (VASP) [35-38] using the Projector Augmented Wave (PAW) approach [39, 40]. The correlated subspace and the orbitals with which it mixes are constructed using maximally localized Wannier functions [41] defined over the full 10 eV range spanned by the p-d band the Appendix A: Metal-insulator transition of Sr 2 VO 4In this appendix, we show that within single-site DMFT and with the p-d separation fixed by the experimental photoemission data, there is a metal-insulator transition in Sr 2 VO 4 with an increasing Hubbard U V (U ′ V is determined by the p-d separation for each given U V ). I. INTRODUCTION Advances in thin film epitaxy growth techniques have made it possible to induce emergent electronic [1][2][3][4][5][6][7][8][9][10], magnetic [11][12][13] and orbital [14,15] states, which are not naturally occurring in bulk constituents, at atomically sharp transition metal oxide interfaces [16][17][18][19]. For example, the interface between the two nonmagnetic band insulators LaAlO 3 and SrTiO 3 [4] has been reported to exhibit both conductance [20] and magnetism [21] (see reviews [22][23][24][25] and references therein). In LaAlO 3 /SrTiO 3 and related heterostructures, the interface electron gas is believed to be produced by the polar catastrophe mechanism, which leads to the transfer of charge from the sample surface to the interface. Here, we consider a different mechanism for controlling the electronic properties of an interface: namely, electronegativity-driven charge transfer. Recently, we have shown that internal charge transfer in a LaTiO 3 /LaNiO 3 superlattice transforms metallic LaNiO 3 into a S = 1 Mott insulator and Mott insulating LaTiO 3 into a S = 0 band insulator [26]. A natural question arises: can we reverse the process and utilize internal charge transfer to induce conductance via oxide interfaces? In this regard, it is very tempting to explore Mott interfaces (one or both constituents are Mott insulators) due to the unusual phenomena (colossal magnetoresistance and high temperature superconductivity) exhibited in certain doped Mott insulators. In this paper we use density functional theory + dynamical mean field theory (DFT+DMFT) to theoretically design a superlattice with emergent metallic behavior. We explore two different types of structure: the perovskite structure (referred to as 113-type) and the n = 1 Ruddlesden-Popper structure (referred to as 214-type). Among the four bulk constituents (SrVO 3 , SrMnO 3 , Sr 2 VO 4 and Sr 2 MnO 4 ), all are correlation-driven insulators except SrVO 3 , which is a moderately correlated metal [27][28][29]. We show that the difference of electronegativity between the elements V and Mn drives internal charge transfer from V to Mn sites, leading to a "self-doping" at the interface and possibly inducing conductance as Mn sites become weakly electron doped and V sites hole doped. The rest of the paper is organized as follows: Section II presents the theoretical methods. A schematic of band alignment is presented in Section III to illustrate the underlying mechanism of charge transfer. All the bulk results from ab initio calculations are in Section IV and the results of vanadate-manganite superlattices are in Section V, both of which provide qual-complex, resulting in an well localized set of d-like orbitals. We approximately find the stationary point of our DFT+DMFT functional by first converging the charge density within DFT and then subsequently converging the DMFT equations. This procedure is found to yield reasonable results in calculations in bulk systems [29,42,43]. For the bulk materials, we consider two structures: the experimental one and the theoretical relaxed structure obtained by the use of DFT within the local density approximation (LDA) [44]. For the superlattice, we use DFT-LDA to obtain the relaxed structure. The simulation cell is illustrated in Fig. 1. The stacking direction of the superlattice is along [001]. We use an energy cutoff 600 eV. A 12 × 12 × 12 Lx Lz (L x and L z are the lattice constants along the x and z directions, and [x] is the integer part of x) Monkhorst-Pack grid is used to sample the Brillouin zone. Both cell and internal coordinates are fully relaxed until each force component is smaller than 10 meV/Å and the stress tensor is smaller than 10 kBar. A higher energy cutoff, a denser k-point sampling, and a smaller force threshold are employed to test the convergence and no significant changes are found in the results. For the vanadates, we treat the empty e g orbitals with a static Hartree-Fock approximation (recent work shows this approximation is adequate to describe the electronic structure of vanadates [29]), while correlations in the V t 2g manifold are treated within single-site DMFT including the Slater-Kanamori interactions using U V = 5 eV and J V = 0.65 eV [45,46]. For manganites, we treat the correlations on all the five Mn d orbitals within single-site DMFT using the Slater-Kanamori interactions with U Mn = 5 eV and J Mn = 1 eV [47]. The DMFT impurity problem is solved using the continuous time quantum Monte Carlo method [48,49]. In order to use the "segment" algorithm [50], we neglect the exchange and pairing terms in the Slater-Kanamori Hamiltonian. All the calculations are paramagnetic and the temperature is set to 232 K. An important outstanding issue in the DFT+DMFT procedure is the "double counting correction" which accounts for the part of the Slater-Kanamori interactions already included in the underlying DFT calculation and plays an important role by setting the mean energy difference between the d and p bands. The p-d separation plays a crucial role in determining the band alignment, which affects the charge transfer. However, currently there is no exact procedure for the double counting correction. We use the U ′ double counting method recently introduced by Park and two of us [43], where the parameter U ′ is the prefactor in the doublecounting which determines the p-d separation and equivalently the number of electrons in in particular we show here they hold also for the conventional fully localized limit (FLL) double counting [51] which is the U ′ = U limit of the method of Ref. [43]. The spectral function presented throughout this work is defined as follows: A i (ω) = − 1 πN k k Im [(ω + µ)I − H 0 (k) − Σ(ω) + V dc )] −1 ii(1) where i is the label of a Wannier function, N k is the number of k-points, I is an identity matrix, H 0 (k) is the DFT-LDA band Hamiltonian in the matrix form using the Wannier basis. Σ(ω) is the self-energy and is understood as a diagonal matrix only with nonzero entries on the correlated orbitals. Local tetragonal point symmetry of the V and Mn sites ensures that Σ(ω) is diagonal within the correlated orbital subspace. V dc is the double counting potential. µ is the chemical potential. III. SCHEMATIC OF BAND STRUCTURE AND BAND ALIGNMENT We consider the following materials as components of the superlattice: SrVO 3 , a moderately correlated metal with nominal d-valence d 1 ; Sr 2 VO 4 , a correlation-driven insulator also with nominal valence d 1 ; and SrMnO 3 and Sr 2 MnO 4 , both of which are d 3 correlation-driven (Mott) insulators. Fig. 1 shows the atomic structure of the bulk phases of the constituent materials and the corresponding superlattices. Fig. 1A, B, and C are bulk SrVO 3 , bulk SrMnO 3 , and SrVO 3 /SrMnO 3 superlattice, respectively. Fig. 1D There is a large energy separation (around 2 eV) between V d and O p states (see Fig. 2A). In the Mn-based materials (see Fig. 2B), the highest occupied states are Mn t 2g -derived and the lowest unoccupied states are Mn e g -derived. Due to the electronegativity difference between V and Mn, visible as the difference in the energy separation of the transition metal d levels from the oxygen p levels, if we align the O p states between vanadates and manganites (see Fig. 2C), the occupied V t 2g states overlap in energy with the unoccupied Mn e g states. The overlap drives electrons from V sites to Mn sites. As the superlattice is formed, a common Fermi level appears across the interface and thus we expect that Mn e g states become electron doped and V t 2g states hole doped. We make two additional points: i) though SrVO 3 is a metal and Sr 2 VO 4 is an insulator with a small energy gap (around 0.2 eV) [27], the near Fermi level electronic structure does not affect the band alignment and therefore the internal charge transfer is expected to occur no matter whether there is a small energy gap in V t 2g states at the Fermi level or not; ii) in our schematic, we assume that the main peak of O p states are exactly aligned between the vanadates and manganites in the superlattices. Of course, real material effects will spoil any exact alignment. We will use ab initio calculations to provide quantitative information on how O p states are aligned between the two materials. Table I, along with the experimental bond lengths and octahedral volumes (in parentheses) for comparison. However, in order to directly compare to the photoemission data, we only present the spectral functions that are calculated using the experimental structures. Sr 2 MnO 4 . The pink dots are the experimental spectra for either SrVO 3 or SrMnO 3 (identical data are plotted alongside the theoretical spectra for the Ruddlesden-Popper structures) [52]. For vanadates, U ′ double counting is employed with U V = 5 eV and U ′ V = 3.5 eV. The red (very thick), blue (thin) and green (thick) curves are V t 2g , V e g and O p projected spectral functions, respectively. For manganites, U ′ double counting is employed with U Mn = 5 eV and U ′ Mn = 4.5 eV. The red (thin), blue (very thick) and green (thick) curves are Mn t 2g , Mn e g and O p projected spectral functions, respectively. The Fermi level is set at zero energy. SrVO 3 SrMnO 3 SrVO 3 /SrMnO 3 l in (V-O) 1.89Å (1.92Å) - 1.88Å l out (V-O) 1.89Å (1.92Å) - 1.85Å Ω VO 6 9.00Å 3 (9.44Å 3 ) - 8.72Å 3 l in (Mn-O) - 1.86Å (1.90Å) 1.88Å l out (Mn-O) - 1.86Å (1.90Å) 1.89Å Ω MnO 6 - 8.58Å 3 (9.15Å 3 ) 8.91Å 3 Sr 2 VO 4 Sr 2 MnO 4 Sr 2 VO 4 /Sr 2 MnO 4 l in (V-O) 1.88Å (1.91Å) - 1.85Å l out (V-O) 1.96Å (1.95Å) - 1.93Å Ω VO 6 9.24Å 3 (9.49Å 3 ) - 8.81Å 3 l in (Mn-O) - 1.82Å (1.90Å) 1.85Å l out (Mn-O) - 1.99Å (1.95Å) 1.99Å Ω MnO 6 - 8.79Å 3 (9.39Å 3 ) 9. A. Bulk vanadates We begin with bulk vanadates: SrVO 3 and Sr 2 VO 4 . SrVO 3 has a cubic structure with a lattice constant a = 3.841Å [53]. Sr 2 VO 4 forms n = 1 Ruddlesden-Popper structure with the in-plane lattice constant a = 3.826Å and the out-of-plane lattice constant c = 12.531Å [54]. We use a Hubbard U V = 5 eV on both vanadate materials to include correlation effects on V d orbitals, which is in the vicinity of previous studies [45,46]. bulk Sr 2 VO 4 (Fig. 3B), along with the experimental photoemission data for bulk SrVO 3 [52]. The threshold of O p states is around 2 eV below the Fermi level. We find that U ′ V = 3.5 eV yields a reasonable agreement between the calculated O p states and experimental photoemission data. At U V = 5 eV, with the p-d separation fixed by the experimental photoemission data, our DFT+DMFT calculations find SrVO 3 to be metallic, consistent with the experiment. However, they do not reproduce a Mott insulating state in Sr 2 VO 4 , as observed in experiment. We show in the Appendix A that a metal-insulator transition does occur in Sr 2 VO 4 with an increasing Hubbard U V and a fixed p-d separation (via U ′ V ). However, the critical U V is larger than typical values employed previously in literature for the vanadates [45,46]. It is possible that the experimentally observed narrow-gap insulating behavior (experimentally observed to persist above the Néel temperature [27,55]) arises from long-range magnetic correlations and spatial correlations that are not captured in our single-site paramagnetic DMFT calculation. These correlations relate to low energy scale physics [56] and are not expected to affect the charge transfer energetics of interest here. B. Bulk manganites Next we discuss the bulk manganites: SrMnO 3 and Sr 2 MnO 4 . For ease of comparison with the superlattice results to be shown in the next section, we study here the cubic phase of SrMnO 3 (isostructural to SrVO 3 ) with the lattice constant of a = 3.801Å (though other structures of SrMnO 3 also co-exist) [57]. Sr 2 MnO 4 forms the n = 1 Ruddlesden-Popper structure with in-plane and out-of-plane lattice constants a = 3.802Å and c = 12.519Å [58]. Consistent with the experimental estimation of Hubbard U from photoemission data [47], we use a Hubbard U Mn = 5 eV on both materials to include correlation effects on Mn d orbitals. Fig. 3 shows the orbitally-resolved spectral function A(ω) of bulk SrMnO 3 (Fig. 3C) and Sr 2 MnO 4 (Fig. 3D) [59]. The threshold of O p states is around 1 eV below the Fermi level. We find that U ′ Mn = 4.5 eV provides a good agreement between the calculated O p states and experimental photoemission data. We observe that for these parameters the occupied Mn t 2g states are visible as a peak slightly above the leading edge of the oxygen band. We will show in Appendix B that modest changes of parameters will move this peak slightly down in energy so that it merges with the leading edge of the oxygen p states. The experimental situation is not completely clear. Published x-ray photoelectron spectroscopy work [60,61] indicates a resolvable t 2g peak at or slightly above the leading edge of the oxygen bands; other studies including recent photoemission measurements [52,62] do not find a separately resolved t 2g peak. The issue is not important for the results of this paper but further investigation of the location of the t 2g states would be of interest as a way to refine our knowledge of the electronic structure of the manganites. With this value of in experiment [27]. This discrepancy may arise because this calculation does not take into account spatial correlation [63]. However, the Mott gap is separated by Mn t 2g and e g states, while the energy difference between O p states and Mn e g states (i.e. p-d separation) is fixed by the experimental photoemission data (via U ′ Mn ). We will show in the next section as well as in the Appendix B that it is the p-d separation that controls the charge transfer and therefore the underestimation of the Mott gap does not significantly affect our main results. V. VANADATE-MANGANITE SUPERLATTICES In this section we discuss vanadate-manganite superlattices. There are two types: we refer to SrVO 3 /SrMnO 3 superlattice as 113-type and refer to Sr 2 VO 4 /Sr 2 MnO 4 superlattice as 214-type. The two types of superlattices have similarities and differences. In both types, the charge transfer from V sites to Mn sites occurs, in which electron dopes the Mn e g states and drains the V t 2g states at the Fermi level. However, in the 214 type, the VO 6 and MnO 6 octahedra are decoupled and the charge transfer arises mainly from the electronegativity difference between V and Mn elements. In the 113 type, in addition to the electronegativity difference between V and Mn, the movement of the shared apical oxygen changes the hybridization and thus also affects the charge transfer. We will show below that due to the movement of the shared apical oxygen atom, the 113-type superlattice generically has a more enhanced charge transfer than the 214-type superlattice. We discuss the phenomena of charge transfer in terms of: 1) structural properties, 2) electronic properties and 3) direct electron counting. Table I shows the DFT-LDA relaxed structure of SrVO 3 /SrMnO 3 and Sr 2 VO 4 /Sr 2 MnO 4 superlattices as well as the bulk materials. We see that the VO 6 octahedron is smaller in the superlattice than in the bulk, while the MnO 6 octahedron is larger. This is suggestive that the VO 6 octahedron loses electrons and the MnO 6 octahedron gains electrons (i.e that internal charge transfer from V to Mn sites occurs), and this will be quantified below. Next, we compare the V t 2g and Mn e g states between the superlattices and bulk materials to show how the Fermi level shifts in the two constituents. Fig. 5A to the schematic (Fig. 2), since electrons are drained out of V t 2g state, both the V t 2g states and O p states of the VO 2 layer are shifted towards the high energy-lying region, compared to their counterparts in bulk vanadates. This shift can be seen ( Fig. 5C and D) i) in the V t 2g states from the bulk (blue or thin dark curves) to the superlattice (red or thick light) and ii) A. Structural properties B. Electronic properties in the O p states of the VO 2 layer from the bulk (turquoise or thin light) to the superlattice (maroon or thick dark). However, since the peak of V t 2g states at the Fermi level is much higher than that of Mn e g states, the shift in the V t 2g states is much smaller than that in the Mn e g states. Fig. 5 reproduces our schematic of how V t 2g and Mn e g states are shifted and re-arranged to reach one common Fermi level in a vanadate-manganite superlattice. A possible consequence is electron (hole) conductance in the MnO 2 (VO 2 ) layer. C. Direct electron counting Now we calculate the occupancy on each orbital by performing the following integral: N i = ∞ −∞ A i (ω)n F (ω)dω(2) where A i (ω) is the spectral function for the ith orbital (defined from the Wannier construction), which is defined in Eq. (1). n F (ω) is the fermion occupancy factor. In order to explicitly display the charge transfer phenomenon, we calculate the V d and Mn d occupancy in both bulk materials and the superlattices. We summarize the results in Table II. We can see that N d (V) decreases and N d (Mn) increases from bulk to the superlattices and an average charge transfer from V to Mn is 0.40e for the 113-type superlattice and 0.25e for the 214-type superlattice. Moreover, due to the strong covalency between transition metal d states and oxygen p states, the occupancy of oxygen p states also changes between bulk materials and the superlattices. For this reason, the change in d occupancy may not be an accurate representation of charge transfer. We also calculate the total occupancy of VO 2 and MnO 2 layers and find that the total charge transfer between the two layers amounts to 0.53 for the 113-type superlattice and 0.38 for the 214-type superlattice. Unlike the 113-type superlattice in which the apical oxygen is shared by two octahedra, the 214-type superlattice has a unique property that each octahedron is decoupled between layers. Therefore in the superlattice, we can count the charge transfer from the VO 6 octahedron to the MnO 6 octahedron. Note that since we only From Table II, we can see that the internal charge transfer is stronger in the 113-type superlattice, compared to the 214-type. We show below that the difference arises because in the 113-type superlattice, the apical oxygen is shared by the VO 6 and MnO 6 octahedra, whereas the octahedra are decoupled in the 214-type. We see from Table I) 7A shows that for U V = 5 eV and U ′ V = 3.5 eV, Sr 2 VO 4 is metallic with mainly V t 2g states at the Fermi level, which is a reproduction of Fig 3B. Fig. 7B shows that with U V increased to 8 eV and U ′ V to 6.8 eV which approximately fixes the p-d separation, a metalinsulator transition occurs and Sr 2 VO 4 is rendered a Mott insulator. However, the critical U V depends on the approximation scheme we employ. A more elaborate cluster-DMFT calculation and/or the inclusion of long range order may find a smaller critical U V [63]. If we increase U Mn to 8 eV and U ′ Mn to 7.5 eV, the Mott gap is correspondingly increased to around 1 eV with the p-d separation fixed by the photoemission data [52]. The Mn t 2g peak and the main peak of O p states now merge together. However, even with U Mn = 8 eV, the Mott gap is still smaller than the optical gap (around 2 eV) from experiment [27]. The difference could be due to spatial correlations not included in our single-site DMFT approximation [63]. SrVO 3 SrMnO 3 SrVO 3 /SrMnO 3 N d (V) N d (Mn) N d (V) N d (N d (V) N d (Mn) N d (V) N d (Mn) ∆N Using the parameters U Mn = 8 eV and U ′ Mn = 7.5 eV, we redo the calculations on Sr 2 VO 4 /Sr 2 MnO 4 superlattices (with U V = 5 eV and U ′ V = 3.5 eV) to test the effects of Mott gap size on charge transfer. As Fig. 8B shows, the key features in electronic structure remain the same as Fig. 4 in the main text: i) Mn e g and V t 2g states emerge at the Fermi level and ii) the main peaks of O p states associated with the MnO 2 and VO 2 layers are approximately aligned. This shows that it is the p-d separation that controls the charge transfer across the interface while the size of Mott gap plays a secondary role. Appendix C: Electronic structure calculated using the fully localized limit double counting In this appendix, we show the electronic structure of both SrVO 3 /SrMnO 3 and Sr 2 VO 4 /Sr 2 MnO 4 superlattices, calculated using the standard fully localized limit (FLL) double counting. The orbitally resolved spectral function is shown in Fig. 9, which is compared to Fig. 4 in the main text. We employ U V = U Mn = 5 eV. We find the FLL double counting does not change the key features of electronic structure, such as the emergence of Mn e g and V t 2g states at the Fermi level. SrVO 3 SrMnO 3 (SrVO 3 ) 1 /(SrMnO 3 ) 1 (SrVO 3 ) 2 /(SrMnO 3 ) 2 N d (V) N d (Mn) N d (V) N d (Mn) N d (V) N d ( FIG. 2 : 2Schematic band structure of A) vanadates and B) manganites. C) is the band alignment of the superlattice before the charge transfer occurs, i.e. two independent Fermi levels. D) is the band structure of the superlattice after the charge transfer occurs, i.e. with one common Fermi level. The dashed red line denotes the Fermi level. the d-manifold. In this study, U ′ is chosen to produce an energy separation between the O p and transition metal d bands which is consistent with photoemission experiments. Our main qualitative conclusions do not depend on the details of the double counting scheme; Fig. 2 2is a schematic of the band structure of bulk vanadates, bulk manganites, and the band alignments in the superlattice (the small insulating gap of Sr 2 VO 4 is not relevant here. 08Å 3 3IV. BULK PROPERTIES This section is devoted to properties of vanadates and manganites in their bulk single crystalline form. We perform DFT+DMFT calculations on both experimental structures and relaxed atomic structures obtained from DFT-LDA. The DFT-LDA relaxed V-O and Mn-O bond lengths, as well as the volume of VO 6 and MnO 6 octahedra, are summarized in FIG. 3 : 3Orbitally resolved spectral function of A) SrVO 3 and B) Sr 2 VO 4 ; C) SrMnO 3 and D) Fig. 3 3shows the orbitally-resolved spectral function A(ω) of bulk SrVO 3(Fig. 3A)and U ′ Mn , the theory produces a small energy gap around 0.5 eV in both SrMnO 3 and Sr 2 MnO 4 . However, the gap value is U Mn -dependent. We show in Appendix B that with the p-d separation fixed, via the adjustment of U ′ Mn , a larger U Mn increases the Mott gap by further separating the Mn lower and upper Hubbard bands. However, for the value of U Mn (around 5 eV) that is extracted from photoemission experiments [47], the size of the Mott gap of Sr 2 MnO 4 is substantially underestimated, compared to the optical gap (around 2 eV) Fig. 4 4shows the orbitally resolved spectral function of the SrVO 3 /SrMnO 3 superlattice (left panels) and the Sr 2 VO 4 /Sr 2 MnO 4 superlattice (right panels). In both superlattices, the Mn e g states emerge at the Fermi level, while in bulk manganites, there is a small gap in the Mn d states (separated by Mn e g and t 2g ) in both materials. In the VO 2 layer, V t 2g states dominate at the Fermi level. Another feature worth noting is the O p states of the MnO 2 and of the VO 2 layers. Though the very first peak of O p states in the MnO 2 layer below the Fermi level is lined up with Mn t 2g states due to strong covalency, the main peak almost exactly overlaps with that of O p states in the VO 2 layer. This supports our hypothesis in the schematic that the main peaks of O p states of the VO 2 and MnO 2 layers are aligned in the superlattices. We need to mention that the general features in electronic structure of the superlattices are robust for different double counting schemes. We show in Appendix C that the standard FLL double counting yields a very similar electronic structure of the superlattices. and B show the comparison of Mn e g and O p states of the MnO 2 layer between the superlattices and bulk manganites (A: 113-type and B: 214-type). The Fermi levels of bulk manganites and of the superlattices are lined up in the same figure. FIG. 4 : 4Orbitally resolved spectral function of vanadate-manganite superlattices. Left panels: SrVO 3 /SrMnO 3 superlattice. Right panels: Sr 2 VO 4 /Sr 2 MnO 4 superlattice. A) and B): Mn t 2g (red thin) and Mn e g (blue thick) states; C) and D): V t 2g (green thick) and V e g (violet thin) states; E) and F): O p states of the MnO 2 layer (turquoise thick) and O p states of the VO 2 layer (maroon thin). U ′ double counting is employed with U V = U Mn = 5 eV and U ′ V = 3.5 eV, U ′ Mn = 4.5 eV. The Fermi level is set at zero point. According to the schematic (Fig. 2), with respect to bulk manganites, both the Mn d and O p states in the MnO 2 layer are shifted towards the low energy-lying region due to the electron doping. Fig. 5A and B clearly reproduce this rigid shift in i) Mn e g states from the bulk (blue or thin dark curves) to the superlattice (red or thick light) and ii) in O p states of the MnO 2 layer from the bulk (turquoise or thin light) to the superlattice (maroon or thick dark). Similarly, Fig. 5C and D show the comparison of V t 2g and O p states (of the VO 2 layer) between the superlattices and bulk vanadates (C: 113-type and D: 214-type). The Fermi 2g (SL) V t 2g (bulk) O p (VO 2 -SL) O p (VO 2 -bulk) FIG. 5: A) Comparison of Mn e g and O p states of the MnO 2 layer between the SrVO 3 /SrMnO 3 superlattice and bulk SrMnO 3 . B) Comparison of Mn e g and O p states of the MnO 2 layer between the Sr 2 VO 4 /Sr 2 MnO 4 superlattice and bulk Sr 2 MnO 4 . C) Comparison of V t 2g and O p states of the VO 2 layer between the SrVO 3 /SrMnO 3 superlattice and bulk SrVO 3 . D) Comparison of V t 2g and O p states of the VO 2 layer between the Sr 2 VO 4 /Sr 2 MnO 4 superlattice and bulk Sr 2 VO 4 . The Fermi level is set at zero energy. "SL" refers to the superlattices. levels of bulk vanadates and of the superlattices are lined up in the same figure. According FIG. 6 : 6Movement of the apical oxygen, corresonding changes in the energy of V d and Mn d states and enhancement of the charge transfer. A) SrVO 3 /SrMnO 3 superlattice and B) Sr 2 VO 4 /Sr 2 MnO 4 superlattice. The green atoms are Sr. The blue and purple cages are VO 6 and MnO 6 octahedra, respectively. The arrows on the oxygen atoms indicate the atom movement. The arrows on the metal d states indicate the trend of energy shift. take into account the p-d band manifold, the V and Mn octahedra include all the Wannier states and therefore in bulk Sr 2 VO 4 , the number of electrons per VO 4 unit is exactly 25e and in bulk Sr 2 MnO 4 , the number of electrons per MnO 4 unit is exactly 27e. We find that relative to the bulk materials, the V octahedron of the 214-type superlattice loses 0.48e and Mn octahedron of the 214-type superlattice gains exactly 0.48e. Comparison of this 0.48e charge transfer to the 0.25e found by only considering d orbitals further confirms that not only the transition metal d states but also oxygen p states participate in the charge transfer. that due to the internal charge transfer, the VO 6 octahedron loses electrons and shrinks; on the other hand, the MnO 6 octahedron gains electrons and expands.Therefore the shared apical oxygen atom moves away from Mn sites and towards V sites (seeFig. 6A). A direct consequence is that the out-of-plane Mn-O hopping decreases and the out-of-plane V-O hopping increases. Since the V d and Mn d states are anti-bonding in nature, the changes in the metal-ligand hopping push the V d states higher in energy and lower the energy of Mn d states and thus enhance the internal charge transfer. In the 214-type superlattice, we have a different situation because the two oxygen octahedra have their own apical oxygen atoms, whose movements are decoupled. From theTable I, the VO 6 shrinks and the apical oxygen atom of VO 6 moves towards the V atom, just like the 113-type superlattice. However, the MnO 6 expands but the apical oxygen does not move (the in-plane Mn-O bond does increase, so does the overall volume of MnO 6 ). Therefore, the energy of V d states is increased due to the enhanced out-of-plane V-O hopping, but the N (VO 2 ) N (MnO 2 ) N (VO 2 ) N (MnO 2 ) ∆N 13.35 (13.36) 15.42 (15.44) 13.01 15.83 0.38 N (VO 4 ) N (MnO 4 ) N (VO 4 ) N (MnO 4 ) ∆N 25.00 (25.00) 27.00 (27.00) 24.52 27.48 0.48 energy of Mn d states remains almost unchanged because the apical oxygen atom does not move (Fig. 6B). As a result, the charge transfer between V and Mn sites is weaker in the 214-type superlattice, compared to the 113-type superlattice. Our discussions in this paper have focussed mainly on the (SrVO 3 ) 1 /(SrMnO 3 ) 1 superlattice. Though m = 1 superlattices (in the notation of (SrVO 3 ) m /(SrMnO 3 ) m ) are easy for theoretical studies, experimentally it is more practical to grow m = 2 or larger m superlattices. We show in the Appendix D that comparing (SrVO 3 ) 1 /(SrMnO 3 ) 1 and (SrVO 3 ) 2 /(SrMnO 3 ) 2 superlattices, the charge transfer is very similar. However, for a large m, we will have inquivalent V sites and eventually the charge transfer will be confined to the interfacial region. Investigating the length scales associated with charge transfer is an important open question. VI. CONCLUSIONS We use DFT+DMFT calculations to show that due to the difference in electronegativity, internal charge transfer could occur between isostructural vanadates and manganites in both 113-type and 214-type superlattices. The charge transfer is enhanced by associated lattice distortions. The moderate electronegativity difference between Mn and V leads to moderate charge transfer, in contrast to the LaTiO 3 /LaNiO 3 superlattice, in which a complete charge transfer fills up the holes on the oxygen atoms in the NiO 2 layer. The partially filled bands imply metallic conductance that could possibly be observed in transport, if the thin film quality is high enough that disorder is suppressed and Anderson localization does not occur [64]. Our study of a superlattice consisting of two different species of transition metal oxides establishes that internal charge transfer is a powerful tool to engineer electronic structure and tailor correlation effects in transition metal oxides. In particular, for vanadatemanganite superlattices, internal charge transfer may serve as an alternative approach to dope Mott insulators without introducing chemical disorder. Finally our work shows that in addition to perovskite structure, Ruddlesden-Popper structures can also be introduced in the design of oxide superlattices with tailored properties [65]. FIG. 7: Orbitally resolved spectral function of Sr 2 VO 4 . U ′ double counting is employed with A) U V = 5 eV, U ′ V = 3.5 eV and B) with U V = 8 eV, U ′ V = 6.8 eV.The red (very thick), blue (thick) and green (thin) curves are V t 2g , V e g and O p projected spectral functions, respectively. The pink dots are experimental photoemission data for SrVO 3[52]. The Fermi level is set at zero energy. Fig. Fig. 7A shows that for U V = 5 eV and U ′ V = 3.5 eV, Sr 2 VO 4 is metallic with mainly V t 2g states at the Fermi level, which is a reproduction of Fig 3B. Fig. 7B shows that with U V Appendix B: Mott gap of Sr 2 MnO 4 and its effects on Sr 2 MnO 4 /Sr 2 VO 4 superlattices In this appendix, we show how the Hubbard U Mn changes the Mott gap of Sr 2 MnO 4 with the p-d separation approximately fixed. Fig. 8A shows the orbitally resolved spectral function of Sr 2 MnO 4 with U Mn = 8 eV and U ′ Mn = 7.5 eV. Note that since Sr 2 MnO 4 is a Mott insulator, the Fermi level in the calculation is shifted at the conduction band edge, i.e. the edge of Mn e g states. In Fig. 3D of the main text, the Mott gap of Sr 2 MnO 4 is around 0.5 eV with U Mn = 5 eV and U ′ Mn = 4.5 eV. Appendix D: (SrVO 3 ) 1 /(SrMnO 3 ) 1 versus (SrVO 3 ) 2 /(SrMnO 3 ) 2 superlattices In this appendix, we compare the (SrVO 3 ) 1 /(SrMnO 3 ) 1 superlattice to the (SrVO 3 ) 2 /(SrMnO 3 ) 2 superlattice. We focus on the d occupancy and the charge transfer from V to Mn sites. Table III shows that N d of V sites and Mn sites are very similar between (SrVO 3 ) 1 /(SrMnO 3 ) 1 and (SrVO 3 ) 2 /(SrMnO 3 ) 2 superlattices. FIG. 8 : 8A) Orbitally resolved spectral function of Sr 2 MnO 4 . U ′ double counting is employed with U Mn = 8 eV, U ′ Mn = 7.5 eV. The red (thin), blue (very thick) and green curves (thick) are Mn t 2g , Mn e g and O p projected spectral functions, respectively. The pink dots are experimental photoemission data of SrMnO 3 [52]. The Fermi level is set at zero energy. B) Orbitally resolved spectral function of Sr 2 VO 4 /Sr 2 MnO 4 superlattices. U ′ double counting is employed with U Mn = 8 eV, U ′ Mn = 7.5 eV and U V = 5 eV, U ′ V = 3.5 eV. B1): Mn t 2g (red thin) and Mn e g (blue thick) states; B2): V t 2g (green thick) and V e g (violet thin) states; B3): O p states of the MnO 2 layer (turquoise thick) and O p states of the VO 2 layer (maroon thin). FIG. 9 : 9Orbitally resolved spectral function of vanadate-manganite superlattices. Left panels: SrVO 3 /SrMnO 3 superlattice. Right panels: Sr 2 VO 4 /Sr 2 MnO 4 superlattice. A) and B): Mn t 2g (red thin) and Mn e g (blue thick) states; C) and D): V t 2g (green thick) and V e g (violet thin) states; E) and F): O p states of the MnO 2 layer (turquoise thick) and O p states of the VO 2 layer (maroon thin). Fully localized limit double counting is employed with U V = U Mn = 5 eV. The Fermi level is set at zero energy. TABLE I : IThe in-plane and out-of-plane V-O and Mn-O bond lengths l of SrVO 3 , SrMnO 3 , Sr 2 VO 4 and Sr 2 MnO 4 . The corresponding VO 6 and MnO 6 octahedral volumes Ω are also calculated. The relaxed structures are obtained from DFT-LDA non-spin-polarized calculations. The values in parentheses are experimental values which are referenced in the main text. TABLE II : IIThe occupancy of V d and Mn d states, as well as VO 2 and MnO 2 layers in vanadates,manganites and the superlattices. All the occupancies without the parentheses are calculated from Wannier basis using the DFT-LDA relaxed structures. 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[ "Fractional backward stochastic differential equations and fractional backward variational inequalities", "Fractional backward stochastic differential equations and fractional backward variational inequalities" ]	[ "Lucian Maticiuc es:[email protected] \nFaculty of Mathematics\n\"Alexandru Ioan Cuza\" University\nCarol I Blvd., no. 11700506IasiRomania\n", "Tianyang Nie [email protected] \nFaculty of Mathematics\n\"Alexandru Ioan Cuza\" University\nCarol I Blvd., no. 11700506IasiRomania\n\nSchool of Mathematics\nShandong University\n250100JinanShandongChina\n\nLaboratoire de Mathématiques\nUMR 6205\nCNR\nUniversité de Bretagne Occidentale\n29285, Cédex 3BrestFrance\n" ]	[ "Faculty of Mathematics\n\"Alexandru Ioan Cuza\" University\nCarol I Blvd., no. 11700506IasiRomania", "Faculty of Mathematics\n\"Alexandru Ioan Cuza\" University\nCarol I Blvd., no. 11700506IasiRomania", "School of Mathematics\nShandong University\n250100JinanShandongChina", "Laboratoire de Mathématiques\nUMR 6205\nCNR\nUniversité de Bretagne Occidentale\n29285, Cédex 3BrestFrance" ]	[]	In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as:where η is a stochastic process given by η(t) = η(0) + t 0 σ(s)δB H (s), t ∈ [0, T ], and B H is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence type integral. Based on Hu and Peng's paper, BDSEs driven by fBm, SIAM J. Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equationwhere ∂ϕ is a multivalued operator of subdifferential type associated with the convex function ϕ.	10.1007/s10959-013-0509-9	[ "https://arxiv.org/pdf/1102.3014v4.pdf" ]	119,697,387	1102.3014	e6a3c8e92ca876ef38ebc2e282857879acddb632	Fractional backward stochastic differential equations and fractional backward variational inequalities 7 May 2012 January 12, 2013 Lucian Maticiuc es:[email protected] Faculty of Mathematics "Alexandru Ioan Cuza" University Carol I Blvd., no. 11700506IasiRomania Tianyang Nie [email protected] Faculty of Mathematics "Alexandru Ioan Cuza" University Carol I Blvd., no. 11700506IasiRomania School of Mathematics Shandong University 250100JinanShandongChina Laboratoire de Mathématiques UMR 6205 CNR Université de Bretagne Occidentale 29285, Cédex 3BrestFrance Fractional backward stochastic differential equations and fractional backward variational inequalities 7 May 2012 January 12, 2013(Tianyang Nie) 1AMS Subject Classification: 60H1047J2060H0560G22 Keywords: Backward stochastic differential equationfractional Brownian motiondi- vergence type integralbackward stochastic variational inequalitysubdifferential operator * Corresponding author In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as:where η is a stochastic process given by η(t) = η(0) + t 0 σ(s)δB H (s), t ∈ [0, T ], and B H is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence type integral. Based on Hu and Peng's paper, BDSEs driven by fBm, SIAM J. Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equationwhere ∂ϕ is a multivalued operator of subdifferential type associated with the convex function ϕ. Introduction General backward stochastic differential equations (BSDEs) driven by a Brownian motion were first studied by Pardoux and Peng in [15], where they also gave a probabilistic interpretation for the viscosity solution of semilinear partial differential equations (PDEs). In 1998 Pardoux and Rȃşcanu [16] studied backward stochastic differential equations involving a subdifferential operator (which are often called backward stochastic variational inequalities, BSVIs), and they used them in order to generalize the Feymann-Kac type formula to represent the solution of multivalued parabolic PDEs (also called parabolic variational inequalities, PVIs). Hu and Peng [10] were the first to study nonlinear BSDEs governed by a fractional Brownian motion (fBm). Our work, based on [10], has the objective to develop a rigorous approach for such BSDEs driven by a fBm and to extend the discussion to fractional BSVIs. Our paper is, to our best knowledge, the first one to study fractional BSVIs. Let us recall that, for H ∈ (0, 1), a (one dimensional) fBm (B H (t)) t≥0 with Hurst parameter H is a continuous and centered Gaussian process with covariance E B H (t)B H (s) = 1 2 (t 2H + s 2H − \|t − s\| 2H ), t, s ≥ 0. For H = 1/2, the fBm is a standard Brownian motion. If H > 1/2, then B H (t) has a longrange dependence, which means that, for r(n) := cov(B H (1) , B H (n + 1) − B H (n)), we have ∞ n=1 r(n) = ∞. Moreover, B H is self-similar, i.e., B H (at) has the same law as a H B H (t) for any a > 0. Since there are many models of physical phenomena and finance which exploit the self-similarity and the long-range dependence, fBms are a very useful tool to characterize such type of problems. However, since fBms are not semimartingales nor Markov processes when H = 1/2, we cannot use the classical theory of stochastic calculus to define the fractional stochastic integral. In essence, two different integration theories with respect to fractional Brownian motion have been defined and studied. The first one, originally due to Young [18], concerns the pathwise (with ω as a parameter) Riemann-Stieltjes integral which exists if the integrand has Hölder continuous paths of order α > 1 − H. But it turns out that this integral has the properties comparable to the Stratonovich integral, which leads to difficulties in applications. The second one concerns the divergence operator (or Skorohod integral), defined as the adjoint of the derivative operator in the framework of the Malliavin calculus. This approach was introduced by Decreusefond andÜstünel [6], and it was very intensely studied, e.g., in Alòs and Nualart [3] for H > 1/2, and in Alòs, Mazet and Nualart [1] for H < 1/2. An equivalent approach consists in defining, for H ∈ (1/2, 1), the stochastic integral based on Wick product (introduced by Duncan, Hu and Pasik-Duncan in [7]), as the limit of Riemann sums. We mention that, in contrast to the pathwise integral, the expectation of this integral is zero for a large class of integrands. Concerning the study of BSDEs in the fractional framework, the major problem is the absence of a martingale representation type theorem with respect to a fBm. For the first time, Hu and Peng [10] overcome this problem, in the case H > 1/2. For this, they used the notion of quasi-conditional expectationÊ (introduced in Hu and Øksendal [9]). In our paper, we consider the BSDE −dY (t) = f (t, η(t), Y (t), Z(t))dt − Z(t)δB H (t) , t ∈ [0, T ], Y (T ) = g(η(T )), driven by a fBm B H and governed by the process η(t) = η(0) + t 0 σ(s)δB H (s), t ∈ [0, T ], where σ : [0, T ] → R is deterministic, continuous function. A special care has to be payed here to the stochastic integral in the BSDE (1). In [10] this stochastic integral is the Wick product one, but the Itô formula and the integration by part formula they used were established for the Itô-Skorohod type integral (see Definition 6.11 [8] ). In our approach we use as stochastic integral the divergence operator. Concerning the coefficient σ of the driving process η, Hu and Peng [10] supposed that there exists c 0 > 0 such that inf t∈[0,T ]σ (t) σ(t) ≥ c 0 , forσ(t) := t 0 φ(t − r)σ(r)dr, t ∈ [0, T ]. Here in our manuscript we work without such a condition. Let us also mention that in [10], it is assumed that η(t) = η(0) + t 0 b(s)ds + t 0 σ(s)δB H (s), t ∈ [0, T ]. However, Proposition 4.5 [10] is not proved for such kind of η. Indeed, in their proof the authors use a quasi-conditional expectation formulâ E[f (η(T ))\|F t ] = P σ 2 T − σ 2 t f (η(t)), which is only applicable for η of the form η(t) = η(0)+ t 0 σ(s)δB H (s), t ∈ [0, T ] (see Theorem 3.8 [10]). That is why we adopt in our paper this latter form of η. Based on the above described framework we make a rigorous approach to prove the existence and the uniqueness for BSDE (1). This approach includes, in particular, first a rigorous discussion of the equation Y (t) = g(η(T )) + After, the existence for BSDE (1) is proved by using a fixed point theorem over an appropriate Banach space. Let us mention that in [10] the fixed point theorem was applied, but the obtained fixed point (Y, Z) was not checked to be a solution and in particular, the existence of the integral T t Z(s)δB H (s) was not discussed. Based on our results on BSDE driven by a fBm and on Pardoux and Rȃşcanu [16] on BSVI governed by a standard Brownian motion, we consider the following fractional BSVI −dY (t) + ∂ϕ(Y (t))dt ∋ f (t, η(t), Y (t), Z(t))dt − Z(t)δB H (t) , t ∈ [0, T ], Y (T ) = g(η(T )), where ∂ϕ is the subdifferential of a convex lower semicontinuous (l.s.c.) function ϕ : R → (−∞, +∞]. The existence of the solution will be proved. Now we give the outline of our paper: In Section 2 we recall some definitions and results about fractional stochastic integrals and the related Itô formula. We present the assumptions and some auxiliary results including the Itô formula w.r.t. the divergence type integral in Section 3. Section 4 is devoted to prove the existence and the uniqueness result for BSDE driven by a fBm. In Section 5, we study the existence for fractional BSVI governed by a fBm. Finally, in the Appendix, we prove a more general Itô formula based on Theorem 8 [3] and a auxiliary lemma. Preliminaries: Fractional stochastic calculus In this section we shall recall some important definitions and results concerning the Malliavin calculus, the stochastic integral with respect to a fBm and Itô's formula. For a deeper discussion we refer the reader to [3], [5], [7], [8] and [13]. Throughout our paper, we assume that the Hurst parameter H always satisfies H > 1/2. Define φ(x) = H(2H − 1)\|x\| 2H−2 , x ∈ R. Let us denote by \|H\| the Banach space of measurable functions f : [0, T ] → R such that f 2 \|H\| := T 0 T 0 φ(u − v)\|f (u)\|\|f (v)\|dudv < +∞. Given ξ, η ∈ \|H\|, we put ξ, η T = T 0 T 0 φ(u − v)ξ (u) η (v) dudv and ξ 2 T := ξ, ξ T .(2) Then ξ, η T is a Hilbert scalar product. Let H be the completion of the space of step functions in \|H\| under this scalar product. We emphasize that the elements of H can be distributions. Moreover, from [11] we have the continuous embedding L 2 ([0, T ]) ⊂ L 1 H ([0, T ]) ⊂ \|H\| ⊂ H. Let P T denote the set of elementary random variables of the form F = f T 0 ξ 1 (t)dB H (t) , . . . , T 0 ξ n (t)dB H (t) , where f is a polynomial function of n variables and ξ 1 , ξ 2 , . . . , ξ n ∈ H. The Malliavin derivative D H of an elementary variable F ∈ P T is defined by D H s F = n i=1 ∂f ∂x i T 0 ξ 1 (t)dB H (t), . . . , T 0 ξ n (t)dB H (t) ξ i (s), s ∈ [0, T ]. We denote by D 1,2 the Banach space defined as the completion of P T w.r.t. the following norm F 1,2 = E(\|F \| 2 ) + E D H s F 2 T 1/2 , F ∈ P T . Hence, D 1,2 consists of all F ∈ L 2 (Ω, F, P ) such that there exists a sequence F n ∈ P T , n ≥ 1, which satisfies F n −→ F in L 2 (Ω, F, P ), D H F n n≥1 is convergent in L 2 (Ω, F, P ; H). Moreover, from Proposition 1.2.1 [13] we have that D H = D H s s∈[0,T ] is a closable operator from L 2 (Ω, F, P ) to L 2 (Ω, F, P ; H). Thus, D H F = lim n→∞ D H G n in L 2 (Ω, F, P ; H), for every sequence G n ∈ P T , n ≥ 1, which satisfies G n −→ F in L 2 (Ω, F, P ), D H G n n≥1 is convergent in L 2 (Ω, F, P ; H). Let us introduce also another derivative D H t F = T 0 φ(t − v)D H v F dv, t ∈ [0, T ].(3) We also need the adjoint operator of the derivative D H . This operator is called divergence operator, it is denoted by δ(·) and represents the divergence type integral with respect to a fBm (see, e.g., [5] for more details). Definition 1 We say that a process u ∈ L 2 (Ω, F, P ; H) belongs to the domain Dom(δ), if there exists δ(u) ∈ L 2 (Ω, F, P ), such that the following duality relationship is satisfied E(F δ(u)) = E( D H · F, u T ), for all F ∈ P T .(4) Remark 2 In (4), the class P T can be replaced by D 1,2 (see [5] [8], which is defined in the spirit of the anticipative Skorohod integral w.r.t. Brownian motion in [14]). Theorem 3 [Proposition 6.25, [8]] We denote by L 1,2 H the space of all stochastic processes u : (Ω, F, P ) → H such that E u 2 T + T 0 T 0 \|D H s u (t) \| 2 dsdt < ∞.(5) If u ∈ L 1,2 H , then the Itô-Skorohod type stochastic integral T 0 u (s) dB H (s) defined by Proposition 6.11 [8] exists and coincides with the divergence type integral (see Theorem 6.23 [8]). Moreover,          E T 0 u (s) dB H (s) = 0, E T 0 u (s) dB H (s) 2 = E u 2 T + T 0 T 0 D H s u(t)D H t u(s)dsdt . Let us finish this section by giving an Itô formula for the divergence type integral. Due to Theorem 8 [3], the following Itô formula holds. Theorem 4 Let ψ be a function of class C 2 (R). Assume that the process (u t ) t∈[0,T ] belongs to D 2,2 loc (\|H\|) and that the integral X t = t 0 u s δB H (s) is almost surely continuous. Assume that E\|u\| 2 1/2 belong to H. Then, for each t ∈ [0, T ], the following formula holds ψ(X t ) = ψ(0) + t 0 ∂ ∂x ψ(X s )u s δB H (s) +H(2H − 1) t 0 ∂ 2 ∂x 2 ψ(X s )u s T 0 \|s − r\| 2H−2 s 0 D r u θ δB H (θ)dr ds +H(2H − 1) t 0 ∂ 2 ∂x 2 ψ(X s )u s s 0 u θ \|s − θ\| 2H−2 dθ ds.X t = X 0 + t 0 g s ds + t 0 f s δB H (s), t ∈ [0, T ], and ψ ∈ C 1,2 ([0, T ] × R), we have ψ(t, X t ) = ψ(0, X 0 ) + t 0 ∂ ∂t ψ(s, X s )ds + t 0 ∂ ∂x ψ(s, X s )dX s + 1 2 t 0 ∂ 2 ∂x 2 ψ(s, X s ) d ds f 2 s ds, t ∈ [0, T ].(6) 3 Assumptions and auxiliary results Assumptions Let us consider the Itô-type process η(t) = η(0) + t 0 σ(s)δB H (s) , t ∈ [0, T ],(7) where the coefficients η(0) and σ satisfy: (H 1 ) η(0) ∈ R is a given constant; (H 2 ) σ : R → R is a deterministic continuous function such that σ(t) = 0, for all t ∈ [0, T ]. Letσ (t) := t 0 φ(t − r)σ(r)dr, t ∈ [0, T ].(8) We recall that (see (2)) σ 2 t = H(2H − 1) t 0 t 0 \|u − v\| 2H−2 σ (u) σ (v) dudv. Remark 6 The functionσ defined by (8) can be written in the following form: σ(t) = H(2H − 1)t 2H−1 1 0 (1 − u) 2H−2 σ (tu) du, t ∈ [0, T ]. Moreover, we observe that σ 2 t is continuously differentiable with respect to t, and (a) d dt σ 2 t = 2σ (t)σ(t) > 0, t ∈ (0, T ], (b) for a suitable constant M > 0, 1 M t 2H−1 ≤σ (t) σ(t) ≤ M t 2H−1 , t ∈ [0, T ] .(9) Our objective is to study the following BSDE driven by the fBm B H and the above introduced stochastic process η: −dY (t) = f (t, η(t), Y (t), Z(t))dt − Z(t)δB H (t) , t ∈ [0, T ], Y (T ) = ξ. Here the stochastic integral is understood as the divergence operator. We make the following assumptions on the function f and the terminal condition ξ: (H 3 ) The function f : [0, T ] × R 3 −→ R belongs to the space C 0,1 pol [0, T ] × R 3 * , and there exists a constant L such that, for all t ∈ [0, T ], x, y 1 , y 2 , z 1 , z 2 ∈ R, \|f (t, x, y 1 , z 1 ) − f (t, x, y 2 , z 2 )\| ≤ L(\|y 1 − y 2 \| + \|z 1 − z 2 \|). (H 4 ) ξ = g(η T ), where g : R → R is a differentiable function with polynomial growth. Before giving the definition of the solution for the above BSDE and investigating its wellposedness (see Section 4), we introduce the following space V T := Y = φ(·, η(·)) : φ ∈ C 1,3 pol ([0, T ] × R) with ∂φ ∂t ∈ C 0,1 pol ([0, T ] × R) as well as its completionV α T under the following α-norm Y α = T 0 t 2α−1 E\|Y (t) \| 2 dt 1/2 = T 0 t 2α−1 E\|φ(t, η (t))\| 2 dt 1/2 ,(10) where α ≥ 1/2. Let us study some auxiliary results concerning these spaces. Remark 7 We should mention that in Hu and Peng's paper [10], the space V T was defined as Y = φ(·, η(·)) : φ ∈ C 1,2 ([0, T ] × R) . However, for the sake of mathematical rigor, we have to restrict their space to that defined above (See the proofs of Lemma 4.2 and Proposition 4.3 in [10]). An Itô formula We begin with the following result concerning the space V T . Lemma 8 We have V T ⊂ L 1,2 H ⊂ Dom(δ). Proof. Let u ∈ V T . In order to show (5), we first prove that E u 2 T < ∞. From L 2 ([0, T ]) ⊂ H we see that it is sufficient to show that E T 0 \|u(s, η(s))\| 2 ds < ∞, where the latter property can be deduced from E\|u(s, η(s))\| 2 ≤ C, s ∈ [0, T ], for some suitable C ∈ R. Indeed, since u ∈ V T , we have, for some C > 0, k ≥ 1, On the other hand, from (7) and Theorem 7.10 [8], we see that for any p ≥ 1, there exists C p > 0 such that E\|η(s)\| p ≤ CE 1 + s 0 σ(v)δB H (v) p ≤ C p + C p σ p/2 s .(11) Hence E u 2 T < ∞. In a second step, we show that D H s u(t, η(t)) exists for all s, t ∈ [0, T ] and E T 0 T 0 \|D H s u(t, η(t))\| 2 dsdt < ∞.(12) In fact, a straight-forward computation shows that D H s u(t, η(t)) = ∂ ∂x u(t, η(t)) t 0 φ(s − v)σ(v)dv. From the polynomial growth of ∂u ∂x and the continuity of σ, we conclude E\|D H s u(t, η(t))\| 2 ≤ C, s, t ∈ [0, T ], for a suitable C ∈ R, which yields (12). Consequently, the process u satisfies (5) and belongs to L 1,2 H . Let us give now a statement for the Itô formula in the framework of the divergence type integral, which is for our purposes better adapted than Theorem 4.5 [7]. We mention that the formula in the following theorem is a particular case of our generalized Itô formula (67) (see Theorem 34 in the Appendix), but here we use a different approach. Theorem 9 Let u ∈ V T and f ∈ C 0,1 pol ([0, T ] × R) . We put f s = f (s, η(s)), s ∈ [0, T ]. Then for the Itô process X t = X 0 + t 0 f s ds + t 0 u s δB H (s), t ∈ [0, T ], we have (i) uXI [0,t] ∈ Dom(δ), t ∈ [0, T ], (ii) X s ∈ D 1,2 , s ∈ [0, T ], (iii) D H s X s s∈[0,T ] ∈ L 2 ([0, T ] × Ω) and X 2 t = X 2 0 + 2 t 0 X s f s ds + 2 t 0 X s u s δB H (s) + 2 t 0 u s D H s X s ds, a.s. t ∈ [0, T ]. Before proving this theorem, we give the following lemma. Lemma 10 Let u ∈ V T and X t = t 0 u s δB H (s), t ∈ [0, T ]. Then uXI [0,t] ∈ Dom(δ), t ∈ [0, T ], D H s X s s∈[0,T ] ∈ L 2 ([0, T ] × Ω) , and X 2 t = 2 t 0 X s u s δB H (s) + 2 t 0 u s D H s X s ds, t ∈ [0, T ]. Proof. Let F ∈ P T . Then, since obviously X t F ∈ D 1,2 , we have from Definition 1 E X 2 t F = E [X t (X t F )] = E T 0 u s I [0,t](s) δB H (s)(X t F ) = E T 0 u s I [0,t](s) D H s (X t F )ds = E t 0 u s D H s F ds X t + E t 0 u s D H s X t ds F . Here we have used X ∈ L 1,2 H and, hence also XF ∈ L 1,2 H . In particular, we observe that D H s X t = t 0 φ(s − r)u r dr + t 0 D H s u r δB H (s), s, t ∈ [0, T ].(13) Moreover, since t 0 u s D H s F ds ∈ D 1,2 , we get again from Definition 1 E t 0 u s D H s F ds X t = E t 0 u s D H s F ds t 0 u s δB H (s) = E t 0 t 0 D H r u s D H s F u r drds . On the other hand, using (13) it follows E t 0 u s D H s X t ds F = E t 0 t 0 φ(s − r)u s u r dsdr · F + t 0 E t 0 D H r (u s F )D H s u r dr ds. Therefore, by combining the above relations we obtain E X 2 t F = E t 0 u s D H s F ds X t + E t 0 u s D H s X t ds F = E t 0 t 0 D H r u s D H s F u r drds + E t 0 t 0 φ(s − r)u s u r dsdr · F +E t 0 t 0 D H r (u s F )D H s u r drds . By noticing that the right-hand side of the above equality is symmetric in (s, r) we deduce E X 2 t F = 2E t 0 s 0 D H r u s D H s F u r drds + 2E t 0 s 0 φ(s − r)u s u r dsdr · F +2E t 0 s 0 D H r (u s F )D H s u r drds := 2I 1 + 2I 2 + 2I 3 .(14) Let us begin with the evaluation of I 1 . Obviously, by using that uI [0,s] ∈ L 1,2 H ⊂ Dom(δ) and u s D H s F ∈ D 1,2 , we have from Fubini's Theorem and Definition 1 I 1 = E t 0 s 0 D H r u s D H s F u r drds = t 0 E T 0 D H r u s D H s F u r I [0,s](r) dr ds = t 0 E T 0 u r I [0,s](r) δB H (r)u s D H s F ds = E t 0 u s X s D H s F ds .(15) On the other hand, since also D H s uI [0,s] ∈ L 1,2 H ⊂ Dom(δ) and u s F ∈ D 1,2 , s ∈ [0, t], we obtain again from Fubini's Theorem as well as Definition 1 that E t 0 s 0 D H r (u s F )D H s u r drds = t 0 E s 0 D H r (u s F )D H s u r dr ds = t 0 E s 0 D H s u r δB H (r)u s · F ds = E t 0 s 0 D H s u r δB H (r)u s · F ds . Thus, due to (13) I 2 + I 3 = E t 0 s 0 φ(s − r)u s u r dsdr · F + E t 0 s 0 D H r (u s F )D H s u r drds = E t 0 u s s 0 φ(s − r)u r dr + s 0 D H s u r δB H (r) ds · F = E F t 0 u s D H s X s ds .(16) Consequently, from (14)- (16), E 2 T 0 u s X s I [0,t] (s)D H s F ds = E X 2 t − 2 t 0 u s D H s X s ds F , for all F ∈ P T .(17) On the other hand, from Theorem 7.10 [8] and the fact that u ∈ V T , it follows that there exists C > 0 such that E t 0 u s δB H (s) 4 ≤ CE u 4 T + CE T 0 T 0 \|D H t u s \| 2 dsdt 2 ≤ CE T 0 \|u s \| 2 ds 2 + CE T 0 T 0 \|D H t u s \| 4 dsdt ≤ C, for all t ∈ [0, T ], as well as E t 0 D H s u r δB H (r) 4 ≤ CE D H s u 4 T + CE T 0 T 0 \|D H t (D H s u r )\| 2 drdt 2 ≤ C + CE T 0 T 0 \|u xx (r, η(r))\| 4 dr ≤ C, for all t ∈ [0, T ]. Taking into account the definition of the process X, we deduce from the above two estimates and Theorem 3 that uX ∈ L 2 (Ω, F, P ; H) and X 2 t − 2 t 0 u s D H s X s ds ∈ L 2 (Ω, F, P ). Therefore, from (17) and Definition 1 it follows that uXI [0,t] ∈ Dom(δ) and 2 t 0 u s X s δB H (s) = X 2 t − 2 t 0 u s D H s X s ds. Proof of Theorem 9. Let Y t := t 0 u s δB H (s) and Z t := X 0 + t 0 f s ds, t ∈ [0, T ]. From the previous lemma we know that uY I [0,t] ∈ Dom(δ), for all t ∈ [0, T ], and Y 2 t = 2 t 0 u s Y s δB H (s) + 2 t 0 u s D H s Y s ds, t ∈ [0, T ]. On the other hand, it is obvious that Z 2 t = X 2 0 + 2 t 0 f s Z s ds, t ∈ [0, T ]. Moreover, we assert that uZI [0,t] ∈ Dom(δ), for all t ∈ [0, T ], and Y t Z t = t 0 u s Z s δB H (s) + t 0 f s Y s ds + t 0 u s D H s Z s ds, t ∈ [0, T ]. Indeed, since Z t F ∈ D 1,2 and D H s (Z t F ) = D H s (Z s F ) + t s D H s (f r F )dr, s ∈ [0, t], we have E [Y t Z t F ] = E T 0 u s I [0,t] (s)δB H (s) Z t F = E t 0 u s D H s (Z t F )ds = E t 0 u s D H s (Z s F )ds + E t 0 t s u s D H s (f r F )drds = E t 0 u s D H s Z s ds · F + E t 0 u s Z s D H s F ds + t 0 E r 0 u s D H s (f r F )ds dr = E t 0 u s D H s Z s ds · F + E t 0 u s Z s D H s F ds + E t 0 Y r f r F dr . Therefore, E T 0 u s Z s I [0,t] (s)D H s F ds = E Y t Z t − t 0 u s D H s Z s ds − t 0 Y r f r dr F , F ∈ P T , and since uZI [0,t] ∈ L 2 (Ω, F, P ; H) as well as Y t Z t − t 0 u s D H s Z s ds − t 0 Y s f s ds ∈ L 2 (Ω, F, P ), we conclude from Definition 1 that uZI [0,t] ∈ Dom(δ) and t 0 u s Z s δB H (s) = Y t Z t − t 0 u s D H s Z s ds − t 0 Y s f s ds. Consequently, using the above notation as well as the linearity of Dom(δ), we have X t = Y t + Z t , uXI [0,t] ∈ Dom(δ) and X 2 t = Y 2 t + 2Y t Z t + Z 2 t = X 2 0 + 2 t 0 X s f s ds + 2 t 0 X s u s δB H (s) + 2 t 0 u s D H s X s ds, t ∈ [0, T ]. Emphasizing that the Itô-Skorohod integral and the divergence type integral coincide for all u ∈ L 1,2 H . Then from Hu and Peng Lemma 4.2 [10] the following lemma holds true: Lemma 11 Let a, b ∈ C 0,1 pol ([0, T ] × R). If t 0 b(s, η(s))ds + t 0 a(s, η(s))δB H (s) = 0, for all t ∈ [0, T ], then b(s, x) = a(s, x) = 0, for all t ∈ [0, T ], x ∈ R. Quasi-conditional expectation In this subsection, we recall the quasi-conditional expectation which was introduced by Hu and Øksendal [9]. For any n ≥ 1, we introduce the set H ⊗n of all real symmetric Borel functions f n of n variables such that f n 2 H ⊗n := R n ×R n n i=1 φ(s i − r i )f n (s 1 , . . . , s n )f n (r 1 , . . . , r n )ds 1 . . . ds n dr 1 . . . dr n < ∞. Then one can define the iterated integral (see [5,8,9] ) I n (f n ) = n! t 1 <...<tn f n (t 1 , . . . , t n )dB H (t 1 ) · · · dB H (t n ). in the sense of Itô-Skorohod. For n = 0 and f = f 0 be a constant we set I 0 (f 0 ) = f 0 and f 0 2 H ⊗0 = f 2 0 . We recall the following theorem, see Theorem 3.9.9 [5] or [7] (Theorem 6.9) or [9] (Theorem 3.22). Theorem 12 Let F ∈ L 2 (Ω, F, P ). Then there exists f n ∈ H ⊗n ,n ≥ 1 such that F = ∞ n=0 I n (f n ). Moreover, E\|F \| 2 = ∞ n=0 n! f n 2 H ⊗n < ∞. The convergence in this chaos expansion of F is understood in the sense of L 2 (Ω, F, P ). Definition 13 If F ∈ L 2 (Ω, F, P ), then the quasi-conditional expectation is defined aŝ E[F \|F t ] = ∞ n=0 I n (f n I ⊗n [0,t] ), t ∈ [0, T ],(18) where I ⊗n [0,t] (t 1 , . . . , t n ) = I [0,t] (t 1 ) · · · I [0,t] (t n ). Remark 14 Observe that (18) converges in L 2 (Ω, F, P ) andÊ Ê [F \|F t ] F s =Ê[F \|F s ], for 0 ≤ s ≤ t ≤ T . Lemma 15 (Lemma 3.3 [10]) For all f ∈ L 1,2 H and all t ∈ [0, T ], P-a.s. E T t f (u)δB H (u) F t = 0. The following lemma is inspired by Theorem 3.9 [10]. Lemma 16 Let F = f (η(T )), where f : R → R is a continuous function of polynomial growth. Then F ∈ L 2 (Ω, F, P ) and E Ê [F \|F t ] = EF, t ∈ [0, T ]. Proof. First, from the polynomial growth of f and E\|η(T )\| p ≤ M E 1 + T 0 σ(v)dB H (v) p ≤ C p + C p σ p/2 T ≤ M p , p ≥ 1, we obtain F ∈ L 2 (Ω, F, P ). We now put p t (x) = 1 √ 2πt e − x 2 2t , t ∈ (0, T ], x ∈ R, and P t f (x) = R p t (x − y)f (y)dy. Applying (6) to P σ 2 T − σ 2 t f (η(t)) and noticing that ∂ ∂t P t f (x) = 1 2 ∂ 2 ∂x 2 P t f (x), we have f (η(T )) = P σ 2 T − σ 2 t f (η(t)) + T t ∂ ∂x P σ 2 T − σ 2 s f (η(s))σ(s)δB H (s),(19) and, hence Ef (η(T )) = E P σ 2 T − σ 2 t f (η(t)) .(20) On the other hand, from the proof of Theorem 3.8 [10], it follows that E[F \|F t ] = P σ 2 T − σ 2 t f (η(t)).(21) For the reader's convenience, we give a justification for (21) here. By taking t = 0 in (19) we obtain f (η(T )) = P σ 2 T f (η(0)) + T 0 ∂ ∂x P σ 2 T − σ 2 s f (η(s))σ(s)δB H (s). Thus, due to Lemma 15 and Remark 4.10 [9], E [F \|F t ] = P σ 2 T f (η(0)) + t 0 ∂ ∂x P σ 2 T − σ 2 s f (η(s))σ(s)δB H (s). On the other hand, by applying (6) to P σ 2 t − σ 2 s f (η(s)) over time interval [0, t], we get f (η(t)) = P σ 2 t f (η(0)) + t 0 ∂ ∂x P σ 2 t − σ 2 s f (η(s))σ(s)δB H (s). Thus, from semigroup property (20) and (21), we have P σ 2 T − σ 2 s f (x) = P σ 2 T − σ 2 t P σ 2 t − σ 2 s f (x), 0 ≤ s ≤ t ≤ T and ∂ ∂x P σ 2 T − σ 2 s f (x) = P σ 2 T − σ 2 t ∂ ∂x P σ 2 t − σ 2 s f (x), it follows thatÊ [F \|F t ] = P σ 2 T − σ 2 t f (η(t)). Consequently, fromE Ê [F \|F t ] = E P σ 2 T − σ 2 t f (η(t)) = EF. BSDEs driven by B H The objective of this section is to study the BSDE −dY (t) = f (t, η(t), Y (t), Z(t))dt − Z(t)δB H (t) , t ∈ [0, T ], Y (T ) = ξ.(22) We now give the definition of the solution for the above BSDE. (a 1 ) Y ∈V 1/2 T and Z ∈V H T (Recall (10)); (a 2 ) Y (t) = ξ + T t f (s, η(s), Y (s), Z(s)) ds − T t Z(s)δB H (s), t ∈ (0, T ], a.s. Let us begin by discussing the existence of a solution for BSDE (22). Existence We begin with considering the following equation: Y (t) = ξ + T t f (s, η(s), χ(s, η(s)), ψ(s, η(s))) ds − T t Z(s)δB H (s), t ∈ [0, T ],(23) where χ, ψ ∈ C 1,3 pol ([0, T ] × R) with ∂χ ∂t , ∂ψ ∂t ∈ C 0,1 pol ([0, T ] × R). Observe that (23) is a special case of BSDE (22). We mention that in Proposition 4.5 [10], the existence problem of a solution for an equation of type (23) was not considered. Therefore we shall give the following proposition in a rigorous manner: Proposition 18 Under the assumptions (H 1 )-(H 4 ), BSDE (23) has a unique solution (Y, Z) ∈ V T × V T of the form (i)Y (t) = u(t, η(t)), Z(t) = v(t, η(t)), (ii)v(t, x) = σ(t) ∂ ∂x u(t, x). where u, v ∈ C 1,3 pol ([0, T ] × R) with ∂u ∂t , ∂v ∂t ∈ C 0,1 pol ([0, T ] × R) . Before giving the proof, we show the following auxiliary result: Lemma 19 Assume that f ∈ C 0,1 pol ([0, T ] × R) and put f s = f (s, η(s)), s ∈ [0, T ]. Then E T t f s ds F t = T tÊ f s F t ds, P − a.s., t ∈ [0, T ]. Proof. Since f ∈ C 0,1 pol ([0, T ] × R), we know that f s ∈ L 2 (Ω, F, P ), for all s ∈ [0, T ]. From Theorem 12 there exist f n,s ∈ H ⊗n , n ≥ 0 such that f s = ∞ n=0 I n (f n,s ), with I 0 (f 0,s ) = Ef s . Recall that the series converges in L 2 (Ω, F, P ). From the proof of Theorem 3.9.9 [5] we deduce that f n,s is measurable w.r.t. s, for n ≥ 0. Similarly, for g := T t f s ds ∈ L 2 (Ω, F, P ) there exist g n ∈ H ⊗n , n ≥ 0, such that, g = E [F · I n (g n )] = E [I n (h n ) · I n (g n )] = E I n (h n ) · T t f s ds = T t E [I n (h n ) · f s ] ds = T t E [I n (h n ) · I n (f n,s )] ds = T t E [F · I n (f n,s )] ds = E F · T t I n (f n,s )ds . It follows that I n (g n ) = Now we are going to show thatÊ T t f s ds F t = T tÊ f s F t ds. In fact E T t f s ds F t =Ê ∞ n=0 I n (g n ) F t = ∞ n=0 I n (I ⊗n [0,t] g n ) = ∞ n=0 I n (I ⊗n [0,t] T t f n,s ds) = ∞ n=0 I n ( T t I ⊗n [0,t] f n,s ds) = ∞ n=0 T t I n (I ⊗n [0,t] f n,s )ds, where, for the later equality in the above equation, we have used the stochastic Fubini Theorem. Similar to (24), we also obtain Obviously,Ê[M (t)\|F s ] = M (s), 0 ≤ s ≤ t ≤ T (see Remark 14). Using (21) and Lemma 19, we obtain M (t) = P σ 2 T − σ 2 t g(η(t)) + T t P σ 2 u − σ 2 t f (u, η(t) , χ(u, η(t)), ψ(u, η(t)))du + t 0 f (u, η(u), χ(u, η(u)), ψ(u, η(u)))du. Recall the definition of the Malliavin derivative, it follows that that if F ∈ D 1,2 is F t - measurable, then D H s F = 0, ds-a.e. on [t, T ]. Therefore, for s ∈ [0, t] D H s M (t) = σ s P σ 2 T − σ 2 t g ′ (η(t)) + σ s T t P σ 2 u − σ 2 tg (u, η(t))du + σ s t sg (u, η(u))du (25) whereg (u, x) = ∂ ∂x f (u, x, χ(u, x), ψ(u, x)) + ∂ ∂y f (u, x, χ(u, x), ψ(u, x))χ x (u, x) + ∂ ∂z f (u, x, χ(u, x), ψ(u, x))ψ x (u, x). Moreover, (21) and the semigroup property of P u,v yieldŝ E[D H s M (t)\|F s ] = σ s P σ 2 T − σ 2 t P σ 2 t − σ 2 s g ′ (η(s)) +σ s T t P σ 2 u − σ 2 t P σ 2 t − σ 2 sg (u, η(s))du + σ s t s P σ 2 u − σ 2 sg (u, η(u))du = σ s P σ 2 T − σ 2 s g ′ (η(s)) + σ s T s P σ 2 u − σ 2 sg (u, η(s))du.(26) Now, we are going to prove that E\|M (t)\| 2 < ∞. Indeed, E\|M (t)\| 2 ≤ 3E Ê [g(η(T ))\|F t ] 2 + 3E Ê T t f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds F t 2 +3E t 0 f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds 2 , and, similarly to Theorem 3.9 [10], we obtain E Ê [g(η(T ))\|F t ] 2 ≤ E\|g(η(T ))\| 2 On the other hand, from Lemma 19, we have E\|f (s, η(s), χ(s, η(s)), ψ(s, η(s)))\| 2 ds. Consequently, E\|M (t)\| 2 < ∞. Then by using fractional Clark formula (see [8] and [9]) we get M (t) = EM (t) + t 0Ê [D H s M (t)\|F s ]δB H (s).(27) From Lemmas 16 and 19 we have E Ê [g(η(T ))\|F t ] = Eg(η(T )) and E Ê T t f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds F t = T t E Ê f (s, η(s), χ(s, η(s)), ψ(s, η(s))) F t ds = E T t f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds. Consequently, E Ê [M (T )\|F t ] = E Ê [g(η(T ))\|F t ] + E t 0 f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds +E Ê T t f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds F t = Eg(η(T )) + E T 0 f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds = EM (T ). On the other hand, from (21) and (25) we obtain E[D H s M (T )\|F s ] = σ s P σ 2 T − σ 2 s g ′ (η(s)) + σ s T s P σ 2 u − σ 2 sg (u, η(s))du, P − a.s., s ∈ [0, T ]. Then the comparison with (26) yieldŝ E[D H s M (t)\|F s ] =Ê[D H s M (T )\|F s ], P − a.s., s ∈ [0, t].(29) We deduce from (27) Moreover, from Remark 4.10 [9] and (21) Y (t) =Ê[Y (t)\|F t ] =Ê g(η(T )) + T t f (s, η(s), χ(s, η(s)), ψ(s, η(s))) ds F t = P σ 2 T − σ 2 t g(η(t)) + T t P σ 2 s − σ 2 t f (s, η(t), χ(s, η(t)), ψ(s, η(t))) ds, so that it can be easily shown that also Y ∈ V T . Consequently, we have constructed a solution (Y, Z) ∈ V T × V T for BSDE (23). Moreover, Y is continuous since Y ∈ V T . Finally, using that Y, Z ∈ V T , we can find u, v ∈ C 1,3 x). Indeed, by applying (6) we have pol ([0, T ] × R) with ∂u ∂t , ∂v ∂t ∈ C 0,1 pol ([0, T ] × R) such that Y (t) = u(t, η(t)), Z(t) = v(t, η(t)), t ∈ [0, T ]. Then v(t, x) = σ(t) ∂ ∂x u(t,du(t, η(t)) = ∂ ∂t u(t, η(t))dt + σ(t) ∂ ∂x u(t, η(t))δB H (t) + 1 2σ (t) ∂ 2 ∂x 2 u(t, η(t))dt = ∂ ∂t u(t, η(t)) + 1 2σ (t) ∂ 2 ∂x 2 u(t, η(t)) dt + σ(t) ∂ ∂x u(t, η(t))δB H (t) , whereσ(t) := d dt ( σ 2 t ). Consequently u(t, η(t)) = ξ − T t ∂ ∂t u(s, η(s)) + 1 2σ (s) ∂ 2 ∂x 2 u(s, η(s)) ds − T t σ(s) ∂ ∂x u(s, η(s))δB H (s) . From (23) it can be concluded that T t ∂ ∂t u(s, η(s)) + 1 2σ (s) ∂ 2 ∂x 2 u(s, η(s)) ds + T t σ(s) ∂ ∂x u(s, η(s))δB H (s) = − T t f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds + T t v(s, η(s))δB H (s) . Using Lemma 11 we deduce that v(t, x) = σ(t) ∂ ∂x u(t, x), for all t ∈ [0, T ], x ∈ R.(30) It remains to prove that the above solution is the unique one in V T × V T for BSDE (23). Indeed, we suppose that there is another solution (Ỹ ,Z) ∈ V T × V T . Then by applying Theorem 9, using (30) and taking expectation, we have E\|Y (t) −Ỹ (t)\| 2 + 2 M T t s 2H−1 E\|Z(s) −Z(s)\| 2 ds ≤ E\|Y (t) −Ỹ (t)\| 2 + 2 T tσ (s) σ(s) E\|Z(s) −Z(s)\| 2 ds = 2E T t Y (t) −Ỹ (t) Z(t) −Z(t) δB H (s) = 0, for all t ∈ [0, T ], where the latter equality follows the fact that (Y −Ỹ )(Z −Z) ∈ V T . Therefore, taking into account the continuity of Y −Ỹ , the uniqueness follows. Proof. From Theorem 9 and Proposition 18 we deduce that, for t ∈ [0, T ], E\|Y (t)\| 2 + 2 M T t s 2H−1 E\|Z(s)\| 2 ds ≤ E\|Y (t)\| 2 + 2 T tσ (s) σ(s) E\|Z(s)\| 2 ds = 2 T t E Y (s)f (s, η(s), U (s), V (s)) ds ≤ 2 T t E [\|Y (s)\| (L\|U (s)\| + L\|V (s)\| + \|f (s, η(s), 0, 0)\|)] ds ≤ 2 T t E \|Y (s)\| 2 1/2 E (L\|U (s)\| + L\|V (s)\| + \|f (s, η(s), 0, 0)\|) 2 1/2 ds.(32) Let x(t) = E \|Y (t)\| 2 1/2 , t ∈ [0, T ]. Then x 2 (t) ≤ 2 √ 3 T t x (s) E L 2 \|U (s)\| 2 + L 2 \|V (s)\| 2 + \|f (s, η(s), 0, 0)\| 2 1/2 ds, t ∈ [0, T ] . (33) To estimate x(t), we will apply the following inequality: Lemma 21 Let a, α, β : [0, T ] → R + be three nonnegative Borel functions such that a is decreasing and α, β ∈ L 1 loc ([0, ∞]). If x : [0, T ] → R + is a continuous function such that x 2 (t) ≤ a (t) + 2 T t α (s) x (s) ds + 2 T t β (s) x 2 (s) ds, t ∈ [0, T ] , then x (t) ≤ a (t) exp T t β (s) ds + T t α (s) exp s t β (r) dr ds, t ∈ [0, T ] . Remark 22 For this Lemma the reader is referred to Corollary 6.61 [17] . Now from (33) and the above lemma, by setting a(t) = 0, β(s) = 0, α(s) = √ 3 E L 2 \|U (s)\| 2 + L 2 \|V (s)\| 2 + \|f (s, η(s), 0, 0)\| 2 1/2 , s ∈ [0, T ], we have x(t) ≤ √ 3 T t E L 2 \|U (s)\| 2 + L 2 \|V (s)\| 2 + \|f (s, η(s), 0, 0)\| 2 1/2 ds, t ∈ [0, T ], and, hence, for any β > 0, E \|Y (t)\| 2 1/2 ≤ √ 3 T t L E\|U (s)\| 2 1/2 + L E\|V (s)\| 2 1/2 + E \|f (s, η(s), 0, 0)\| 2 1/2 ds ≤ √ 3L T t e −βs e 2βs E\|U (s)\| 2 1/2 + e −βs s H−1/2 s 2H−1 e 2βs E\|V (s)\| 2 1/2 ds + √ 3 T t e −βs e 2βs E \|f (s, η(s), 0, 0)\| 2 1/2 ds ≤ √ 3L T t e −2βs ds 1/2 T t e 2βs E\|U (s)\| 2 ds 1/2 + √ 3L T t e −2βs s 2H−1 ds 1/2 T t s 2H−1 e 2βs E\|V (s)\| 2 ds 1/2 + √ 3 T t e −2βs ds 1/2 T t e 2βs E \|f (s, η(s), 0, 0)\| 2 ds 1/2 . (34) Let us use the following notations: β > 0, e 2βt T t e −2βs s 2H−1 ds ≤ T t (2β(s − t)) −α s 2H−1 ds ≤ 1 (2β) α T 0 1 s α+2H−1 ds < ∞. This allows to conclude from (34), that e 2βt E \|Y (t)\| 2 ≤ 9L 2 2β A 2 t + 9L 2 (2β) α T t (s − t) −α s 2H−1 ds B 2 t + 9 2β C 2 t .(35) Consequently, there exists C (β) with lim β→∞ C (β) = 0, s.t. e 2βt E \|Y (t)\| 2 dt ≤ C(β) A 2 t + B 2 t + C 2 t , t ∈ [0, T ].(36) Applying the Itô formula to \|Y (t)\| 2 , taking the expectation E\|Y (t)\| 2 and then determining the function d e 2βt E\|Y (t)\| 2 and using (35) we obtain (Recall (9) for the definition of M ) e 2βt E\|Y (t)\| 2 + 2β T t e 2βs E\|Y (s)\| 2 ds + 2 M T t s 2H−1 e 2βs E\|Z(s)\| 2 ds ≤ 2 T t e 2βs E [\|Y (s)\| (L\|U (s)\| + L\|V (s)\| + \|f (s, η(s), 0, 0)\|)] ds ≤ 2L T t E e 2βs \|Y (s)\| 2 1/2 E e 2βs \|U (s)\| 2 1/2 ds +2L T t E e 2βs s 2H−1 \|Y (s)\| 2 1/2 E e 2βs s 2H−1 \|V (s)\| 2 1/2 ds +2 T t E e 2βs \|Y (s)\| 2 1/2 E e 2βs \|f (s, η(s), 0, 0)\| 2 1/2 ds ≤ 2L T t C(β) A 2 s + B 2 s + C 2 s 1/2 E e 2βs \|U (s)\| 2 1/2 ds +2L T t 1 s 2H−1 C(β) A 2 s + B 2 s + C 2 s 1/2 E e 2βs s 2H−1 \|V (s)\| 2 1/2 ds +2 T t C(β) A 2 s + B 2 s + C 2 s 1/2 E e 2βs \|f (s, η(s), 0, 0)\| 2 1/2 ds ≤ 2L C(β) (A t + B t + C t ) √ T − t A t + T 2−2H − t 2−2H 2 − 2H B t + √ T − t C t . Thus, the above inequality and (36) allow to conclude inequality (31). Theorem 23 Let the assumptions (H 1 )-(H 4 ) be satisfied. Then the BSDE Y (t) = ξ + T t f (s, η(s), Y (s), Z(s)) ds − T t Z(s)δB H (s), t ∈ [0, T ] has a solution (Y, Z) ∈V 1/2 T ×V H T . Remark 24 Let us mention that it is not clear here, if the solution Y has continuous paths or not. Indeed, since Z does not necessarily belong to L H 1,2 , the divergence integral T t Z(s)δB H (s) can eventually be discontinuous in t. Proof. The existence of the solution is obtained by the Banach fixed point theorem. Let us consider the mapping Γ : V T × V T → V T × V T given by (U, V ) −→ Γ (U, V ) = (Y, Z), where (Y, Z) is the unique solution in V T × V T for the BSDE Y (t) = ξ + T t f (s, η(s), U (s), V (s)) ds − T t Z(s)δB H (s) , t ∈ [0, T ]. First, we remark that Γ is well defined (see Proposition 18). Let us show that Γ is a contraction w.r.t. the norm (u, v) 1/2,H := u 1/2 + v H , for (u, v) ∈ V 1/2 T × V H T (for the definition of · α , see (10)). Taking β large enough such that C(β) ≤ 1/2, then Γ becomes a strict contraction on V T × V T w.r.t. the norm (·, ·) 1/2,H . For (U, V ), (U ′ , V ′ ) ∈ V T × V T and (Y, Z) = Γ(U, V ), (Y ′ , Z ′ ) = Γ(U ′ , V ′ ), we set ∆Y = Y − Y ′ , ∆Z = Z − Z ′ , ∆U = U − U ′ and ∆V = V − V ′ .Now we define {(Y k , Z k )} k∈N recursively by putting Y 0 = χ(t, η(t)), Z 0 = ψ(t, η(t)) for χ, ψ ∈ C 1,3 pol ([0, T ]×R) with ∂χ ∂t , ∂ψ ∂t ∈ C 0,1 pol ([0, T ]×R), and by defining (Y k+1 , Z k+1 ) ∈ V T ×V T through the BSDE Y k+1 (t) = ξ + T t f (s, η(s), Y k (s), Z k (s)) ds − T t Z k+1 (s)dB H (s), t ∈ [0, T ],(38) k ≥ 0. From Proposition 18, we know that for all k ≥ 0, Y k (t) = u k (t, η (t)), Z k (t) = v k (t, η(t)), for suitable u k , v k ∈ C 1,3 pol ([0, T ] × R) with ∂u k ∂t , ∂v k ∂t ∈ C 0,1 pol ([0, T ] × R) such that Z k (t) = σ(t) ∂ ∂x u k (t, η(t)), t ∈ [0, T ]. Since Γ is contraction, which means that {(Y k , Z k )} k∈N is a Cauchy sequence inV 1/2 T ×V H T , then there exists (Y, Z) ∈V 1/2 T ×V H T such that (Y k , Z k ) converges to (Y, Z) inV 1/2 T ×V H T . Moreover, we can prove that (Y, Z) satisfies Y (t) = ξ + T t f (s, η(s), Y (s), Z(s)) ds − T t Z(s)δB H (s) , t ∈ (0, T ] . Indeed, from (37) lim k→∞ E\|Y k (t) − Y (t)\| 2 = 0, t ∈ [0, T ], and lim k→∞ E T 0 \|Y k (s) − Y (s)\| 2 ds + E T 0 s 2H−1 \|Z k (s) − Z(s)\| 2 ds = 0. and Z k 1 [t,T ] → Z1 [t,T ] in L 2 (Ω, F; H). Therefore, using Definition 1, (38) and (40), we see that for all F ∈ P T , E D H · F, Z(·)1 [t,T ] (·) T = lim k→∞ E D H · F, Z k+1 (·)1 [t,T ] (·) T = lim k→∞ E F T t Z k+1 (s)δB H (s) = E(F θ(t)) . From the definition of the divergence operator δ, it follows that Z1 [t,T ] ∈ Dom(δ) and δ(Z1 [t,T ] ) = θ(t). Consequently, we have Y (t) = ξ + T t f (s, η(s), Y (s), Z(s)) ds − T t Z(s)δB H (s), for all t ∈ [ρ, T ], a.s. Considering that ρ is arbitrary, we complete our proof. Proposition 25 Let (Y, Z) ∈ V T × V T be the solution of BSDE (22) constructed in Theorem 23. Then for almost t ∈ (0, T ], D H t Y (t) =σ (t) σ(t) Z(t). Proof. From (38) we know that (Y k , Z k ) ∈ V T × V T satisfies Y k+1 (t) = ξ + T t f (s, η(s), Y k (s), Z k (s)) ds − T t Z k+1 (s)δB H (s), t ∈ [0, T ], k ≥ 1. We recall that Y k (t) = u k (t, η(t)), Z k (t) = v k (t, η(t)), t ∈ [0, T ] and Z k (t) = σ(t) ∂ ∂x u k (t, η(t)). Since (Y k , Z k ) → (Y, Z) inV 1/2 T ×V H T , there exists a subsequence, by convenience still denoted by {(Y k , Z k )} k∈N , such that for arbitrary ρ > 0, we have that As a process with the parameter r, D r Y k (t) = ∂ ∂x u(t, η(t))σ(r)1 [0,t] (r) = σ(r) σ(t) Z k (t)1 [0,t] (r) L 2 ([0,T ]×Ω) −−−−−−−−→ σ(r) σ(t) Z(t)1 [0,t] (r), as k → ∞, for almost all t ∈ [ρ, T ]. On the other hand, since L 2 ([0, T ]) ⊂ H, we conclude that the convergence also holds in L 2 (Ω, F, P ; H). Consequently, in L 2 (Ω, F, P ; H) D r Y (t) = lim k→∞ D r Y k (t) = lim k→∞ σ(r) σ(t) Z k (t)1 [0,t] (r) = σ(r) σ(t) Z(t)1 [0,t] (r), a.e. t ∈ [ρ, T ], and, thus, D H t Y (t) = T 0 φ(t − r)D r Y (t)dr =σ (t) σ(t) Z(t), a.e. t ∈ [ρ, T ], whereσ(t) is defined by (8). Considering that ρ > 0 is arbitrary, we have D H t Y (t) =σ (t) σ(t) Z(t), a.e. t ∈ (0, T ], which completes the proof. Uniqueness Before giving our uniqueness result, we introduce the following spaces: M = X X(t) = X(0) − t 0 v s ds − t 0 u s δB H (s), t ∈ [0, T ] with u ∈ V T , v s = v(s, η(s)), where v ∈ C 0,1 pol ([0, T ] × R) and S f , the set of (Y, Z) ∈V T ×V H T . Applying the Itô formula to Y 2 k+1 (see Theorem 9) and taking the expectation we have 1/2 T ×V H T such that (i) E\|Y (t)\| 2 + 2E T tσ (s) σ(s) \|Z(s)\| 2 ds = E\|ξ\| 2 + 2E T t Y (s)f (s, η(s), Y (s), Z(s))ds, t ∈ [0, T ], (ii) E [Y (t)X(t)] + E T t σ(s) σ(s) u s + D H s X(s) Z(s)ds = E [ξX(T )] +E T t [Y (s)v s + X(s)f (s, η(s),E\|Y k+1 (t)\| 2 + 2E T tσ (s) σ(s) \|Z k+1 (s)\| 2 ds = E\|ξ\| 2 + 2E T t Y k+1 (s)f (s, η(s), Y k (s), Z k (s))ds. (41) Moreover, from (39) we know lim k→∞ E\|Y k (t) − Y (t)\| 2 = 0, t ∈ [0, T ], and lim k→∞ E T t \|Y k (s) − Y (s)\| 2 ds + E T t s 2H−1 \|Z k (s) − Z(s)\| 2 ds = 0, t ∈ [0, T ]. Letting k → ∞ in (41), it follows that, for arbitrary ρ > 0, On the other hand, for any X ∈ M, we deduce from Theorem 9, E\|Y (t)\| 2 + 2EE [Y k+1 (t)X(t)] + E T t u s D H s Y k+1 (s) + Z k+1 (s)D H s X(s) ds = E [ξX(T )] + E T t [Y k+1 (s)v s + X(s)f (s, η(s), Y k (s), Z k (s))] ds. Letting k → ∞ in the above equation and recalling that D H s Y k+1 (s) =σ (s) σ(s) Z k+1 (s), we obtain for arbitrary ρ > 0, E [Y (t)X(t)] + E T t σ(s) σ(s) u s + D H s X(s) Z(s)ds = E [ξX(T )] + E T t [Y (s)v s + X(s)f (s, η(s), Y (s), Z(s))] ds, t ∈ [ρ, T ].(43) Consequently, (42) and (43) yield that (Y, Z) ∈ S f . Now it remains to show the uniqueness in the class S f . We suppose that (Ỹ ,Z) ∈ S f is another solution of BSDE (22). Then for arbitrary ρ > 0, E\|Ỹ (t)\| 2 + 2E T tσ (s) σ(s) \|Z(s)\| 2 ds = E\|ξ\| 2 + 2E T tỸ (s)f (s, η(s),Ỹ (s),Z(s))ds, t ∈ [ρ, T ]. (44) and E Ỹ (t)Y k+1 (t) + E T t 2σ (s) σ(s) Z k+1 (s)Z(s)ds = E\|ξ\| 2 + E T t Ỹ (s)f (s, η(s), Y k (s), Z k (s)) + Y k+1 (s)f (s, η(s),Ỹ (s),Z(s)) ds, t ∈ [ρ, T ], and letting k → ∞, we have E Ỹ (t)Y (t) + E T t 2σ (s) σ(s) Z(s)Z(s)ds = E\|ξ\| 2 + E T t Ỹ (s)f (s, η(s), Y (s), Z(s)) + Y (s)f (s, η(s),Ỹ (s),Z(s)) ds, t ∈ [ρ, T ]. (45) Thus, from (42), (44) and (45) as well as where M is the constant introduced in Remark 6. Then, using Remark 6 E\|Y (t)−Ỹ (t)\| 2 = E\|Y (t)\| 2 −2E Y (t)Ỹ (t) +E\|Ỹ (t)\| 2 we have, for t ∈ [ρ, T ], E\|Y (t) −Ỹ (t)\| 2 + 2EE\|Y (t) −Ỹ (t)\| 2 + 1 M E T t s 2H−1 \|Z(s) −Z(s)\| 2 ds ≤ E T t (L + L 2 M s 1−2H )\|Y (s) −Ỹ (s)\| 2 ds and Gronwall's inequality yields that E\|Y (t) −Ỹ (t)\| 2 + 1 M E T t s 2H−1 \|Z(s) −Z(s)\| 2 ds = 0, t ∈ [ρ, T ]. Since ρ > 0 is arbitrary, our proof is complete now. Fractional backward stochastic variational inequality Let us now consider the following BSVI driven by a fBm: −dY (t) + ∂ϕ(Y (t))dt ∋ f (t, η(t), Y (t), Z(t))dt − Z(t)δB H (t), t ∈ [0, T ], Y (T ) = ξ,(46) where the coefficients satisfy (H 1 )-(H 4 ) and ∂ϕ is the subdifferential of the function ϕ : R → (−∞, +∞] satisfying (H 5 ) ϕ is a lower semicontinuous (l.s.c.) function with ϕ(x) ≥ ϕ(0) = 0, for all x ∈ R and E\|ϕ(ξ)\| < ∞ (Recall that ξ = g(η(T ))). Let us introduce the following notations: Dom ϕ = {u ∈ R : ϕ(u) < ∞}, ∂ϕ(u) = {u * ∈ R : u * (v − u) + ϕ(u) ≤ ϕ(v), for all v ∈ R}, Dom(∂ϕ) = {u ∈ R : ∂ϕ(u) = ∅}, (u, u * ) ∈ ∂ϕ ⇔ u ∈ Dom(∂ϕ), u * ∈ ∂ϕ(u). We know that the multivalued subdifferential operator ∂ϕ is a monotone operator, i.e., (u * − v * )(u − v) ≥ 0, for all (u, u * ), (v, v * ) ∈ ∂ϕ. Now we give the definition of the solution for BSVI (46). In this section, our objective is to show the following existence result: Theorem 28 Let the assumptions (H 1 )-(H 5 ) be satisfied. There exists a solution of BSVI (46). A priori estimates We consider the penalized BSDE by using the Moreau-Yosida approximation of ϕ: Y ε (t) + T t ∇ϕ ε (Y ε (s))ds = ξ + T t f (s, η(s), Y ε (s), Z ε (s)) ds − T t Z ε (s)δB H (s) . (47) Recall that the regularization ϕ ε of ϕ is defined by: ϕ ε (u) := inf 1 2ε \|u − v\| 2 + ϕ(v) : v ∈ R , u ∈ R, ε > 0. It is well-known that ϕ ε is a convex function of class C 1 on R and its gradient ∇ϕ ε is a Lipschitz function with Lipschitz constant 1/ε. Let J ε u = u − ε∇ϕ ε (u), u ∈ R. For all u, v ∈ R and ε, δ > 0, the following properties hold true (see [4] and [16]). (a) ϕ ε (u) = ε 2 \|∇ϕ ε (u)\| 2 + ϕ(J ε u), (b) \|J ε u − J ε v\| ≤ \|u − v\|, (c) ∇ϕ ε (u) ∈ ∂ϕ(J ε u), (d) 0 ≤ ϕ ε (u) ≤ u∇ϕ ε (u), (e) (∇ϕ ε (u) − ∇ϕ δ (v)) (u − v) ≥ − (ε + δ) ∇ϕ ε (u)∇ϕ δ (v).(48) Theorem 29 Let the assumptions (H 1 )-(H 5 ) be satisfied. Then, for all ε > 0, the penalized BSDE (47) has a solution (Y ε , Z ε ) ∈V 1/2 T ×V H T such that, for t ∈ (0, T ], E\|Y ε (t)\| 2 + 2E T tσ (s) σ(s) \|Z ε (s)\| 2 ds = E\|ξ\| 2 + 2E T t Y ε (s)f (s, η(s), Y ε (s), Z ε (s))ds −2E T t Y ε (s)∇ϕ ε (Y ε (s))ds.(49) Proof. In order to use Theorem 26, we mollify ∇ϕ ε in a standard way: (∇ϕ ε ) α (x) := R ∇ϕ ε (x − αu)λ(u)du, x ∈ R, where λ(u) = 1 √ 2π e − u 2 2 , u ∈ R. Considering that ϕ ε is convex and ∇ϕ ε is Lipschitz continuous with Lipschitz constant 1/ε, (∇ϕ ε ) α has the following properties for x 1 , x 2 ∈ R and α, α 1 , α 2 > 0: (i) (∇ϕ ε ) α belongs to C 1 pol (R), and is convex; (ii) \|(∇ϕ ε ) α 1 (x 1 ) − (∇ϕ ε ) α 2 (x 2 )\| ≤ 1 ε \|x 1 − x 2 \| + 1 ε 2 π \|α 1 − α 2 \|. Now we consider the following mollified BSDE Y ε,α (t)+ T t (∇ϕ ε ) α (Y ε,α (s))ds = ξ+ T t f (s, η(s), Y ε,α (s), Z ε,α (s)) ds− T t Z ε,α (s)δB H (s) . (50) From Theorem 26, we obtain that (50) admits a unique solution (Y ε,α , Z ε,α ) in S f,ε,α := S f −(∇ϕε) α ⊂V 1/2 T ×V H T , This solution (Y ε,α , Z ε,α ) can be approximated by the sequence (Y k,ε,α , Z k,ε,α ) ∈ V T × V T , k ≥ 0 constructed by the following method: Define (Y k,ε,α , Z k,ε,α ), k ≥ 0 recursively: Y 0,ε,α = χ(t, η(t)), Z 0,ε,α = ψ(t, η(t)) for χ, ψ ∈ C 1,3 pol ([0, T ] × R) with ∂χ ∂t , ∂ψ ∂t ∈ C 0,1 pol ([0, T ] × R), and let (Y k+1,ε,α , Z k+1,ε,α ) ∈ V T × V T be the unique solution of the BSDE Y k+1,ε,α (t) + T t (∇ϕ ε ) α (Y k,ε,α (s))ds = ξ + T t f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds − T t Z k+1,ε,α (s)δB H (s), t ∈ [0, T ].(51) Similar to (39), we have lim k→∞ E\|Y k,ε,α (t) − Y ε,α (t)\| 2 = 0, t ∈ [0, T ], and lim k→∞ E T 0 \|Y k,ε,α (s) − Y ε,α (s)\| 2 ds + E T 0 s 2H−1 \|Z k,ε,α (s) − Z ε,α (s)\| 2 ds = 0.(52) Moreover, analogously to (42), we show that, for arbitrary ρ > 0, E\|Y ε,α (t)\| 2 + 2E T tσ (s) σ(s) \|Z ε,α (s)\| 2 ds = E\|ξ\| 2 − 2E T t Y ε,α (s) (∇ϕ ε ) α (Y ε,α (s))ds +2E T t Y ε,α (s)f (s, η(s), Y ε,α (s), Z ε,α (s))ds, t ∈ [ρ, T ],(53) and E\|∆Y ε,α 1 ,α 2 (t)\| 2 + 2E T tσ (s) σ(s) \|∆Z ε,α 1 ,α 2 (s)\| 2 ds = 2E T t ∆Y ε,α 1 ,α 2 (s)∆f ε,α 1 ,α 2 (s)ds −2E T t ∆Y ε,α 1 ,α 2 (s) ((∇ϕ ε ) α 1 (Y ε,α 1 (s)) − (∇ϕ ε ) α 2 (Y ε,α 2 (s))) ds, t ∈ [ρ, T ], where ∆Y ε,α 1 ,α 2 (s) = Y ε,α 1 (s) − Y ε,α 2 (s), ∆Z ε,α 1 ,α 2 (s) = Z ε,α 1 (s) − Z ε,α 2 (s) and ∆f ε,α 1 ,α 2 (s) = f (s, η(s), Y ε,α 1 (s), Z ε,α 1 (s)) − f (s, η(s), Y ε,α 2 (s), Z ε,α 2 (s)), s ∈ [0, T ]. Then E\|∆Y ε,α 1 ,α 2 (t)\| 2 + 2E T tσ (s) σ(s) \|∆Z ε,α 1 ,α 2 (s)\| 2 ds ≤ E T t (2L + L 2 M s 1−2H )\|∆Y ε,α 1 ,α 2 (s)\| 2 ds +E T t 1 M s 2H−1 \|∆Z ε,α 1 ,α 2 (s)\| 2 ds + 3 ε E T t \|∆Y ε,α 1 ,α 2 (s)\| 2 ds + 2T π \|α 1 − α 2 \| 2 , t ∈ [ρ, T ], (by using the property for (∇ϕ ε ) α ) where M is the constant given by Remark 6. Then, using (9) we obtain E\|∆Y ε,α 1 ,α 2 (t)\| 2 + 1 M E T t s 2H−1 \|∆Z ε,α 1 ,α 2 (s)\| 2 ds ≤ 2T π \|α 1 − α 2 \| 2 + E T t (2L + 3 ε + L 2 M s 1−2H )\|∆Y ε,α 1 ,α 2 (s)\| 2 ds, t ∈ [ρ, T ], and Gronwall's inequality yields that E\|∆Y ε,α 1 ,α 2 (t)\| 2 + 1 M E T t s 2H−1 \|∆Z ε,α 1 ,α 2 (s)\| 2 ds ≤ M ε,L,T \|α 1 − α 2 \| 2 , t ∈ [ρ, T ], where M ε,L,T is a constant depending only on ε, L, T but independent of ρ > 0. Consequently, taking into account the arbitrariness of ρ > 0, there exists a couple of processes (Y ε , Z ε ) with Y ε 1 [ρ,T ] ∈V 1/2 T , Z ε 1 [ρ,T ] ∈V H T for all ρ > 0, such that lim α→0 E\|Y ε,α (t) − Y ε (t)\| 2 = 0, for all t ∈ [ρ, T ], lim α→0 E T t \|Y ε,α (s) − Y ε (s)\| 2 ds + E T t s 2H−1 \|Z ε,α (s) − Z ε (s)\| 2 ds = 0, for all t ∈ [ρ, T ],(54) Now let α → 0, and by using (50) and a similar discussion as in Theorem 23, we obtain that Z ε 1 [t,T ] ∈ Dom(δ), t ∈ (0, T ], and Y ε (t)+ T t ∇ϕ ε (Y ε (s))ds = ξ + T t f (s, η(s), Y ε (s), Z ε (s)) ds − T t Z ε (s)δB H (s), t ∈ (0, T ]. Moreover, taking α → 0 in (53) yields that E\|Y ε (t)\| 2 + 2E T tσ (s) σ(s) \|Z ε (s)\| 2 ds = E\|ξ\| 2 + 2E T t Y ε (s)f (s, η(s), Y ε (s), Z ε (s))ds −2E T t Y ε (s)∇ϕ ε (Y ε (s))ds, t ∈ (0, T ]. The next three propositions provide a priori estimates for the sequence (Y ε , Z ε ), ε > 0. Proposition 30 Let the assumptions (H 1 )-(H 5 ) be satisfied. Let (Y ε , Z ε ) be the solution constructed in the proof of Theorem 29. Then there exists a positive constant C independent of ε > 0, such that, for all t ∈ [0, T ], E\|Y ε (t)\| 2 + E T t s 2H−1 \|Z ε (s)\| 2 ds ≤ C Γ 1 (T ), where Γ 1 (T ) = E \|ξ\| 2 + Proof. From (49), (9-b) and u∇ϕ ε (u) ≥ 0, for all u ∈ R, we have, for t ∈ (0, T ], E \|Y ε (t)\| 2 + 2 M E T t s 2H−1 \|Z ε (s)\| 2 ds ≤ E \|ξ\| 2 + 2 T t E [Y ε (s)f (s, η(s), Y ε (s), Z ε (s))] ds. On the other hand, from assumption (H 3 ) and Schwartz's inequality we obtain 2Y ε (s)f (s, η(s), Y ε (s), Z ε (s)) ≤ 2 \|Y ε (s)\| \|f (s, 0, 0, 0)\| + L \|η(s)\| + L \|Y ε (s)\| + L \|Z ε (s)\| ≤ \|f (s, 0, 0, 0)\| 2 + \|η(s)\| 2 + 1 + L 2 + 2L + L 2 M 1 s 2H−1 \|Y ε (s)\| 2 + 1 M s 2H−1 \|Z ε (s)\| 2 . Then, E \|Y ε (t)\| 2 + 1 M E T t s 2H−1 \|Z ε (s)\| 2 ds ≤ E\|ξ\| 2 + T t E \|f (s, 0, 0, 0)\| 2 ds + T t E \|η(s)\| 2 ds + T t 1 + L 2 + 2L + L 2 M 1 s 2H−1 E \|Y ε (s)\| 2 ds. Therefore, by Gronwall's inequality we deduce that E \|Y ε (t)\| 2 + 1 M E T t s 2H−1 \|Z ε (s)\| 2 ds ≤ Γ 1 (T ) exp 1 + L 2 + 2L (T − t) + L 2 M T 2−2H − t 2−2H 2 − 2H , which completes the proof. Proposition 31 Let the assumptions (H 1 )-(H 5 ) be satisfied. Then there exists a positive constant C such that, for all t ∈ [0, T ], (i) E T t s 2H−1 \|∇ϕ ε (Y ε (s))\| 2 ds ≤ C Γ 2 (T ), (ii) t 2H−1 E [ϕ (J ε (Y ε (t)))] ≤ C Γ 2 (T ), (iii) t 2H−1 E \|Y ε (t) − J ε (Y ε (t))\| 2 ≤ εC Γ 2 (T ), where Γ 2 (T ) = E \|ξ\| 2 + ϕ(ξ) + T 0 \|η(s)\| 2 ds + T 0 \|f (s, 0, 0, 0)\| 2 ds . In order to obtain the above propsition it is essential to use the following fractional stochastic subdifferential inequality: Lemma 32 Let ψ : R → R + be a convex C 1 function which derivative ∇ψ is a Lipschitz function (with Lipschitz constant denoted by K). Then, for all t ∈ (0, T ], P -a.s. Proof. We first show that t 2H−1 E ψ Y k+1,ε,α (t) + E T t s 2H−1 ∇ψ Y k+1,ε,α (s) (∇ϕ ε ) α (Y k,ε,α (s))ds ≤ T 2H−1 E [ψ (ξ)] + E T t s 2H−1 ∇ψ Y k+1,ε,α (s) f (s, η(s), Y k,ε,α (s), Z k,ε,α (s))ds, where (Y k,ε,α , Z k,ε,α ) ∈ V T ×V T is defined through (51). We mollify the function ψ by setting, for θ > 0, ψ θ (x) := R ψ(x − θu)λ(u)du, x ∈ R, where λ(u) = 1 √ 2π e − u 2 2 , u ∈ R. From the convexity of ψ follows that ψ θ is convex. Moreover ψ θ ≥ 0. The generalized Itô formula (see (67) in Remark 35) yields T 2H−1 ψ θ (ξ) = t 2H−1 ψ θ (Y k+1,ε,α (t)) + (2H − 1) t 2H−1 E [ψ (Y ε (t))] + E T t s 2H−1 ∇ψ (Y ε (s)) ∇ϕ ε (Y ε (s))ds ≤ T 2H−1 E [ψ(ξ)] + E T t s 2H−2 ψ θ (Y k+1,ε,α (s))ds − T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s))f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds + T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds + T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s))Z k+1,ε,α (s)δB H (s) + T t s 2H−1 D 2 xx ψ θ (Y k+1,ε,α (s))Z k+1,ε,α (s)D H s Y k+1,ε,α (s)ds. (56) Now taking the expectation in (56), by using ψ θ ≥ 0, the convexity of ψ θ and the fact that D H s Y k+1,ε,α (s) =σ (s) σ(s) Z k+1,ε,α (s), we have t 2H−1 E ψ θ (Y k+1,ε,α (t)) + E T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds ≤ T 2H−1 Eψ θ (ξ) + E T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s))f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds. Considering that ∇ψ θ (x) − ∇ψ(x) = ∇ R ψ(x − θu)λ(u)du − ∇ψ(x) ≤ R \|∇ψ(x − θu) − ∇ψ(x)\| λ(u)du ≤ 2 π K\|θ\|, we have E T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds − T t s 2H−1 ∇ψ(Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds ≤ T 2H−1 2 π K\|θ\|E T t (∇ϕ ε ) α (Y k,ε,α (s)) ds → 0, as θ → 0. Similarly, we get E T t s 2H−1 ∇ψ θ (Y k+1,ε,α (s))f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds → E T t s 2H−1 ∇ψ(Y k+1,ε,α (s))f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds → 0, as θ → 0. Moreover, using Fatou's Lemma (recalling that ψ ≥ 0), we obtain E ψ(Y k+1,ε,α (t)) = E lim inf θ→0 ψ θ (Y k+1,ε,α (t)) ≤ lim inf θ→0 E ψ θ (Y k+1,ε,α (t)) . On the other hand, we know that ψ is quadratic growth, therefore there exists a suitable constant C, such that \|ψ θ (x)\| ≤ R \|ψ(x − θu)\|λ(u)du ≤ C(1 + x 2 + θ 2 ). From (H 4 ) and (11), it follows sup θ≤1 E \|ψ θ (ξ)\| 2 < ∞, which implies that {ψ θ (ξ)} θ≤1 is uniformly integrable. Then considering that ψ θ (ξ) Consequently, letting θ → 0 in (57) we have t 2H−1 E ψ(Y k+1,ε,α (t)) + E T t s 2H−1 ∇ψ(Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds ≤ T 2H−1 E [ψ(ξ)] + E T t s 2H−1 ∇ψ(Y k+1 ,ε,α (s))f s, η(s), Y k,ε,α (s), Z k,ε,α (s) ds. Recalling that (see (52) ψ(Y ε,α (t)), for all t ∈ [0, T ]. Thus, using Fatou's Lemma once again (recalling that ψ ≥ 0), we obtain E [ψ(Y ε,α (t))] ≤ lim inf k→∞ E ψ(Y k+1,ε,α (t)) . Now we are able to give the proof of Proposition 31. From Proposition 31 (iii), we deduce that lim ε→0 E \|Y ε (t) − J ε (Y ε (t))\| 2 = 0 for all t ∈ [0, T ], lim ε→0 J ε (Y ε ) = Y inV H T .(64) For each ε > 0, let U ε (t) = ∇ϕ ε (Y ε (t)), t ∈ [0, T ]. The process U ε belongs to the spaceV H T (see Lemma 37 in the Appendix E b a s 2H−1 1 A U ε (t) V (t) − J ε (Y ε (t)) dt + E b a s 2H−1 1 A ϕ(J ε (Y ε (t)))dt ≤ E b a s 2H−1 1 A ϕ(V (t))dt . Considering that ϕ is a proper convex l.s.c. function, hence (64) and (65) yield that E b a s 2H−1 1 A U (t)(V (t) − Y (t))dt + E b a s 2H−1 1 A ϕ(Y (t))dt ≤ E b a s 2H−1 1 A ϕ(V (t))dt , for all A × [a, b] ⊂ Ω × [0, T ]. Therefore, U (t)(V (t) − Y (t)) + ϕ(Y (t)) ≤ ϕ(V (t)) dP ⊗ dt a.e. on Ω × [0, T ], which means that (Y (t), U (t)) ∈ ∂ϕ, dP ⊗ dt a.e. on Ω × [0, T ]. This together with (66) complete the proof. Appendix Theorem 34 Let ψ be a function of class C 1,2 ([0, T ] × R). Assume that u is a process in V T and f ∈ C 0,1 pol ([0, T ] × R). Let s)δB H (s), t ∈ [0, T ]. E\|u(s, η(s))\| 2 ≤ CE(1 + \|η(s)\| + . . . + \|η(s)\| k ), s ∈ [0, T ]. * C k,l pol ([0, T ] × R m ) is the space of all C k,l -functions over [0, T ] × R m , which together with their derivatives, are of polynomial growth. Definition 17 A pair (Y, Z) is called a solution of BSDE (22), if the following conditions are satisfied: (g n ) with I 0 (g 0 ) = Eg. Let us show that we can choose g n = n (f n,s )\| 2 ds = E T t \|f s \| 2 ds < ∞, and hence I n (f n,s ) is square integrable w.r.t. s. Now, for arbitrary F ∈ L 2 (Ω, F, P ) with the chaos expansion F = ∞ n=0 I n (h n ), we have from Fubini's Theorem (f n,s )ds, n ≥ 0, and the stochastic Fubini Theorem (see Theorem 1.13.1[12]) yields I n (g n ,s ds . Thus, we can indeed choose g n =T t f n,s ds, n ≥ 0 and I ⊗n [0,t] g n = I ⊗n [0,t] T t f n,s ds. Consequently, we have lim N →∞ N n=0 I n ( T t f n,s ds) = lim N →∞ N n=0 I n (g n ) = g = T t f s ds, in L 2 (Ω, F, P ). Iff n (I ⊗n [0,t] f n,s )ds, P − a.s. and the right hand side of above equation is nothing else but T tÊ [f s \|F t ] ds, which completes our proof.After the above auxiliary result we can now prove our Proposition 18.Proof of Proposition 18. Using similar arguments to those of the proof of Lemma 8(s, η(s), χ(s, η(s)), ψ(s, η(s)))ds ∈ L 2 (Ω, F, (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds F t , t ∈ [0, T ]. (T )\|F s ]δB H (s). Let us now introduce the process Z(s) =Ê[D H s M (T )\|F s ], s ∈ [0, T ]. Similarly as Proposition 4.5 [10], using the property of the operator P σ 2 T − σ 2 s , we can prove that Z ∈ V T . Moreover, in virtue of the latter relation we have M (t) = EM (T ) + t 0 Z(s)δB H (s), P − a.s. t ∈ [0, T ]. f (s, η(s), χ(s, η(s)), ψ(s, η(s)))ds, t ∈ [0, T ].Then,Y (T ) − Y (t) = M (T ) − M (t) Proposition 20 s 20Let the assumptions (H 1 )-(H 4 ) be satisfied. For (U, V ) ∈ V T × V T , let (Y, Z) ∈ V T × V T be the unique solution of the following BSDE Y (t) = T t f (s, η(s), U (s), V (s)) ds − T t Z(s)δB H (s), t ∈ [0, T ].Then, for all β > 0, there exists C (β) ∈ R (depending also on L and T 2H−1 e 2βs E\|V (s)\| 2 ds + T 0 e 2βs \|f (s, η(s), 0, 0)\| 2 ds .(31) Moreover, C(β) can be chosen such that lim β→∞ C (β) = 0. e −2βt − e −2βT we have for α > 0 with 0 < α < 2 − 2H < 1 and Using Proposition 20 we know that there exists C(β) which can depend on L and T , such that lim s 2H−1 e 2βs E\|∆V (s)\| 2 ds . arbitrary ρ > 0 and for all t ∈ [ρ, T ], lim k→∞ −Y k+1 (t) + ξ + T t f (s, η(s), Y k (s), Z k (s)) ds = −Y (t) + ξ + T t f (s, η(s), Y (s), Z(s)) ds := θ(t), in L 2 (Ω, F, P ), lim k→∞ E\|Y k (s) − Y (s)\| 2 = 0 and lim k→∞ E\|Z k (s) − Z(s)\| 2 = 0, for almost all s ∈ [ρ, T ]. Y (s), Z(s))] ds, X ∈ M.Theorem 26 Let the assumptions (H 1 )-(H 4 ) be satisfied. Then BSDEY (t) = ξ + T t f (s, η(s), Y (s), Z(s)) ds − T t Z(s)δB H (s) has a unique solution (Y, Z) ∈ S f .Proof. We show first that the solution (Y, Z) we constructed in the proof of Theorem 23 belongs to S f . Indeed, the sequence {(Y k , Z k )} k∈N introduced in the proof of Theorem 23 is in V T × V T and converges to (Y, Z) inV 1/2 \|Z(s)\| 2 ds = E\|ξ\| 2 + 2E T t Y (s)f (s, η(s), Y (s), Z(s))ds, t ∈ [ρ, T ]. (42) Y (s) −Ỹ (s) f (s, η(s), Y (s), Z(s)) − f (s, η(s),Ỹ (s),Z(s)) ds ≤ E T t L\|Y (s) −Ỹ (s)\| 2 + L 2 M s 1−2H \|Y (s) −Ỹ (s)\| 2 + 1 M s 2H−1 \|Z(s) −Z(s)\| 2 ds. Definition 27 A triple (Y, Z, U ) is a solution for BSVI (46), if: (a 1 ) Y, U ∈V H T and Z ∈V 2 ) (Y (t), U (t)) ∈ ∂ϕ, dP ⊗ dt a.e.on Ω × [0, T ], s)δB H (s), t ∈ (0, T ], a.s. s)\| 2 ds + T 0 \|f (s, 0, 0, 0)\| 2 ds . s 2H−1 ∇ψ (Y ε (s)) f (s, η(s), Y ε (s), Z ε (s))ds.(55) ξ), we have E[ψ θ (ξ)] → E[ψ(ξ)]. s k,ε,α (t) − Y ε,α (t)\| 2 = 0, t ∈ [0, T ]2H−1 \|Z k,ε,α (s) − Z ε,α (s)\| 2 ds = 0, it follows that Y k,ε,α (t) k→∞ − −−− → P −a.s. Y ε,α (t), for all t ∈ [0, T ] and then also ψ(Y k,ε,α (t)) k→∞ − −−− → P −a.s. D δB H (s), s ∈ [0, T ]Then for all t ∈ [0, T ], the following formula holdsψ(t, X t ) = ψ(0, X 2 ψ(s, X s )u s D H s X s ds.(67)Remark 35 Since u ∈ V T , we know that u is adapted. Then, due to the definition of D r f (θ, η(θ))dθ dr +T 0 φ(s − r) s 0 D r u θ δB H (θ) dr + s 0 φ(s − r)u θ dθ. Let us recall a result about a sufficient condition for the existence of the divergence type integral. For this we use the Itô-Skorohod type stochastic integral introduced in Definition 6.11Definition 2.2.2 and 2.2.3). If u ∈ Dom(δ), δ(u) is unique, and we define the divergence type integral of u ∈ Dom(δ) w.r.t. fBm B H by putting T 0 u(s)δB H (s) := δ(u). ). From Proposition 31 (i), we obtain that \|\|U ε \|\| 2 H = E and we deduce that, for all A × [a, b] ⊂ Ω × [0, T ], A ∈ F, t i+1 t i u θ δB H (θ) 2 . AcknowledgementThe authors wish to express their thanks to Rainer Buckdahn, Yaozhong Hu, Shige Peng and Aurel Rȃşcanu for their useful suggestions and discussions.Considering that ∇ψ and (∇ϕ ε ) α are Lipschitz with the Lipschitz constant K and 1/ε, respectively, we get E T t s 2H−1 ∇ψ(Y k+1,ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s))ds − T t s 2H−1 ∇ψ(Y ε,α (s)) (∇ϕ ε ) α (Y ε,α (s))ds ≤ T 2H−1 E T t ∇ψ(Y k+1,ε,α (s)) − ∇ψ(Y ε,α (s)) (∇ϕ ε ) α (Y k,ε,α (s)) dsIndeed, using that the functions ∇ψ and (∇ϕ ε ) α are of linear growth, E s 2H−1 ∇ψ(Y ε,α (s))f (s, η(s), Y ε,α (s), Z ε,α (s)) ds → 0, as k → ∞.Again with the help of Fatou's Lemma we see thatConsequently, letting k → ∞ in (58) yields thatA similar argument allows to take the limit α → 0 in (59) (using (54)), it followsand the statement is proven.Proof of Proposition 31. We consider ψ(x) = ϕ ε (x), x ∈ R, and applying (55) we haveConsidering Proposition 30, we see thatTherefore, since ϕ(J ε u) ≤ ϕ ε (u), u ∈ R, we have proven (i) and (ii) of the proposition. Finally, in order to obtain (iii) is suffices to remark that \|u − J ε (u)\| 2 = \|∇ϕ ε (u)\| 2 ≤ 2εϕ ε (u), u ∈ R.Proposition 33 Let the assumptions (H 1 )-(H 5 ) be satisfied. Then there exists a positive constant C such that, for all ε, δ > 0,Proof. Similarly to (49), we have, for t ∈ (0, T ],whereas well as (60), we deduce that for t ∈ (0, T ],SinceBy using the following inequality (see (48-e))and Proposition 31 (i) as well as Gronwall's inequality, we conclude from (62) that there exists C > 0 such thatand the proof is complete.Proof of the existence of the solutionProof of Theorem 28. For arbitrary ρ > 0, by Proposition 33, there exist (Y,Proof of Theorem 34. We follow the similar discussion as in the proof of Theorem 8[3].Here, we only give the sketch of the proof. First we mention that since u ∈ V T , we haveSimilar to the discussion as in the proof of Theorem 8[3], we can assume that ψ, ∂ψ ∂t ,(for ·, · T , see(2)). Observe that from our assumption, ∂ ∂x ψ(t i , X t i )u ∈ D 1,2 (\|H\|) and all the terms in the above inequality are square integrable. Moreover,Now we use the following several steps to proof our theorem.Step 1 The term 1 2converges to 0 in L 1 (Ω, F, P ) as n → ∞. In fact,Then our result holds because of Proposition 7[3]as well asE\|f (θ, η(θ))\| 2 ds → 0, as n → ∞.Step 2 The termby the dominate convergence theorem and the continuity of ∂ψ ∂t .Step 3 The termby the dominate convergence theorem and the continuity of ∂ψ ∂x .Step 4 The terms 0 D r f (θ, η(θ))dθ dr ds in L 1 (Ω, F, P ) as n → ∞. IndeedD H r f (θ, η(θ))dθ dr ds → 0, as n → ∞ by the dominate convergence theorem and the continuity of\|D H r f (θ, η(θ))\|dθ dr ds < ∞Step 5 Analogously to the Step 4, we getin L 1 (Ω, F, P ) as n → ∞.Step 6 The termin L 1 (Ω, F, P ), as n → ∞. Indeed, we can adapt the discussion of step 3 in the proof of Theorem 8[3], by using E\|u s \| 2 + E\|D H τ u s \| 2 + E\|D H τ 1 D H τ 2 u s \| 2 ≤ C, for all s, τ, τ 1 , τ 2 , where C is a suitable constant.Step 7 The termIn factwhich converges to 0 by the dominated convergence theorem and the continuity of ∂ψ ∂x . Note that for u ∈ V T and ∂ψ ∂x , ∂ 2 ψ ∂x 2 bounded, we have ∂ψ ∂x u ∈ D 1,2 (\|H\|) ⊂ Dom(δ). Consequently, for F ∈ P T , we haveOn the other hand, from steps 1-6, we know thatas n → ∞, which allows to complete the proof.In particular, we have the following corollary.Corollary 36 Let f : [0, T ] → R and g : [0, T ] → R be deterministic continuous functions. Ifand ψ ∈ C 1,2 ([0, T ] × R), then we haveProof. Since f, g are deterministic function, they satisfy the condition of Theorem 34. Then from (67) and Remark 35 we haveOn the other handwhich completes our proof.Lemma 37 If Y ∈V α T and ψ is a Lipschitz function, then ψ(Y ) ∈V α T .Proof. We remark first that, since Y ∈V α T , there exists a sequence (Y n ) n ⊂ V T such that limwhere δ > 0 and ρ(u) = 1 √ 2π e − u 2 2 , u ∈ R. We know thatwhere K is a Lipschitz constant of ψ. Now we haveConsequently, T 0 t 2α−1 E\|ψ δ (Y n (t)) − ψ(Y (t))\| 2 dt → 0, for n → ∞, δ → 0, and the proof is completed by showing that ψ δ (Y n ) ∈ V T , for all n and δ > 0. First it is obvious that ψ δ ∈ C ∞ and \|ψ δ (x) − ψ δ (y)\| ≤ R \|ψ(x − δu) − ψ(y − δu)\|ρ(u)du ≤ K\|x − y\|, which implies that d dx ψ δ (x) ≤ K.Moreover, a straight-forward computation shows that the derivatives d 2 dx 2 ψ δ (x) and d 3 dx 3 ψ δ (x) have polynomial growth. Recalling that Y n belongs to V T , we complete the proof. 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[ "EQUILIBRIUM RESTRICTIONS AND APPROXIMATE MODELS: PRICING MACROECONOMIC RISK", "EQUILIBRIUM RESTRICTIONS AND APPROXIMATE MODELS: PRICING MACROECONOMIC RISK" ]	[ "Andreas Tryphonides \nHUMBOLDT UNIVERSITY\n\n" ]	[ "HUMBOLDT UNIVERSITY\n" ]	[]	Economists are often confronted with the choice between a completely specified, yet approximate model and an incomplete model that only imposes a set of behavioral restrictions that are believed to be true. We offer a reconciliation of these approaches and demonstrate its usefulness for estimation and economic inference. The approximate model, which can be structural or statistical, is distorted such that to satisfy the equilibrium conditions which are deemed as credible. We provide the relevant asymptotic theory and supportive simulation evidence on the MSE performance in small samples. We illustrate that it is feasible to do counterfactual experiments without explicitly solving for the equilibrium law of motion. We apply the methodology to the model of long run risks in aggregate consumption(Bansal and Yaron, 2004), where the auxiliary model is generated using the Campbell and Shiller (1988) approximation. Using US data, we investigate the empirical importance of the neglected non-linearity. We find that the distorted model is strongly preferred by the data and substantially improves the identification of the structural parameters. More importantly, it completely overturns key qualitative predictions of the linear model, such as the absence of endogenous time variation in risk premia and level effects, which is crucial for understanding the link between asset prices and macroeconomic risk.	null	[ "https://arxiv.org/pdf/1805.10869v1.pdf" ]	62,835,868	1805.10869	c55f5323cb497abc9dda80529b6c5afa33133647	EQUILIBRIUM RESTRICTIONS AND APPROXIMATE MODELS: PRICING MACROECONOMIC RISK Date: 27/05/2018. 28 May 2018 Andreas Tryphonides HUMBOLDT UNIVERSITY EQUILIBRIUM RESTRICTIONS AND APPROXIMATE MODELS: PRICING MACROECONOMIC RISK Date: 27/05/2018. 28 May 2018Parts of this paper draw from chapter 3 of my PhD thesis (EUI). I thank Fabio Canova, Peter Reinhard Hansen, Giuseppe Ragusa and Frank Schorfheide for useful comments and suggestions. Earlier versions of this paper (circulated with different titles) greatly benefited from discussions with Raffaella Giacomini, George Tauchen and comments from the participants at the 23rd MEG (Bloomington), the 1st IAAE (London), the 68th ESEM (Toulouse), the 4th International Conference in memory of Carlo Giannini (Pavia), the Econometrics Study Group (Bristol), the Economic Risk seminar (Humboldt U), the EUI Econometrics Working Group and the University of Wisconsin Madison lunch seminar. Any errors are my own. 1Approximate modelsInformation projectionsSmall SamplesRisk pre- miumNon-linearityStochastic volatility JEL Classification: C10E44 Economists are often confronted with the choice between a completely specified, yet approximate model and an incomplete model that only imposes a set of behavioral restrictions that are believed to be true. We offer a reconciliation of these approaches and demonstrate its usefulness for estimation and economic inference. The approximate model, which can be structural or statistical, is distorted such that to satisfy the equilibrium conditions which are deemed as credible. We provide the relevant asymptotic theory and supportive simulation evidence on the MSE performance in small samples. We illustrate that it is feasible to do counterfactual experiments without explicitly solving for the equilibrium law of motion. We apply the methodology to the model of long run risks in aggregate consumption(Bansal and Yaron, 2004), where the auxiliary model is generated using the Campbell and Shiller (1988) approximation. Using US data, we investigate the empirical importance of the neglected non-linearity. We find that the distorted model is strongly preferred by the data and substantially improves the identification of the structural parameters. More importantly, it completely overturns key qualitative predictions of the linear model, such as the absence of endogenous time variation in risk premia and level effects, which is crucial for understanding the link between asset prices and macroeconomic risk. Introduction The use of estimated structural models has become pervasive in both academia and economic policy institutions. In order to answer quantitative questions within a data coherent framework, practitioners have resorted to a variety of full or limited information methods. Nevertheless, while economic theory provides a set of equilibrium conditions, it rarely dictates the complete probability distribution of observables. The latter is necessary to perform full information analysis i.e. counter-factual experiments and probabilistic forecasts, and this forces users to make several auxiliary assumptions. For example, one has to choose which solution concept to use and type (and degree) of approximation to consider. Although approximations make computation of the solution of the model easier, this can possibly cause a form of misspecification with respect to the exact model. Approximations to non linear models might not necessarily work well, as they can distort the dynamics implied by the model (e.g. Haan and Wind (2012)). Distorting the dynamics can lead to severely wrong inference about parameters, policy recommendations and the relative importance of different mechanisms. Moreover, as shown by Canova and Sala (2009), approximation and model solution can introduce further uncertainties like loss of identification. For all of the aforementioned reasons and despite the significant advances in computing power, researchers often resort to employing incomplete models 1 . The most prominent approach to estimating models that are not completely specified is the Generalized Method of Moments (GMM) and its variants (Hansen, 1982). Nevertheless, the performance of GMM is distorted in small samples (Hansen, Heaton, and Yaron, 1996). In addition, as we already hinted to, incomplete models are not easily amenable for full information exercises. This paper offers a possible reconciliatory approach which enables estimating the parameters of a dynamic structural model, does not require the equilibrium decision rules and still produces an estimated probability model for the observables. To obtain the latter, we use what we refer to as a "base" conditional probability measure with density f pX\|Z, ϕq where Z is conditioning information. This measure can be generally interpreted as an approximate model for the observables, statistical or structural. The latter can obviously include models that are obtained using conventional approximation methods e.g. log-linearization. Utilizing a variation of the method of information projections (Kitamura and Stutzer, 1997;I.Csiszar, 1975) we obtain a probability distribution that satisfies the conditional restrictions of the economic model, that is EpmpX, ϑq\|zq " 0, and is as close as possible to the base measure. This is also related to the work of Giacomini and Ragusa (2014); Robertson, Tallman, and Whiteman (2005) in a forecasting context. This paper explores the econometric properties of this approach by explicitly acknowledging that the benchmark model can be locally misspecified. More importantly, it illustrates that this approach can be used to account for losses due to approximations in the case when the benchmark model is indeed an approximation to the true economic structure 2 . We develop the corresponding frequentist inference, while we limit the analysis to the case of finite dimensional ϕ. However, extensions under suitable assumptions are possible 3 . Furthermore, we deal with correctly specified or locally misspecified classes 2 Linearized Dynamic Stochastic Equilibrium models is a prime example. 3 Independent work by Shin (2014) proposes Bayesian algorithms to implement the exponential tilting estimation using flexible mixtures of densities. Our contribution is mostly on the frequentist properties of exponential tilting for a general parametric family of densities and our results are therefore complementary of f pX\|Z, ϕq. More interestingly, an explicit form of the asymptotic variance of the estimator is provided. Under the condition that there exists an admissible parameter of f pX\|Z, ϕq such that the moment conditions are satisfied asymptotically,θ attains the semi-parametric lower bound (see Chamberlain (1987)). Moreover, simulation comparisons of the Mean Squared Error (MSE) show that local misspecification of the density in the form of improper finite dimensional restrictions leads to efficiency gains and therefore a bias -variance trade-off in finite samples. We thus conclude that the approximate model can actually be helpful when samples are small. In addition, we illustrate that it is feasible to do counter-factual experiments without explicitly solving for the equilibrium law of motion. This is very important as we can indeed make use of equilibrium restrictions to investigate counterfactual paths without specifying the complete non-linear probability model. We apply the method to pricing long run risks in aggregate consumption (Bansal and Yaron, 2004), where the auxiliary density is generated using the Campbell and Shiller (1988) approximation. Using US data, we investigate the empirical importance of the neglected non-linearity by re-imposing the original equilibrium restriction. We find that the distorted model is strongly preferred by the data and substantially improves identification. More importantly, it completely overturns key qualitative predictions of the linear model, such as the absence of endogenous time variation in risk premia and level effects. The latter are crucial for understanding the link between asset prices and macroeconomic risk. The strand of literature that is closer to the econometrics of methodology considered in this paper is the literature on Exponential Tilting i.e. Schennach (2007); Kitamura and Stutzer (1997) ;Imbens, Spady, and Johnson (1998), and Generalized Empirical Likelihood criteria i.e. Newey and Smith (2004) in a conditional moment restrictions framework. Formally, our estimator is not an extension of GEL criteria, in the same way the ETEL estimator (Schennach (2007)) cannot be obtained as a particular version of GEL estimator. The reason is that, as in ETEL, the first step uses a different objective function. We depart from this literature by considering a generalized version of exponential tilting in the "first step", where the form of f pX\|Z, ϕq is parametrically specified. We also briefly comment on the main difference, from an econometric point of view, to the approach of Gallant and Tauchen (1989) (including extensions thereafter). Since the information projection in the first step (i.e. imposing the conditional moment restrictions) is done using a different divergence measure than the estimation objective, it is an immediate consequence that the first order conditions of our estimator are mathematically (and substance-wise) different than a constrained estimator. Moreover, Ai (2007) illustrates the main differences of the Empirical Likelihood (EL) approach to the Semi-Non Parametric (SNP) approach of Gallant and Tauchen (1989), which amount to the concentrated objective not having a density interpretation 4 . Our approach successfully overcomes these issues and is therefore expected to have similar statistical performance to SNP. The rest of the paper is organized as follows. In Section 2, we introduce information projections and we provide an asset pricing example. In Section 3 we present the large sample properties, the corresponding analysis for the case when the base model is a structural approximation, and supportive simulation evidence. Section 4 applies the methodology to pricing macroeconomic risk and Section 5 concludes. Appendix A provides analytical details for the example and application, discusses the computational aspect of the method, and contains the main proofs. Appendix B contains auxiliary proofs and another Monte Carlo exercise. Finally, a word on notation. Let N denote the length of the data and N s the length of simulated series. X is an n xˆ1 vector of the variables of interest while Z is an n zˆ1 vector of conditioning variables. Both X and Z induce a probability space pΩ, F, Pq. In dynamic models, Z will be predictable from past values of X. In the paper three different probability measures are used, the true measure P (with P N the corresponding empirical measure ), the base measure F ϕ which is indexed by parameters ϕ and the H pϕ,ϑq measure which is obtained after the information projection. Moreover, these measures are considered absolutely continuous with respect to a dominating measure v, where v in most interesting cases is the Lebesgue measure. All these measures possess the corresponding density functions p, f and h. We denote the conditional measures and densities by an additional index that specifies the conditioning variable, i.e. F ϕ,z . The set of parameters ψ is decomposed in ϑ P Θ, the set of structural (economic) parameters and ϕ the parameters indexing the density f pX\|Z, ϕq. In addition, P s is the conditional distribution where s can be a variable or a parameter. Furthermore, q j pX, Z, ψq is a general X b Z measurable function and qpX, Z, ψq is an n qˆ1 vector containing these functions. Moreover, q ψ abbreviates the Jacobian matrix of q and q ψψ 1 the Hessian with respect to ψ. For any (matrix) function the subscript i denotes the evaluation at datum px i , z i q. Similarly, subscript j is for simulated data using the base density. The operator Ñ p signifies convergence in probability and Ñ d convergence in distribution; N p., .q signifies the Normal distribution with certain mean and variance. In terms of norms, \|\|.\|\| signifies the Euclidean norm unless otherwise stated. In addition \|\|.\|\| T V is the Total Variation distance 5 . E P and is the mathematical expectations operator with respect to measure P . V P pxq signifies the variance of variable x under the P´measure while V P,s pxq is the second moment of a particular functionsp.q. If P " P then V P " V. r.s ll 1 signifies the pl, l 1 q component of a matrix, that is, v ll :" V ll for any matrix V . 5 \|\|.\|\| T V " sup BPΩ´B \|f´p\|dv Information Projections as Distortions to the Approximate Model For completeness, we present below the formal problem of an information projection. Given a class of candidate base densities f pX, Z\|ψq, a conditional information projection is equivalent to solving for hpX, Z, ψq in the following program: min hpX\|Z,ψqPHˆh pX\|Z, ψqlogˆh pX\|Z, ψq f pX\|Z, ϕq˙h pZqdpX, Zq (1) H :" h P L p :ˆhpX\|Z, ψqmpX, Z, θqdX " 0,ˆhpX\|Z, ψqdX " 1, Z´a.e. ( In the information projections literature the minimization problem in 1 is called exponential tilting as the distance metric minimized is the Kullback-Leibler distance, whose convex conjugate has an exponential form. The set H is the set of admissible conditional densities i.e. the densities that by construction satisfy the conditional moment conditions. Denoting the optimal density by h ‹ pX\|Z, ψq we perform extremum estimation using the log likelihood function as follows: max ψPΨ´l ogph ‹ pX\|Z, ψqqdPpX, Zq(2) The above problem can be conveniently rewritten 6 such that the choice of density hpX\|Z, ϑq is equivalent to the choice of a perturbation MpX, Z, θq to the prior density, that is hpX\|Z, ϑ, ϕq " f pX\|Z, ϕqMpX, Z, ϑq. The perturbation factor MpX, Z, ϑq will be a function of the sufficient information to estimate ϑ and is in general not unique. min MPM E f pX\|Z,ϕqhpZq MpX, Z, ϑq log MpX, Z, ϑq M :" M P L p : E f pX\|Z,ϕq MpX, Z, θqmpX, Z, θq " 0 E f pX\|Z,ϕq MpX, Z, θq " 1 ( Selecting hpX\|Z, ϑ, ϕq by minimizing the Kullback-Leibler distance to the prior density is one way of selecting a unique factor M. The optimal perturbation factor is therefore M ‹ " exp pλpZq`µpZq 1 mpX, Z, ϑqq which implies the choice of the following family of distributions: (3) hpX\|Z, ψq " f pX\|Z, ϕq exp pλpZq`µpZq 1 mpX, Z, ϑqq where µ is the vector of the Lagrange multiplier functions enforcing the conditional moment conditions on f pX\|Z, ϕq and λ is a scaling function. Had we used an alternative objective function to 1, e.g. another particular case from the general family of divergences in Cressie and Read (1984) (Schennach, 2007). Moreover, in the case in which f pX\|Z, ϕq belongs to the exponential family and the moment conditions are linear, exponential tilting is the natural choice. We present an illustrative example of projecting on densities that satisfy moment conditions that arise from economic theory. In this simple case, due to linearity, the resulting distribution after the change of measure implied by the projection is conjugate to the prior. Economic theory therefore imposes additional structure to the moments of the prior density. 2.1. An Example from Asset Pricing. The consumption -savings decision of the representative household implies an Euler equation restriction on the joint stochastic process of consumption, C t , and gross interest rate, R t , where F t is the information set of the agent at time t and E P signifies rational expectations : E P pβR t`1 U c pC t`1 q´U c pC t q\|F t q " 0 Suppose that a prior statistical model is a bivariate VAR for consumption and the interest rate which, for analytical tractability, are not correlated. Their joint density conditional on F t is therefore: c t`1 R t`1 \|F t‚ " N¨¨ρ c c t ρ R R t‚ ,¨1 0 0 1‚‚ For a quadratic utility function, that is U pC t q " C 2 t , the Euler equation is a covariance restriction as EpR t`1 C t`1 \|F t q " ct β and thus CovpR t`1 , C t`1 \|F t q " Ct β p1´R t βρ c ρ R q. The distorted density hpC t`1 , R t`1 \|F t q is therefore: C t`1 R t`1 \|F t‚ " N¨¨ρ c C t ρ R R t‚ ,¨1 Ct β p1´R t βρ c ρ R q 1‚‚ Since we know the new density in this case, the perturbation MpX, Z; ϑq, can be reverse engineered as follows: M " » -N¨¨ρ c C t ρ R R t‚ , I 2‚ fi fl´1 N¨¨ρ c C t ρ R R t‚ ,¨1 Ct β p1´R t βρ c ρ R q 1‚‚ " exp¨´1 2¨C t`1´ρc C t R t`1´ρR R t‚ 1¨1 Ct β p1´R t βρ c ρ R q 1‚¨C t`1´ρc C t R t`1´ρR R t‚‹ ‚ In Appendix A, we illustrate how the same expression for M can be obtained formally using a conditional density projection 7 , that is, solving 1. Note that in this example, the fact that the Euler equation is a direct restriction on the parameters of the base density is an artifact of the form of the utility function assumed, and is therefore a special case. In more general examples an analytical solution cannot be easily obtained and we therefore resort to simulation. Details of the algorithm are provided in Appendix A. 7 More precisely, what is obtained is the density conditional on Z " z. In the next section we analyze the frequentist properties of using the tilted density to estimate ψ " pϑ, ϕq. The main challenge is the fact that we project on a possibly locally misspecified density. Explicitly acknowledging for estimating the parameters of the density yields some useful insight to the behavior of the estimator. Large Sample Theory Below we present asymptotic results, that is consistency and asymptotic distribution for ψ. The properties of the estimator, as expected, depend crucially on the distance between the prior and the true population conditional density. Before stating the main results, we make certain assumptions that are fairly standard in parametric extremum estimation and are necessary and sufficient for the results to be valid. For a stationary ergodic sequence tx i , z i u N i"1 , we assume the following: ASSUMPTIONS I. (1) (COMP) Θ Ă R n ϑ , Φ Ă R nϕ are compact. (2) (ID)D!ψ 0 P intpΨq : ψ 0 " arg max Ψ E log hpX\|Z, ψ 0 q (3) (BD-1a)@l P 1..n m and for d ě 4, P P tF ϕ , Pu : E Pz sup ψ \|\|m l px, ϑq\|\| d , E Pz sup ψ \|\|m l ϑ px, ϑq\|\| d , and E Pz sup ψ \|\|m l ϑϑ px t , ϑq\|\| d are finite, Ppzq´a.s. (4) (BD-1b)sup ψ E Pz \|e µpzq 1 \|mpx,z,ϑq\| \|\| 1`δ ă 8 for δ ą 0, @µpzq ą 0, Ppzq´a.s 8 (5) (BD-2)sup ψ Eplog hpx\|z, ψqq 2`δ ă 8 whereδ ą 0. (6) (PD-1) For any non zero vector ξ and closed B δ pψ 0 q , δ ą 0, and P P pF ϕ , Pq, 0 ă inf ξˆB δ pψ 0 q ξ 1 E P mpx, ϑqmpx, ϑq 1 ξ ă sup ξˆB δ pψ 0 q ξ 1 E P mpx, ϑqmpx, ϑq 1 ξ ă 8 Assumptions (1)-(2) correspond to typical compactness and identification assumptions found in Newey and McFadden (1994) while (3) assumes uniform boundedness of conditional moments and their first and second derivatives, up to a set of measure zero. Assumption (4) assumes existence of exponential absolute 2`δ moments and (5) boundedness of the population objective function 9 . Finally, (6) assumes away pathological cases of perfect correlation between moment conditions. Note that the assumptions above correspond to the case of estimation of a density with finite dimensional parameters ϕ. In case ϕ is infinite dimensional, the conditions have to be sufficiently generalized. Such a generalization involves additional conditions that control for parametric or semi-non parametric estimators for f pX\|Zq. Although we abstract from the above generalizations, the characterization of the asymptotic distribution using the high level assumption of asymptotically correctly specified f pX\|Zq is sufficient to illustrate the main properties of the estimator. Recall that we maximize the empirical analogue to (2), which, abstracting from simulation error, is equivalent to the following: max pϑ,ϕqPΘˆΦ Q N pϑ, ϕq " 1 N ř i"1..N log pf px i \|z i , ϕq exppµ 1 i mpx i , z i , ϑq`λ i q where @i " 1..N, µ i :ˆf pX\|z i , ϕq exppµ 1 i mpX, z i , ϑqmpX, z i , ϑqdX " 0 λ i :ˆf pX\|z i , ϕq exppµ 1 i mpX, z i , ϑqdX " 1 where for notational brevity we substituted Z " z i for z i and µ i " µpz i q, λ i " λpz i q. Comparing our objective function with that of Kitamura, Tripathi, and Ahn (2004), apart from using exponential tilting, we also do not smooth using local values for the instrument Z. Accounting for local smoothing would complicate the analysis in an unnecessary way without apparent gain. Most importantly, as the relevant applications are in macroeconomics, instruments will be lagged values of X t , whose distribution is already pinned down by f p.q. In other non-time series applications, Z is treated as fixed. The corresponding first order conditions of the estimator are going to be useful in order to understand both the asymptotic but also the finite sample results. Denoting the Jacobian of the moment conditions by M, the first order conditions are the following, where µ ψ pz i q, λ ψ pz i q denote derivatives with respect to the corresponding parameter vector: ϑ : 1 N ř i pMpx i , z i , ϑq 1 µpz i q`µ ϑ pz i q 1 mpx i , z i , ϑq`λ ϑ pz i qq " 0 (4) ϕ : 1 N ř i pspx i , z i , ϕq`µ ϕ pz i q 1 mpx i , z i , ϑq`λ ϕ pz i qq " 0 (5) where: µpz i q " arg min µPR nm´f pX\|z i , ϕq exppµ 1 mpX, z i , ϑqdX λpz i q "´log`´f pX\|z i , ϕq exppµpz i q 1 mpX, z i , ϑqdXȃ nd sp.q is the score of the base density. Regarding the existence of µpZq, or equivalently, the existence of the conditional density projection, Komunjer and Ragusa (2016) provide primitive conditions for the case of projecting using a divergence that belongs to the φ´divergence class and moment restrictions that have unbounded moment functions. Assumptions BD-1a and BD-1b are sufficient for their primitive conditions (Theorem 3) 10 . In Appendix B we provide expressions for the first and second order derivatives of pµpz i q, λpz i qq which determine the behaviour ofψ in the neighborhood of ψ ‹ 0 . More interestingly, these expressions will be useful for the characterization of the properties of the estimator in the case when the total variation distance between the prior density and the true density shrinks to zero at a certain rate. We first outline certain Lemmata which are systematically applied in the proofs of all propositions, and they are also useful in understanding the the source of the differences to traditional GEL estimation. We present auxiliary Lemmata in Appendix B. Lemma 1. For an auxiliary conditional density F z , (a) µ i " O p pT V pF z i , P z i qq (b) max i sup ϑ \|µ 1 i mpϑ, x i q\| " O pz pmax i T V pF z i , P z i qN 1 d q Proof. See Appendix A A specific case of the above result is that of Newey and Smith (2004), where the total variation distance between the two densities is T V pF z i , P z i q " O pz pN´ξq . Therefore µ i " o pz p1q, and if 1 d ă ξ ă 1 2 , max i sup ϑ \|µ 1 i mpϑ, x i q\| " o p p1q. Given the above results, we show consistency for both the case of misspecification and correct specification, and the asymptotic distribution which is invariant under local misspecification (N´1 2 localizations). The latter is true as local misspecification does not affect the estimating equations up to first order. 3.1. Consistency, Asymptotic Normality and Efficiency. The uniform consistency of the estimator is shown by first proving pointwise consistency and then stochastic equicontinuity of the objective function. Details of the proof are in Appendix A. Under misspecification, the estimator is consistent for ψ ‹ 0 , which we define below: Definition 2. The pseudo-true value ψ ‹ 0 is the value of ψ P Ψ that minimizes the Kullback-Leibler pKLq distance between HpX\|Z, ϕq and PpX\|Zq, which is decomposed as follows: (6) Correspondingly, since µ ‹ pψq :" arg min`E F pϕ,zq exppµpψq 1 mpX, ϑqq˘, ϕ ‹ 0 is the value of ϕ P Φ such that F pX\|Z, ϕq is as close as possible (in KL) to PpX\|Z, ϕq and 0 ď E logˆd P dF pϕ,zq˙´µ 1‹ pψqEmpX, ϑq`log`E F pϕ,zq exppµ 1‹ pψqmpX, ϑqqsatisfies E F pϕ,zq exppµ 1‹ pψqmpX, ϑ ‹ 0 qqmpX, ϑ ‹ 0 q " 0 where ϑ ‹ 0 is the value of ϑ P Θ such that E F pϕ ‹ ,zq exppµ 1‹ pψqmpX, ϑqqmpX, ϑq " 0. The interpretation of Definition 2 is straightforward. H pψ ‹ 0 ,zq is the closest parametric distribution to P z , while both distributions satisfy a common moment restriction, E H pψ ‹ ,zq mpX, ϑ ‹ q " E P mpX, ϑ 0 q " 0 11 . The smaller KLpF, Pq is, the closer to zero are the second and third terms in Definition 2. If KLpF, Pq can be indexed by the sample size, then so can ψ ‹ 0 . A trivial choice of F N is the empirical distribution, P N , which uses no prior information and assigns equal weight to all data points. From an economic point of view, the above interpretation becomes useful when one considers equilibrium models that are approximated. F can be thought to represent this approximation, which by construction does not satisfy the original first order equilibrium conditions. The tilted distribution will satisfy those restrictions, and will be -by construction-closer to the distribution implied by the economic model. Therefore, ϑ ‹ is closer to ϑ 0 than the pseudo-true value implied by the approximated model. Approximations are therefore less detrimental to economic inference. We elaborate more on this at the end of this section. Having defined the pseudotrue values, we present below the asymptotic results. 11 There is a similarity between our definition of ψ ‹ 0 to the definition of Hellerstein and Imbens (1999), but in our case the moment restriction is satisfied by the sampled population, asymptotically. Theorem 3. Consistency for ψ ‹ 0 Under Assumptions I : pθ,φq Ñ p pϑ ‹ 0 , ϕ ‹ 0 q Proof. See Appendix A As expected, under correct specification, consistency is for ϑ 0 . This leads to the following corollary: Corollary 4. Consistency for ϑ 0 If F pX\|Z,φq is consistent for PpX\|Zq or correctly specified, then ϑ ‹ 0 " ϑ 0 . Proof. See Appendix A For the limiting distribution of the estimator, we use the usual first order approximation around ψ 0 . Below, we present the main result for a general, asymptotically correct density 12 . Denoting by Gpψ, .q the matrix of first order derivatives with respect to pϑ, ϕq, the asymptotic distribution is regular. Theorem 5. Asymptotic Normality Under Assumption I and for N s ,N Ñ 8 such that N Ns Ñ0 : N 1 2 pψ´ψ 0 q Ñ d N`0,V pψ 0 qP roof. See the Appendix A. In Appendix A we derive the exact form of the variance covariance matrix of the estimator. Given a finite number of conditional moment restrictions and the specified density, under correct specification and local misspecification, Gpψq "´V g pψq where Gpψq is the Jacobian and V g pψq is the variance of the first order conditions of the estimator. Thenθ attains the the semi-parametric lower bound, i.e. Chamberlain (1987). What this implies is that lack of knowledge of ϕ does not affect the efficiency of estimating the structural parameters ϑ, at least asymptotically, and estimation is therefore fully adaptive, within the context of regular models. The asymptotic variance is block diagonal as follows: V pψ 0 q "¨`E z M i pϑq 1 V´1 m pϑqM i pϑq˘´1 0 0 pEs i pϕqs i pϕq 1´E B i pψqV m,i pϑqB i pψq 1 q´1‚ where B 1 i " V´1 m,i m i s 1 i is the coefficient of projecting the scores on the moment conditions. Notice also that the upper left component is the same as the information matrix corresponding to ϑ when the conventional optimally weighted GMM criterion is employed. Interestingly, the expressions above have an intuitive interpretation. If the moment conditions we use span the same space spanned by the scores of the density, thenV trivially attains the Cramer -Rao bound and the covariance matrix becomes singular as both m and s give the same information 13 . Conversely, the less predictable is the score from the additional moment conditions used (that is, }B i } is close to zero), the higher the efficiency attained for estimating ϕ, where pEs i pϕqs i pϕq 1 q´1, is the lowest variance possible under regularity 14 . An interesting observation can be made when the model is solved accurately and f pX\|Zq is derived. In this case, f pX\|Zq is pinned down by a unique parametric sub-model that automatically satisfies the moment conditions and pµ i , λ i q are zero 13 The same result follows if one re-computes the variance of the estimator by exploiting the cross equation restrictions that link ϕ and ϑ, and thus the dimension of the matrix is n ϑˆnϑ and no singularity appears in the covariance. 14 This finding is also in line with the results of Imbens, Spady, and Johnson (1998) in the context of testing unconditional moment restrictions, who find that exponential tilting utilizes "efficient" estimates of probabilities rather than the inefficient 1 N weight used in the empirical likelihood literature. for all i, which implies that the second term in the variance of ϕ vanishes. More- over, Es i pϕqs i pϕq 1 " E Bϑ Bϕ 1 Es i pϑqs i pϑq 1 Bϑ Bϕ ą E Bϑ Bϕ 1 M 1 V´1 m,i M Bϑ Bϕ . Thus, the moment restrictions are trivial and add nothing to the information already embedded in the scores. What is also evident from the proof of Theorem 5 is that the first order conditions of the estimator are equivalent to the optimal GMM type of first order conditions up to an error, which is function of the discrepancy between F pX\|ϕ, Zq and PpX\|Zq. We have parameterized this discrepancy by the total variation norm, which is of order κ´1 N . As long as κ´2 N ă N´1 2 or equivalently κ N ą N 1 4 , the discrepancy has no first order effects. While this paper has not formally dealt with non or semi parametric estimation of the conditional density of the observations, we can gauge that the behavior of the estimator will be unaffected up to first order as long as the the auxiliary parameters i.e. the bandwidth are chosen such that the rate of convergence is faster than N 1 4 . If this is not true, then we should expect slower rates of convergence for ϑ. Another interesting case arises when the model is solved using approximations, which generate the corresponding F pX\|ϑ, Z, ∆q where ∆ parameterizes the approximation error, i.e. F pX\|θ, Z, ∆q Ñ ∆Ñ0 PpX\|ϑ 0 , Zq. Ackerberg, Geweke, and Hahn (2009) have shown that as long as N 1 2 ∆ Ñ 0, the approximation error 15 to the true conditional density does not affect the asymptotic distribution. As mentioned above, in our case, we can accommodate slower rates of convergence, as long as the parameters of the approximated density (θ) are treated as distinct to ϑ, although they both converge to ϑ 0 asymptotically. Of course, for the rate suggested by Ackerberg, Geweke, and Hahn (2009), this distinction does not matter. We finish this section by illustrating that within the class of approximated equilibrium models, the information projection alleviates, at least partially, the misspecification caused by the local approximation, both at ϑ 0 and at the pseudo-true values under an N 1 2 localization. logC t`1 logR t`1‚ " N¨¨ρ C ρ CR ρ RC ρ R ,‚¨l ogC t logR t‚ ,¨σ 2 C σ CR σ RC σ 2 R‚‚ Moreover, assuming a quadratic utility for the representative agent, U pC t q :" αC tγ C 2 t and that βR ss " 1 the Euler equation becomes as follows: E tˆβ C t`1 R t`1 C t´1˙" 0 For the DGP we use the following parameterization: ρ C " ρ RC " 0, ρ R " 0.95, ρ CR " 0.05, β " 0.75 and Σ " r0.05, 0.002; 0.002, 0.05s. Below, we plot the MSE comparisons for estimating the discount factorβ across typical sample (and sub-sample) sizes for quarterly macroeconomic data sets i.e. N " t20..220u for two experiments. In the first experiment ( Figure 1), we compare the performance of the CU-GMM estimator to our estimator, both in the case of knowing the density and when estimating σ CR . As evident, the performance of CU-GMM is much worse than the other two cases, as we use the empirical distribution function rather than the correctly specified density. Interpreting GMM as a plug-in estimator using the empirical CDF, where the latter is the most basic infinite dimensional model for the true CDF, it is not surprising that a low dimensional but locally misspecified density performs better in terms of MSE in small samples. What this implies is that although asymptotic results under local misspecification do not justify efficiency gains, they latter are entirely possible in small samples. In Figure 2 we present the same results but we focus on the relative difference between incorrectly restricting σ CR to zero and using the correct density: the efficiency gain does not overcome the resulting bias. However, as we increase the dimension of the estimated parameters, the MSE gains from imposing incorrect local restrictions become noticeable. In Figure 3 we present the case when we estimate pρ RC , ρ R , σ CR q subject to the restriction ρ CR " 1´ρ R , that is the true model, and compare it to only estimating σ CR and imposing local misspecification (T´1 2 h) on pρ RC , ρ R q for h=0.01 (and ρ CR " 1´ρ R ). The latter is the approximate model as it features local misspecification (restrictions in the autoregressive coefficients ρ). What we observe is that the bias -variance trade-off holds for a moderately sized samples, indicating that our estimator can be potentially useful for estimating models in small subsamples Moreover, it is tempting to interpret the MSE evidence in terms of the theoretical results of Newey and Smith (2004) where it is shown that GEL estimators have lower bias than the CU estimator as the correlation between the Jacobian and the moment restrictions is removed. Since the estimator we propose has some similarities with GEL, we could expect a lower bias too, but showing this is beyond this paper. Counterfactual Distributions. An additional advantage of the method used in this paper, is that although the model is not solved for the equilibrium decision rules, we can still perform counterfactual experiments. What is more important is that this method readily gives a counterfactual distribution, while the distribution of the endogenous variables is hardly known in non-linear equilibrium models. Knowing the distribution of outcomes is extremely important for policy analysis, especially when non linear effects take place, and therefore the average effect is not a sufficient statistic to make a decision. We present below an example which is based on a modification of Example 1, where the only difference is that the utility function is of the Constant Relative Risk Aversion form. The counterfactual experiment consists of increasing the CRRA coefficient. We plot the contour maps of the conditional joint density of pR t`1 , c t`1 q with a change in the risk aversion coefficient. An increase in risk aversion is consistent with higher mean interest rate, and lower mean consumption. Moreover, consumption and interest rates are less negatively correlated. This is also consistent what the log -linearized Euler equation implies, c t "´1 σ r t . It is important to stress that even if the underlying density is an approximation, as we show in Proposition 6, tilting the density to satisfy the nonlinear condition indeed improves the approximation. Therefore, the pseudo-true value will be closer to the 'true' value and the counterfactual experiment will be closer to the ideal counterfactual experiment one would wish to implement. plaining several asset pricing facts. The long run risk model of Bansal and Yaron (2004) and its subsequent variations impose cross equation restrictions that link asset prices to consumption growth, where Epstein and Zin (1989) preferences differentiate between between risk aversion and the intertemporal elasticity of substitution (IES). These restrictions can be summarized by the Euler equation that involves the unobserved aggregate consumption dividend R α,t`1 , consumption growth G c and stochastic variation in the discount factor, G l as in Rui, Martin, Xi, and Sergio (2016): E t δ θ G θ l,t`1 G´θ ψ c,t`1 R´p 1´θq a,t`1 R i,t`1 " 1 (7) where θ " 1´γ 1´1 ψ , γ is the coefficient of relative risk aversion, ψ the IES and δ the discount factor. Relatively recent attempts to estimate this model using standard non-durable consumption data have stressed several issues that need to be taken into account. As argued by Bansal, Kiku, and Yaron (2016), time aggregation is an important source of bias when low frequency data is used. In fact, Bansal, Kiku, and Yaron (2016) find empirical support for a monthly decision interval. Quarterly or yearly data are therefore likely to be responsible for the downward bias to estimates of the IES and upward bias in risk aversion, which have been puzzling in the literature, as they imply that asset prices are increasing in uncertainty. Nevertheless monthly data are contaminated by measurement error and a recent paper (Schorfheide, Song, and Yaron, 2018), SSY hereafter, provides a mixed frequency approach to make optimal use of a long span of consumption data while keeping measurement error under control. In this paper we investigate the empirical implications of an equally important aspect of empirical macro-finance, which is the quality of the underlying approximation to the equilibrium value of the unobserved R α,t`1 . What has been standard up to now was to use the Campbell and Shiller (1988) log linear approximation, which has been recently criticized by Pohl, Schmedders, and Wilms (2018) as being too crude when the underlying dynamics are persistent. We take the linear approximation as given, and we impose (7) using the methodology in this paper.To isolate the informational content of imposing the Euler equation, we employ a similar specification to Schorfheide, Song, and Yaron (2018) but we only use monthly data on non durable consumption growth and the risk free rate. The underlying approximating model is summarized as follows: ∆c t`1 " µ c`xt`σc,t η c,t`1 (8) r f,t " B 0`B1 x t`B1,l x l,t`B2,x σ 2 x,t`B 2,c σ 2 c,t (9) x t`1 " ρx t`a 1´ρ 2 σ x,t η x,t`1 (10) x l,t`1 " ρ l x t`σl η l,t`1 (11) σ c,t " σe vx,t (12) σ x,t " σχ x e vx,t (13) v x,t`1 " ρ vx v x,t`σv,x w x,t`1(14) where pB 0 , B 1 , B 2,x , B 2,c q are functions of the deep parameters pγ, ψ, δq, the risk aversion, the elasticity of intertemporal substitution and discount factor respectively 16 . Regarding the observation equation, we calibrate the measurement error to the values estimated by SSY 17 . 16 For details on the underlying solution mapping we encourage the reader to consult Schorfheide, Song, and Yaron (2018). 17 For monthly consumption growth, we set σ 2 me " 2pσ 2 `σ 2 q q where σ 2 and σ 2 q are the variances of the measurement error in monthly and quarterly consumption respectively, as estimated by SSY. This specification is approximately equal to SSY's specification of the measurement error in consumption growth from the third month to the first month of the next quarter, so σ 2 me is actually an upper bound to the measurement errors of the rest of the months. We perform estimation in two steps. The reduced form dynamics, that is, equations 8,10,12-15, are identified without using the long run risk model. We therefore estimate the cash flow parameters φ " pρ, χ x , σ, ρ v,x , σ v,c q by using Markov Chain Monte Carlo and the particle filter 18 . We then estimate the deep parameters pψ, γq conditional on the posterior mode, φ ‹ post . Since the moment condition 7 does not provide a measurement density for x l,t , we would have to rely on the approximate model to identify the process parameters 19 . We thus calibrate the time preference risk parameters ρ l and σ l to the posterior median estimates of SSY. The resulting posterior distributions for pψ, γq do not reflect the posterior uncertainty about φ. We make use of the asymptotic results presented in the previous sections to neglect uncertainty about φ 20 . The base density is therefore the predictive density of the non-Gaussian state space model (Gaussian conditional on the identified volatility states) for p∆c t`1 , r f,t`1 q which we construct using the particle filter 21 . Correspondingly, the conditionally Gaussian model is a bivariate Normal distribution for p∆c t`1 , r f,t`1 q with conditional means µ c`xt and B 0`B1 x t`1`B2,x σ 2 x,t`1`B 2,c σ 2 c,t`1 respectively. Conditional linearity is achieved by using the Campbell and Shiller (1988) approximation to asset returns r a,t`1 and solving for the price consumption ratio. In what follows we investigate the usefulness of tilting the approximating density to satisfy the non linear condition 7, both in terms of parameter identification and model 18 As in Chen, Christensen, and Tamer (2016), we rely on quantiles of the posterior draws to construct confidence sets. 19 A Bayesian approach has been recently proposed to deal with the lack of measurement density by Gallant, Giacomini, and Ragusa (2017). 20 Recall that as long as ID is true, by the Bernstein-von Mises Theorem, inference around the posterior mode will be similar to the maximum likelihood estimate. Since uncertainty over φ does not show up in the variance of the structural parameters we can potentially neglect uncertainty about φ when conducting inference about the latter. 21 The are obviously alternative more efficient algorithms to deal with stochastic volatility i.e. Metropolis within Gibbs algorithm. prediction. We find that this is both empirically relevant, and economically significant 22 . In Figure 5, we report the Posterior Distributions using the approximated model (AM), and the tilted model (TM). As evident, correcting for the underlying non-linearity leads to improved identification, in the sense that posteriors are much narrower, and the mode (and MLE estimates) are closer to what are considered more plausible values. A direct implication is that measurement error is not the only source of upward bias in the estimates of risk aversion. Approximation errors is clearly another one. We also report the Maximum Likelihood estimates for the tilted model, to give a sense of how much the prior information matters for both exercises. Having demonstrated the superior performance of the tilted model in terms of statistical fit and identifying power, we next turn to economic inference, which is equally important from the perspective of understanding what determines the risk premium once we allow for non-linearity. 4.1. Qualitative Inference. What we show below is that the tilted model uncovers relationships between asset prices and macroeconomic risk that are hidden by the approximate model. This is crucial from an economic point of view, as any conclusions drawn from the linear model are likely to be misguiding. We use importance sampling to generate the conditional risk premium for the tilted model, while we also use simulation for the approximate model for comparability reasons. More particularly, for every t " 1..N , we use the approximate model to produce a conditional simulation, estimate pµ i , λ i q, and use the perturbation w i " exppµ 1 i m i`λi q to re-weight the sample, where m i is the moment function in 7. The risk premium is then computed using the conditional covariance of the stochastic discount factor, δ θ G θ λ,t`1 G´θ ψ a,t`1 , and the return on the consumption claim, R a,t . For the approximate model, we use the corresponding estimates and no re-weighting of the simulated sample. In general, due to the resulting parameter estimates and the non-linearity of the TM, the risk premium for total wealth predicted by the TM is lower than the AM, 0.58% versus 1.99% on average. This observation points to the possibility that satisfactory predictions for risk premia for certain markets coming from linearized models is an artefact of the ignored non-linearities, and not the underlying mechanism. Figure 6 re-confirms the analytical insights that come from the linearized model, which implies that when the representative agent has preference for early resolution of risk, the risk premium is increasing with long run risk. Although the sensitivity is different, the qualitative insight is similar. Nevertheless, this is no longer true once we look at the relation between the risk premium and long run fluctuations, x t . As evident from Figure 7, the predicted risk premium using the approximate model does not depend on the level of x t,t . This is expected though, as in the linearized model the risk premium varies because of stochastic volatility, σ 2 c,t and σ 2 x,t , an arguably unsatisfactory result. On the other hand, using the tilted model, there is a strong positive association between the realized risk premium and the level of long run fluctuations. To rationalize this observation, we use the basic insight of the long run risk model, which is that the risk premium is a linear combination of pricing using the CAPM and the Consumption-CAPM as follows: rp t "´Cov t pm t,t`1 , r i,t`1 q " θ ψ Cov t p∆c t`1 , r i,t`1 q`p1´θqCov t pr a,t`1 , r i,t`1 q where r a,t`1 is the (log) return on total wealth and r i,t`1 the (log) return of the asset to be priced. Since we study returns to total wealth, we substitute for r i,t`1 " r a,t`1 , which implies that the risk premium will vary as long as the Cov t p∆c t`1 , r a,t`1 q and V ar t pr a,t`1 q vary over time. Under the approximate model, both conditional moments are functions of volatilities v x,t and v c,t but not the levels. Under the tilted model, this is not true. It turns out that the component that depends on x t is V ar t pr a,t`1 q. ln Figure 8, we plot on the left the recovered relation between V ar t pr a,t`1 q andx t,t using the tilted model. The relationship is quadratic, which implies strong non-linearities in the risk premium. For low values of x t , the conditional variance of r a,t`1 is increasing in x t , and for high values of x t the conditional variance is decreasing. On the right hand side, we plot the kernel density estimate of tx t,t u tďN as identified by the reduced form. The distribution is skewed to the left, which implies that x t has visited the upward sloping part of the domain more frequently. This essentially rationalizes the positive relation in Figure 7 asθ ă 0. It is also important to stress that the positive association of the risk premium with x t becomes more apparent when we do not allow for stochastic volatility. Stochastic volatility increases the realized domain for x t , which leads to a less predictable relation between risk premia and x t . The inverse-U relation between V ar t pr a,t`1 q and x t can be explained as follows. Recall that, abstracting from time preference risk, the return on total wealth can be written as a function of consumption growth and the ratio of the continuation value, U t`1 , to its certainty equivalent, R t pU t`1 q. Taking logs, it reads as follows: r a,t`1 "´logpδq`1 ψ ∆c t`1`p 1´1 ψ qv t`1 where v t`1 " log´U t`1 RtpU t`1 q¯. The corresponding conditional variance is therefore V ar t pr a,t`1 q " 1 ψ 2 V ar t p∆c t`1 q`ˆ1´1 ψ˙2 V ar t pv t`1 q`ψˆ1´1 ψ˙C ov t p∆c t`1 , v t`1 q The first and last term do not depend on x t , as the latter is independent from short run fluctuations in consumption growth, σ c,t η c,t`1 . Regarding the second term, by utilizing the fact that the wealth -consumption ratio, Wt Ct , is equal to 1 1´δ´U t Ct¯ψ´1 ψ then v t`1´Et v t`1 can be expressed as a function of wealth and consumption: v t`1´Et v t`1 " ψ ψ´1 pw t`1´Et w t`1 q´1 ψ´1 pc t`1´Et c t`1 q Since the last term does not depend on x t , then variation in V ar t pv t`1 q must come from variations in V ar t pw t`1 q. This can be verified, by expressing W t as the present discount value of future wealth, that is, W t " C t`Et M t`1 W t`1 " E t M t`1 W t`1ˆ1`C t E t M t`1 W t`1ẇ here M t`1 is the Epstein-Zin stochastic discount factor. Taking logs, we have that w t " log pE t M t`1 W t`1 q`logˆ1`C t E t M t`1 W t`1Ṫ hus, V ar t pw t`1 q " V ar t´l og´1`C t`1 EtM t`2 W t`2¯¯" V ar t´l og´1`e c t`xt`σc,t η c,t`1 EtM t`2 W t`2¯¯, which is a function of x t . Once we condition on information at time t, the conditional variance fluctuates even if σ c,t is constant. In Appendix A, we derive the following characterization for the gradient of V ar t pw t`1 q with respect to x t : B Bx t V ar t pw t`1 q "´κ t B Bx t E t M t`2 W t`2 "´κ t E tˆWt`2 B Bx t M t`2`Mt`2 B Bx t W t`2"´κ t E tˆWt`2 B Bx t M t`2`Mt`2ˆp W t`1´Ct`1 q B Bx t R α,t`2´Rα,t`2 C t`1˙3 where κ t is a positive random variable. This expression has a fairly straightforward economic interpretation. For very low values of x t , and thus low values of C t`2 C t`1 , the pricing kernel (M t`2 ) is very sensitive to (positive) marginal changes in consumption growth. Correspondingly, the return to the consumption claim R α,t`2 , which is inversely related to the stochastic discount factor, does not increase as much since the agent experiences a large reduction in marginal utility and an increase in continuation utility respectively. This implies that while the change in the stochastic discount factor is large and negative, the corresponding change in future wealth is likely to be of smaller order. This generates a positive slope for the conditional variance. The exact converse holds for large values of consumption growth, i.e. the stochastic discount factor falls but not as much, while the corresponding change in future wealth is likely to be positive, as returns increase to compensate for the small reduction in marginal utility. This observation comes as no surprise. The Campbell and Shiller (1988) approximation to the return, which in our case is the return to the consumption claim, has to do with the dynamics of wealth. Thus, non-linearities in the stochastic behaviour of wealth must explain the shortcomings of this approximation. We conjecture that this will be the case for pricing other assets too. In addition, what our analysis suggests is that setting ψ ‰ 1 is quite important for non-linear effects to arise. This is a quite common approximation that has been used to facilitate the analysis for the long run risk model (e.g. Hansen, Heaton, and Li (2008)). If ψ " 1, then the wealth consumption ratio is no longer time varying, which implies that the only source of variation in the moments of wealth is the source of variation in consumption moments i.e. stochastic volatility. For completeness, we plot in Appendix A the relation between the risk premium and short run fluctuations. Again, the tilted model implies a weak positive relation, while the approximate model does not predict such a relation, as expected. Risk Premia and the Great Recession. We next investigate the relative contribution of stochastic volatility to the risk premium once we allow for state dependence in pricing the aggregate consumption claim. We construct the counterfactual by shutting down stochastic volatility (in v x,t and v c,t ). In Figure 9, we plot the predicted risk premium in the tilted model with constant volatility (CV) and stochastic volatility (SV), and the predicted risk premium in the approximate model with constant volatility (CV). As expected, the latter is constant over time, as volatilities are constant. Nevertheless, once we tilt the model, even with constant volatility, the risk premium exhibits significant fluctuations. Therefore, the difference between the blue line and the yellow line can be attributed to state dependence of risk valuation. A very interesting observation arises by noticing that there is almost no difference in the prediction for the risk premium between the end of 2007 and mid 2009. Since we focus on the risk premium for aggregate wealth, which includes human wealth 23 , this is not a statement about the equity premium per se. Nevertheless, this observation matters for how we interpret the aftermath of the recent financial crisis. In Figure 10 we plot the time series of aggregate consumption in the US (the picture for GDP looks almost the same). The last crisis has had a significant impact on consumption growth, and thus a negative effect on consumption in the long run. Correspondingly, the large drop in consumption (and output) which manifested significant losses in human and business capital, see for example the analysis of Hall (2015), increased uncertainty about aggregate wealth. This explanation of the rise in the risk premium is more in line with macroeconomic research on the effects of the financial crisis than the exogenous increase in consumption volatility. Conclusion In this paper we have proposed an alternative approach to estimate and analyse the implications of economic models defined by conditional moment restrictions. The approach employs an approximate but complete probability model while it utilizes the information coming from the moment restrictions. It therefore combines the advantages of complete and incomplete models, while it circumvents to a certain extent their corresponding drawbacks. Employing an approximate density improves the finite sample behaviour of the estimates of the structural parameters, while it maintains our ability to perform probabilistic predictions and counterfactual exercises. More importantly, we have demonstrated that if the approximate model is indeed an economic approximation, tilting the approximation to satisfy the original restrictions clearly improves on critical issues like identification and qualitative inference. Our paper also contributes to the macro-finance literature by demonstrating that employing the widely used linear approximations to returns can hinder our ability to understand the nature of the link between movements in asset prices and macroeconomic risk. In the context of the last financial crisis, we find that the contribution of the large drop in the US aggregate consumption to the rise in the risk premium on total wealth is significant, while stochastic volatility plays a minor role. This obviously has wider implications for the way we think about the relative roles of non-linearity and time varying volatility in explaining complex economic outcomes. An interesting avenue for future research is to investigate further the treatment of unobservables in the moment conditions. While we rely on the approximate model to estimate the unobservable components, tilting distorts the joint distribution of the variables that appear in the moment conditions to the right direction. Further work could explore the gains from combining the approach of Gallant, Giacomini, and Ragusa (2017) with this paper's methodology. 6. Appendix A 6.1. Analytical derivations for the Asset Pricing Example. Suppressing λ, the perturbation, exppµ 1 mpx, ϑq`λq is proportional to exp¨´1 2¨¨c t`1´ρc c t R t`1´ρR R t‚ 1¨0´µ t µ t 0‚¨c t`1´ρc c t R t`1´ρR R t‚‹ ‚´µt ct β p1´R t ρ c ρ R βq‹ ‚ The trick here is that we can get the representation by rearranging terms, and dropping terms that do not depend on µ, and then do the minimization. Therefore, for t`1 :"¨ 1,t`1 2,t`1‚ "¨c t`1´ρc c t R t`1´ρR R t‚ the problem becomes as follows : min µ´e xp¨´1 2¨¨ 1,t`1 2,t`1‚ 1¨1´µ t µ t 1‚¨ 1,t`1 2,t`1‚`2 µ t ct β p1´R t ρ c ρ R βq‹ ‚‹ ‚dpR, Cq " min µ´e xp¨´1 2 1 t`1¨1 p1´µ 2 t q µt p1´µ 2 t q µ p1´µ 2 t q 1 p1´µ 2 t q‹ ‚´1 t`1`2 µ t ct β p1´R t ρ c ρ R βq‹ ‚dpR, Cq We therefore have that the F.O.C is: exp´1 2¨ 1 t`1¨1 p1´µ 2 t q µt p1´µ 2 t q µ p1´µ 2 t q 1 p1´µ 2 t q‚´1 t`1´2 µ t c t β p1´R t ρ c ρ R βq‹ ‚ˆ... ...ˆp´p 1,t`1 2,t`1`c t β p1´R t ρ c ρ R βqqdpR, Cq " 0 Then, for the Normal scaling constant C, C´N¨¨0 0‚ ,¨1 p1´µ 2 t q µt p1´µ 2 t q µ p1´µ 2 t q 1 p1´µ 2 t q‚‚ p 1,t`1 2,t`1´c t β p1´R t ρ c ρ R βqqdpR, Cq " 0 which also reads as µt p1´µ 2 t q´c t β p1´R t ρ c ρ R βq " 0. Therefore, µ t is the solution of the latter equation. Computational Considerations for the Information Projection. This section comments on the computational aspects of using information projections to estimate models defined by moment restrictions. In the case of conditional moment restrictions, the projection involves computing Lagrange multipliers which are both functions defined on ΨˆZ. Therefore, the projection has to be implemented at all the points of z i and at every proposal for the vector ψ. Nevertheless, since the identifiability of ϕ (the reduced form) does not depend on ϑ, ϕ can be pre-estimated. Moreover, is can be more efficient to estimate the unknown functions µpX, Z, ψq and λpX, Z, ψq by simulating at different points of the support and use function approximation methods i.e. splines. In case the model admits a Markov structure, the information set is substantially reduced, making computation much easier. The general algorithm for the inner loop is therefore as follows: (1) Given proposal for pϕ, ϑq, simulate N s observations from F px; z, ϕq (2) For a finite set tz 1 , z 2 , ..z k ..z K u compute : ‚ µpx; z k , ϑq " arg min 1 Ns ř j"1:Ns exppµpx j ; z k , ϑq 1 mpx j ; z k , ϑqq and ‚ λpx; z k , ϑq " 1´logp 1 Ns ř j"1:Ns exppµpx j ; z k , ϑq 1 mpx j ; z k , ϑqqq (3) Evaluate log-likelihood: Lpx\|z, ψq " 1 N ř i"1:N plog hpx i , z i ϑqq In order to facilitate the quick convergence for the inner minimization and avoid indefinite solutions, one can transform the objective function with a one to one mapping, and add a penalizing quadratic function i.e. for T pµq :" 1 Ns ř i"..Ns e mpx j ;z k ϑq , useT pµq " logpF pµq`1q`τ \|\|µ\|\| 2 where τ is the regularization parameter. Regularization becomes important when the simulation size is smaller, something that makes sense only if we want to reduce computational time. Additional Analytical and Empirical Results for Application. Proof. Derivative of V ar t pw t`1 q with respect to x t Recall that V ar t pw t`1 q " V ar t´l og´1`C t`1 EtM t`2 W t`2¯¯. We compute the derivative by using a Taylor expansion of log´1`C t`1 EtM t`2 W t`2¯a round the steady state value, which is zero. For the interest of brevity, let y " C t`1 EtM t`2 W t`2 . Therefore, V ar t plogp1`y t qq " V ar t pyq`1 4 V ar t py 2 q`1 9 V ar t py 3 q`...`Covariances Denoting the j´th central moment of C t`1 byC j , it can be shown that for j ě 1, V ar t py j q "C 2j pEtM t`2 W t`2 q 2j and Cov t py j , y k q " CovtpC j t`1 ,C k t`1 q pEtM t`2 W t`2 q j`k . Correspondingly, B Bx t V ar t py j q "´2jC 2j B Bxt pE t M t`2 W t`2 q pE t M t`2 W t`2 q j`1 and B Bx t Cov t py j , y k q "´pj`kqCov t pC j t`1 , C k t`1 q B Bxt pE t M t`2 W t`2 q pE t M t`2 W t`2 q j`k`1 Therefore, the derivative can be written compactly as follows: B Bx t V ar t plogp1`y t qq "´ÿ j,k"1..8˜j`k jk Cov t pC j t`1 , C k t`1 q pE t M t`2 W t`2 q j`k`1¸B Bx t pE t M t`2 W t`2 q "´κ t B Bx t E t M t`2 W t`2 Finally, κ t is finite as long as savings, W t´Ct satisfy S t " pCov t pC j t`1 , C k t`1 qjq pj`k`1q for some positive constant , i.e. savings grow at a sufficient rate. Empirical Results ř j"1..s e j,i , κ j.i "´p e µ 1 i m j,i pϑq´1 q µ i m j,i pϑq 1 , s j,i :" B Bϕ log f px j \|ϕ, z i q and s j,i :" s j,i f j,i . Proof. of Lemma 1. (a) Recall that µ i satisfies the moment conditions under the F ϕ p., z i q measure, whose simulation sample version is 1 Ns ř j"1..s e j,i m j,i pϑq " 0. We make use of the following implicit map to characterize the stochastic properties of µ i : µ i "˜N´1 s ÿ j"1..s κ j,i m j,i pϑqm j,i pϑq 1¸´1 N´1 s ÿ j"1..s m j,i pϑq where κ j.i " 1´e µ 1 i m j,i pϑq m j,i pϑq 1 µ i . Given the end result, max i sup ϑ \|\|µ 1 i m j,i pϑq\|\| ă 8, so we assume that κ i,j is bounded. Letting v ll 1 κ,j :" rκ j,i m l j,i pϑqm l 1 j,i pϑq 1 s ll 1 , it follows that v ll 1 κ,j ă sup ψ \|v ll 1 κ,j \|. Using BD-1a and Cauchy Schwarz (CS), we conclude that Esup ψ \|v κ,j \| ă 8, Ppzq´a.s. Therefore, the denominator of µ i is O pz p1q and the stochastic order of µ i will be determined by the numerator as follows: 1 N s ÿ j"1..s m j,i pϑq "ˆm i pϑqdF Ns,z i "ˆm i pϑqpdF Ns,z i´d F z i`d F z i´d P z i`d P z i q " o Pz p1q`ˆm i pϑqpdF z i´d P z i q where the last equality is due to the convergence of the empirical and simulation measures and correct specification of the moment condition (under P z ). Applying Corollary 10 we have that 1 N s ÿ j"1..s m j,i pϑq " o Pz p1q`O Pz pT V pF z i , P z i qq " O Pz pT V pF z i , P z i qq and thus µ i " O Pz pT V pF z i , P z i qq (for every element of the vector µ i and all i). (b) ForM ă 8 and d ě 4 (see BD´1) Ppmax i sup ϑ \|\|m i pϑq\|\| ąM N 1 d q " Pp ď iďN tsup ϑ \|\|m i pϑq\|\| ąM N 1 d uq ď ÿ i Ppsup ϑ \|\|m i pϑq\|\| ąM N 1 d q ď ř i Epsup ϑ \|\|m i pϑq\|\| d 1psup ϑ \|\|m i pϑq\|\| d ąM d N q M d N "M Epsup ϑ \|\|m i pϑq\|\| d 1psup ϑ \|\|m i pϑq\|\| d ąM d N q Ñ 0 max i sup ϑ \|µ 1 i m i pϑq\| ď max i sup ϑ \|\|µ i \|\| max i sup ϑ \|\|m i pϑq\|\| ď n 1{2 m max i max l"1..nm sup ϑ pµ l,i q max i sup ϑ \|\|m i pϑq\|\| " O p pmax i T V pF z i , P z i qN 1 d q provided that the number of moment conditions n m is bounded. Proof. of Theorem 3. : Consider the sets V µ,δ " tµ P M : \|\|µ´µ 0 \|\| ă δuand V pϑ,ϕq,δ " tϑ P Θ : \|\|ϑ´ϑ 0 \|\| ă δ, ϕ P Φ : \|\|ϕ´ϕ 0 \|\| ă δu and the objective functions they optimize respectively. (1) (Component-wise) Convergence ofμ i : Proofs for (a)μ i´µi,0 " o p p1q and (b) Q N pψ,μq " Q N pψ, µq`o pz p1q. (a) Using the definition ofμ nmˆ1 pϕ, ϑq " arg inf T pz i , µq where T pµ, z i q " 1 Ns ř j"1..Ns e µ 1 i m i px j ,ϑq and assumptions BD-1, µ exists for all ϑ, ϕ and is unique. Fix Z " z i ,@δ ą 0. Using a Taylor expansion of T pµ, z i q around µ 0 with Lagrange remainder, we have that: T pµ 0 , z i q`T 1 µ pµ 0 , z i qpµ´µ 0 q`1 2 T 2 µ pμ, z i qpµ´µ 0 q 2 . Since T pµ 0 , z i q ě T pµ, z i q, 1 2 T 2 µ pμ, z i qpµ´µ 0 q 2`T 1 µ pµ 0 , z i qpµ´µ 0 q ď 0, and therefore \|T 1 µ pµ 0 , z i q\| ą C\|\|µμ 0 \|\|. We next show that T 1 µ pµ, z i q " o pz p1q andμ i´µi,0 " o pz p1q. By (BD-1a), the sequence te µ 1 i m j,i pϑq m j,i pϑqu j"1..Ns is uniformly integrable with respect to the F´measure , and by the WLLN for U.I sequences, we have that 1 Ns ř j"1..Ns e µ 1 i mpx j ,z i ,ϑq`λ i mpx j , ϑ 0 q u.p Ñ E h\|ϕ,z i mpx j , ϑ 0 , z i q " 0 and therefore T 1 µ pµ, z i q " o pz p1q andμ i´µi,0 " o pz p1q. Moreover, using similar arguments, 1 Ns ř 1..Ns e µ 1 i mpx j ,ϑq m i px j , ϑ 0 qm i px j , ϑ 0 q 1 u.p Ñ E Hϕ,z i m i pϑqm i pϑq 1 . The above result can be strengthened. Applying the classic Central Limit Theorem, we have thatμ i " µ i,0`op pN´1 2 s q. (b) Defining Q N pψ,μq " 1 N ř i"1..N log pf px i \|z i , ϕq exppμ 1 i mpx i , z i , ϑqqq, we have that Q N pψ,μq " Q N pψ, µq`o pz p1q. (2) Uniform Convergence for Q N pψ, µq By Theorem 1 in Andrews (1992), we need to show (i) BD (Total Boundedness) of the metric space in which pϕ, ϑq lie together with (ii) PC (Pointwise consistency) and (iii) SE (Stochastic Equicontinuity). Regarding (i), Assumption COMP implies total boundedness. For (ii), P˜\| 1 N ÿ i plogphpx i ; z i , ψqq´E logphpx i ; z i , ψqqq\| ą ḑ 1 N 2 V˜ÿ i \| logphpx i ; z i , ψqq´E logphpx i ; z i , ψqqq\|¸Ñ 0 using the Markov Inequality, BD-2 and that autocovariances are summable by ergodicity. Regarding (iii), Stochastic equicontinuity for the objective function can be verified by the "weak" Lipschitz condition in Andrews (1992): \|Q N pψ, µq´Q N pψ 1 , µq\| ď B Ng pdpψ, ψ 1 qq, @pψ, ψ 1 q P Ψ where B N " O p p1q andg:lim yÑ0g pyq " 0. To verify this condition, since Q N pψ, µq is differentiable, it suffices to use the mean value theorem: \|Q N pψ, µq´Q N pψ 1 , µq\| " \|`∇ ψ Q N pψ, µq´∇ ψ Q N pψ, 1 µq˘1 pψ´ψ 0 q\| ď \|\|`∇ ψ Q N pψ, µq´∇ ψ Q N pψ, 1 µq˘\|\|\|\|ψ´ψ 0 \|\| where \|\|ψ´ψ 0 \|\| satisfies the definition ofg. Regarding B N :" \|\|`∇ ψ Q N pψ, µq´∇ ψ Q N pψ, 1 µq˘\|\|, since ∇ ψ Q N pψ, µq are the first order conditions in (5), it suffices to consider whether all the relevant sums are bounded in probability. First, notice that in (5) ∇ ψ Q N pψ, µq is a composition of (matrix) functions of tλ ϑ,i , M 1 i µ i , µ 1 ϑ,i m i , µ 1 ϕ,i m i , s i , λ ϕ,i u. A sufficient condition for B N " Op1q is E\|B N \| ă 8 and thus E\|\|∇ ψ Q N pψ, µq\|\| ă 8. Correspondingly, using the Cauchy-Schwarz inequality, it is sufficient that the variances and covariances of tλ Proof. Corollary 4: Consistency or correct specification of f pX\|Z, ϕq imply that there exists a ϕ 0 P ϕ : f pX\|Z, ϕ 0 q " PpX\|Zq. By Lemma 1, λpZ i q " µpZ i q " 0@i and therefore hpX\|Z, ψq " f pX\|Z, ϕq. By construction, the moment condition holds under the H measure, E H pz,ψq mpX, Z, ϑ ‹ 0 q " 0, and thus´PpX, ZqmpX, Z, ϑ ‹ 0 qdpX, Zq " 0. Since it is also true that´PpX, ZqmpX, Z, ϑ 0 qdpX, Zq " 0, by ID, ϑ 0 " ϑ ‹ 0 . ϑ,i , M 1 i µ i , µ 1 ϑ,i m i , µ 1 ϕ,i m i , s i , λ ϕ, Proof. of Theorem 5 (Asymptotic Normality): We show asymptotic Normality by looking at the first order expansion around the true value, that is, N 1 2 pψ´ψ 0 q " G´1 N N 1 2 g N where g N pn ϑ`nϕ qˆ1 " pg 1 , g 2 q 1 is the vector of the first order conditions. We first analyze the convergence in distribution of N 1 2 g N . We will then show the convergence of the Jacobian term and by the continuous mapping theorem we will conclude. We drop dependence of quantities on coefficients. We denote any function q whose mean is computed under measure P by q P . Systematically applying Lemma 1 and the auxiliary Lemmata in Appendix B to each average computed under the approximating density, we first show that only certain terms matter asymptotically at the N´1 2 rate. As in Corollary 10, κ´1 N parameterizes the distance between the true and the approximating density. We show below that this rate does not influence g 1,N to first order. Regarding the first term of g 1,N : › › › › 1 N ÿ i µ 1 i,ϑ m i › › › › " › › › › 1 N ÿ i˜1 N s ÿ j M j,i¸1˜1 N s ÿ j e j,i m j,i m 1 j,i¸´1 m i › › › › " › › › › 1 N ÿ i˜1 N s ÿ j M j,i¸1ˆV´1 f i ,m`O pz pN´1 2 s q˙m i › › › › " › › › › 1 N ÿ i`O pz pκ´1 N q`M f i˘1 V´1 f i ,m m i`1 N ÿ i`O pz pκ´1 N q`M f i˘1 O pz pN´1 2 s qm i › › › › ď › › › › 1 N ÿ i`O pz pκ´1 N q`M f i˘1 V´1 f i ,m m i › › › ›`s up i sup ψ › › › › O pz pN´1 2 s κ´1 N q › › › › › › › › 1 N ÿ i m i › › › › sup i sup ψ › › › › M f i › › › › sup i sup ψ › › › › O pz pN´1 2 s q › › › › › › › › 1 N ÿ i m i › › › › " › › › › 1 N ÿ i`O pz pκ´1 N q˘1 V´1 f i ,m m i`1 N ÿ i M 1 f i V´1 f i ,m m i › › › ›`o p pκ´1 N q ď sup i sup ψ › › › › O pz pκ´1 N q › › › › sup i sup ψ › › › › V´1 f i ,m › › › › › › › › 1 N ÿ i m i › › › ›`s up i sup ψ › › › › M 1 f i V´1 f i ,m › › › › › › › › 1 N ÿ i m i › › › ›`o p pκ´1 N q " O p pκ´1 N qˆO p p1qˆO p pN´1 2 q`O p p1qO p pN´1 2 q " O p pN´1 2 q Similarly, for the second term of g 1,N : › › › › 1 N ÿ i µ 1 i˜M i´ÿ jẽ i,j M i,j¸› › › › " › › › › 1 N ÿ i µ 1 i˜M i´Mh i`M h i´ÿ jẽ i,j M i,j¸› › › › ď › › › › 1 N ÿ i µ 1 i pM i´Mh i q › › › ›`s up i sup ψ › › › › µ i › › › › sup i sup ψ › › › › M h i´ÿ jẽ i,j M i,j › › › › ď sup i sup ψ › › › › µ 1 i › › › › › › › › 1 N ÿ i pM i´Mh i q › › › ›`s up i sup ψ › › › › µ i › › › › sup i sup ψ › › › › M h i´ÿ jẽ i,j M i,j › › › › " O p pκ´2 N q`O p pκ´1 N qO p pN s´1 2 q With regard to the first order condition with respect to ϕ, › › › › 1 N ÿ i˜s i´ÿ jẽ i,j s j,i`µ 1 i,ϕ m i¸› › › › " › › › › 1 N ÿ i˜s i´sh i`s h i´ÿ jẽ i,j s j,i`µ 1 i,ϕ m i¸› › › › ď › › › › 1 N ÿ i ps i´sh i q › › › ›`s up i sup ψ › › › › s h i´ÿ jẽ i,j s j,i › › › ›`› › › › 1 N ÿ i µ 1 i,ϕ m i › › › › ď › › › › 1 N ÿ i s i › › › ›`s up i sup ψ › › › › s h i › › › ›`s up i sup ψ › › › › s h i´ÿ jẽ i,j s j,i › › › ›`s up i sup ψ › › › › µ 1 i,ϕ › › › › › › › › 1 N ÿ i m i › › › › " O p pN´1 2 q`O p pκ´1 N q`sup i sup ψ › › › › O pz pN´1 2 s q › › › ›`O p pN´1 2 q A key driver of the results is BD-1a, as conditional moments are bounded for all z P Z, and are therefore bounded random variables. Multiplying the first order conditions by the parametric rate, N 1 2 , N 1 2 g N " N 1 2 A i,0ò p p1q where the terms in A i,0 are those terms in the above derivations that converge at this rate. Therefore, N 1 2 g N " » - - N´1 2 ř i M 1 f i V´1 f i ,m m i N´1 2 ř i`s i´sf i`µ 1 i,ϕ m i˘fi ffi fl`opp1q To show asymptotic normality, we make use of the Cramer-Wold device. Let ξ be a pˆ1 vector of real valued numbers where ξ 1 pˆ1 "ˆξ 1 1 dimpϑq , ξ 1 2 dimpϕq˙n ormalized such that \|\|ξ\|\| " 1. Then: N 1 2 ξ 1 pˆ1 g N " N´1 2 ÿ i ξ 1 1 M 1 f i V´1 f i ,m m i``N´1 2 ÿ i ξ 1 2`s i´sf i`µ 1 i,ϕ m i˘`op p1q "Ξ 1`Ξ2`o p1q What we need to show is that the variance ofΞ 1 andΞ 2 is finite. We do not need to actually compute the covariances of the above terms as we can further bound them by their variances using C-S inequality. With regard toΞ 1 , EV z pξ 1 M 1 f i V´1 f i ,m m i q " ξ 1 1 EpM 1 f i V´1 f i ,m V m V´1 f i ,m M f i qξ 1 ă 8 as all conditional expectations are bounded almost surely. Similar argument is followed for Ξ 2 . Combining the above results, using the CLT for Martingale Difference Sequences (CLT-MDS) : N 1 2 ξ 1 pˆ1 g N " N´1 2 ξ 1 pˆ1 Ξ N`op p1q Ñ N p0, ξ 1 V g ξq and therefore N 1 2 pg N pψ 0 qq Ñ N p0, V g q 6.5. Efficiency. From the set of first order conditions, G N pθ,φq " 0, using the mean value theorem, 0 " g N pψ 0 q`G N pψqpψ´ψ 0 q Using Lemma 7 we next investigate the exact form of the non random limits of both the Jacobian term and the variance covariance matrix of the moment conditions. Under correct specification, by the WLLN, averages converge pointwise to a constant. is straightforward. Letting κ N " N 1 2 , we have that: VpN 1 2ĝ 1 pϑqq " EM 1 f i V´1 f i ,m m i m 1 i V´1 f i ,m M f i`o p p1q " EM 1 P,i V´1 P,i,m M P,i`op p1q VpN 1 2ĝ 2 pϑqq " Es i s 1 i`E s i m 1 i µ i,ϕ`E µ 1 i,ϕ m i s 1 i`E µ 1 i,ϕ m i m 1 i µ i,ϕ`op p1q " Es i s 1 i´E s i m 1 i V´1 m,i m i s 1 i`o p p1q CovpN 1 2ĝ 1 pϑq, N 1 2ĝ 2 pϑqq " E´M 1 f i V´1 f i ,m m i¯`si´sf i`µ 1 i,ϕ m i˘1`op p1q " EM 1 f i V´1 f i ,m m i s 1 i´E M 1 f i V´1 f i ,m m i m 1 i µ i,ϕ`op p1q " o p p1q In the case of autocorrelated moment conditions, the derivation follows exactly the same steps. Proof. of Proposition 6 a) Similar to Giacomini and Ragusa (2014), E Pz logˆd P z dH z pϑ 0 q˙´E Pz logˆd P z dF z pϑ 0 q" E Pz log f z pϑ 0 q´E Pz log h z pϑ 0 q "´λpZq By construction, λpZq ą 0 as 0 ď E hzpϑq log´h z pϑq fzpϑq¯" E hzpϑq mpϑq`λpZq " λpZq. Appendix B Lemma 7. Limits of derivatives of pµ, λq with respect to pϑ, ϕq: Under correct specification, the unconditional moments of all derivatives are as follows: ‚ First order derivatives whereẽ j,i,ϑ "ẽ j pµ 1 ϑ m j,i`M 1 j,i µ i´ř jẽ j M 1 j,i µ i q. We have already established that as long as the base density is asymptotically correctly specified, then µ i Ñ p 0 for almost all z i . Therefore, e j,i Ñ p 1, and κ j,i Ñ p´1 and the unconditional expectation of the derivatives of pµ, λq with respect to ψ have well defined and interpretable weak limits. E Pn µ i,ϑ Ñ´V´1 m M P E Pn λ i,ϑ Ñ 0 E Pn µ i,ϕ Ñ´V´1 m Epms 1 q E Pn λ i,ϕ Ñ 0 ‚( Applying the Portmanteau Lemma, we conclude. Lemma 8. Influence function for plug-in estimator (Wasserman, 2006) For a general function W px, zq, conditional density Qpx\|zq and Lpx, zq " W px, zq5 ´W px, zqdP z px\|zq W Q N´W P "ˆW px, zqdpQpx\|zqPpzqq´ˆW px, zqdpPpx\|zqPpzqq "ˆˆLpx, zqdQpx\|zqPpzq Corollary 9. Parametric Density. For any px, zq -measurable function W p.q and P " P pϕq, Ppϕq 1-differentiable in ϕ, the following statement holds: W P pϕ 0`h N´1 2 q´W P " N´1 2 hˆδ W pzqdPpzq for some integrable function δ W pzq. Proof. In the parametric case within the class of smooth densities, we can rewrite dQpx\|zq " dP px\|ϕ`N´1 2 h, zq. Therefore, using a Taylor expansion of around ϕ 0 dP px\|ϕ`N´1 2 h, zq " dP px\|ϕ, zq`s ϕ px, zqN´1 2 h`opN´1 2 hq Evaluating´´Lpx, zqdQpx\|zqPpzq in Lemma 8 gives the result: w Q N´w P "ˆwpx, zqps ϕ px, zqN´1 2 h`opN´1 2 hqqdPpzq " N´1 2 hˆδ w pzqdPpzq Corollary 10. Non Parametric Bounded Density. For any z -measurable and integrable function W p.q and density q N px\|zq : sup x \|\|p px\|zq5 q N px\|zq\|\| " O pz pκ´1 N q, the following statement holds: \|w Q N´w P \| " O pz pκ´1 N ∆ w pZqq where ∆ w pzq "´W px, zqdx Proof. \|w Q N´w P \| " › › › ›ˆw px, zqq N px\|zqdx´ˆwpx, zqppx\|zqdx › › › › ď sup x \|\|q N px\|zq´ppx\|zq\|\|ˆW px, zqdx " O pz pκ´1 N ∆ w pZqq Lemma 11. For some invertible matrix C f " 1 Ns ř Ns j C j , denoteC´1 f :" E F pNs,zq C f (1) C´1 f "C´1 f`O pz pN´1 2 s q. (2) More generally, for some integrable density g : sup x \|\|f´g\|\| " O pz pκ´1 N q, C´1 g "C´1 f`O pz pκ´1 N ∆ c pZqq where ∆ c pzq "´Cpx, zqdx. Proof. of Lemma 11 (1) C´1 f "C´1 f´C´1 f pC f´Cf qC´1 f`O pz pN´1 s q "C´1 f`O pz pN´1 2 s q (2)C´1 g "C´1 f´C´1 f pC g´Cf qC´1 f`O pz p\|\|g´f \|\| 2 T C q. Therefore, C´1 g "C´1 f´C´1 f pC g´Cf qC´1 f`O pz p\|\|g´f \|\| 2 T C q "C´1 f´C´1 f pC g´Cf qC´1 f`O pz p\|\|g´f \|\| 2 T C q "C´1 f`O pz pκ´1 N ∆ c pZqq where the last equality uses Corollary 10 7.1. Second MC Experiment. The true data generating process (DGP) for the vector of observables pX, Y q, pM 1 q, is as follows: y i " δ 1`ui u i " ε i`δ2 x i`δ3 x 2 i ε i " iidD 1 pα 1 , α 2 q x i " iidD 2 pγ 1 , γ 2 q The above model satisfies the moment restriction for some β 0 : Epexppyq´β 0´β 0 yxq " 0. For the base model, we experiment with variations between M 1 and the following model (M 2 ) : y i " δ 1`ui u i " iidD 3 pα 1b , α 2b q x i " iidD 4 pγ 1b , γ 2b q We use the following variations as base models: (1) M 2 with D 1 ‰ D 3 (2) M 1 with D 1 ‰ D 3 (3) M 1 with D 1 " D 3 and D 2 " D 4 Table 3. Distributions Used Case D 1 pα 1 , α 2 q, D 2 pγ 1 , γ 2 q D 3 pα 1b , α 2b q, D 4 pγ 1b , γ 2b q 1 tp7),Γp2, 5q N p0, 4),Γp2, 5q 2 tp7),Γp2, 5q N p0, 4),Γp2, 5q 3 tp7),Γp2, 5q tp7),Γp2, 5q Below, we plot in the left panel the MSE for estimating β 0 when using the true density, the misspecified density and GMM, and in the right panel the implied true and misspecified densities of u t . What we notice is that when the base density is very misspecified (Cases 1-3), the estimator is clearly dominated, apart from very small samples. In cases of slight misspecification i.e. Case 2, the estimator performs better both in absolute and in relative terms, compared to GMM. Since misspecification is not local, in larger samples the bias dominates the variance and performance reverses as expected. Overall, getting the conditional mean right seems to be important for performance, and this is something that can be tested a priori. Figure 1 . 1β for low dimensional Φ vs optGMM Figure 2 . 2β for low dimensional Φ vs True model (1000 MC replications)Figure 3.β for "higher" dimensional Φ vs Restricted model (500 MC replications) Figure 4 . 4Increase in Risk Aversion Coefficient 4. Application: Pricing Macroeconomic Risk Low frequency fluctuations in consumption have been shown to be important in ex- Figure 5 . 5Bayesian and Frequentist estimatesIn terms of relative fit, the tilted model is strictly preferred by the data. In table 1, we report the values of the posterior and the likelihood evaluated at their respective modes (ψ ‹ ). The tilted model dominates by 659.1 log-posterior and 659.8 log-likelihood units respectively. Figure 6 .Figure 7 . 67Risk Premium on Aggregate Wealth versus Long Run Risk in Consumption using the Tilted Model (Left) and Approximate Model (Right) Risk Premium on Aggregate Wealth versus Long Run Fluctuation in Consumption using the TM (Left) and AM (Right) Figure 8 . 8Conditional Variance of r a,t (L) and Density forx t,t (R) Figure 9 .Figure 10 . 910Stochastic Volatility versus Non-Linearity using ψ T M US Consumption and the Financial Crisis aftermath i u are finite 24 . By BD´1a we conclude. Given the definition of the estimating equation i.e. the estimator ofψ is an extremum estimator,weak uniform convergence, assumptions ID, COMP, and BD´2( which guarantees continuity of the population objective), consistency follows by standard arguments (i.e. Newey and McFadden (1994), Theorem 2.1). Relevant) Second order derivativesE Pn λ i,ϕ l ϕ 1 Ñ Eps l j m 1 j V´1 m ms 1 q E Pn λ i,ϑ l ϑ 1 Ñ M 1 V´1 m M l E Pn λ i,ϕ l ϑ 1 Ñ´Eps l m 1 qV´1 m M Proof. of Lemma 7Defining the following quantities :e j,i " e µ 1 i m j,i pϑq ,ẽ j,i " 1..s e j,i , κ j.i "´p e µ 1 i m j,i pϑq´1 q µ i m j,i pϑq 1 , s j,i :" B Bϕ log f px j \|ϕ, z i q and s j,i :" s j,i f j,i , the derivatives of pµ, λq with respect to ψ are as follows:Bϑ 1 µ i λ i,ϕ l ϕ 1 "´ÿ jẽ j s l j m 1 j µ i,ϕ λ i,ϕ l ϑ 1 "´ÿ jẽ j pµ 1 ϑ m j,i`M 1 j,i µ i´ÿ jẽ j M 1 j,i µ i qs l 1 j Figure 12 .Carlo Case 1 Figure 13 .Carlo Case 2 Figure 14 . 12113214Monte Monte Monte Carlo Case 3 , this would result to a different form for h ‹ pX\|Z, ψq. Under correct specification for f pX\|Z, ϕq, this choice does not matter asymptotically, while it matters in finite samples. Exponential tilting ensures a positive density function h ‹ while it has been shown that it is robust under misspecification of the moment conditions Proposition 6. E Pz log´d Pz dHzpϑ 0 q¯ă E Pz log´d Pz dFzpϑ 0 qP roposition 6 implies that if one obtains an approximate solution, tilting the density to satisfy the non-linear conditions implies a more accurate approximation (in KL units). A better approximation implies better decision making based on the more accurate model. Since the estimator is first order equivalent, and as efficient as the optimal GMM estimator asymptotically, in the next section, we provide simulation evidence for the corresponding finite sample performance of this method. In the context of this paper, what is useful is to look at the extent to which estimates can be biased when the base density is slightly misspecified, when it is in principle observed and estimable, but we have limited sample size. Prior information on what could be a good reduced form density can be potentially used. 3.1.1. Choice of approximate density. The choice of approximate density can be informed in different ways. First, as we already suggested, the approximate density can be constructed by looking at simpler i.e. linearized conditions and the corresponding likelihood function. The latter can be easily generated even in the presence of unobservables, using i.e. the Kalman filter. Obviously, higher order approximations or other forms of non-linearity can also be accommodated with the use of an appropriate filter, i.e. the particle filter. What is more is that we can utilize our possible knowledge of the reduced form of the structural model and directly use such a form in constructing the base density without explicitly solving the model. In the linear case, for example, this corresponds to using a V ARppq or V ARM App, qq where pp, qq can increase with the sample size, or a state space model in general. These reduced forms are known to correspond to linear DSGE models. More generally, an approximate density can be constructed by using prior information on which model can be best fitting, in sample or out of sample in previous exercises. This is information that is routinely used by practitioners even if it is not explicitly acknowledged. In the next section (and in the Appendix ) we investigate the performance of the estimator for different choices of base densities. 3.2. Monte Carlo Experiments. We conducted two Monte Carlo (MC) experiments ; in this section we present the MC experiment for the consumption Euler equation while the rest of the exercises are in Appendix B . 3.2.1. Estimating the Consumption Euler equation. We investigate the performance of our estimator in terms of M SEpβq in the case of locally and non-locally misspecified base densities. Similar to the analytical example we used in previous sections, the DGP is a Bivariate log-Normal VAR for the (demeaned) consumption and interest rate : Table 1 . 1Comparison of Relative Fit at ψ ‹Model TM AM log-Likelihood 7001.0 6341.2 log-Posterior 6997.2 6338.1 Table 2 . 2Robust Confidence Set for Reduced form ParametersParameter q 2.5% mode q 95% ρ 0.8678 0.9757 0.9934 χ x 0.1321 0.1539 0.2101 σ 0.0001 0.0008 0.0024 ρ l - 0.9560 - σ l - 0.0004 - ρ v,x 0.8747 0.9078 0.9555 σ v,x 0.0011 0.0078 0.0099 6.3.2. Additional Figures: Figure 11. Risk Premium on Aggregate Wealth versus Short Run Fluctuations in Consumption using the Tilted Model (Left) and Ap- proximate Model (Right) We mainly refer to probablistically incomplete models, that is models that do not pin down a unique probability distribution for the variables of interest. The EL weights can be negative, do not satisfy the restrictions for arbitrary parameters and do not provide an estimated density. Note that BD-1a and BD-1b imply that sup ψ E Pz \|\|e µpzq 1 mpx,z,ϑq`λpz,ϑq mpx, z, ϑ 0 q\|\| 1`δ ă 8 for δ ą 0 and @z. The additional subtlety here is that it has to hold for the base measure and the true measure. Given absolute continuity of dPpX\|Zq with respect to dF pX\|Zq, the existence of moments under PpX\|Zq is sufficient for the existence of moments under F pX\|Zq. Note that in the main part of the paper we replaceμ zi andλ zi , the simulation based estimates, with µ zi p" µ i q and λ zi p" λ i q respectively for the interest of brevity. The same holds for their derivatives with respect to pϑ, ϕq. We formally deal with simulation in the Appendix. We define an asymptotically correct density as the density f pX\|Z,φq that converges to the true density as N Ñ 8 i.e. the total variation distance in Lemma 1 converges to zero. They define ∆ to be the Sobolev Norm. This also corroborates the numerical results ofPohl, Schmedders, and Wilms (2018), who have shown that this approximation can be too crude when consumption growth is persistent. The share of Human wealth in aggregate wealth has been estimated, using a linear model, to by roughly 92% on averageLustig, Van Nieuwerburgh, and Verdelhan (2013). We postpone complete analytical derivations for the variance and covariance terms to the proof of Theorem 5, where boundedness of the variances and covariances oftλ ϑ,i , M 1 i µ i , µ 1 ϑ,i m i , µ 1 ϕ,i m i , s i , λ ϕ,i u is illustrated. Furthermore, given that all of these quantities are functions of mpx, zq, M px, zq using measure F or P , we can obtain dominating functions by taking the supremum over Ψ. Then, By assumption BD1´a they are bounded. Uniform convergence follows. 6.5.1. Form of Jacobian G N pψq: The population Jacobian matrix is the following:where superscript l denotes the l th column.6.5.2. Form of V g . 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[ "Shape transformations of a model of self-avoiding triangulated surfaces of sphere topology", "Shape transformations of a model of self-avoiding triangulated surfaces of sphere topology" ]	[ "Hiroshi Koibuchi [email protected] \nDepartment of Mechanical and Systems Engineering\nIbaraki National College of Technology\n866, 312-8508Nakane, Hitachinaka, IbarakiJapan\n" ]	[ "Department of Mechanical and Systems Engineering\nIbaraki National College of Technology\n866, 312-8508Nakane, Hitachinaka, IbarakiJapan" ]	[]	We study a surface model with a self-avoiding (SA) interaction using the canonical Monte Carlo simulation technique on fixed-connectivity (FC) triangulated lattices of sphere topology. The model is defined by an area energy, a deficit angle energy, and the SA potential. A pressure term is also included in the Hamiltonian. The volume enclosed by the surface is well defined because of the self-avoidance. We focus on whether or not the interaction influences the phase structure of the FC model under two different conditions of pressure ∆p; zero and small negative. The results are compared with the previous results of the self-intersecting model, which has a rich variety of phases; the smooth spherical phase, the tubular phase, the linear phase, and the collapsed phase. We find that the influence of the SA interaction on the multitude of phases is almost negligible except for the evidence that no crumpled surface appears under ∆p = 0 at least even in the limit of zero bending rigidity α → 0. The Hausdorff dimension is obtained in the limit of α → 0 and compared with previous results of SA models, which are different from the one in this paper.	10.1142/s0129183110015889	[ "https://arxiv.org/pdf/1009.5724v1.pdf" ]	119,177,452	1009.5724	b1334fca2073cbdc0e972a2db7284737d43f8a4d	Shape transformations of a model of self-avoiding triangulated surfaces of sphere topology 29 Sep 2010 Hiroshi Koibuchi [email protected] Department of Mechanical and Systems Engineering Ibaraki National College of Technology 866, 312-8508Nakane, Hitachinaka, IbarakiJapan Shape transformations of a model of self-avoiding triangulated surfaces of sphere topology 29 Sep 2010Triangulated surface modelSelf-avoiding interactionMonte CarloShape transformationsPhase transitions We study a surface model with a self-avoiding (SA) interaction using the canonical Monte Carlo simulation technique on fixed-connectivity (FC) triangulated lattices of sphere topology. The model is defined by an area energy, a deficit angle energy, and the SA potential. A pressure term is also included in the Hamiltonian. The volume enclosed by the surface is well defined because of the self-avoidance. We focus on whether or not the interaction influences the phase structure of the FC model under two different conditions of pressure ∆p; zero and small negative. The results are compared with the previous results of the self-intersecting model, which has a rich variety of phases; the smooth spherical phase, the tubular phase, the linear phase, and the collapsed phase. We find that the influence of the SA interaction on the multitude of phases is almost negligible except for the evidence that no crumpled surface appears under ∆p = 0 at least even in the limit of zero bending rigidity α → 0. The Hausdorff dimension is obtained in the limit of α → 0 and compared with previous results of SA models, which are different from the one in this paper. Introduction Over the past few decades, a considerable number of studies have been conducted on the surface models. The model was constructed for strings and membranes 1,2,3,4,5,6,7 , and it was defined on the basis of the differential geometric notion of curvatures 8,9,10,11 . The so-called crumpling transition is a shape transformation between the smooth phase at sufficiently large bending rigidity α and the collapsed phase at α → 0, and it has long been studied both theoretically 12,13,14,15 and numerically 16,17,18,19,20,21 . While the transition is considered as a continuous one 14,15 , a possibility that it is of first-order is pointed out 13 , and renormalization group studies 22 and recent numerical studies 23,24,25,26 predict that the transition is of first order. The transition was observed in the canonical surface model on relatively large sized surfaces 25 . In addition to the smooth and the collapsed phases, a variety of phases including the tubular phase are observed in surface models 27 , which are defined by a onedimensional bending energy on the cytoskeletal structure. A planar phase and an oblong linear phase can be seen in a model 28 , which is defined by a one-dimensional bending energy and the Nambu-Goto area energy on the fixed-connectivity (FC) surface. A surface model defined by a deficit angle energy also has a rich variety of phases including a tubular phase 29 . It must be noted that these phase transitions can be observed on relatively smaller surfaces in contrast to the above mentioned crumpling transition of the canonical surface model. To construct a surface model, the self-avoiding (SA) property should be taken into account if we focus on membranes 31,32,33,34,35,36,37,38,39,40,41,42 . However, numerical studies of the SA surfaces are very time consuming because of the nonlocal property of the interactions. The simulations on such large surfaces like those in Refs. 25,26 are still not feasible on currently available computers. Nevertheless, the numerical studies on the above mentioned variety of phases in those specific models are considered to be feasible on the SA surfaces. Therefore, it is very interesting to study whether or not the SA interaction influences the phase structure in those models without SA interactions (phantom surface models). It is possible that the multitude of phases is strongly influenced by the SA interactions. In fact, no completely-collapsed phase is observed in FC SA surfaces 33,34,35,36,37,38,42 . Moreover, the SA interaction is expected to play a non-trivial role in the membrane morphology even at the smooth phase. It was recently reported that SA property is essential for a variety of shapes of the so-called excess cone at high bending regime 43 . In this paper, we study the surface model in Ref. 29 with a SA interaction on FC triangulated surfaces by using the canonical Monte Carlo (MC) simulation technique. The smooth spherical phase, the tubular phase, the linear phase, and the collapsed phase are seen in the FC phantom surface model 29 . Our interests are focused on whether or not such a variety of phases, including the collapsed phase, are influenced by the SA interaction. Two different values of pressure ∆p are assumed such that ∆p is zero and small negative. This paper is organized as follows: in Section 2, we make a brief outline of the current results of the numerical studies of phantom surface models and SA surface models on triangulated surfaces. In Section 3, we define the model with a SA interaction, which is slightly different from the currently well-known SA interactions for numerical studies. The Monte Carlo simulation technique is shown in Section 4, and the numerical results are presented in Section 5. We summarize the results in the final Section 6. 2. Triangulated surface models 2.1. Phase structure of phantom surface models In this subsection, we give a brief outline of the phantom surface models on triangulated lattices in R 3 and the current numerical results. We start with the continuous model, which is given by the continuous Hamiltonian S = S 1 +αS 2 , where S 1 = √ gd 2 xg ab ∂ a X µ ∂ b X µ and S 2 = (1/2) √ gd 2 xg ab ∂ a n µ ∂ b n µ . S 1 is just iden-tical with the action of Polyakov string 9,10 , where X µ denotes a mapping from a two-dimensional surface M to R 3 and represents the surface position in R 3 , g ab is the inverse of the metric tensor g ab of M , and g is the determinant of g ab . The variables (x 1 , x 2 ) represent a local coordinate of M . The image X(M )(⊂ R 3 ) is the surface, which is triangulated in numerical studies. The symbol n µ in S 2 denotes a unit normal vector of X(M ), and S 2 is called the bending energy, and α is the bending rigidity. If g ab is fixed to the Euclidean metric δ ab and X(M ) is triangulated by piecewise linear triangles, then we have S = S 1 + αS 2 , S 1 = ij (X i −X j ) 2 , S 2 = ij (1−n i · n j ), where X i (∈ R 3 ) in S 1 is the position of the vertex i, n i in S 2 is a unit normal vector of the triangle i. The FC model is statistical mechanically defined by the partition function Z fix = ′ N i=1 dX i exp [−S(X)] , (fixed),(1) where the prime in ′ N i=1 dX i denotes that the three-dimensional multiple integrations are performed by fixing the center of mass of the surface at the origin of R 3 to remove the translational zero mode. We call the FC model defined by the energies S 1 and S 2 as the "canonical" surface model. It was reported that the canonical model on surfaces of sphere topology undergoes a first-order transition at finite α c between the smooth phase at α → ∞ and the collapsed phase at α → 0 25 . The role of the Gaussian bond potential S 1 is to make the mean bond length constant and, hence, S 1 can be replaced by a Lennard-Jones type potential 23 and also by a hard-wall potential 26 . In a surface model on triangulated lattices of the seminal paper Ref. 18 of Kantor and Nelson, S 1 is given by a hard-core and hard-wall potential. This type of potential can be used as a SA potential, which is described in the following sebsection. A variation of the canonical model is obtained by replacing S 1 with the Nambu-Goto area energy S ∆ = ∆ A ∆ , where A ∆ is the area of the triangle ∆. S ∆ is also obtained from the above mentioned continuous Hamiltonian S 1 by fixing g ab as the induced metric g ab = ∂ a X µ ∂ b X µ of the mapping X. We call a model as the Nambu-Goto surface model if the Hamiltonian includes S ∆ as the bond potential term. It is well-known that the Nambu-Goto model with the canonical bending energy S 2 = ij (1−n i · n j ) is ill-defined in the sense that no equilibrium configuration is obtained in the numerical simulations 44 . The ill-definedness comes from the fact that the area A ∆ is totally independent of the shape of ∆, and the oblong and very thin triangles, which are considered as singular triangles, dominate the surface configurations in the whole range of α. However, if the canonical bending energy S 2 is replaced by a deficit angle energy such as S int 46,47 , the model turns to be well defined except in the limit of α → 0 29 . The symbol δ i in S int 2 is the sum of internal angles of triangles meeting at the vertex i, and φ 0 is a constant and fixed to φ 0 = 2π if the surface is closed. The deficit angle energy S int 2 = − i log (δ i /2π) is possible on closed surfaces such as a sphere 29,48 . Those deficit angle energies are called as the intrinsic curvature energy. The reason of the variety of phases in the Nambu-Goto model with S int 2 29 seems that both S ∆ and S int 2 are insensitive to the surface shape. In fact, S ∆ is independent of whether or not the surface is composed of almost-regular triangles or oblong triangles. S int 2 is also independent of whether the surface is planar or cylindrical. Nevertheless, both of the smooth phase and the collapsed phase are stable on the disk surface 49 and on the torus 50 . In this, paper we study the Nambu-Goto surface model with the intrinsic curvature S int 2 . We should comment on the reason why we use S int 2 = − i log (δ i /2π) as the intrinsic curvature energy. The origin of S int 2 = i (δ i − φ 0 ) 2 or S int 2 = i \|δ i −φ 0 \| 45,2 = − i log (δ i /2π) is the measure factor q σ i in the integrations ′ N i=1 dX i q σ i in Z, where q i is the coordination number of the vertex i and σ(= 3/2) is a constant 51 . By identifying q i with δ i and extending the constant σ to the variable coefficient α, we have the expression −α i log (δ i ). Including the normalization factor 2π, we have the curvature energy S int 2 = − i log (δ i /2π). We should also comment on the fact that the mean value of S 1 is constant such that S 1 /N = 3/2 even in the limit of b → 0. The reason for S 1 /N = 3/2 is understood from the scale invariant property of the partition function 7 . In fact, by rescaling the integration variable in Z such that X → λX, we obtain Z(λ) = λ 3(N−1) ′ N i=1 dX i exp [−S(λX)], where S(λX) = λ 2 S 1 + bS 2 . The scale invariance of Z indicates that Z(λ) is independent of λ and, therefore, is represented by ∂Z(λ)/∂λ\| λ=1 = 0. Thus, we have S 1 /N = 3(N −1)/2N ≃ 3/2. A variety of phases can also be seen in a model, which is obtained by replacing S 1 and S 2 of the canonical model with S ∆ and the one-dimensional bending energy S 1−d 2 , respectively 28 . In this case, S 1−d 2 is sensitive to the surface shape, while S ∆ is not as mentioned above. A variation of the canonical model is obtained also by including fluidity, which represents a lateral diffusion of vertices 6,16,17,19,20,21 . This two-dimensional fluidity is defined on dynamically triangulated surfaces, where the triangulation T is considered as a dynamical variable of the model. The partition function of the model with fluidity is thus given by Z flu = T ′ N i=1 dX i exp [−S(X, T )] ,(fluid) where S(X, T ) represents that S is dependent on the variables X and T , and T represents the sum over all possible triangulations. In the fluid model corresponding to the canonical model, we cannot see the transition, which is seen in the canonical model on FC spherical surfaces. This is expected from the phase structure of compartmentalized fluid surfaces 52 , where the lateral diffusion is allowed only inside the compartment, which is a sublattice structure on the surface. In this compartmentalized model, a first-order transition, which is considered to be identical to the one in the canonical FC model, disappears if the compartment size L C is increased. The homogeneous fluid surface is obtained from the compartmentalized fluid surface by maximizing L C such that the surface is composed of a single compartment or the surface has no compartment. Thus, we understand that the transition, which is observed on the compartmentalized fluid surfaces at relatively small L C , cannot be observed on the homogeneous fluid surfaces. The Nambu-Goto model with the intrinsic curvature energy is well-defined even on the fluid surfaces and has a variety of phases 30 . By combining two different sets of ball-spring systems, Boal and Seifert introduced a fluid surface model with cytoskeletal structures, which is a two-components network model for red cells 53 . If a curvature energy is introduced on the compartment in place of the canonical bending energy S 2 in the compartmentalized fluid surface model in Ref. 52 , we have also fluid surface models with cytoskeletal structures 27 . A large variety of shape transformations are observed in such inhomogeneous fluid surface models, where the bond potential S 1 is the Gaussian bond potential, and the curvature energy S 2 is the one-dimensional bending energy S 1−d 2 defined only on the compartments, which are one-dimensional objects linked with junctions 27 . The phase structure depends on the elasticity at the junctions; a planar phase, and a tubular phase are observed in those models. The reason for such a variety of phases is closely connected to the cytoskeletal structure and the lateral diffusion of vertices. In fact, S 1−d 2 is considered to be insensitive to the surface shape, because Self-avoiding surface models The current studies that have been conducted on SA surfaces are considered to be still in the pioneer stage. In this subsection, we briefly comment on the existing SA surface models and the results of the numerical studies. The SA surface model is defined by a SA interaction, which is an extension of the Hamiltonian of the Edward model for polymers 3,4 . The phase structure of the SA models has been extensively studied 31,32,33,34,35,36,37,38,39,40,41,42 , although the total number of studies are currently considered to be far smaller than those of the phantom surfaces. We have two types of SA models for numerical studies: the ball-spring (BS) model and the impenetrable plaquette (IP) model. The BS model is defined on twodimensional networks, which are composed of vertices and bonds connecting two nearest neighbor vertices by a hard-core and hard-wall potential 31,32 . The size of ball as the vertices and the length of spring as bonds are constrained such that no vertex can move from one side of a triangle to the other side. This SA potential of the BS model is defined between all pairs of vertices, however, the simulations are slightly less time-consuming than those of the IP model. The SA interaction of the IP model is defined such that the triangles are constrained to avoid intersecting. Although the simulations of the IP model are relatively time consuming, the IP model seems advantageous to the BS model. In fact, two neighboring triangles i and j of the IP model can completely bend such that 1−n i · n j = 2, while in the BS model the bending angle θ ij is constrained such that θ ij < θ 0 , where cos θ ij = n i · n j , and θ 0 (< π) is determined by the SA potential. The crumpling transition is reported to disappear from the SA FC surfaces 33,34,35,36,37,38,42 ; (1), although both models are defined on the disk surface. As mentioned in the Introduction, it is possible that the SA interaction plays a non-trivial role in membrane shapes in the smooth phase 43 . Thus, we should study the SA model more extensively. To summarize the comments including those in the previous subsection, we have several phantom surface models, which have a multitude of phases. The models are considered as non-trivial variations of the canonical surface model. The phase structures of almost all models have not yet been studied on the SA surfaces. The current understanding of the phase structure of SA surfaces are as follows: the crumpling transition disappears from the FC model, because the SA interaction prohibits the surfaces from collapsing in both of the BS model and the IP model. The smooth phase of the SA surfaces are considered to be almost identical to the smooth phase of the phantom surfaces, while the membrane shapes are expected to be influenced by the SA interaction under some specific conditions. Model In this section, we define a SA model, which corresponds to the phantom surface models in Refs. 29,30 . Triangulated lattices of sphere topology are assumed to define the model, and the lattices are constructed using the icosahedron. By splitting the edges and faces of the icosahedron, we have a lattice of size N = 10ℓ 2 +2, where ℓ is the devision number of an edge of the icosahedron. The coordination number q of vertices is q = 6 almost everywhere excluding 12 vertices of q = 5. The lattice is characterized by the three numbers N , N B (= 3N−6 = 30ℓ 2 ), and N T (= 2N−4 = 20ℓ 2 ), which are the total number of vertices, the total number of bonds, and the total number of triangles, respectively. The lattices used in Ref. 29 are random lattices, of which the coordination number is not always uniform and, they are slightly different from the lattices constructed as above. However, the phase structure of FC surface models is expected to be independent of the lattice structure 26 . The dynamical variable of the FC model is the position X i (∈ R 3 ) of the vertex i(= 1, · · · , N ). The partition functions of the model are given by Eq. (1). The Hamiltonian S(X) is defined by a linear combination of the area energy S 1 , a curvature energy S 2 , the pressure term −∆p V , and a SA potential U , such that S(X) = S 1 + αS 2 − ∆p V + U, S 1 = ∆ A ∆ , S 2 = − i log (δ i /2π) ,(3)U = ∆,∆ ′ U (∆, ∆ ′ ), U (∆, ∆ ′ ) = ∞ (triangles ∆, ∆ ′ intersect), 0 (otherwise). S 1 is the sum over the area A ∆ of triangle ∆. The symbol δ i in S 2 is the sum of internal angle of triangles meeting at the vertex i. S 2 can be called a deficit angle energy, although S 2 is different from the sum of the deficit angle δ i − 2π of the vertex i. If S 2 is defined without "log" and is given by i (δ − 2π), then S 2 depends only on the surface topology and is a constant on piece-wise linearly triangulated surfaces. However, S 2 in Eq. (3) is well-defined as a curvature energy because of the log function as mentioned in Section 2.1. The symbol α[kT ] denotes the bending rigidity, where k is the Boltzmann constant and T is the temperature. V is the volume enclosed by the surface, and ∆p is the pressure which is defined by ∆p = p in − p out , where p out (p in ) is the pressure outside (inside) the surface. If p out is assumed to be p out = 0, then the positive (negative) ∆p implies that p in is positive (negative). We should note also that the volume V is well defined only if the surface is self-avoiding. V is bounded below such that V ≥ 0 in the SA surfaces, while V can be negative in non SA surfaces. ∆,∆ ′ in the SA potential U denotes the sum over all pairs of non nearest neighbor triangles ∆ and ∆ ′ . The potential U (∆, ∆ ′ ) is defined such that any pairs of non nearest neighbor triangles ∆ and ∆ ′ should not be intersecting. Figure 1(a) shows two pairs of intersecting triangles, in which the triangle ABC penetrates the triangle DEF or in other words the bonds AB and BC intersect with the triangle DEF . On the contrary in Fig. 1(b), the triangle ABC and the triangle DEF intersects with each other, or in other words a bond of one triangle intersects with the other triangle and vise versa. We describe the numerical implementation of the SA interaction U in detail in the following section. The SA potential U in Eq. (3) is not identical to the one assumed in the SA model in Ref. 42 and, hence, the surface is completely self-avoiding under the potential U . In fact, the triangles are allowed to intersect with finite energy in Ref. 42 , while those in the model of this paper are prohibited to intersect with each other because of the infinite energy assumed in U . Finally in this section, we comment on how to compute the volume V enclosed by the surface. The initial value of V in the simulations is assumed such that V = 4πr 3 /3, where r is the radius of the initial configuration of sphere lattice. This initial value V = 4πr 3 /3 is slightly larger than the real volume, because the surface is linearly triangulated. However, it is almost evident that the deviation can be negligible in the limit of N → ∞. The volume V changes during the simulations according to the rule V → V + ∆V every update of vertex, where ∆V is the volume of small tetrahedra, such as the one shown in Fig. 1(c). ∆V is positive or negative, which is determined according to whether the new position X ′ i is outside or inside the surface, in which the orientation is uniquely fixed by a normal vector of each triangle. We should note that ∆V is well defined only when the surface is selfavoiding. It is apparent that ∆V is not well defined when some part of volume element of ∆V is shared by some other ∆V ′ , i.e., the surface is allowed to self intersect. The enclosed volume V can also be computed by using the divergence theorem applying the position vector r i of the center of mass of the triangle i. Not only ∆V but also V is exactly identical to the one obtained by the above mentioned technique. A very small deviation can be seen in the total volume V , however, it is less than 1% even in the cup like phase on the N = 1442 surface during the simulations. This small deviation of V is the one between V = 4πr 3 /3 and V of the initial triangulated sphere. Monte Carlo technique The canonical Metropolis Monte Carlo (MC) technique is employed for simulating the integrations of the variables X in Z fix of Eq. (1). The three-dimensional random move X → X ′ = X+δX is accepted with the probability Min[1, exp(−δS)], where δS is given by δS = S(new)−S(old) under the constraint of the potential U . The symbol δX is randomly chosen in a small sphere, whose radius is fixed in the simulations such that the acceptance rate r X of X ′ should be approximately r X = 50%. The constraint of U (∆, ∆ ′ ) in Eq. (3) is composed of two different constraints on a new vertex position as follows: let X i and X ′ i denote the current position and the new position of the vertex i as shown in Fig. 1(c). The shaded triangle in Fig. 1(c) forms a new surface. One of the constraint imposed on X ′ i is that the new triangle i ′ jk has no intersection with the disjoint bonds, where "disjoint bonds" are the edges of triangles disconnected with the triangle ijk. The other constraint is that every new bond, such as the bond i ′ j in Fig. 1(c), has no intersection with the disjoint triangles, where "disjoint triangles" are those disconnected with the bond ij. These two constraints imposed on X ′ i make the surface self-avoiding in the sense that any two disjoint triangles have no intersection with each other. The first constraint prohibits the new triangle i ′ jk shown in Fig. 1(c) from being penetrated by disjoint triangles. The second constraint imposed on X ′ i prohibits the new triangle i ′ jk from penetrating some other triangles. The intersection of the triangles shown in Fig. 1(b) is prohibited by both of the constraints, while the intersection in Fig. 1(a) is prohibited only by one constraint and is not prohibited by the other constraint. This is the reason why two constraints are necessary to make the surface self-avoiding by checking an intersection of a bond and a triangle. We assume a sphere of radius R 0 at the center of mass of the triangle i ′ jk shown in Fig. 1(c), and check whether or not the triangle intersects with disjoint bonds inside the sphere. The check of intersection in the second constraint is also performed assuming the sphere of size R 0 at the center of the bond i ′ j. The radius R 0 is assumed to be R 0 = 6 L , where L is the mean bond length. As a consequence, the computational time is reduced by 20% ∼ 60% or more, which depends on α. The bond length L and the triangle area A ∆ are bounded below such that L > 1 × 10 −7 and A ∆ > 0.5 × 10 −7 in the simulations. The final results of the simulations are considered to be independent of these lower bounds, because these bounds are sufficiently small and almost all bond lengths and triangle areas are larger than these values. The total number of MC sweeps (MCS) after the thermalization MCS is about 1 × 10 7 ∼ 2 × 10 7 on the N = 1442 surface, and relatively small number of MCS is assumed on the smaller surfaces. The total number of the thermalization MCS is about 0.5×10 6 . The thermalization MCS in the collapsed tubular phase is very large; it is sometimes 1×10 7 or more at the phase boundary close to the cup like phase on the N = 1442 surface. Intersection of bonds with triangles is checked every 5×10 5 MCS throughout the simulation; the check is performed between every disjoint pair of bond and triangle. No intersection is observed at every assumed value of α including α = 0 and ∆p. Results The snapshots of FC surfaces at ∆p = 0 are shown in Figs spherical phase. The surface in Fig. 2(a) can be called a collapsed surface because the surface is highly fluctuating, however, it encloses empty space inside the surface and, therefore, the surface is not always crumpled in the limit of α → 0. The spherical surfaces in Figs. 2(b) and 2(c) look slightly smooth, however, they are apparently different from the surface at sufficiently large α shown in Fig. 2 Fig. 3(c) is also collapsed. The cup like surfaces in Figs. 3(d) and 3(e) are new and typical of the condition ∆p = −0.5, therefore, we call the new phase as the cup like phase. The smooth phase in Fig. 3(f) corresponds to the smooth phase in Fig. 2(f) at ∆p = 0. The phase structure at α → ∞ is understood to be independent of ∆p. We also see that almost all parts of the surfaces in Figs. 2(d) and 2(e) consist of oblong triangles and are locally smooth along one specific direction and wrinkled along the direction vertical to the smooth direction. This is also expected in the linear phase shown in Fig. 3(c) at ∆p = −0.5. To the contrary, the surface in the wrinkled phase shown in Fig. 2(a) consist of almost regular triangles and locally wrinkles along any directions. In the case of smooth phase in Fig. 2(f), the surface is smooth along any directions. Thus, the surface is symmetric under the threedimensional rotations both in the limit of α → ∞ and α → 0, while the rotational symmetry is spontaneously broken at intermediate region of α. This observation is independent of the two values of ∆p. This symmetry breakdown or restoration is considered to be closely connected to the structural change of the constituent triangles; the symmetric surfaces are composed of almost regular triangles, while the non-symmetric surfaces are composed of oblong triangles. (1), S ′ 1 /N is expected to be S ′ 1 /N = 3/2 at sufficiently large N . We see that all of the results are consistent with the prediction. This implies that the volume V is welldefined and that the SA interaction is correctly implemented in the simulations. We note that it is straightforward to prove that S ′ 1 /N = 3/2 7 . As described in Section 2.1, the scale invariance of Z is represented by ∂Z(λX)/∂λ\| λ=1 = 0. Because of the scale transformation X → λX, S 1 and V change to λ 2 S 1 and λ 3 V while S 2 and U remain unchanged. Since the integration i dX i also changes to λ 3(N−1) i dX i , then we have the relation S ′ 1 /N = 3/2 in the limit of N → ∞. S ′ 1 is identical with S 1 , which is the total area of surface, in the case ∆p = 0, and therefore, the scale invariance implies that the surface area remains unchanged in the whole range of α. To the contrary, the surface area S 1 discontinuously changes at the transition points in the case ∆p = −0.5 at least, because V discontinuously changes at the transitions as we will see below, while S ′ 1 remains unchanged. This implies that the internal property of surface is significantly influenced by the external condition ∆p. The volume V enclosed by the surface should be bounded below such that V ≥ 0, which is satisfied only if the surface is self-avoiding. The model in this paper is strictly self-avoiding, and hence V is expected to be well defined even when ∆p is large negative. Figures 5(a) and 5(b) show the dependence of V on α under ∆p = 0 and ∆p = −0.5, respectively. The vertical dashed lines in the figures represent the phase boundaries between two different phases just like in Fig. 4. The name of the phases corresponds to the surface shape, which can be visualized as snapshots just like those in Figs. 2 and 3. The detailed informations such as the order of the transitions are not obtained. It is possible to perform the finite-size scaling analyses to see the order of the transitions by performing the simulations at the transition region more extensively, however, we confine ourselves of the phase structure in the wide range of α and, as a consequence, the order of the transitions is not fully examined. Thus, it remains unclear whether or not the smooth spherical phase and the tubular phase (or the cup like phase) are separated by a first-order transition, although the volume V discontinuously changes at the phase boundary. We see that the volume V in the collapsed phase is larger than that in the cup like phase under ∆p = −0.5 at least, while V at α → 0 is smaller than that in the tubular phase under ∆p = 0. The mean square size X 2 is defined by X 2 = 1 N i X i −X 2 ,X = 1 N i X i ,(4) whereX is the center of mass of the surface. The value of X 2 changes depending on the distribution of the vertices in R 3 , and hence X 2 as well as V can reflect shape transformations. However, the quantity X 2 does not always show the same behavior against α as that of V . Figures 6(a) and 6(b) show X 2 vs. α under ∆p = 0 and ∆p = −0.5. We see in Fig. 6(a) that X 2 discontinuously changes at the phase boundary between the smooth spherical phase and the tubular phase. It is also easy to see from Fig. 6(b) that X 2 discontinuously changes at the phase boundaries between the smooth spherical phase, the cup like phase, and the collapsed tubular phase. X 2 ∼ N 2/H (N → ∞).(5) By fitting the data X 2 obtained at α = 0 to Eq. (5), we draw straight lines in Fig. 7(c), and the values of H are shown in Table 1 including H at α = 0.1 and α = 1 under ∆p = 0, and at α = 2 under ∆p = −0.5. The fitting is performed using the largest three data points under each condition of ∆p. We have H ≃ 2.6 at ∆p = −0.5 and H ≃ 2.33 at ∆p = 0 in the limit of α → 0. The value of H ≃ 2.33 is compatible with the one H ≃ 2.3 in Ref. 38 , while H ≃ 2.6 at ∆p = −0.5 is slightly larger than the Flory estimate 2.5 and compatible with the fact that the surface is almost crumpled as we see in the snapshot in Fig. 3(a). We should note that the value H ≃ 2.3 seems independent of the details of the model, the SA interaction, and the surface topology. However, the result H ≃ 2.33 is larger than the one H = 2.1(1) of Ref. 42 , thus it is also possible that H depends on the model on the SA surfaces. We comment on the size effect of the results in Table 1. As mentioned above, the data obtained on the small sized surfaces, such as N = 162 and N = 362, were excluded from the fitting. By including the small two data in the fitting, we have H = 2.50(5) for ∆p = 0, α = 0 and H = 2.93(7) for ∆p = −0.5, α = 0. Both of H are slightly larger than H = 2.33 (8) and H = 2.60(17) shown in Table 1. Thus, the size effect is not negligible at least on the surfaces N ≤ 362. The surface size can also be reflected in the maximum linear extension L E , which is defined by the maximum distance between two vertices on the surface: where X i and X j are not always connected by a bond. The phase transition of shape transformation is also reflected in the structure of triangles; we see in the snapshots in Figs. 2 and 3 that the surface consists of almost regular triangles in the smooth spherical phase while it includes oblong triangles in the tubular phase, where L E expected to be very large. We expect that this structural change is reflected in L E . The two-dimensional bending energy S 3 /N B is shown in Figs. 9(a) and 9(b), where S 3 is defined by using a unit normal vector n i of the triangle i such that L E = Max{\|X i − X j \| \| (i, j = 1, · · · , N )},(6)S 3 = (ij) (1 − n i · n j ).(7) We write the two-dimensional bending energy as S 3 to distinguish it with the deficit angle energy S 2 in Eq. (3). We see that S 3 /N B discontinuously changes at the phase boundaries, where the physical quantities such as V , X 2 and L E discontinuously change. To the contrary, the deficit angle energy S 2 defined in Eq. (3), which is not shown in the figures, appears to vary almost smoothly in the whole range of α. At the boundary between the smooth spherical phase and its neighboring phase, S 2 /N is expected to change discontinuously like the other physical quantities. However, the discontinuity is very small and it is almost invisible just as in the case of the self-intersecting model in Ref. 29 . Summary and Conclusion We have numerically studied a self-avoiding (SA) surface model on fixedconnectivity (FC) triangulated lattices of sphere topology. The self-avoidance of the model in this paper is not identical to those of the well-known SA models; the ball spring model and the impenetrable plaquette (IP) model. However, the SA model in this paper belongs to the IP models, because the intersection of disjoint triangles are prohibited by the SA interaction. The phase structure of the FC model under ∆p = 0 is found to be almost identical to that of the phantom surface model in Ref. 29 except for the evidence that the collapsed phase disappears from the SA model. Thus, the influence of the SA interaction on the phase structure is very small contrary to the expectation that the SA interaction can suppresses the multitude of phase transitions in the phantom surface model. To be more precise, the model in this paper is a Nambu-Goto surface model with a deficit angle energy. The SA interaction is defined such that all possible pairs of non-nearest neighbor triangles are prohibited from intersecting. Because the volume enclosed by the SA surface is well defined, the pressure term −∆p V can be included in the Hamiltonian. The simulations are performed under ∆p = 0, and ∆p = −0.5 on the FC surfaces, where ∆p = −0.5 implies that the pressure inside the surface is lower than the pressure outside the surface. Our observations on the FC surfaces are as follows: the smooth spherical phase, the tubular phase, and the collapsed phase can be seen under those two conditions of ∆p, and the cup like phase is seen under ∆p = −0.5. Thus, the phase structure of the model under ∆p = 0 is almost identical to that of the phantom surface model, although the collapsed phase is slightly different from each other; the collapsed surfaces are completely shrunk in the phantom surface model, while the SA surfaces are not completely shrunk at ∆p = 0 at least. The Hausdorff dimension H = 2.33 (8), obtained at α = 0 under ∆p = 0, is independent of the curvature energy and is considered as the Hausdorff dimension of the Nambu-Goto SA surface. This result is consistent with the known result of H ≃ 2.3 of the IP model in Ref. 38 , where the model, the SA interaction, and the surface topology are different from those in this paper. In this sense, it is possible that the value H ≃ 2.3 depends only on the self-avoidance, although the surface size of the simulation in Ref. 38 is relatively smaller than those assumed in this paper. To the contrary, H = 2.33 (8) is larger than the result H = 2.1(1) of Ref. 42 , and therefore, it is also possible that H of the SA surface depends on the model. The SA surface models should be studied more extensively. It is also interesting to study whether or not the multitude of phases in the fluid surface models with cytoskeletal structures in Ref. 27 is observed under a SA interaction. The SA interaction assumed in the model of this paper can also be assumed in those fluid surface models even when ∆p is negative. This remains to be a future study. only on the compartments in contrast to the model in Ref. 28 , where S 1−d 2 is defined all over the lattice. The surface shape is not always uniquely determined if the curvature is given only at small part of the surface, and moreover large surface fluctuations are expected in the compartmentalized model in Ref. 27 due to the lateral diffusion of vertices inside the compartments. Fig. 1 . 1(a),(b) Two intersecting triangles, and (c) an intersection of the bond I and the triangles with the vertices X ′ i , X j and X k , where X ′ i is a new position of the vertex i. In (a), the bonds AB and BC of the triangle ABC intersect with the triangle DEF , while no bond of the triangle DEF intersects with the triangle ABC. In (b), the bond BC of the triangle ABC intersects with the triangle DEF , and the bond DE of the triangle DEF intersects with the triangle ABC. . 2(a)-2(f). The surface size is N = 1442. The assumed bending rigidities are (a) α = 0, (b) α = 100, (c) α = 500, (d) α = 1000, (e) α = 1×10 4 , and (f) α = 2×10 4 . The scales of the figures are all different from each other. The surfaces shown in the figure are considered to be in (a),(b),(c) the wrinkled phase, (d),(e) the tubular phase, and (f) the smooth Fig. 2 . 2The snapshots of FC surfaces and the surface sections of size N = 1442 obtained under ∆p = 0 at (a) α = 0 (wrinkled), (b) α = 100 (wrinkled), (c) α = 500 (wrinkled), (d) α = 1000 (tubular), (e) α = 1×10 4 (tubular), and (f) α = 2×10 4 (smooth spherical). (f). The surfaces in Figs. 2(d) and 2(e) can be called a tubular surface. The surface in Fig. 2(f) is very smooth and can be called the smooth spherical surface. All of the phases, excluding the wrinkled phase, correspond to those of the same model without the SA interaction in Ref. 29 . The collapsed phase can be seen in the model in Ref. 29 , while it is not in the SA model at least under ∆p = 0. Snapshots of the FC surfaces and the surface sections are shown in Figs. 3(a)-3(f), where a negative pressure ∆p = −0.5 is assumed. The snapshots are slightly different from those at ∆p = 0 shown in Figs. 2(a)-2(f). The snapshots in Figs. 3(a) and 3(b) indicate that the surfaces in the collapsed phase are almost crumpled. We see that the tubular surface in Fig. 3 . 3The snapshots of FC surfaces and the surface sections of size N = 1442 obtained under ∆p = −0.5 at (a) α = 0 (collapsed), (b) α = 100 (collapsed), (c) α = 1.5×10 4 (collapsed tubular), (d) α = 2×10 4 (cup like), (e) α = 3×10 5 (cup like), and (f) α = 4×10 5 (smooth spherical). Fig. 4 . 4[S 1 − (3/2)∆p V ]/N vs. α under (a) ∆p = 0 and (b) ∆p = −0.5. The error bars on the symbols denote the standard errors. The solid lines connecting the symbols are drawn as a guide to the eyes. In the following presentations, we show how the shape transformation transitions and/or the SA interaction are reflected in the physical quantities including the Hausdorff dimension H in the limit of α → 0. First of all, we show [S 1 − (3/2)∆p V ]/N , denoted by S ′ 1 /N , in Figs. 4(a) and 4(b). Because of the scale invariant property of the partition function Z fix of Eq. Fig. 5 . 5The volume V vs. α under (a) ∆p = 0 and (b) ∆p = −0.5. The vertical dashed lines denote the phase boundaries of the N = 1442 surface. The solid lines connecting the symbols are drawn as a guide to the eyes. Fig. 6 . 6The mean square size X 2 vs. α. The vertical dashed lines denote the phase boundaries of the N = 1442 surface. Fig. 7 . 7Figures 7(a) and 7(b) show X 2 obtained under ∆p = 0 and ∆p = −0.5 at small The mean square size X 2 vs. α at small α region under (a) ∆p = 0 and (b) ∆p = −0.5, and (c) X 2 vs. N in a log-log scale obtained at α = 0 under ∆p = 0 and ∆p = −0.5. The straight lines in (c) are drawn by fitting the largest three data points to Eq.(5). 2.33±0.08 H = 2.34±0.08 H = 2.33±0.08 -∆p = −0.5 H = 2.60±0.17 --H = 2.59±0.17 α region. The Hausdorff dimension H of the surface is defined by Fig. 8 . 8The maximum linear extension L E vs. α. The dashed lines denote the phase boundaries of the N = 1442 surface. Figures 8(a) and 8(b) show L E vs. α under ∆p = 0 and ∆p = −0.5. The discontinuous change of L E at the phase boundaries shown in the figure implies that the phase transitions are accompanied by the structural change of surfaces. This structural change is typical of the Nambu-Goto surface model 28,29,30 . Fig. 9 . 9The two-dimensional bending energy S 3 /N B vs. α under (a) ∆p = 0 and (b) ∆p = −0.5. this is because no completely-collapsed phase appears in the SA FC surfaces. In fact, no crumpled phase is observed in both of the BS model 33,34,37 and the IP model 35,36,38,42 . To the contrary, the smooth phase is expected to remain unchanged from that of the phantom surfaces. 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[]	[ "Erhan Bayraktar ", "Moshe A Milevsky [email protected] ", "S David Promislow ", "Virginia R Young [email protected] ", "\nDepartment of Mathematics\nSchulich School of Business\nDepartment of Mathematics and Statistics\nUniversity of Michigan Ann Arbor\nYork University\n48109, M3J 1P3TorontoMichigan, Ontario\n", "\nDepartment of Mathematics\nYork University\nM3J 1P3TorontoOntario\n", "\nUniversity of Michigan Ann Arbor\n48109Michigan\n" ]	[ "Department of Mathematics\nSchulich School of Business\nDepartment of Mathematics and Statistics\nUniversity of Michigan Ann Arbor\nYork University\n48109, M3J 1P3TorontoMichigan, Ontario", "Department of Mathematics\nYork University\nM3J 1P3TorontoOntario", "University of Michigan Ann Arbor\n48109Michigan" ]	[]	We develop a theory for valuing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is identical to the upper good deal bound of Cochrane and Saá-Requejo(2000)and ofBjörk and Slinko (2006)applied to our setting. A second result of our paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure (as inBlanchet-Scalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market's price of mortality risk.	10.2139/ssrn.1335476	[ "https://arxiv.org/pdf/0802.3250v1.pdf" ]	15,360,342	0802.3250	68f253f6107c94cb6b347b6e7c1817d50c97a4c6	22 Feb 2008 Version: 20 February 2008 Erhan Bayraktar Moshe A Milevsky [email protected] S David Promislow Virginia R Young [email protected] Department of Mathematics Schulich School of Business Department of Mathematics and Statistics University of Michigan Ann Arbor York University 48109, M3J 1P3TorontoMichigan, Ontario Department of Mathematics York University M3J 1P3TorontoOntario University of Michigan Ann Arbor 48109Michigan 22 Feb 2008 Version: 20 February 2008Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life Annuities Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life AnnuitiesStochastic mortalitypricingannuitiesSharpe rationon-linear We develop a theory for valuing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is identical to the upper good deal bound of Cochrane and Saá-Requejo(2000)and ofBjörk and Slinko (2006)applied to our setting. A second result of our paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure (as inBlanchet-Scalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market's price of mortality risk. Introduction A basic textbook assumption on pricing life insurance and annuity contracts is that mortality risk is completely diversifiable and, therefore, should not be priced by capital markets in economic equilibrium. Indeed, under the traditional insurance pricing paradigm, the law of large numbers is invoked to argue that the standard deviation per policy vanishes in the limit and in practice a large enough portfolio is sufficient to eliminate mortality risk from the valuation equation. Although this only applies to a large number of claims that are independent random variables, it is commonly accepted that, in fact, typical real-world portfolios of life insurance and pension annuities satisfy the requisite assumptions. Thus, for example, if actuaries estimate that 50% of 65-year-olds in the year 2007 will live to see their 85th birthday in the year 2027, then a sufficiently homogeneous group of 65-year-olds should be charged 50 cents per present value of $1 dollar of a pure endowment contract (ignoring investment income). The underlying assumption is the existence of a well-defined population survival curve, which determines the fraction of the population living to any given time. This is also the probability of survival for any given individual within the group. Thus, if the insurance company sells enough of these policies and charges each policyholder the discounted value, on average the company will have enough to pay the $1 to each of the survivors. Using the language of modern portfolio theory, the idiosyncractic risk, as measured by the standard deviation per policy, will go to zero if insurers sell enough policies, so the mortality risk is not priced. However, a number of recent papers in the insurance and actuarial literature challenge this traditional approach within the framework of financial economics. This research falls under the general title of valuation of stochastic mortality. In this paper, we provide a unique contribution to this literature by valuing stochastic mortality using techniques that have been traditionally applied to portfolio management. Within the topic of stochastic mortality, recent papers by Milevsky and Promislow (2001), Dahl (2004), DiLoernzo and Sibillo (2003), and Cairns, Blake, and Dowd (2006) proposed specific models for the evolution of the hazard rate. Others, such as Blake and Burrows (2001), Biffis and Millossovich (2006), Boyle and Hardy (2004), Cox and Wang (2006), proposed and analyzed mortality-linked instruments. Few, if any, have focused on the actual equilibrium compensation for this risk. Milevsky, Promislow, and Young (2006) provided some simple discrete-time examples, while Denuit and Dhaene (2007) used comonotonic methods to analyze this risk. A practitioneroriented paper by Smith, Moran, and Walczak (2003) used financial techniques to justify the valuing of mortality risk, although their approach is quite different from ours. What this literature essentially argues is that uncertainty regarding the evolution of the instantaneous force of mortality induces a mortality dependence that cannot be completely diversified by selling more contracts. This phenomenon induces a mortality risk premium that is valued by the market and whose magnitude will depend on a representative investor's risk aversion or demanded compensation for risk. In other words, if there is a positive probability that science will find a cure for cancer during the next thirty years, aggregate mortality patterns will change. Insurance companies must charge for this risk since it cannot be diversified. The question we confront is: how much should they charge? From a technical point of view, in this paper, we value mortality-contingent claims by assuming that the insurance company issuing the claims will be compensated for aggregate mortality risk via the instantaneous Sharpe ratio of a suitably defined hedging portfolio. Specifically, we assume that the insurance company chooses a hedging strategy that minimizes the local variance, as in Schweizer (2001a), then picks a target ratio α of net expected return to standard deviation, and finally determines the corresponding value for any given mortality-contingent claim that leads to this pre-determined α. See Bayraktar and Young (2008) for further discussion of this methodology from the standpoint of the mean-variance efficient frontier. The main contribution of this paper is to develop a financial theory of valuing mortalitydependent contracts under stochastic hazard rates. We apply our method to value life annuities. Arguably, the primary result of the paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure, and from this representation, one can interpret the instantaneous Sharpe ratio as the annuity market's price of mortality risk. Although our valuation mechanism is very different-we use a continuous version of the standard deviation premium principle-it turns out that the value of the annuity corresponds to the upper good deal bound of Cochrane and Saá-Requejo (2000) and of Björk and Slinko (2006). The no-arbitrage price interval of a contingent claim is a wide interval, and Cochrane and Saá-Requejo (2000) find a reasonably small interval for prices by ruling out so-called good deals by putting a bound on the absolute value of the market price of risk of the Radon-Nikodym derivative of valuation measures with respect to the physical measure. The lower and upper prices for the resulting interval are obtained through solving stochastic control problems, although this is not precisely stated in Cochrane and Saá-Requejo (2000). Our valuation method provides a new interpretation of the upper good deal bound of a life annuity as the value to a seller under the instantaneous Sharpe ratio α when the hedging strategy is chosen to minimize the local variance. The lower good deal bound is the value to a buyer of a life annuity. The remainder of this paper is organized as follows. In Section 2, we present our financial market, describe how to use the instantaneous Sharpe ratio to value the life annuity, and derive the resulting partial differential equation (PDE) that the value solves. We also present the PDE for the value a (n) of n conditionally independent and identically distributed life annuity risks. In Section 3, we compare our valuation method with two that are common in the literature. Specifically, (1) we consider the good deal bounds of Cochrane and Saá-Requejo (2000) and of Björk and Slinko (2006). We show that for the issuer of a life annuity, our value is identical to the upper good deal bound; for the buyer of a life annuity, our value is identical to the lower good deal bound. Therefore, we give an alternative derivation of the good deal bounds for our setting. Then, (2) we consider indifference pricing via expected utility, as described in Zariphopoulou (2001), for example. We observe that the relationship between our valuation method and indifference pricing is similar to the relationship between the standard deviation principle and variance principle in insurance pricing (Bowers et al., 1986). In Section 4, we present several properties of a (n) and find the limiting value of 1 n a (n) . We show that this limiting value solves a linear PDE and can be represented as an expectation with respect to an equivalent martingale measure. Section 5 concludes the paper. Instantaneous Sharpe Ratio In this section, we describe a life annuity and present the financial market in which the issuer of this contract invests. We obtain the hedging strategy for the issuer of the life annuity. We describe how to use the instantaneous Sharpe ratio to value the life annuity and derive the resulting PDE that the value solves. We also present the PDE for the value a (n) of n conditionally independent and identically distributed life annuity risks. Mortality Model and Financial Market We use the stochastic model of mortality of Milevsky, Promislow, and Young (2005). We assume that the hazard rate λ t (or force of mortality) of an individual follows a diffusion process such that if the process begins at λ 0 > λ for some nonnegative constant λ, then λ t > λ for all t ≥ 0. From a modeling standpoint, λ could represent the lowest attainable hazard rate remaining after all causes of death such as accidents and homicide have been eliminated; see, for example, Gavrilov and Gavrilova (1991) and Olshansky, Carnes, and Cassel (1990). Specifically, we assume that dλ t = µ(λ t , t) (λ t − λ) dt + σ(t) (λ t − λ) dW λ t , λ 0 > λ, (2.1) in which {W λ t } is a standard Brownian motion on a filtered probability space (Ω, F , (F t ) t≥0 , P). The volatility σ is either identically zero, or it is a continuous function of time t bounded below by a positive constant κ on [0, T ]. The drift µ is such that there exists ǫ > 0 such that if 0 < λ − λ < ǫ, then µ(λ, t) > 0 for all t ∈ [0, T ]. This condition on the drift ensures that λ t > λ with probability 1 for all t ∈ [0, T ]. In general, we assume that µ satisfies requirements such that (2.1) has a unique strong solution; see, for example, Karatzas and Shreve (1991, Section 5.2). Note that if σ ≡ 0, then λ t is deterministic. Example 2.1 Suppose µ(λ, t) = g + 1 2 σ 2 + m ln(λ 0 − λ) + mgt − m ln(λ − λ) and σ(t) ≡ σ in which g ≥ 0, m ≥ 0, and σ > 0 are constants. The solution to (2.1) is given by λ t = λ + (λ 0 − λ) exp (g t + Y t ) , dY t = −m Y t dt + σ dW λ t , Y 0 = 0,(2. 2) a mean-reverting Brownian Makeham's law (Milevsky and Promislow, 2001). Note that Y t = σ t 0 e −m(t−s) dW λ s , which reduces to σW λ t when m = 0. As an aside, µ(λ, t) satisfies the conditions in Assumption 4.1 below. Suppose an insurer issues a life annuity to an individual that pays at a continuous rate of $1 per year until the individual dies or until time T , whichever occurs first. Throughout this paper, we assume that the horizon T is fixed. In Section 2.2.2, to determine the value of the life annuity, we will create a portfolio composed of (1) the obligation to pay this life annuity, of (2) default-free, zero-coupon bonds that pay $1 at time T , regardless of the state of the individual, and of (3) money invested in a money market account earning at the short rate. Therefore, we require a model for bond prices, and we use a model based on the short rate and the bond market's price of risk. The dynamics of the short rate r, which is the rate at which the money market increases, are given by dr t = b(r t , t) dt + c(r t , t) dW t ,(2.3) in which b and c ≥ 0 are deterministic functions of the short rate and time, and {W t } is a standard Brownian motion with respect to the probability space (Ω, F , (F t ) t≥0 , P), independent of {W λ t }. As for equation (2.1), we assume that b and c ≥ 0 are such that r > 0 almost surely and such that (2.3) has a unique solution; see, for example, Karatzas and Shreve (1991, Section 5.2). From the principle of no-arbitrage in the bond market, there is a market price of risk process {q t } for the bond that is adapted to the filtration generated by {W t }; see, for example, Lamberton and Lapeyre (1996) or Björk (2004). Moreover, the bond market's price of risk at time t is a deterministic function of the short rate and of time; that is, q t = q(r t , t). Thus, the time-t price F of a default-free, zero-coupon bond that pays $1 at time T is given by F (r, t; T ) = E Q e − T t r s ds r t = r , (2.4) in which Q is the probability measure with Radon-Nikodym derivative with respect to P given by dQ dP F t = exp − t 0 q(r s , s) dW s − 1 2 t 0 q 2 (r s , s) ds . It follows that {W Q t }, with W Q t = W t + t 0 q(r s , s) ds, is a standard Brownian motion with respect to Q. From Björk (2004), we know that the price F of the T -bond solves the following PDE:    F t + b Q (r, t)F r + 1 2 c 2 (r, t)F rr − rF = 0, F (r, T ; T ) = 1, (2.5) in which b Q = b − qc. For the remainder of Section 2, we drop the notational dependence of F on T because T is understood; whenever the bond pays at a time other than T , then we specify that time. We can use the PDE (2.5) to obtain the dynamics of the T -bond's price F (r s , s), in which we think of r t = r as given and s ∈ [t, T ]. Indeed, dF (r s , s) = (r s F (r s , s) + q(r s , s) c(r s , s) F r (r s , s))ds + c(r s , s) F r (r s , s) dW s , F (r t , t) = F (r, t). (2.6) We use (2.6) in the next section when we develop the dynamics of a portfolio containing the obligation to pay the life annuity and a certain number of T -bonds. Valuing via the Instantaneous Sharpe Ratio In this section, we first describe our method for valuing life annuities in this incomplete market. Then, we fully develop the valuation for a single life annuity, that is, n = 1, and we present the valuation for the case when n ≥ 1. Recipe for valuation The market for insurance is incomplete; therefore, there is no unique pricing mechanism. To value contracts in this market, one must assume something about how risk is "priced." For example, one could use the principle of equivalent utility (see Zariphopoulou (2001) for a review) or consider the set of equivalent martingale measures (Blanchet-Scalliet, El Karoui, and Martellini, 2005;Dahl and Møller, 2006) to value the risk. We use the instantaneous Sharpe ratio because of its analogy with the bond market's price of risk and because of the desirable properties of the resulting value; see Sections 3.1 and 4. Also, in the limit as the number of contracts approaches infinity, the resulting value can be represented as an expectation with respect to an equivalent martingale measure. Because of these properties, we anticipate that our valuation methodology will prove useful in pricing risks in other incomplete markets; for example, see Bayraktar and Young (2008). Our method for valuing contingent claims in an incomplete market is as follows: A. First, define a portfolio composed of two parts: (1) the obligation to underwrite the contingent claim (life annuity, in this case), and (2) a self-financing sub-portfolio of T -bonds and money market funds to (partially) hedge the contingent claim. B. Find the investment strategy in the bonds so that the local variance of the total portfolio is a minimum. If the market were complete, then one could find an investment strategy so that the local variance is zero. However, in an incomplete market, there will be residual risk as measured by the local variance. This control of the local variance is called local risk minimization by Schweizer (2001a). C. Determine the value of the contingent claim so that the instantaneous Sharpe ratio of the total portfolio equals a pre-specified value. This amounts to setting the drift of the portfolio equal to the short rate times the portfolio value plus the Sharpe ratio times the local standard deviation of the portfolio. Cochrane and Saá-Requejo (2000) and Björk and Slinko (2006) apply the idea of limiting the instantaneous Sharpe ratio to restrict the possible prices of claims in an incomplete market, and in Section 3.1, we further discuss how our method relates to theirs. Also, Schweizer (2001b) derives a financial counterpart to the standard deviation premium principle. His was a global standard deviation premium principle, whereas ours can be viewed as a local standard deviation premium principle. Valuing life annuities Consider the time-t value of the life annuity that pays at the continuous rate of $1 per year until the individual dies or until time T , whichever comes first. Denote the value of this annuity by a = a(r, λ, t), in which we explicitly recognize that the value of the annuity depends on the short rate r and the hazard rate λ at time t. (As an aside, by writing a to represent the value of the annuity, we mean that the buyer of the annuity is still alive. If the individual dies before time T , then the value of the policy jumps to 0. Also, the value at time T of the annuity is 0.) As prescribed in Step A of Section 2.2.1, suppose the insurer creates a portfolio with value Π t at time t. The portfolio contains (1) the obligation to underwrite the life annuity, with value −a, including the continuous flow of $1 out of the portfolio to pay the life annuity, and (2) a selffinancing sub-portfolio of T -bonds and money market funds with value V t at time t to (partially) hedge the risk of the annuity. Thus, while the individual is alive, Π t = −a(r t , λ t , t) + V t with a continuous outflow of money at the rate of $1 per year for t ≤ T . Let π t denote the number of T -bonds in the self-financing sub-portfolio at time t, with the remainder, namely V t − π t F (r t , t), invested in a money market account earning the short rate r t . The annuity risk cannot be fully hedged because of the randomness inherent in the individual living or dying. If the individual dies at time t < T , then the value of the annuity jumps to 0; therefore, the value of the portfolio Π t jumps from −a(r t , λ t , t) + V t to V t . In other words, the value of the portfolio instantly changes by a(r t , λ t , t). Note that the portfolio that includes the value of the annuity is not self-financing, although the sub-portfolio with value V t is. For t ≥ 0, let τ (t) denote the future lifetime of the individual; that is, t + τ (t) is the time of death of the individual given that the individual is alive at time t. Alternatively, τ (t) = inf{u ≥ 0 : N (t + u) = 1 N (t) = 0}. Proposition 2.1 When r t = r and λ t = λ, the drift and local variance of the portfolio Π t are given by, respectively, lim h→0 1 h E Π t+(h∧τ (t)) − Π t F t = −D b a(r, λ, t) − 1 + π t q(r, t) c(r, t) F r (r, t) + r Π, (2.7) and lim h→0 1 h Var(Π t+(h∧τ (t)) \|F t ) = c 2 (r, t)(π t F r (r s , s)−a r (r, λ, t)) 2 +σ 2 (t)(λ−λ) 2 a 2 λ (r, λ, t)+λa 2 (r, λ, t). (2.8) Proof. We first specify the dynamics of a(r t , λ t , t) and V t . By Itô's lemma (see, for example, Protter (1995)), the dynamics of the value of the life annuity are given as follows: da(r t , λ t , t) = a t dt + a r dr t + 1 2 a rr d[r, r] t + a λ dλ t + 1 2 a λλ d[λ, λ] t = a t dt + a r (b dt + c dW t ) + 1 2 a rr c 2 dt + a λ (λ t − λ)(µ dt + σ dW λ t ) + 1 2 a λλ σ 2 (λ t − λ) 2 dt, (2.9) in which we suppress the dependence of the functions a, b, etc. on the variables r, λ, and t. Also, [r, r], for example, denotes the quadratic variation of r. Because the sub-portfolio with value V t = π t F (r t , t) + (V t − π t F (r t , t)) is self-financing, its dynamics are given by dV t = π t dF (r t , t) + r t (V t − π t F (r t , t))dt = (π t q c F r + r t V t ) dt + π t c F r dW t , (2.10) in which the second equality follows from (2.6), and we again suppress the dependence of the functions on the underlying variables r and t. From the dynamics in equations (2.9) and (2.10), from the fact that the portfolio continually pays at the rate of $1 while the individual is alive, and from the jump in the portfolio value when the individual dies (see the discussion preceding Proposition 2.1), it follows that the value of the portfolio at time t + (h ∧ τ (t)) with h > 0, namely Π t+(h∧τ (t)) , equals Π t+(h∧τ (t)) = Π t − t+(h∧τ (t)) t da(r s , λ s , s) + t+(h∧τ (t)) t dV s − t+(h∧τ (t)) t ds + t+(h∧τ (t)) t a(r s , λ s , s) dN s = Π t − t+(h∧τ (t)) t (D b a(r s , λ s , s) + 1)ds + t+(h∧τ (t)) t c(r s , s)(π s F r (r s , s) − a r (r s , λ s , s)) dW s − t+(h∧τ (t)) t σ(s)(λ s − λ)a λ (r s , λ s , s) dW λ s + t+(h∧τ (t)) t a(r s , λ s , s)(dN s − λ s ds) + t+(h∧τ (t)) t π s q(r s , s) c(r s , s) F r (r s , s) ds + t+(h∧τ (t)) t r s Π s ds, (2.11) in which D b is an operator defined on the set of appropriately differentiable functions on R + × (λ, ∞) × [0, T ] by D b v = v t + bv r + 1 2 c 2 v rr + µ(λ − λ)v λ + 1 2 σ 2 (λ − λ) 2 v λλ − (r + λ)v. (2.12) {N t } denotes a counting process with stochastic parameter λ t at time t that indicates when the individual dies. Note that we adjust dN t in (2.11) so that N t − t 0 λ s ds is a martingale. The expressions in (2.7) and (2.8) follow from (2.11). In this single-life case, the process Π is "killed" when the individual dies. If we were to consider the value a (n) of n conditionally independent and identically distributed lives (conditionally independent given the hazard rate), then N would be a counting process with stochastic parameter nλ t until the first death such that Π jumps by a (n) − a (n−1) upon that death, and N becomes a counting process with stochastic parameter (n − 1)λ t , etc. We consider a (n) later in this section and continue with the single-life case now. As stated in Step B in Section 2.2.1, we choose the number of T -bonds π t to minimize the local variance of this portfolio, namely lim h→0 1 h Var(Π t+(h∧τ (t)) \|F t ), a dynamic measure of risk of the portfolio. Note that this investment strategy is the same one advocated by Schweizer (2001a) in local risk minimization. We have the following corollary of Proposition 2.1. Corollary 2.2 The optimal investment strategy π * t that minimizes the local variance is given by π * t = a r (r t , λ t , t)/F r (r t , t), and under this assignment, the drift and local variance become, respectively, lim h→0 1 h E Π t+(h∧τ (t)) F t − Π = −D b Q a(r, λ, t) − 1 + rΠ, (2.13) and lim h→0 1 h Var Π t+(h∧τ (t)) F t = σ 2 (t)(λ − λ) 2 a 2 λ (r, λ, t) + λa 2 (r, λ, t), (2.14) in which the operator D b Q is defined in (2.12) with b replaced by b Q = b − qc. Now, we come to valuing this annuity via the instantaneous Sharpe ratio, as in Step C of Section 2.2.1. Because the minimum local variance in (2.14) is positive, the insurer is unable to completely hedge the risk of the life annuity. Therefore, the value should reimburse the insurer for its risk, say, by a constant multiple α ≥ 0 of the local standard deviation of the portfolio. It is this α that is the instantaneous Sharpe ratio. We restrict the choice of α to be bounded above by √ λ to avoid values that violate the principle of no arbitrage; see Section 3.1 for further discussion of this point. To determine the value a of the life annuity, we set the drift of the portfolio equal to the short rate times the portfolio plus α times the local standard deviation of the portfolio. Thus, from (2.13) and (2.14), we have that a solves the equation −D b Q a − 1 + rΠ = rΠ + α σ 2 (t)(λ − λ) 2 a 2 λ + λa 2 . (2.15) We summarize the above discussion in the following proposition. Proposition 2.3 The value of the life annuity a = a(r, λ, t) solves the non-linear PDE given by          a t + b Q a r + 1 2 c 2 a rr + µ(λ − λ)a λ + 1 2 σ 2 (λ − λ) 2 a λλ − (r + λ)a + 1 = −α σ 2 (λ − λ) 2 a 2 λ + λa 2 , a(r, λ, T ) = 0, (2.16) in which the terminal condition arises from the assumption that the annuity only pays until time T . If we had been able to choose the investment strategy π so that the local variance in (2.14) were identically zero (that is, if the risk were hedgeable), then the right-hand side of the PDE in (2.16) would be zero, and we would have a linear differential equation of the Black-Scholes type. To end Section 2.2.2, we present the PDE solved by the value a (n) of n conditionally independent and identically distributed life annuity risks. Specifically, we assume that all the individuals are of the same age and are subject to the same hazard rate as given in (2.1); however, given that hazard rate, the occurrences of death are independent. As discussed in the paragraph following the proof of Proposition 2.1, when an individual dies, the portfolio value Π jumps by a (n) − a (n−1) . By paralleling the derivation of (2.16), one obtains the following proposition. Proposition 2.4 The value a (n) = a (n) (r, λ, t) of n conditionally independent and identically distributed life annuity risks solves the non-linear PDE given by              a (n) t + b Q a (n) r + 1 2 c 2 a (n) rr + µ(λ − λ)a (n) λ + 1 2 σ 2 (λ − λ) 2 a (n) λλ − ra (n) − nλ a (n) − a (n−1) + n = −α σ 2 (λ − λ) 2 a (n) λ 2 + nλ a (n) − a (n−1) 2 , a (n) (r, λ, T ) = 0 (2. 17) The initial value in this recursion is a (0) ≡ 0, and the value a defined by (2.16) is a (1) . Our model includes the special case for which α = 0; this corresponds to the local risk minimization pricing and hedging model of Schweizer (2001a). Therefore, our methodology is more general than that of local risk minimization. Corollary 2.5 When α = 0, then a (n) = n a α0 solves (2.17), in which a α0 is given by a α0 (r, λ, t) = T t F (r, t; s) E λ,t e − s t λ u du ds, (2.18) in which F is the time-t price of a default-free zero-coupon bond that pays $1 at time s as in (2.4) with T replaced by s. Relationship with the Literature In Section 3.1, we show that our valuation of a life annuity for the seller is identical to the upper good deal bound of Cochrane and Saá-Requejo (2000), generalized by Björk and Slinko (2006) to the case of a jump diffusion. The lower good deal bound is obtained as the buyer's valuation of the life annuity. Then, in Section 3.2, we compare our valuation with that given by indifference pricing via expected utility, as in Zariphopoulou (2001), for example. Good Deal Bounds Our work is this section is motivated by similar results of Bayraktar and Young (2008); see the remark at the end of their Section 2.5. It is straightforward to show that one can write (2.16) as          a t + b Q a r + 1 2 c 2 a rr + 1 2 σ 2 (λ − λ) 2 a λλ + 1 + max δ 2 +λγ 2 ≤α 2 , γ≥−1 (µ + δσ)(λ − λ)a λ − (r + λ(1 + γ))a = 0, a(r, λ, T ) = 0. (3.1) Note that the optimal values for δ and γ are given by, respectively, δ * = ασ(λ − λ)a λ σ 2 (λ − λ) 2 a 2 λ + λa 2 ,(3.2) and γ * = − αa σ 2 (λ − λ) 2 a 2 λ + λa 2 . (3.3) Because we restrict α to lie between 0 and √ λ inclusively, it follows that γ * > −1 because λ > λ. Therefore, the restriction that γ ≥ −1 in (3.1) is automatically satisfied at the maximum but we need it for the representation of a in (3.4) below. From ( Here, {r t } and {λ t } follow the processes dr u = b Q du+c dW u and dλ u = (µ+δσ)(λ u −λ)du+σ(λ u − λ)dW λ u , respectively, withW u = W u + u 0 q(r s , s)ds andW λ u = W λ u − u 0 δ u ds. The processes {W t } and {W λ t } are independent standard Brownian motions with respect to the filtered probability space (Ω, F , (F t ) t≥0 ,P), in which dP dP F t = L t , with L t given by dL t = L t− −q(r t , t) dW t + δ t dW λ t + γ t (dN t − λ t dt) , L t = 1. (3.5) E denotes expectation with respect toP. The expression for a in (3.4) is identical to the upper good deal bound given by Björk and Slinko (2006; Theorem 2.2). As mentioned following the expression for γ * in (3.3), we have γ * > −1 because α ≤ √ λ. This fact ensures that the expression in (3.4) is no greater thanĒ r,t T t F (r, t; s) ds , which is required by no arbitrage. Also, note that γ * s ≥ −1 is required in order forP defined via (3.5) to be nonnegative. Insurance actuaries are familiar with expressions similar to those in the expectation of (3.4) because in valuing life annuities, they effectively discount for interest and mortality often modifying the hazard rate to account for longevity risk (Wang, 1995). To gain further understanding of (3.4), we can write it as follows: a(r, λ, t) = T t F (r, t; s) sup {{δ s ,γ s }: δ 2 s +λ s γ 2 s ≤α, γ s ≥−1, t≤s≤T }Ē λ,t e − s t λ u (1+γ u )du ds,(3.6) in which we slightly abuse notation by writingĒ here. Because λ s (1 + γ s ) > 0 with probability 1, we can think of it as a modified hazard rate in (3.6). Note that if α = 0, then γ * ≡ 0 and δ * ≡ 0, and the expectation in the integral of (3.6) reduces to ϕ α0 (λ, t; s) := E λ,t exp(− s t λ u du) , the physical probability that a person survives from time t to s, and the corresponding value of a equals a α0 as in (2.18). One can repeat the derivation in Section 2.2 to show that the value a b for the buyer of a life annuity solves (2.16) with α replaced by −α. Equivalently, a b solves (3.1) with max replaced by min. Therefore, the buyer's value of the annuity is identical to the lower good deal bound of Björk and Slinko (2006;Theorem 2.2). It follows that we have the following bid-ask interval for life annuity values under our methodology, which we have just shown is equal to the one under good deal bounds: inf {δ s ,γ s }Ē r,λ,t T t e − s t r u du 1 {τ (t)>s} ds F t , sup {δ s ,γ s }Ē r,λ,t T t e − s t r u du 1 {τ (t)>s} ds F t , (3.7) in which we take the sup, respectively inf, over {{δ s , γ s } : δ 2 s + λ s γ 2 s ≤ α, γ s ≥ −1, t ≤ s ≤ T }. Therefore, we have provided an alternative derivation of the good deal bounds for our setting as the result of a local risk minimization investment strategy combined with a (local) risk loading expressed as a multiple of the local standard deviation of the portfolio. The interval in (3.7) is a subinterval of the interval of no-arbitrage prices inf Q∈M E Q T t e − s t r u du 1 {τ (t)>s} ds F t , sup Q∈M E Q T t e − s t r u du 1 {τ (t)>s} ds F t , (3.8) in which M is the set of equivalent martingale measures. Since this no-arbitrage pricing interval is too wide and practically useless, it was Cochrane and Saá-Requejo (2000)'s idea to find a tighter subinterval. From the representation of the seller's price in (3.4), one can see that our valuation method results in a coherent risk measure (Artzner et al., 1999). Indifference Pricing via Expected Utility Ludkovski and Young (2008) apply indifference pricing via expected utility to value pure endowments and life annuities in the presence stochastic hazard rates. They use exponential utility so that the value of the insurance contracts are independent of the wealth of the seller. With absolute risk aversion η > 0, one can follow the derivation of equation (2.41) in Ludkovski and Young (2008) to show that, in the setting of this paper, the indifference price a IP = a IP (r, λ, t) solves the following PDE:              a IP t + b Q a IP r + 1 2 c 2 a IP rr + µ(λ − λ)a IP λ + 1 2 σ 2 (λ − λ) 2 a IP λλ − (r + λ)a IP + 1 = − η 2F σ 2 (λ − λ) 2 (a IP ) 2 λ − λF η e −ηa IP /F − 1 + ηa IP F , a IP (r, λ, T ) = 0. (3.9) Here F = F (r, t; T ). Note that when η = 0, then a IP = a α0 , as defined in (2.18), which is also the price under the local risk minimization method of Schweizer (2001a). The choice of α in (2.16) reflects the seller's risk aversion, just as the choice of η in (3.9) reflects the seller's risk aversion in the setting of expected utility. Parallel to equation (3.1), one can rewrite (3.9) as follows:                      a IP t + b Q a IP r + 1 2 c 2 a IP rr + 1 2 σ 2 (λ − λ) 2 a IP λλ + 1 + max δ, γ − F 2η δ 2 + λF η (γ − (1 + γ) ln(1 + γ)) + (µ + δσ)(λ − λ)a IP λ − (r + λ(1 + γ))a IP = 0, a IP (r, λ, T ) = 0. (3.10) From (3.10), it follows that a IP can be represented as a IP (r, λ, t) = sup {δ s ,γ s }Ē r,λ,t T t e − s t (r u +λ u (1+γ u )) du F (r s , s; T ) η − δ 2 s 2 + λ s (γ s − (1 + γ s ) ln(1 + γ s )) ds . (3.11) As in (3.4), {r t } and {λ t } follow the processes dr u = b Q du + c dW u and dλ u = (µ + δσ)(λ u − λ)du + σ(λ u − λ)dW λ u , respectively, withW u = W u + u 0 q(r s , s)ds andW λ u = W λ u − u 0 δ u ds. Note that (3.11) is similar to the representation of a in (3.4) with the constraint on the controls {δ s , γ s } replaced by the penalty term (F (r s , s; T )/η) −δ 2 s /2 + λ s (γ s − (1 + γ s ) ln(1 + γ s )) . What follows is a formal discussion to show how the PDE in (3.9) is further related to the one in (2.16). For ηa IP /F small enough, one can "approximate" this PDE by expanding e −ηa IP /F and ignoring terms of the third power and higher. When one does this, the resulting PDE is given by          A t + b Q A r + 1 2 c 2 A rr + µ(λ − λ)A λ + 1 2 σ 2 (λ − λ) 2 A λλ − (r + λ)A + 1 = − η 2F σ 2 (λ − λ) 2 A 2 λ + λA 2 , A(r, λ, T ) = 0. (3.12) Compare this PDE with the one in (2.16). Note that they are quite similar with α in (2.16) replaced by η/(2F ) in (3.12) and with the removal of the square root in (2.16). Milevsky, Promislow, and Young (2005) show that valuing insurance risks via the instantaneous Sharpe ratio is closely related to the standard deviation principle, which is used by actuaries to price insurance (Bowers et al., 1986). Similarly, Pratt (1964) shows that indifference pricing is closely related to the variance principle, under which the risk load is the square of the risk load under the standard deviation pricing principle. Pratt's result is consistent with (3.12). Thus, the valuation method that we propose in this paper is related to indifference pricing in a way parallel to how the standard deviation principle is related to the variance principle, both of which are used in insurance pricing. Properties of a (n) , the Value of n Life Annuities To demonstrate important properties of a (n) , we rely on a comparison principle (Walter, 1970), and we present that in Appendix A. In Section 4.1, we list some qualitative properties of a (n) . Then, in Section 4.2, we prove the limiting result for 1 n a (n) . Assumption 4.1 Henceforth, we assume that the volatility on the short rate c satisfies the growth condition in the hypothesis of Theorem A.1 and that the drifts b Q and µ satisfy the growth conditions in the hypothesis of Lemma A.2. For later purposes (specifically for Property 3 in Proposition 4.1 below), we also assume that µ λ satisfies the growth condition \|µ λ \|(λ − λ) + \|µ\| ≤ K 1 + (ln(λ − λ)) 2 . Also, we assume that 0 ≤ α ≤ √ λ, as stated in Section 3.1. Qualitative Properties of a (n) In this section, we list and discuss properties of the risk-adjusted value a (n) of n life annuity contracts. We state the following proposition without proof because its proof is similar to those in Milevsky, Promislow, and Young (2005). Proposition 4.1 Under Assumption 4.1, a (n) = a (n) (r, λ, t) satisfies the following properties on G = R + × (λ, ∞) × [0, T ] for n ≥ 1 : (1) No arbitrage: 0 ≤ a (n) ≤ n T t F (r, t; s) ds. (2) Increasing in n: a (n) ≥ a (n−1) . (3) Decreasing in λ: a (n) λ ≤ 0. (4) Increasing in α: Suppose 0 ≤ α 1 ≤ α 2 ≤ √ λ, and let a (n),α i be the solution of (2.17) with α = α i , for i = 1, 2 and n ≥ 0. Then, a (n),α 1 ≤ a (n),α 2 . (5) Lower bound: na α0 ≤ a (n),α , in which a α0 is defined in (2.18). (6) Decreasing in µ: Suppose µ 1 ≤ µ 2 , and let a (n),µ i denote the solution of (2.17) with µ = µ i , for i = 1, 2. Then, a (n),µ 1 ≥ a (n),µ 2 . (7) Increasing in σ if convex in λ: Suppose 0 ≤ σ 1 (t) ≤ σ 2 (t) on [0, T ], and let a (n),σ i denote the solution of (2.17) with σ = σ i , for i = 1, 2. If a (n),σ 1 λλ ≥ 0 for all n, or if a (n),σ 2 λλ ≥ 0 for all n, then a (n),σ 1 ≤ a (n),σ 2 . (8) Subadditive: For m, n nonnegative integers, a (m+n) ≤ a (m) + a (n) . (9) Decreasing value per risk: 1 n a (n) decreases with respect to n ≥ 1. (10) Scaling: The value of n annuities that pay at a rate of k ≥ 0 is k a (n) . Remarks: 4.1 Because the payoff under a life annuity is nonnegative, we expect its value to be nonnegative. Also, if we hypothesize that the individuals will not die, then we get the upper bound given in Property 1. In Section 4.2, we sharpen the bounds given in Property 1 considerably. 4.2 One uses Property 2 to prove Property 3. However, Property 2 is interesting in its own right because it confirms our intuition that the marginal value of adding an additional policyholder, namely a (n) − a (n−1) , is nonnegative. 4.3 Property 3 makes sense because we expect the value of life annuities to decrease as the hazard rate increases, that is, as individuals are more likely to die. 4.4 Property 4 states that as the parameter α increases, the risk-adjusted value a (n),α increases. This result justifies the use of the phrase risk parameter when referring to α. Also, by referring to the expression in (2.18), we see that Property 5 follows from Property 4. Therefore, the lower bound of 1 n a (n),α (as α approaches zero) is the same as the lower bound of a α = a (1),α , namely, a α0 . We call the difference 1 n a (n) − a α0 ≥ 0 the risk charge per annuity for a portfolio of n life annuities. Because 1 n a (n) − a α0 ≥ 0, the strategy we propose results in a net profit on average, but this margin is necessary to compensate the issuer for the undiversifiable mortality risk, as in indifference pricing (Section 3.2). 4.5 In words, Property 6 tells us that as the drift of the hazard rate increases, then the value of the life annuities decreases. This occurs for essentially the same reason that the value decreases with the hazard rate; see Property 3. 4.6 Subadditivity is a reasonable property because if it did not hold, then buyers of annuities could buy their annuities separately and thereby save money. Subadditivity follows as a corollary from Property 9, which is interesting in itself, namely that the average value per risk decreases as the number of risks increases. In other words, the risk charge per annuity, namely 1 n a (n) − a α0 ≥ 0, decreases as n increases. In the next section, we will determine the limiting value of the risk charge per annuity as n goes to infinity, and from this result, we will subdivide the risk charge into the portion due to the finite size of the portfolio and the portion due to randomness of the hazard rate. 4.7 The same ten properties hold for the value of n pure endowment risks (given in Appendix B), as shown in Milevsky, Promislow, and Young (2005), with Property 1 appropriately modified. These properties-which also hold in simpler, static settings (that is, for typical actuarial premium principles)-help to demonstrate that our valuation mechanism is reasonable. What we gain in going to our more complicated setting is a dynamic pricing mechanism that compensates the issuer of the annuity for the undiversifiable mortality risk and shows the issuer how to hedge using a local risk minimization strategy. Limiting Result for 1 n a (n) In this section, we determine and interpret the limit lim n→∞ 1 n a (n) . In the following two lemmas, we apply Theorem A.1 and Lemma A.2 to show that on G = R + × (λ, ∞) × [0, T ], we have n T t F (r, t; s) β(λ, t; s) ds ≤ a (n) (r, λ, t) ≤ T t F (r, t; s) ϕ (n) (λ, t; s) ds, in which β and ϕ (n) are defined in (B.4) and (B.2), respectively, in Appendix B. Thus, if we divide by n, we have that lim n→∞ 1 n a (n) (r, λ, t) = T t F (r, t; s) β(λ, t; s) ds. See Appendix C for the proofs of the two following lemmas. Lemma 4.2 Under Assumption 4.1, we have a (n) (r, λ, t) ≤ T t F (r, t; s) ϕ (n) (λ, t; s) ds, (4.1) on G for n ≥ 0, in which ϕ (n) is defined in (B.2). From Lemma 4.2, we learn that if one were to treat ϕ (n) as a modified expected number of survivors in a traditional actuarial computation of an annuity value, that is, the right-hand side of (4.1), then we overvalue the annuity. Inequality (4.1) makes sense because a (n) takes into account the reduction in overall risk when valuing an annuity (left-hand side of (4.1)) versus a series of essentially independent infinitesimal pure endowments (right-hand side of (4.1)). Lemma 4.3 Under Assumption 4.1, we have n T t F (r, t; s) β(λ, t; s) ds ≤ a (n) (r, λ, t), (4.2) on G for n ≥ 0, in which β is defined in (B.4), or equivalently, in (B.6). Lemma 4.3 implies that the value a (n) of a life annuity is greater than n times the value of a life annuity computed by using the limiting value of 1 n ϕ (n) , namely β. In particular, a (n) is greater than n times the traditional actuarial value of a life annuity, namely T t F (r, t; s) ϕ α0 (λ, t; s) ds because β is greater than or equal to the physical probability of survival ϕ α0 . Recall from Appendix B that the value of n conditionally independent and identically distributed pure endowments is given by F ϕ (n) . We combine these two lemmas with a result from Milevsky, Promislow, and Young (2005) to obtain the main result of this paper. Proof. From Milevsky, Promislow, and Young (2005), we know that the sequence 1 n ϕ (n) decreases uniformly to β on (λ, ∞) × [0, s] for all 0 ≤ s ≤ T . Also, recall from Property 9 in Proposition 4.1 that 1 n a (n) decreases with respect to n. Because T t F ϕ ds ≤ T t ϕ ds ≤ 1 λ−α √ λ < ∞, it follows from the Lebesgue (dominated) convergence theorem (Royden, 1988, page 267) and Lemmas 4.2 and 4.3 that From the discussions accompanying (2.4) and (B.6), we can represent the limit in (4.3) as an expectation. Corollary 4.5 For (r, λ, t) ∈ G, lim n→∞ 1 n a (n) (r, λ, t) =Ê r,λ,t T t e − s t (r u +λ u )du ds , (4.5) in which {r t } and {λ t } follow the processes dr u = b Q du + c dŴ u and dλ u = (µ − ασ)(λ u − λ)du + σ(λ u − λ)dŴ λ u , respectively, withŴ u = W u + u 0 q(r s , s) ds andŴ λ u = W λ u + αu. The processes {Ŵ t } and {Ŵ λ t } are independent standard Brownian motions with respect to the filtered probability space (Ω, F , (F t ) t≥0 ,P), in which dP dP F t = exp − t 0 q(r s , s) dW s − 1 2 t 0 q 2 (r s , s) ds e −αW λ t − 1 2 α 2 t . E denotes expectation with respect toP. Remarks: 4.8 The expression in (4.5) is identical to that of the value of a coupon bond that pays coupons continuously at the rate of $1 under the discount rate of r t + λ t , instead of simply r t as for a bond that cannot default. Therefore, one can think of λ t as the default rate. For this reason, we anticipate that our methodology will prove useful in valuing credit risk derivatives. 4.9 The expression in (4.5) is an expectation with respect to a measure that is equivalent to the physical measure P. Therefore, in the limit, we obtain an arbitrage-free value for the life annuity. Thereby, one can think of our valuation method as generalizing the one of Blanchet-Scalliet, El Karoui, and Martellini (2005). 4.10 If the hazard rate is deterministic, then in the limit, there is no mortality risk. Specifically, if σ ≡ 0, then lim n→∞ 1 n a (n) = a α0 on G, in which a α0 is the net premium given in (2.18). In other words, if the hazard rate is deterministic, then as the number of contracts approaches infinity, the value of a life annuity collapses to the discounted expected payout using the physical probability measure to value the mortality risk, regardless of the target value of the Sharpe ratio. Alternatively, one can see that in the limit the average cash flow is certain, hence, the value becomes that of a "certain" (or non-life) annuity with a rate of discount modified to account for the rate of dying. 4.11 On the other hand, if the hazard rate is truly stochastic, then in the limit, mortality risk remains. Specifically, if σ is uniformly bounded below by κ > 0, then lim n→∞ 1 n a (n) ≥ a α0 on G, with equality only when t = T . 4.12 We are now prepared to subdivide the risk charge per annuity into the portion due to the finite portfolio and the portion due to the stochastic hazard rate. From the proof of Theorem 4.4, we know that 1 n a (n) (r, λ, t) ≥ T t F (r, t; s) β(λ, t; s) ds, with equality in the limit. Therefore, define 1 n a (n) − T t F β ds as the risk charge (per risk) for holding a finite portfolio of n annuities, and define T t F β ds − a α0 as the risk charge for the stochastic hazard rate. Thus, we have 1 n a (n) − a α0 = 1 n a (n) − (4.6) in which the risk charge for the stochastic hazard rate is zero if σ ≡ 0 and is positive otherwise. T t F β ds + T t F β ds − a α0 , 4.13 The representation of the limiting value of 1 n a (n) in (4.3) is quite similar to a net single premium as defined in courses on Life Contingencies (Bowers et al., 1986). Indeed, F is the monetary discount function, and β plays the role of the survival probability. Additionally, we can express the limit T t F (r, t; s) β(λ, t; s) ds as the solution p of the following PDE:    p t + b Q p r + 1 2 c 2 p rr + (µ − ασ)(λ − λ)p λ + 1 2 σ 2 (λ − λ) 2 p λλ − (r + λ)p + 1 = 0, p(r, λ, T ) = 0. (4.7) Note that α is analogous to the bond market's price of risk q in (2.4); therefore, we refer to α as the (annuity) market's price of mortality risk. 4.14 The proof of Theorem 4.4 suggests that we can use other reasonable models for bond pricing. Additionally, one could use a different sort of default-free bond, such as a consol bond (or perpetuity). In fact, in work not shown here, the authors obtained the same PDEs as in (2.16) and (2.17) when replacing the T -bonds with consol bonds. The latter may be more appropriate in the setting of whole life annuities. Alternatively, an insurer could continually roll money into longer-term bonds as the shorter-term bonds mature. 4.15 By comparing Corollary 4.5 with the expression in (3.4) and the discussion afterwards, we see that as n approaches infinity, the controls in the expectation in (3.4) are further restricted so that γ s ≡ 0 with probability 1, that is, δ s ≡ −α with probability 1. 4.16 Only in the limit does our valuation method result in a linear pricing rule, as evidenced by the linear PDE in (4.7) as compared with the non-linear PDE in (2.16). Pricing via an equivalent martingale measure (EMM) results in a linear pricing rule; therefore, our general valuation method is not equivalent to choosing an EMM. 4.17 When valuing life insurance using the method of this paper, Young (2007) proves results parallel to those in Sections 4.1 and 4.2. In particular, it is encouraging to learn that the limiting results are robust to the type of insurance contract (life insurance versus life annuities). Also, in that paper, there is a numerical algorithm for valuing life insurance. An interested reader can easily modify that numerical algorithm to value life annuities. Conclusion A number of recent research papers focused attention on the valuation of longevity risk from a variety of perspectives. In this paper, we returned to basics and showed how the uncertainty in the hedging portfolio translates into a non-zero standard deviation per policy. We draw upon the financial analogy of the Sharpe ratio to develop a methodology for valuing the non-diversifiable component of aggregate mortality risk. Our main qualitative insight is that similar to the financial economic approach to analyzing stock market risk, the uncertainty embedded within mortalitycontingent claims can be decomposed into idiosyncractic (diversifiable) and non-diversifiable components. We developed a theoretical foundation for valuing mortality risk by assuming that the risk is "priced" via the instantaneous Sharpe ratio. Because the market for life annuities is incomplete, one cannot assert that there is a unique price. However, we believe that the price that our method produces is a valid one because of the many desirable properties that it satisfies and because it is the upper good deal bound of Cochrane and Saá-Requejo (2000), generalized to the jump diffusion setting by Björk and Slinko (2006). Therefore, we have the additional contribution of providing another derivation of the good deal bounds for the setting of this paper, one that includes a hedging strategy along with a actuarially-inspired risk loading in the form of the local standard deviation. We also studied properties of the value of n conditionally independent and identically distributed life annuities. The value is subadditive with respect to n, and the risk charge per person decreases as n increases. We proved that if the hazard rate is deterministic, then the risk charge per person goes to zero as n goes to infinity. Moreover, we proved that if the hazard rate is stochastic, then the risk charge person is positive as n goes to infinity, which reflects the fact that the mortality risk is not diversifiable in this case. Additionally, in Remark 4.12, we decomposed the per-risk risk charge into the finite portfolio and stochastic mortality risk charges. Appendix A. Comparison Principle In this appendix, we present a comparison principle from Walter (1970, Section 28) on which we rely extensively in the proofs of the properties of a (n) given in Section 4. We begin by stating a relevant one-sided Lipschitz condition along with growth conditions. Suppose a function g = g(r, λ, t, v, p, q) satisfies the following one-sided Lipschitz condition: For v > w, g(r, λ, t, v, p, q ) − g(r, λ, t, w, p ′ , q ′ ) ≤ f 1 (r, λ, t)(v − w) + f 2 (r, λ, t)\|p − p ′ \| + f 3 (r, λ, t)\|q − q ′ \|, (A.1) with growth conditions on f 1 , f 2 , and f 3 given by 0 ≤ f 1 (r, λ, t) ≤ K(1 + (ln r) 2 + (ln(λ − λ)) 2 ), (A.2a) 0 ≤ f 2 (r, λ, t) ≤ Kr(1 + \| ln r\| + \| ln(λ − λ)\|), (A.2b) and 0 ≤ f 3 (r, λ, t) ≤ K(λ − λ)(1 + \| ln r\| + \| ln(λ − λ)\|), (A.2c) for some constant K ≥ 0, and for all (r, λ, t) ∈ R + × (λ, ∞) × [0, T ]. To prove properties of a (n) , we use the following comparison principle, which we obtain from Walter (1970, Section 28). For the proof of a similar result, see Milevsky, Promislow, and Young (2005). T ], and denote by G the collection of functions on G that are twice-differentiable in their first two variables and once-differentiable in their third. Define a differential operator L on G by Theorem A.1 Let G = R + × (λ, ∞) × [0,Lv = v t + 1 2 c 2 (r, t)v rr + 1 2 σ 2 (t)(λ − λ) 2 v λλ + g(r, λ, t, v, v r , v λ ), (A.3) in which g satisfies (A.1) and (A.2). Suppose v, w ∈ G are such that there exists a constant K ≥ 0 with v ≤ e K{(ln r) 2 +(ln(λ−λ)) 2 } and w ≥ −e K{(ln r) 2 +(ln(λ−λ)) 2 } for large (ln r) 2 + (ln(λ − λ)) 2 . Also, suppose that there exists a constant K ≥ 0 such that c(r, t) ≤ Kr for all r > 0 and 0 ≤ t ≤ T . Then, if (a) Lv ≥ Lw on G, and if (b) v(r, λ, T ) ≤ w(r, λ, T ) for all r > 0 and λ > λ, then v ≤ w on G. As a lemma for results to follow, we show that the differential operator associated with our problem satisfies the hypotheses of Theorem A.1. Lemma A.2 Define g n , for n ≥ 1, by g n (r, λ, t, v, p, q) = b Q (r, t)p + µ(λ, t)(λ − λ)q − rv + n − nλ v − a (n−1) + α σ 2 (t)(λ − λ) 2 q 2 + nλ v − a (n−1) 2 , (A.4) in which a (n−1) solves (2.17) with n replaced by n − 1. Then, g n satisfies the one-sided Lipschitz condition (A.1) on G. Furthermore, if \|b Q (r, t)\| ≤ Kr(1 + \| ln r\|) and \|µ(λ, t)\| ≤ K(1 + \| ln(λ − λ)\|), then (A.2) holds. Proof. Suppose v > w, then g n (r, λ, t, v, p, q) − g n (r, λ, t, w, p ′ , q ′ ) = b Q (r, t)(p − p ′ ) + µ(λ, t)(λ − λ)(q − q ′ ) − (r + nλ)(v − w) + α σ 2 (t)(λ − λ) 2 q 2 + nλ v − a (n−1) 2 − α σ 2 (t)(λ − λ) 2 (q ′ ) 2 + nλ w − a (n−1) 2 ≤ \|b Q (r, t)\|\|p − p ′ \| + (\|µ(λ, t)\| + ασ(t)) (λ − λ)\|q − q ′ \| − r + nλ − α √ nλ (v − w) ≤ \|b Q (r, t)\|\|p − p ′ \| + (\|µ(λ, t)\| + ασ(t)) (λ − λ)\|q − q ′ \|. (A.5) In the first inequality, we use the fact that if A ≥ B, then √ C 2 + A 2 − √ C 2 + B 2 ≤ A − B. For the second inequality, recall that 0 ≤ α ≤ √ λ. Thus, (A.1) holds with f 1 (r, λ, t) = 0, f 2 (r, λ, t) = \|b Q (r, t)\|, and f 3 (r, λ, t) = \|µ(λ, t)\|+ασ(t). Note that f 2 satisfies (A.2b) if \|b Q (r, t)\| ≤ Kr(1+\| ln r\|), and f 3 satisfies (A.2c) if \|µ(λ, t)\| ≤ K(1 + \| ln(λ − λ)\|). Appendix B. Limit Result for Valuing n Pure Endowments In this appendix, we review some of the results of Milevsky, Promislow, and Young (2005). Via the instantaneous Sharpe ratio, they value a pure endowment that pays 1 at time s if an individual is alive at that time. They also study properties of the time-t value P (n) = P (n) (r, λ, t; s) of n conditionally independent and identically distributed pure endowment risks with 0 ≤ t ≤ s. By paralleling the derivation of (2.17), one can show that P (n) solves the non-linear PDE given by              P (n) t + b Q P (n) r + 1 2 c 2 P (n) rr + µ(λ − λ)P (n) λ + 1 2 σ 2 (λ − λ) 2 P (n) λλ − rP (n) − nλ P (n) − P (n−1) = −α σ 2 (λ − λ) 2 P (n) λ 2 + nλ P (n) − P (n−1) 2 P (n) (r, λ, s; s) = n, (B.1) with P (0) ≡ 0. We can multiplicatively separate the variables r and λ in P (n) . Indeed, P (n) (r, λ, t; s) = F (r, t; s) ϕ (n) (λ, t; s), in which ϕ (n) solves the recursion              ϕ (n) t + µ(λ − λ)ϕ (n) λ + 1 2 σ 2 (λ − λ) 2 ϕ (n) λλ − nλ ϕ (n) − ϕ (n−1) = −α σ 2 (λ − λ) 2 ϕ (n) λ 2 + nλ ϕ (n) − ϕ (n−1) 2 , ϕ (n) (λ, s; s) = n, (B.2) with ϕ (0) ≡ 0. We can interpret ϕ (n) as a risk-adjusted expected number of survivors to time s from the n individuals alive at time t. Indeed, Milevsky, Promislow, and Young (2005) show that nE λ,t e − s t λ u du ≤ ϕ (n) ≤ ne −(λ−α √ λ)(s−t) , and ϕ (n) λ ≤ 0. The main result of that paper is given in the following theorem: uniformly on R + × (λ, ∞) × [0, s]. Here β is the solution of    β t + (µ − ασ)(λ − λ)β λ + 1 2 σ 2 (λ − λ) 2 β λλ − λβ = 0, β(λ, s; s) = 1. (B.4) More precisely, Milevsky, Promislow, and Young (2005) show that β(λ, t; s) ≤ 1 n ϕ (n) (λ, t; s), (B.5) with equality in the limit. Note that β solves a linear PDE; therefore, one can represent it as an expectation via the Feynman-Kac Theorem (Karatzas and Shreve, 1991): β(λ, t; s) =Ẽ λ,t e − s t λ u du , (B.6) in which {λ t } follows the process dλ u = (µ − ασ)(λ u − λ)du + σ(λ u − λ)dW λ u , withW λ u = W λ u + αu. The process {W λ t } is a standard Brownian motion with respect to the filtered probability space (Ω, F , (F t ) t≥0 ,P), in which dP dP F t = exp −αW λ t − 1 2 α 2 t .Ẽ denotes expectation with respect tõ P. C. Proofs of Lemmas 4.2 and 4.3 In this appendix, we supply the proofs of Lemmas 4.2 and 4.3. C.1 Proof of Lemma 4.2 We proceed by induction. The inequality is clearly true for n = 0 because both sides equal zero in that case. Assume that inequality (4.1) holds for n − 1 in place of n and show that it holds for n. Define a differential operator L on G by (A.3) with g = g n in (A.4). Because a (n) solves (2.17), it follows that La (n) = 0, and L T t F (r, t; s)ϕ (n) (λ, t; s)ds = −F (r, t; t)ϕ (n) (λ, t; t) + T t F t ϕ (n) + F ϕ (n) t ds + b Q T t F r ϕ (n) ds + 1 2 c 2 T t F rr ϕ (n) ds + µ(λ − λ) T t F ϕ (n) λ ds + 1 2 σ 2 (λ − λ) T t F ϕ (n) λλ ds − r T t F ϕ (n) ds − nλ T t F ϕ (n) ds − a (n−1) + n + α σ 2 (λ − λ) 2 T t F ϕ (n) λ ds 2 + nλ T t F ϕ (n) ds − a (n−1) 2 . (C.1) Because F (r, t; t) = 1 and ϕ (n) (λ, t; t) = n, the first term on the right-hand side of (C.1) cancels the n at the end of the third line. Also, we use the fact that F solves F t +b Q (r, t)F r + 1 2 c 2 (r, t)F rr −rF = 0 and that ϕ (n) solves (B.2) to simplify (C.1) and thus obtain L T t F (r, t; s)ϕ (n) (λ, t; s)ds = −nλ where we use the assumption that 0 ≤ α ≤ √ λ, which in turn implies that α √ nλ ≤ nλ for n ≥ 1 and λ > λ. Recall that the Minkowski inequality tells us that if 0 < p < 1, then \|\|f \|\| p + \|\|g\|\| p ≤ \|\|f + g\|\| p , in which \|\|f \|\| p = \|f \| p dν 1/p for some measure ν, (Hewitt and Stromberg, 1965, page 192). Let p = 1/2, f = σ 2 (λ − λ) 2 ϕ (n) λ 2 , g = nλ ϕ (n) − ϕ (n−1) 2 , and dν = F (r, t; s) ds. Then, the Minkowski inequality implies T t F σ 2 (λ − λ) 2 ϕ (n) λ 2 + nλ ϕ (n) − ϕ (n−1) 2 ds 2 ≥ σ 2 (λ − λ) 2 σ 2 (λ − λ) 2 T t F ϕ (n) λ ds 2 + nλ T t F ϕ (n) ds − a (n−1) 2 − √ nλ T t F ϕ (n−1) ds − a (n−1) ≤ σ 2 (λ − λ) 2 T t F ϕ (n) λ ds 2 + nλ T t F ϕ (n) − ϕ (n−1) ds 2 . (C.5) If we define B λ = σ(t)(λ − λ) T t F ϕ (n) λ ds, A = √ nλ T t F ϕ (n) ds, C = √ nλ T t F ϕ (n−1) ds, and D = √ nλ a (n−1) , then (C.5) is equivalent to B 2 λ + (A − D) 2 − (C − D) ≤ B 2 λ + (A − C) 2 . (C.6) From Milevsky, Promislow, and Young (2005), we know that A ≥ C. From the induction assumption, we have that C ≥ D. Thus, the left-hand side of (C.6) is positive, and demonstrating inequality (C.6) is a straightforward matter of squaring both sides and simplifying. Thus, we have that L T t F ϕ (n) ds ≤ 0 = La (n) . Also, both T t F ϕ (n) ds and a (n) equal 0 when t = T . It follows from Theorem A.1 and Lemma A.2 that a (n) ≤ T t F ϕ (n) ds on G. C.2 Proof of Lemma 4.3 We proceed by induction. The inequality is clearly true for n = 0 because both sides equal zero in that case. Assume that inequality (4.2) holds for n − 1 in place of n and show that it holds for n. Define a differential operator L on G by (A.3) with g = g n in (A.4). Because a (n) solves (2.17), it follows that La (n) = 0, and L T t nF (r, t; s) β(λ, t; s) ds = −nF (r, t; t)β(λ, t; t) + Because F (r, t; t) = 1 and β(λ, t; t) = 1, the first term on the right-hand side of (C.7) cancels the n at the end of the third line. Also, we use the fact that F solves F t +b Q (r, t)F r + 1 2 c 2 (r, t)F rr −rF = 0 and that β solves (B.4) to simplify (C.7) and thereby obtain L T t nF (r, t; s)β(λ, t; s)ds = nλ a (n−1) − (n − 1) in which the inequality follows from the induction assumption. Also, both T t F β ds and a (n) equal 0 when t = T . It follows from Theorem A.1 and Lemma A.2 that n T t F β ds ≤ a (n) on G. 3.1), it follows that a can be represented as a(r, λ, t) = sup {{δ s ,γ s }: δ 2 s +λ s γ 2 s ≤α, γ s ≥−1, t≤s≤T }Ē u +λ u (1+γ u )) du ds = sup {{δ s ,γ s }: δ 2 s +λ s γ 2 s ≤α, γ s ≥−1, t≤s≤T }Ē du 1 {τ (t)>s} ds . 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Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio: Theorems and proofs, working paper. V R Young, Department of Mathematics, University of MichiganYoung, V. R. (2007), Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio: Theorems and proofs, working paper, Department of Mathematics, University of Michi- gan, available at http://arxiv.org/abs/0705.1297 Stochastic control methods in asset pricing, Handbook of Stochastic Analysis and. T Zariphopoulou, Applications, D. Kannan and V. LakshmikanthamMarcel DekkerNew YorkZariphopoulou, T., (2001), Stochastic control methods in asset pricing, Handbook of Stochastic Analysis and Applications, D. Kannan and V. Lakshmikantham (editors), Marcel Dekker, New York.	[]
[ "April 24, 2015 1:34 NC˙mBH˙as˙DM˙arXiv NONCOMMUTATIVITY INSPIRED BLACK HOLES AS DARK MATTER CANDIDATE", "April 24, 2015 1:34 NC˙mBH˙as˙DM˙arXiv NONCOMMUTATIVITY INSPIRED BLACK HOLES AS DARK MATTER CANDIDATE" ]	[ "Samuel Kováčik [email protected] \nFaculty of Mathematics, Physics and Informatics\nMlynská dolina Bratislava\nComenius University Bratislava\n842 48Slovakia\n" ]	[ "Faculty of Mathematics, Physics and Informatics\nMlynská dolina Bratislava\nComenius University Bratislava\n842 48Slovakia" ]	[]	We study black holes with a source that is almost point-like (blurred), rather than exactly point-like, which could be caused by the noncommutativity of 3-space. Depending on its mass, such object has either none, one or two event horizons. It possesses new properties, which become important on microscopic scale, in particular the temperature of its Hawking radiation does not increase infinitely as its mass goes to zero, but vanishes instead. Such frozen, extremely dense pieces of matter are good dark matter candidate. In addition, we introduce an object oscillating between frozen black hole and naked (softened) singularity, such objects can serve as constituents of dark matter too. We call it gravimond.	null	[ "https://arxiv.org/pdf/1504.04460v6.pdf" ]	119,318,967	1504.04460	deab09f90a3a628f169ff213c46caa84979e9a0c	April 24, 2015 1:34 NC˙mBH˙as˙DM˙arXiv NONCOMMUTATIVITY INSPIRED BLACK HOLES AS DARK MATTER CANDIDATE Samuel Kováčik [email protected] Faculty of Mathematics, Physics and Informatics Mlynská dolina Bratislava Comenius University Bratislava 842 48Slovakia April 24, 2015 1:34 NC˙mBH˙as˙DM˙arXiv NONCOMMUTATIVITY INSPIRED BLACK HOLES AS DARK MATTER CANDIDATE Noncommutative quantum mechanicsmicroscopic black holesdark matter PACS numbers: We study black holes with a source that is almost point-like (blurred), rather than exactly point-like, which could be caused by the noncommutativity of 3-space. Depending on its mass, such object has either none, one or two event horizons. It possesses new properties, which become important on microscopic scale, in particular the temperature of its Hawking radiation does not increase infinitely as its mass goes to zero, but vanishes instead. Such frozen, extremely dense pieces of matter are good dark matter candidate. In addition, we introduce an object oscillating between frozen black hole and naked (softened) singularity, such objects can serve as constituents of dark matter too. We call it gravimond. Introduction Quantum theory allowed us to merge three of the four (known) forces of nature within one unified theory. However, its relation with the last one -gravity is, to put it mildly, questionable. At least some of the problems with it are caused by infinitely large energies or equivalently, by zero distances. If the space we live in has some shortest possible distance, those problems would vanish. Noncommutative (NC) theories are formulated in spaces whose coordinates do not commute with each other and therefore one cannot localize their points (this is similar to ordinary quantum mechanics where one cannot exactly know the phase space position of a particle). They could be viewed as effective theories to some higher theory which fuses quantum physics with gravity, yet they already possess a natural energy cut-off a . Black holes are important objects in both classical and quantum gravity which also posses a high-energy ill behavior. As discovered by Hawking, they radiate with a temperature inversely proportional to their mass, thus as they become infinitely small, they also turn infinitely hot. a For example in [13] it has been shown that the spectrum of free Hamiltonian in a NC space has not only a lower boundary but also an upper one When a black hole forms, its matter shrinks into a singular point. However, in NC theories there is nothing like a separate point, and hence the singularity cannot presumably arise in the course of the collapse. This restriction has only a negligible effect on huge black holes, however a question is whether it can modify the behavior of microscopic ones. The aim of this paper is to answer this question. Instead of using a complete NC description of black holes we follow a method used in [16] -NC theory is used only to obtain the energy density of the black hole, rest of the study is done using the classical theory (this is dubbed as NC -inspired black holes). More details on NC inspired cosmology and gravity could be found in [3,4,15,17,18,20,24]. Outline of the paper This paper is organized as follows. At first we briefly demonstrate construction of 3 dimensional NC space and derive a NC point-like ("blurred") density b . Such matter density is completed into the stress-energy tensor T µ ν , for which we write down and solve the Einstein field equations. Afterwards we analyze the solution, mostly focusing on the event horizons and temperature of Hawking radiation. Finally we point out some physical consequences of our theory for λ ∼ l Planck (and provide the scaling of results for different choices of λ). NC inspired [x,p] = i ,(1) which states that one cannot exactly identify a phase space position of a particle. The idea of noncommutative (NC) theories is to have a space in which one cannot determine the exact position of a point -the smooth structure of space is abandoned. Therefore, NC theories are built upon a relation defining how the position operators do not commute [x i ,x j ] = 0 .(2) By choosing the RHS of this equation we define the properties of the corresponding NC space, including symmetries. A popular choice for the RHS is iθ ij , where θ ij b Something as close to point-like density aj one can get to in NC space is some constant antisymmetric matrix. This option however lacks the rotational invariance of our space. A more appropriate option is [x i ,x j ] = 2iλx k ε ijk ,(3) where ε ijk is the Levi-Civita symbol and λ is a constant with the dimension of length, defining the length scale on which NC effects become significant. λ is not fixed within our model, but since it might be an artifact of quantum gravity, it is expected to be equal approximately the Planck's length, λ ∼ l Planck ∼ 10 −35 m. There are several ways how to satisfy (3) [5,6,7,9,14,19], different approaches are equivalent and one is encouraged to switch between them whenever it is comfortable and makes calculations easier. We will employ the bosonic operator approach which was previously used in [8,9,10,13]. Let us define two sets of bosonic creation and annihilation operators satisfying [â α ,â + β ] = δ αβ ; α, β = 1, 2 ,(4) and acting in an auxiliary Fock space F spanned on normalized states \|n 1 , n 2 >= (â + 1 ) n1 (â + 2 ) n2 √ n 1 !n 2 ! \|0, 0 > .(5) where \|0, 0 >= \|0 > is the vacuum state annihilated by bothâ α . NC coordinates defined with the help of Pauli matrices σ i aŝ x i = λσ i αβâ + αâβ ,(6) satisfy (3) (their noncommutativity is inherited from the bosonic operators). The radial coordinate is defined asr = λ(â + αâα + 1) ,(7) note thatr 2 =x 2 + λ 2 . Every \|n 1 , n 2 > is an eigenstate ofr with an eigenvalue λ(n 1 + n 2 + 1). The vacuum state \|0, 0 >≡ \|0 > is the state with the minimal eigenvalue, so it should correspond to the origin of the coordinate system. This is as far as we need to go into the construction of NC space, for more details= about constructing (NC) QM on it see the aforementioned references. Coherent states play an important role in ordinary quantum mechanics and they have a crucial role in NC theories as well [12,21,22,23,25]. A coherent state is well localized wave packet which minimizes the uncertainty relation and is defined as annihilation operator eigenstate (â + \|α >= α\|α >). Such states can be generated as \|α >= e − \|α\| 2 2 e αâ + \|0 > ,(8) We can use them as a useful overcomplete sets of states in F, [1]. The overlap of two coherent states is < α\|β >= e − \|α\| 2 +\|β\| 2 2 +ᾱβ .(9) We are interested in the overlap of a general coherent state and the vacuum state (which corresponds to the origin of the coordinates), ρ(α) = \| < α\|0 > \| 2 = e −\|α\| 2 .(10) This represents a well localized state in the origin of coordinates, which however contains no information about the length scale λ. To overcome this we define new bosonic operators (no longer dimensionless) aŝ z α = √ λâ α ,ẑ + α = √ λâ + α .(11) With these operators, the entire construction (4) -(10) can repeated [ẑ α ,ẑ + β ] = λδ αβ ,(12) x i = σ i αβẑ + αẑβ , r =ẑ + αẑα + λ =ẑ 2 + λ . The overlap of coherents states, now defined as eigenstates of z α , with the state localized at the origin is ρ(z) = \| < z\|0 > \| 2 = e − \|z\| 2 λ = e − r−λ λ .(13) Let us pause for a moment to make a few remarks. First of all, we define λ → 0 as the commutative limit (RHS of (3) vanishes, as in the ordinary QM). It is easy to see that in this limit the RHS of (13) vanishes everywhere but at the point r = 0, it becomes a point-like (particle matter) density. It is therefore natural to callρ ∝ e − r λ an almost point-like density or a blurred point-like density. Note thatρ in (13) is dimensionless. The matter density with proper dimension will be denoted ρ (without a tilde). Since the rest of the calculations will be done using ordinary (not NC) calculus, we will normalize ρ with respect to the ordinary integration instead of a trace norm. This yields an almost point-like mass density ρ(r) = M 8πλ 3 e − r λ .(14) In the paper by P. Nicolini [16], which served as a main inspiration for ours, a similar line of reasoning was used. The starting point in [16] was a two dimensional NC space and the resulting density was generalized into three dimensional only afterwards, yielding ρ ∝ e − r 2 λ 2 . As we have shown, a direct three dimensional derivation based on (3) leads to a different result. Stress-energy tensor and energy conditions The plan is to complete ρ into a full stress-energy tensor, write down Einstein field equations, solve them and analyze their solution. Most of the work will be done analytically, yet some of the equations will be transcendent, so we will have to settle for less and find only a numerical approximation of the solutions. We focus here only on uncharged nonrotating black holes, so we expect all of our results to recover the ordinary Schwarzschild black hole behavior in the λ → 0 limit. This requirement also encourages us to use a "Schwarzschild-like" ansatz for the metric tensor g 00 = −g −1 rr , therefore our goal will be to find a single function f (r) such that g µν =      f (r) 0 0 0 0 − 1 f (r) 0 0 0 0 r 2 0 0 0 0 r 2 sin 2 θ      .(15) We will label the coordinates by (0, r, θ, ϕ) and use the metric tensor signature (−, +, +, +). We will also often omit writing (any of) arguments of functions. We are expecting a diagonal T µ ν , our starting point being the energy density component T 0 0 = −ρ(r) (we put c = 1 so that the mass and energy density coincide). Because of our ansatz (15), T r r = T 0 0 is fixed as well (this can be seen from the Einstein field equations). The other two components follow from the conservation law T µν ;ν = 0. For µ = θ we get T θ θ = T ϕ ϕ =: p ⊥ , for µ = r we get p ⊥ = − r 2 (∂ r ρ + 2 r ρ) = −ρ − r 2 ∂ r ρ ,(16) other equations are trivial. Our stress energy tensor therefore is T µ ν =     −ρ 0 0 0 0 p r 0 0 0 0 p ⊥ 0 0 0 0 p ⊥     , p r = −ρ, p ⊥ = −ρ − r 2 ∂ r ρ .(17) Is such stress-energy tensor realistic or not? To decide on this we can use weak and strong energy conditions : weak T µν X µ X ν ≥ 0 ,(18) strong (T µν − 1 2 T g µν )X µ X ν ≥ 0 , where X µ is a timelike vector. The weak condition can be interpreted as "energy is always positive" and the strong condition can be regarded as "matter gravitates towards matter" [2]. The weak energy condition reduces to the inequalities ρ + p r ≥ 0,(19)ρ + p ⊥ ≥ 0 , which are in our case always satisfied. However the strong energy condition, which takes the form ρ + p r + 2p ⊥ ≥ 0 .(20) is violated for r < 2λ. This could be expected since the noncommutativity generates some sort of quantum repulsion which prevents the matter from collapsing into a singularity. Einstein field equations and their solution Now it is time to write down the Einstein field equations. In fact, because of the form of our ansatz (15) we only need one of them (and from now on, we set G = 1). We can choose G 0 0 = 8πT 0 0 since our choice p r = −ρ r ensures that the equation G r r = 8πT r r is identical to it. The3reads 1 + f + rf r 2 = M λ 3 e − r λ ,(21) and has a solution f (r) = −1 − e − r λ M r r 2 λ 2 + 2r λ + 2 + C r .(22) Recall that g 00 (r) = f (r), therefore if we want the solution to approach Schwarzschild solution for r λ, we need to set C = 2M . For the rest of this paper we will need only the time component of the metric tensor, g 00 (r; λ, M ) = −1 + 2M r − e − r λ M r r 2 λ 2 + 2r λ + 2(23) Event horizon(s) and Hawking radiation Event horizons are solutions of the equation g 00 (r) = 0 .(24) For an ordinary Schwarzschild black hole the solution is r = 2M , however for our metric there are two, one or zero solutions, depending on the value of M . This can be seen in Fig. 1 and one can easily prove it by doing a little mathematical analysis. When the mass is large (M λ), there are two horizons, one near the origin (r − ≈ 0) and the other near the classical horizon (at r + ≈ 2M , see Fig. 2). As M gets smaller, these two surfaces move towards each other and meet for some M =: M 0 at r =: r 0 . We call a black hole with the mass M 0 and a single horizon at the radial coordinate r 0 extremal, since for any smaller M there is no horizon at all, extremal black hole is the smallest possible black hole. Obviously both M 0 , r 0 depend on λ and as can be seen from their physical dimensions the dependence is linear (without the absolute term, since they both vanish as λ → 0). Eq. (24) is transcendental so we can obtain the linear coefficients only numerically, r 0 . = 3.38λ .(25) What happens to the Hawking radiation [11] as a black hole approaches the extremal mass M 0 ? The Hawking temperature is given as T = κ 2π , where κ is the surface gravity at the horizon r + which is equal to κ = − g 00 (r+) 2 . For an extremal black hole the function g 00 (r; M 0 , λ) only touches the horizontal axis at r = r 0 (otherwise there would be two horizons), therefore r = r 0 is the point where it reaches its maximum and its first derivative vanishes there. Because of that there is no surface gravity at the horizon of an extremal black hole and the black hole has zero temperature -it becomes frozen and stops evaporating. Note that infinite temperatures are avoided (Fig. 3). An interesting question is how does the maximal temperature depend on λ. From dimensional analysis we can see that T max ∝ λ −1 and to get an (almost) exact relation let us first factorize out the mass from g 00 , g 00 (r; λ, M ) = −1 + Mg(r; λ) . (26) whereg(r) does not depend on M . At the (outer) horizon g 00 (r + ) = 0, so that g(r + ) = 1 M , and g 00 (r + ) = Mg (r + ) =g (r + ) g(r + ) . (27) This is, up to a multiplicative constant, equal to the Hawking temperature. We may now ask for what size of the (outer) horizon r + does this achieve maximum. To answer this we need to solve one of the two following equations ∂ r+ g 00 (r + ) = 0 ⇔g(r + )g (r + ) =g (r + ) 2 . We choose λ = 1 and solve the numerical numerically to find that the extremal value is g 00 (r + . = 6.54) . = −0.12. The maximal temperature is (we recover all constants for a moment, τ . = 0.18 × 10 −3 mK ) T max . = c 4πk B τ 0.12 λ .(29) As we have seen already, microscopical black holes (mBH) do not evaporate entirely, but stay frozen with the mass M 0 instead. When such extremal black hole consumes a particle with non-zero mass its own mass becomes larger then M 0 and the black hole is reignited (since for M > M 0 is the Hawking temperature nonzero). If we throw a particle with a small mass δM M 0 into an extremal black hole, how much will its radius grow and at what temperature will it radiate? To answer this question we use the decomposition (26). Let us denote the increment in radius δr. We can write down two conditions, one before and one after adding the mass δM , − 1 + M 0g (r 0 ; λ) ! = 0 ,(30)−1 + (M 0 + δM )g(r 0 + δr; λ) ! = 0 .(31) Truncating the Taylor expansion of (31) we obtaiñ g(r 0 + δr; λ) =g(r 0 ; λ) M −1 0 + δr∂ rg (r 0 ; λ) 0 + 1 2 δr 2 ∂ 2 rg (r 0 ; λ) ,(32) and inserting this back into (31) we get δr = ± −2δM M 0 (M 0 +¨δM )∂ 2 rg (r 0 ; λ) . = ± −2δM M 2 0 ∂ 2 rg (r 0 ; λ) .(33) This expression might look a little hideous, but evaluating it for M 0 and r 0 as given in (25) we arrive at a simple equation δr . = ±2.54 √ λδM .(34) We have two symmetric solutions because we have truncated the Taylor expansion after the quadratic term. Now we can determine the temperature T (r 0 + δr) = 0 T (r 0 ) +∂ r T (r 0 )δr . = (35) . = − (M 0 +¨δM )g (r 0 ) 4π δr . = 1 4π 1 M 0 2δM δr . = √ δM 2π6.53λ 3/2 , where we have used (33) first, then (34). If we recover the constants again we get T (M 0 + δM ) . = √ δM 41.01λ 3/2 c k B .(36) It is useful to express this with respect to T max T (M 0 + δM ) T max . = 2.55 δM λ . = 4.09 δM M 0 .(37) We can see that for δM M 0 the black hole does not reach its maximum temperature, only a small fraction of it. The last question of this section will be whether will the temperature reach the maximum value if we merge two extremal black holes together? As can be seen in Figure 3, if we have M = 2M 0 we are in a region where we can safely take r + = 2M = 4M 0 . = 10.28λ. This is larger then the value r + = 6.54λ for which the temperature reaches maximum, therefore the maximum will be reached when the new black hole evaporates from the radius 10.28λ to 6.54λ. Physical implications As we have seen, all of our results depend on λ -the scale of the space noncommutativity. The problem is that we do not know how large λ really is, we can only say it is beyond our experimental reach so far. However, since it should be an artifact of the spacetime structure, one expects that it could be approximated by the Planck's length λ ∼ l Planck . = 1.62 × 10 −35 m. In this section we provide some possible physical implications of the existence of "blurred" microscopic black holes (mBH) c . The calculations are done assuming λ = l Planck and accompanied with a scaling rule for different choices of λ. According to (25) an extremal mBH should have the radius r 0 . = 5.48 × 10 −35 m (we can take the cross section to be σ = πr 2 . = 9.43 × 10 −69 m 2 ) and the mass M 0 . = 5.59 × 10 −8 kg. Thus, such black holes are indeed minuscule, but still quite massive when compared to elementary particles (r 0 , M 0 scale as λ). Furthermore T max is 1.33 × 10 30 K which is two orders below the Planck's temperature (this scales as λ −1 ). Considering these numbers, mBH are perfect cold dark matter candidates. They are cold and absolutely dark (since their radiation froze out), extremely small and heavy enough so there does not need to be too many of them. To make up for the dark matter mass density ρ DM . = 2.38 × 10 −27 kg m −3 we would need mBH concentration n mBH . = 4.25 × 10 −20 m −3 in the universe (this scales as λ −1 ), that means approximately one mBH in every cube with edges almost 3000 km long. Dark matter density is uniform only on cosmological scales, but there seems to be more of it in galaxies than in between them (by the factor 10 5 − 10 6 , see [26], possibly even more within the solar systems). The cross section of extremal mBH is small enough for them not to interact with each other, however it is still possible for them to be hit by some other particles. Let us assume that a mBH gets hit by a proton and swallows it, what would happen? Since the mass of the proton is significantly smaller than M 0 we can use eq. (37), for this example δM M0 . = 2.98 × 10 −20 . The resulting (non extremal) mBH will warm up just to 7.06 × 10 −10 of it the maximum temperature, that is 9.39 × 10 20 K or 8.09 × 10 16 eV (this scales as λ − 3 2 ), two orders below the energy of ultra-high-energy cosmic rays. Had the λ been shorter than the Planck's length, a mBH radiation after consuming a proton could explain these rays. It should be noted here that it might be more correct to consider mBH-electron or mBH-quark collision instead d , since proton is significantly larger than mBH. It is important to note that the energy of radiation exceeds the energy of the consumed particle. The possible scenario is that the energy will be radiated in one or t quanta and the mBH will end with M < M 0 , it will have no horizons and stops being a black hole. Then it will be moving through space as an extremely dense chunk of matter and collect additional mass until it reaches the mass M 0 and becomes black hole again. This object, let us name it a gravimond, lives in cycles: first it is an extremal black hole with mass M 0 , and then, after it consumes a particle its radiation and is reignited as M > M 0 , then it stops being a black hole since so much energy has been radiated that M < M 0 , and it becomes an extremely dense object (almost a black hole) which needs to capture some mass to become (extremal) black hole again. The period of this cycles is unknown and probably largely depends on the location of such object (how often does it get to interact with other matter). Conclusion The paper is devoted to (microscopic) black holes with almost point-like (blurred) mass density, instead of singular one. Such mass density could be due to the noncommutativity of space on some small length scale λ, however all calculations have been done using the ordinary calculus and general relativity e Let us sum up the d Interesting questions about the confinemnt arise in that case e This is why the objects in question are sometimes referred to as NC-inspired black holes instead of just NC black holes important results • Depending on the mass M there are none, one or two event horizons. The black hole with one event horizon (extremal black hole) has the mass M 0 and the radius of event horizon r 0 both equal to the NC constant λ, multiplied by dimensionless constants of order unity, see (25). • The Hawking radiation of an extremal black hole has zero temperature so it does not evaporate anymore. Such frozen black holes are a good dark matter candidate. To make up for the observed mean dark matter energy density ρ DM there needs to be one such mBH in every volume of order 10 19 m 3 . • If an extremal black hole gains additional mass and therefore stops being extremal, for example by consuming a particle or colliding with another mBH, its radiation is reignited and it starts to emit extremely energetic quanta. The resulting radiation might be a possible candidate for ultra high cosmic rays, yet it seems to be 1-2 orders of magnitude too low. We also lack a better understanding of this mechanism. During the radiation stage more energy is emitted than has been absorbed and the mBH ends up with M < M 0 . Having no event horizons it is no longer a black hole, therefore we hypothesize that it turns into an object which we have called a gravimond ) undergoing (theoretically infinite number of) the following life cycles: • M = M 0 , being a frozen extremal microscopic black hole, • after consuming/collision with another particle the object acquires M > M 0 and a rapid radiation begins, • emission of huge quanta of energy leads to M < M 0 , the object has no event horizons anymore and stops being black hole. While moving through space, it gathers mass until it reaches M = M 0 again. The time period of these cycles is unknown and depends on the local density of other particles. Figure 1 . 1g 00 (r) for λ = 1 and different values of M . Figure 2 . 2Radius of the outer horizon r + as a function of M , compared to the Schwarzschild value 2M . Figure 3 . 3The Hawking temperature as a function of black holes mass. Black Holes 2.1. 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[ "WHERE THE NUCLEAR PIONS ARE", "WHERE THE NUCLEAR PIONS ARE" ]	[ "G E Brown \nDepartment of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York\n", "M Buballa \nDepartment of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York\n", "Zi Bang Li \nDepartment of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York\n", "S A \nIL61801, U.S.A. and IKP (Theorie)\nUniversity of Illinois\nForschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany\n", "J Wambach \nIL61801, U.S.A. and IKP (Theorie)\nUniversity of Illinois\nForschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany\n" ]	[ "Department of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York", "Department of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York", "Department of Physics\nPhysics Department\nState University of New York\n11794Stony BrookNew York", "IL61801, U.S.A. and IKP (Theorie)\nUniversity of Illinois\nForschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany", "IL61801, U.S.A. and IKP (Theorie)\nUniversity of Illinois\nForschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany" ]	[]	Three recent experiments, which looked at pionic effects in nuclei have concluded that there are no excess pions. This puts into serious question the conventional meson-exchange picture of the nucleon-nucleon interaction. Based on arguments of partial restoration of chiral symmetry with density we propose a resolution to this problem.	10.1016/0375-9474(95)00330-4	[ "https://arxiv.org/pdf/nucl-th/9410049v1.pdf" ]	17,989,706	nucl-th/9410049	f4f12db3dd5524ae7240a8d6082197e4dd24fd20	WHERE THE NUCLEAR PIONS ARE October 1994 G E Brown Department of Physics Physics Department State University of New York 11794Stony BrookNew York M Buballa Department of Physics Physics Department State University of New York 11794Stony BrookNew York Zi Bang Li Department of Physics Physics Department State University of New York 11794Stony BrookNew York S A IL61801, U.S.A. and IKP (Theorie) University of Illinois Forschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany J Wambach IL61801, U.S.A. and IKP (Theorie) University of Illinois Forschungszentrum JülichD-52425Urbana, JülichFed. Rep. Germany WHERE THE NUCLEAR PIONS ARE October 1994arXiv:nucl-th/9410049v1 31 Oct 1994 Three recent experiments, which looked at pionic effects in nuclei have concluded that there are no excess pions. This puts into serious question the conventional meson-exchange picture of the nucleon-nucleon interaction. Based on arguments of partial restoration of chiral symmetry with density we propose a resolution to this problem. Introduction In an article entitled "Where are the Nuclear Pions?", Bertsch et al [1] discussed three recent experiments which looked for pionic effects in nuclei. One is a ( p, n) quasielastic polarization transfer experiment at LAMPF which, more or less, directly determines the ratio of spin-longitudinal to spin-transverse response functions in the nucleus. At energies below the quasielastic peak conventional models predict this ratio to be significantly larger than unity, while experiment finds a ratio slightly below one. This puts into focus the strength of the tensor interaction in nuclei. In fact, experiment would suggest that at the measured momentum transfer V tensor ∼ 0 while for free nucleon-nucleon interactions it is large. The second is a new deep inelastic muon scattering experiment [2] which no longer sees a significant enhancement in the EMC ratio F A 2 /F D 2 for 0.1 ≤ x ≤ 0.3. Since this region is most sensitive to the virtual pion field in the nucleus it was concluded that there are no excess pions. Very recently it was realized that the enhancement of the pion field was strongly overestimated in the past because of incorrectly normalized wave functions. Introducing the proper normalization factors there is no contradiction between a conventional model calculation and the measured EMC ratio anymore in the kinematical regime which is dominated by the pion [3]. However, the normalization factors remove only about half of the discrepancy found between a recent Drell-Yan experiment at Fermi Lab [4] and the theoretical predictions. With appropriate choice of kinematics, this experiment directly probes modifications of the sea quarks in the nucleus and is therefore more sensitive to the pion field than the deep inelastic scattering experiment. A pion excess in nuclei would predict a strong A-dependence of the Drell-Yan ratio and none was observed. Together with the ( p, n) data this calls into serious question the conventional meson-exchange picture of the nuclear interaction. The authors of ref. [1] suggest that the answer might be found in the modification of gluon properties in the nucleus, suppressing the pion field at distances below 0.5 fm. In the present communication we shall argue that a reasonable explanation lies elsewhere, namely in the partial restoration of chiral invariance with density. Basically, our explanation will involve the fact that at finite density the hadronic world is "swelled". More precisely, masses of hadrons made up out of light up and down quarks decrease with density, all at about the same rate [5] thus: m * N m N ∼ = m * ρ m ρ ∼ = m * ω m ω · · · ∼ = f * π f π , (1.1) where f π is the pion decay constant, but here its more relevant meaning is that of the order parameter for chiral symmetry breaking. There are approximate signs in the equalities because these were shown [5] to hold only at mean field level, and loop corrections will enter in higher order. The mass of the pion m π is exempted from this scaling, because it originates from a higher scale than QCD, possibly the electroweak scale. In fact, from the study of pionic atoms [6] we know that due to many-body effects, the pion mass increases slightly, by ∼ 5 MeV, in going to the saturation density, ρ 0 , of nuclear matter. To proceed, we should recall some recent developments in the description of the nucleon-nucleon potential. Some time ago Thomas [7] showed that the data on the sea quark content of the proton can be used to obtain restrictions on the t-dependence of the πNN vertex function Γ πN N . Frankfurt, Mankiewicz and Strikman [8] extended this analysis, finding that the cut-off, Λ π , in a monopole parameterization of Γ πN N should be less than 0.5 GeV. Inclusion of more mesons in the nucleon cloud allowed Hwang, Speth and Brown [9] to raise this value to ∼ 950 MeV. Clearly such values are too low to correctly describe the NN-scattering data and the binding properties of the deuteron. In various versions of the Bonn potential, Λ π is typically 1.2-1.3 GeV. In an effort to reconcile the deep-inelastic scattering data on the proton with the two-nucleon properties Holinde and Thomas [10] chose Λ π = 0.8 GeV but had to introduce an additional pseudoscalar meson (which they call π ′ ) of mass 1.2 GeV. Assuming a hard form factor of Λ π ′ = 2 GeV, the π ′ NN coupling constant was adjusted to fit the NN data. More recently it has been recognized [11] that there are at least two objects with pionic quantum numbers, one the elementary pion and the other the correlated (ρπ)-system coupled to the quantum numbers of the pion. The latter may explain the properties of the Holinde-Thomas π ′ [10]. We find this scenario quite convincing and shall employ the Holinde-Thomas (HT) interaction in our calculations. Theoretical Development The enhancement of the pion field is driven by the longitudinal spin-isospin part of the NN interaction [12]. Employing the HT interaction [10] it is given by V (q, ω) = [V π (q, ω) + V π ′ (q, ω)]σ 1 ·qσ 2 ·q τ 1 ·τ 2 . (2.1) where V π (q, ω) = f 2 πN N m 2 π Γ 2 π (q, ω)q 2 ω 2 − (q 2 + m 2 π ) (2.2) and similarly for the π ′ meson. In the nuclear medium, this interaction acquires an additional repulsive contribution, usually expressed by the Migdal parameter g ′ : V = V + f 2 πN N m 2 π g ′ N N σ 1 ·qσ 2 ·q τ 1 ·τ 2 (2.3) due to short-range correlations induced by the core for the NN potential. As Baym and Brown have shown [13] g ′ N N receives a significant contribution from ρ-meson exchange, which generates the spin-isospin transverse interaction, V ⊥ . This observation will be important to our discussion. The Migdal parameter g ′ can be calculated by a momentumspace convolution of the central part of the spin-isospin interaction (V central = 1/3(V + 2V ⊥ ) with a two-nucleon correlation function g: V central (k, ω) = d 3 k (2π) 3 g(k − q)V central (q, ω). (2.4) To a good approximation g(q) = (2π) 3 δ(q) − (2π 2 )δ(\|q\| − q c )/q 2 , where q c is of the order of the omega-meson mass (q c = 3.94 fm −1 [14]). The resulting g ′ N N is ω and q dependent and is displayed in the static limit (ω = 0) as the full line in Fig. 1. For small q the value is somewhat lower than those extracted from Gamow-Teller systematics [15] g ′ N N = 0.7 − 0.8). On the other hand it agrees well with G-matrix calculations [16]. The increase in g ′ N N at larger q arises from the ρ-meson exchange tensor interaction. Another important ingredient in the description of the virtual pion field is the strong pionic p-wave coupling of the nucleon to ∆(1232) isobar. The corresponding spin-isospin correlations are described via transition potentials of the form (2.1) with suitable modifications for the coupling constants and spin-isospin operators. Also in this case, short-range correlations have to be included. Thies [17] pointed out that g ′ N ∆ (0) must be close to the classical Lorentz-Lorenz value 1/3 in order to explain the absence of multiple scattering in pion-nucleus interaction. Johnson [18] finds g ′ ∆∆ (0) = 0.40 ± 0.13 which is also consistent with the classical Lorentz-Lorenz value. Therefore we choose q c = 8.66f m −1 so that for the NN → N∆ and N∆ → N∆ transition potentials we reproduce g ′ N ∆ (0) = g ′ ∆∆ (0) = 1/3. The resulting momentum dependence is also indicated in Fig. 1. The physical origin of the difference in g ′ N N , g ′ N ∆ and g ′ ∆∆ lies in the role of the Pauli principle as was shown by Delorme and Ericson [19] and by Arima et al [20] some time ago. GivenṼ N N and the corresponding transition potentials all pionic properties, relevant to the experiments under discussion, can be evaluated in linear response theory within the Random-Phase-Approximation. This is the standard scenario employed by many people. With dropping masses there are several modifications. In medium, the nucleon acquires an effective mass, the mass that enters into the quasiparticle velocity v QP = p m * N , (2.5) where p is the quasiparticle momentum. By itself this is not unconventional and it is often incorporated in the standard treatment. The crucial point, as was shown by Brown and Rho [5], is that m * N is related to the chiral order parameter f * π as m * N m N ∼ = g * A g A f * π f π (2.6) once loop corrections are included which bring in the axial vector coupling constant (g * A denotes the in-medium coupling constant). The scaling relation m * ρ /m ρ = f * π /f π then directly links the in-medium mass of the ρ meson to m * N as m * ρ m ρ = g A g * A m * N m N . (2.7) A drop of the rho-meson mass with density will increase the range of the spin-transverse interaction,Ṽ ⊥ , as well as its coupling constant. We note that [16] f ρN N m ρ = g ρN N (1 + κ ρ V ) 2m N , (2.8) where κ ρ V is the anomalous ρ-meson tensor coupling to the nucleon. In accordance with [5] g ρN N will not depend on density which follows naturally if one considers the ρ meson as the gauge particle of the hidden symmetry [21]. With (2.6) we then obtain that f * ρN N f ρN N = g A g * A , (2.9) the f * ρN N being the in-medium coupling. There will also be a change in the πNN coupling constant g πN N . In analogy to (2.8) we have f πN N m π = g πN N 2m N . (2.10) As noted, m π changes but little with density, increasing by ∼ 5 MeV for ρ ∼ ρ 0 [6]. In chiral perturbation theory, a change in f πN N enters first in one-loop calculations, four powers higher in the Weinberg counting rules [22] than the basic pion exchange interaction. Consequently, changes in f πN N with density are expected to be small and we neglect them. Thus, the ratio f πN N /m π is taken not to change with density. From the Goldberger-Treiman relation which can be written as g πN N m N = g A f π = 2f πN N m π (2.11) this requires g * πN N to scale as m * N with density. This will turn out to be quite important for the deep inelastic experiments. The cut-off parameter Λ π , which determines the extent of the pion in π-nucleon interactions and boson exchange models will also be modified in the medium. We can see this in the following way. Consider the πNN vertex and the mass dispersion in the pion channel. Structures, other than elementary pion, will set in with the correlated (ρ, π)state (see Fig. 2). The mass Λ π involved in this new structure will be determined by the integral which involves cutting horizontally through the ρ and π lines, and putting them on shell: Λ π ∼ m ρ + m π + T ρπ (2.12) where T ρπ is the summed kinetic energy of the ρ and π. Carrying out the integral in the dispersion relation self-consistently will involve inserting the πNN vertex function appropriately. This has been done many times, most recently by Janssen et al [11]. The net result is that Λ * π Λ π ∼ = m * ρ m ρ . (2.13) Whereas the kinetic energies T ρπ in eq. (2.12) would seem to increase Λ π , in fact in our picture there are two pions, the π and π ′ . The latter, when treated as a correlated (ρ, π) system, coupled to the quantum numbers of the π, has a broad mass distribution starting at m ρ + m π . The elementary π and the π ′ mix pushing the π down and the π ′ up, in energy. It is clear that the main player in the pion vertex function is the mass of the ρ-meson. We do not yet have a detailed description of the π ′ -meson, but much of its mass must come from that of the ρ-meson. We take m π ′ and Λ π ′ to scale as in eq. (1.1), although the scaling of the Λ π ′ has little effect. The scaling of f π ′ N N is assumed to be the same as for f ρN N . Finally we have to scale the mass of the ∆(1232) isobar. It is known, e.g. from inclusive electron scattering experiments, that the mass difference between the nucleon and the ∆ does not significantly change in the nuclear medium. Therefore we keep this difference constant in our calculations: m * ∆ − m * N = m ∆ − m N . (2.14) From the above discussion, once the density dependence of m * N is known, the medium modification of all the other quantities can be derived. For m * N we shall assume a linear density dependence as m * N (ρ) m N = 1 − 0.3 ρ ρ 0 (2.15) which is adjusted to a value of 0.7 at saturation density as shown in Fig. 3 (for an extensive discussion of the nucleon effective mass in nuclei see [23]). Brown and Rho [24] find, from the measured isovector exchange current in 209 Bi, a value of 0.75 for m * N /m N averaged over the nucleus, which is consistent with (2.15). Whereas g A = 1.26, the in-medium coupling is roughly g * A ≃ 1 (2.16) as is inferred from the quenching of magnetic moments and M1 transitions, as well as the missing Gamow-Teller strength [25,15]. The renormalization of g A has two sources: a screening through virtual ∆-hole states [26] and second-order mixing of nucleonic excitations, chiefly through the tensor force [27]. In our model the density dependence of g * A is determined, however, from the assumption that the ratio f πN N /m π remains fixed. Then using eq. (2.6) as well as the Goldberger-Treiman relation (2.11) gives that Fig. 3). At saturation density this yields g * A /g A = 0.99 in close agreement with (2.16). From eq. (2.7) the density dependence of the ρ-meson mass is also determined (Fig. 3) and we obtain a value of 0.79 for m * ρ (ρ 0 )/m ρ . This is close to QCD sum rule calculations [28,29]. * * Hatsuda and Lee [29] give 0.82 as their central value. Chanfray and Ericson [30] have shown that exchange current type processes involving the virtual pion field decrease the quark condensate by a factor 10 − 20% over that of Hatsuda and Lee. Birse and McGovern [31] show that with proper calculation of the symmetry breaking matrix element, those result in an enhancement of chiral symmetry breaking in nuclei. g * A = g A m * N m N 2/3 (2.17) ( The Quasifree Polarization-Transfer in the (p,n) Reaction at 495 MeV The ( p, n) polarization experiments aim at extracting the ratio of the spin-longitudinal to spin-transverse response function R (q, ω) R ⊥ (q, ω) = n =0 \| 0\|O \|n \| 2 δ(ω − E n ) n =0 \| 0\|O ⊥ \|n \| 2 δ(ω − E n ) (3.1) where O = A i=1 σ i ·qτ − i e iq·r i ; O ⊥ = A i=1 σ i ×qτ − i e iq·r i (3.2) from a combination of spin-transfer coefficients. Measurements have been performed in 12 C and 40 Ca at a fixed angle of 18 • , corresponding to a peak three-momentum transfer of 1.72 fm −1 , so as to maximize the difference between R and R ⊥ expected from the standard treatment of the response functions. The ratio, however, is found to be essentially unity for projectile energy losses below the quasielastic peak [32]. In our picture of dropping masses this can be largely explained. We will sketch below the recent work of Brown and Wambach [33]. When looking at differences in the two-body interaction that cause a deviation of R /R ⊥ from unity, it is clear that these can only come from the tensor part, since V = V central + 2V tensor V ⊥ = V central − V tensor . (3.3) For simplicity, let us consider the π+ρ exchange neglecting form factors (the full HT interaction leads to very similar conclusions). In the static limit V tensor (q) = − f 2 πN N m 2 π q 2 (q 2 + m 2 π ) + f 2 ρN N m 2 ρ q 2 (q 2 + m 2 ρ ) S 12 (q)τ 1 · τ 2 . (3.4) where S 12 (q) = σ 1 ·qσ 2 ·q − 1/3(σ 1 · σ 2 ). A way of interpreting the experiment is to say that the tensor force is essentially zero at the momentum transfer q = 1.72f m −1 . We adopt as ratio of coupling constants [16] f 2 ρN N m 2 ρ = 2 f 2 πN N m 2 π (3.5) which, within a few percent, is that given by the HT interaction at the relevant momentum transfer. Since m 2 π ≪ q 2 ≪ m 2 ρ , the requirement V tensor = 0 implies that f 2 ρN N /m 2 ρ f 2 πN N /m 2 π q 2 m 2 ρ ∼ = 1. (3.6) Using m π and m ρ , however, from the Particle Data book, this ratio is 0.4 instead of unity. From eqs. (2.9) and (3.6) we find that the condition for zero net tensor force at q = 1.72f m −1 becomes † m ρ m * ρ 4 g A g * A = 2.5,(3.7) or, assuming g * A = 1, m ρ m * ρ 4 = 2. (3.8) This gives m * ρ /m ρ = 0.84 (3.9) which is slightly larger than the value of 0.79 at saturation density, found above. One should note that the (p, n) reaction at 500 MeV is strongly surface dominated and therefore the nucleus is probed at a lower density. In the calculations of ref. [33] this was taken into account in a semiclassical description, including distortion effects, which amounts to an averaging of the local nuclear matter response functions with a weight factor F as R A ,⊥ (q, ω) = d 3 rR NM ,⊥ (q, ω; ρ(r))F (r). (3.10) The weight factor can be evaluated using the Eikonal approximation [34]. The average density ρ = d 3 rρ(r)F (r)/ d 3 rF (r) turns out to be 0.35ρ 0 at which m * ρ /m ρ = 0.93. It is therefore expected that the tensor interaction is reduced but does not completely vanish. As shown in Fig. 4 this is born out of the more complete calculation using the full HT interaction as well as all the medium modifications discussed in sect. 2. The results (full lines) for 40 Ca at q = 1.72 fm −1 (left panel) and a more recent measurement for 12 C at q = 1.2 fm −1 [36] (right panel) give an improvement for energies below the quasielastic † The loop correction g A was not introduced in ref. [33]. peak as compared to a standard RPA treatment (dashed-dotted lines). Our model should be most reliable, however, near the quasielastic peaks of ω ≃ 82MeV for q = 1.72f m −1 and ω ≃ 55MeV for q = 1.2f m −1 [36], including a Q-value of ∼ 20 MeV for the (p, n) reaction. For substantially lower ω the finite-nucleus collective excitations have to be treated explicitly. For excitation energies above the quasielastic peak, on the other hand, one should bear in mind that contributions from two-step processes become significant [35]. Double scattering contributions to the spin observables are not well understood at present. For q = 2.5f m −1 Brown and Wambach [33] predicted a ratio below unity, which has been experimentally confirmed very recently by Taddeucci et al. [36]. However, at this large momentum transfer our results are very sensitive to the exact scaling behavior of the π ′ which is presently worked out within the microscopic model of ref. [11]. We therefore postpone the discussion of the q = 2.5f m −1 data until these investigations are finished. The authors of ref. [36] come to the conclusion that the ratio R /R ⊥ remains small because of an unexpected enhancement of the transverse response rather than a nonenhancement of the longitudinal one. There is no theoretical explanation for such an effect. However, the data analysis strongly depends on the treatment of the distortion which enters in terms of an effective number of neutrons. Because of the short range character of the interaction in the transverse channel it is quite reasonable that the distortion is much larger in this channel than in the longitudinal one. Of course, this would also change the ratio. Our model contains as an essential ingredient the change of the nucleon effective mass m * N with density. It is useful to single out this effect in order to make contact with the recent relativistic calculations of Horowitz and Piekarewicz [37], since the nucleon effective mass enters much in the same way as in the relativistic treatment. A drop of m * N has two effects: (1) the density of particle-hole states is decreased. This shifts the position of the quasielastic peak to higher energies and broadens it. At the same time the longitudinal and transverse correlations are weakened because of an increase of the particle-hole energies. (2) more importantly g ′ N N is increased. Recall that g ′ N N receives a significant contribution from ρ exchange. At zero density and momentum transfer we find the contribution to be g ′(ρ) N N = 0.28 while π and π ′ contribute with 0.08 and 0.21 respectively. By using eq. (2.8) one can relate the increase of the ρ contribution to m * N : g ′(ρ) N N (ρ) ∼ = g ′(ρ) N N (ρ = 0) m N m * N 2 . (3.11) Using m * N /m N = 0.7 appropriate for ρ 0 we find g ′(ρ) N N (ρ 0 ) = 0.57. This would increase g ′ N N to a value of 0.86 at nuclear matter density which is very similar to the g ′ N N = 0.9 of Horowitz and Piekarewicz although they assign only a value of 0.3 from ρ-exchange. However, in our model the π ′ scales in the same way as the ρ and this brings g ′ N N up to 1.08 (see Fig. 5). On the other hand this additional short range correlation due to the π ′ is canceled or even overcompensated by an increase of the π ′ contribution to the tensor force. Therefore, compared with the results of Horowitz and and Piekarewicz we find a somewhat weaker effect of the scaling on the longitudinal-transverse ratio. Preliminary results by Janssen [11] seem to indicate that the π ′ NN coupling constant coming out of a more microscopic calculation will be much weaker than the HT value. This would considerably improve our results. The (Lack of ) EMC and Drell-Yan Effects We shall discuss here the region of 0.1 < x < 0.3 where pion enhancement effects were supposed to be and where they weren't [1]. Below x = 0.1 there is shadowing, an interesting phenomenon in its own right. In any description which fits the shadowing, (see, e.g. the 'reggeized' discussion of Brodsky and Lu [38]) and preserves the momentum sum rule, there will be some small overshoot above x = 0.1. It is argued that this overshoot concerns valence quarks [39], i.e. it enters only into the EMC effect. We also will not discuss the dip in the EMC effect in the region of x ∼ 0.5 − 0.6. There are parameterizations of this dip [40]- [42] in terms of rescaling. In the region 0.1 < x < 0.3 the EMC as well as the Drell-Yan ratio are sensitive to the sea quark distributions in the nucleus. We consider a two phase model where the nucleon is made up of a bare quark core and a second component consisting of virtual meson-baryon pairs. Therefore the structure function F 2 reads: F N 2 (x) = Z N {F core 2 (x) + i (δF B i /N 2 (x) + δF M i /N 2 (x))}. (4.1) Here Z N denotes a wave function renormalization constant, on which we comment below. The sum in the bracket may run over all meson-baryon decompositions of the nucleon. In deep-inelastic scattering processes the virtual photon couples to the core as well as to the recoil baryon (described by δF B i /N 2 ) and the meson (described by δF M i /N 2 ). The most important example for the latter case is the Sullivan process [43] where the photon couples to the pion cloud ( Fig. 6(a)). The corresponding contribution to the structure function of the nucleon is: Z N δF π/N 2 (x) = Z N 1 x dyf π/N (y)F π 2 ( x y ) (4.2) with f π/N (y) = 3 16π 2 g 2 πN N y ∞ m 2 N y 2 /(1−y) dt t \|Γ πN N (t)\| 2 (t + m 2 π ) 2 (4.3) being the probability of finding a pion in the nucleon which carries the plus-momentum fraction y = p o π + p 3 π m N . (4.4) The renormalization constant Z N (which is missing in the original paper by Sullivan [43]) normalizes the total probability of finding the nucleon in one of the two phases to unity. The importance of this constant in connection with number sum rules has been shown by Szczurek and Speth [44]. It is given by Z N = (1 + i 1 0 dyf M i /N (y)) −1 , (4.5) where f M i /N is the distribution function for the meson M i , analogous to eq. (4.3). Taking into account a large set of processes the authors of ref. [44] find Z N ≃ 0.6 for a cut-off which roughly corresponds to a monopole form factor with Λ = 800MeV . We adopt this value for our calculations. In the nuclear medium analogous relations hold. We assume that the structure function (x) = A x dzf N/A (z)F N 2 ( x z ) + A x dy(Z A f π/A (y) − Z N f π/N (y)) F π 2 ( x y ). (4.6) The function f N/A (z) in the first integral describes the nucleon distribution due to Fermi motion. Since Fermi motion is not very important at small values of x the main effect comes from the change of the pion distribution function which gives rise to the second integral. Up to this point everything is like in the conventional pion excess model. However, in eq. (4.6) the pion distribution functions f π/N and f π/A are multiplied by the normalization factors Z N and Z A , respectively. Assuming that we can neglect the change in the distribution functions for mesons other than the pion it follows from eq. (4.5) and ‡ For simplicity we discuss only isospin averaged structure functions. In nuclei with neutron excess, like 56 F e, the pion cloud contains more π − than π + mesons, i.e. a small amount of negative charge is transferred from the nucleons to the pion cloud. This effect is properly taken into account in our numerical calculations although it is almost negligible. For details see ref. [3]. the analogous equation for Z A : Z −1 A = Z −1 N + 1 0 dy(f π/A (y) − f π/N (y)), (4.7) i.e. the normalization factor Z A decreases with an increasing distribution function f π/A . Thus the second integral in eq. (4.6) is much smaller (about a factor of Z 2 N ) than it would be without normalization factors. In our model the in-medium distribution function is given by f π/A (y) = 3 16π 2 g 2 πN N y ∞ m 2 N y 2 dt t−m 2 N y 2 2m N y 0 dω t \|Γ πN N (t)\| 2 R (ω, √ t) (t + m 2 π ) 2 ,(4.8) with R (ω, q) being the non-relativistic spin-longitudinal response function for nuclear matter. It describes Pauli blocking as well as rescattering corrections (see Fig. 6(b)). Pauli blocking leads to a depletion in the mean number of pions per nucleon as compared to the free nucleon which is overcompensated by the rescattering, chiefly through ∆-hole excitations, giving a net pion excess. As in eq. (4.3), t is the (space-like) four-momentum transfer t = q 2 − ω 2 . The integration limits for t and ω follow directly from eq. (4.4) (with p o π = −ω and \| p π \| = \| q\|). This is different from the distribution functions which can be found in the literature [46] [47], where the three-momentum transfer is the integration variable (leading to ω max = \| q\| − m N y) as well as the second argument of R . In the non-relativistic regime, i.e. when the main contributions to the integral come from regions with ω ≪ \| q\|, both prescriptions become identical. In addition, however, eq. (4.8) has the correct relativistic low density limit: lim k F →0 f π/A (y) = f π/N (y),(4.9) which is not the case for the function given in refs. [46] and [47]. Since both functions, f π/A and f π/N , enter into eq. (4.6) we prefer the more consistent prescription eq. (4.8). In our final results this enhances the pion contribution by a few percent. For the Fe nucleus we choose an average density of ρ = 0.87ρ 0 corresponding to k F = 260 MeV [48] § . With no medium modified masses we obtain for the first moment of the pion distribution function M π/A 1 = dyf π/A (y) a value of 0.70 at this density. This has to be compared with the free value of M π/N 1 = 0.41 (see also Fig. 7). However, because of the normalization factors this enhancement has much less influence on the EMC and Drell-Yan ratios than expected in the past. The corresponding predictions are displayed as the dashed lines in Figs. 8 and 9. In the region 0.1 ≤ x ≤ 0.3 there is no significant deviation of the predicted EMC ratio from the data. Of course, this does not mean that the pion field is amplified. Rather, since the more sensitive Drell-Yan data remain strongly overestimated, we are still led to the conclusion that the standard picture fails. This cannot be reconciled by changes in the key parameter g ′ N ∆ which would have to be chosen unrealistically large. It also seems implausible that more sophisticated many-body approaches will cure this problem. With dropping masses, coupling constants and formfactors several modifications occur. Brown, Li and Liu [49] pointed out that the nucleon effective mass m * N rather than m N should be used at the soft πNN vertices in Fig. 6. The nucleon mass enters at two places into the derivation of the Sullivan formula (eq. (4.3)). The first place is the spin-isospin current which mixes the large and the small spinor components of the nucleon. Secondly the energy of the virtual pion, as a function of its momentum, is determined from the on-shell condition for the initial and the final nucleon. In both cases the nucleon effective mass should be used. f N π * (y) = 3 16π 2 g * 2 πN N ( m N m * N ) 2 y ∞ m 2 N y 2 /(1− m N m * N y) dt t \|Γ * πN N (t)\| 2 (t + m 2 π ) 2 . (4.11) Note that the coupling constant g * πN N comes together with a factor m N m * N . As we have argued in sect. 2 this combination is density independent as long as we keep f πN N /m π constant. Compared with eq. (4.3), eq. (4.11) leads to a reduced number of pions: The first reason is the smaller cutoff Λ * πN N in the πNN form factor. The second reason is the enhanced lower limit of the t-integration. Including nucleon-hole and ∆-hole rescattering diagrams ( Fig. 6(b)) amplifies the pion field again. Because of the stronger short-range repulsion the effect is smaller than without scaling but it is still present. The general behavior can be seen from Fig. 7 where the first moment of the pion distribution function, M As can be expected from these results we almost produce a null effect in the Drell-Yan as well as in the EMC experiments (full lines of Fig. 8 and Fig. 9). This can be seen by comparison with the dotted lines, which show the result of a calculation with the pion contribution (second integral of eq. (4.6)) being switched off. The EMC data in the xregion of interest are even somewhat underestimated. To obtain the change in the quark distributions and the nucleon structure function (eq. (4.6)) we have employed the quark distributions of the free nucleon and the pion by Owens [45]. Other parameterizations [50]- [52] yield basically the same result. In the EMC calculation nuclear separation energy effects and Fermi motion have been put in as in refs. [49] and [53]. As discussed by Li, Liu and Brown [53], only part of the dip at larger x ∼ 0.6 is explained by binding energy effects, once the proper normalization for the baryon number is used. As mentioned above, the enhancement in the region of x = 0.1 to 0.2 seems to be in the valence quarks and such an enhancement can sensibly come from antishadowing or any description of the shadowing which preserves the momentum sum rule. The pion enhancement in both Drell-Yan and EMC experiments involves convolutions over a fairly wide region of momenta. In this sense, they are less specific than the Los Alamos polarization transfer experiments. While in the former the dropping nucleon mass and the resulting density dependence of g ′ N N is the key physical effect, in agreement with the findings by of Horowitz and Piekarewicz [37], the deep inelastic experiments are much more sensitive a change of the ρ-meson mass, chiefly through the softening of the πNN vertex in the medium. The Drell-Yan data cannot be described if this softening is not taken into account. Summary We have analyzed three recent experiments which have looked at effects of an enhancement of the virtual pion field in nuclei. The negative outcome can be understood from a perspective of partial restoration of chiral symmetry with density which reflects itself in a drop of hadron masses, especially the nucleon and ρ-meson mass. Following recent developments in the nucleon-nucleon interaction which try to reconcile the sea quark distribution in the nucleon with the low-energy properties of the two-nucleon system, as well as employing microscopic calculations of the πNN form factor we are able to explain the apparent lack of pion enhancement in nuclei. Thus the large discrepancies [1] between the conventional theory and experiment are removed. The momentum dependence of the Fermi liquid parameters g ′ N N , g ′ N ∆ and g ′ ∆∆ at zero density deduced from the potential [10] by using a realistic two-body correlation function. For g ′ N ∆ (0) and g ′ ∆∆ (0) the classical Lorentz-Lorenz value was taken in agreement with the data analysis by Thies [17] and Johnson [18]. The density dependence of the key quantities in our description. Assuming a linear dependence for m * N (ρ)/m N such that m * N (ρ 0 )/m N = 0.7 the effective ρ-meson mass is fixed by eq. (2.7) while g * A (ρ) is determined by the requirement that the πNN coupling f πN N /m π in the medium is the same as in free space. The empirical value of m * N is taken from ref. [23] while that of g * A is estimated from [15]. The data (measured at a fixed angle of 18 • corresponding to q = 1.7 fm −1 at and below the quasielastic peak ω = 80MeV ) were taken from refs. [32] (open circles) and [36] (solid circles). The right panel displays the predictions for a momentum transfer of 12 C at 1.2 fm −1 recently measured at LAMPF [36] (θ = 12.5 • ). The labeling of the curves is the same as in the left panel. are also given, including their uncertainties. Figure Captions right panel: the momentum dependence of the Fermi liquid parameters g ′ at saturation density, ρ 0 . So far only the drop of the nucleon mass entered into our argumentation. As can be seen by comparing the solid and dashed lines in Fig. 4 dropping m ρ and the other properties as described in section 2 in addition to m N does not have a large effect. One concludes that the quasielastic polarization transfer experiments are most sensitive to the change in the nucleon effective mass. In the deep-inelastic scattering experiments the situation is quite different. of the mesons and the baryon cores remain unchanged whereas the meson distribution functions f M i and the corresponding ones for the recoil baryon have to be modified. As shown in ref. [3] the modification of the baryon part can be absorbed in the Fermi motion of the nucleons. Furthermore, in the kinematical region we are interested in (x ≤ 0.3), the only relevant contribution comes from the pion and we can neglect the change of the other meson distribution functions. Thus the structure function F 2 of a nucleon in the nuclear medium becomes: The modification of the Sullivan formula due to Brown/Rho scaling can be obtained most easily by replacing all properties on the r.h.s. of eq. (4.2) and in eq. (4.3) by the scaled ones. This also includes the variables x and y which have to be replaced by x * = ) § Of course, the Drell-Yan (and EMC) experiments see a higher ρ than the polarization transfer, because in the latter case the projectile is affected by the strong interactions. enhanced in the standard RPA calculation (dasheddotted line) the scaled result (solid line) comes quite close to the free nucleon value (dotted line). At ρ = .87ρ 0 which corresponds to the averaged density of the F e for the free nucleon . The dashed line shows the result of the scaled first-order calculation (eq. (4.9)). Fig. 1 1Fig. 1 The momentum dependence of the Fermi liquid parameters g ′ N N , g ′ N ∆ and g ′ ∆∆ at zero density deduced from the potential [10] by using a realistic two-body correlation Fig. 2 2The vertex function for the πNN interaction. Here we have displayed only the simplest contribution. Because of self interactions, a large number of higher-order diagrams are possible[11]. Fig. 3 3Fig. 3 The density dependence of the key quantities in our description. Assuming a linear Fig. 4 4The ratio of spin-longitudinal to -transverse response functions in 40 Ca as a function of excitation energy and at fixed momentum transfer q = 1.7 fm −1 (left panel). The dashed line gives the result of a standard RPA treatment with only nucleon effective mass, while the full line includes the effects of medium-dependent nucleon mass and meson masses. The dashed-dotted line displays the result without any effective mass. Fig. 5 5left panel: the density dependence of the Fermi liquid parameters g ′ (0) after inclusion of medium-modified masses. The empirical values for g ′ N N and g ′ ∆∆ (see text) Fig. 6 6(a) Deep inelastic scattering off a pion in lowest order. (b) Rescattering correction to (a). The shaded areas in the bubbles are vertex corrections which introduce the local field correction g ′ ; nucleon-hole and ∆-hole intermediate states are included similarly as in the works of refs.[46] and[47]. Fig. 7 7The π/A (y) of the pion distribution function as a function of density. The dotted line indicates the free nucleon value M π/A 1 = .41. The standard RPA result without Brown/Rho scaling is represented by the dasheddotted line while the solid line corresponds to the calculation with scaling. The dashed line shows the result of the scaled first-order calculation (no rescattering), corresponding to eq. (4.11). Fig. 8 8The EMC ratio: The dashed line gives the result of a conventional RPA treatment without medium modifications of the hadron masses, while the full line includes those effects. Wave function renormalization constants have been used in both cases as described in the text. Switching off the pion contribution (second integral of eq. (4.6)) one obtains the result represented by the dotted line. The calculations have been performed at ρ = 0.87ρ o which corresponds to the average density of 56 F e. The data were taken from refs. [2] ( 40 Ca) and [54] ( 56 F e). Fig. 9 9The Drell-Yan ratio: The dashed line gives the result of a conventional RPA treatment without medium modifications of the hadron masses, while the full line includes those effects. The dotted line represents the result of a calculation where the pion AcknowledgementWe would like to thank Leonid Frankfurt, Gerry Garvey and Mark Strikman for many stimulating discussions. 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[ "Computability logic: Giving Caesar what belongs to Caesar", "Computability logic: Giving Caesar what belongs to Caesar" ]	[ "Giorgi Japaridze [email protected] \nVillanova University and Institute of Philosophy Russian Academy of Sciences\n800 Lancaster Avenue19085VillanovaPAUSA\n", "Giorgi Japaridze \nVillanova University and Institute of Philosophy Russian Academy of Sciences\n800 Lancaster Avenue19085VillanovaPAUSA\n" ]	[ "Villanova University and Institute of Philosophy Russian Academy of Sciences\n800 Lancaster Avenue19085VillanovaPAUSA", "Villanova University and Institute of Philosophy Russian Academy of Sciences\n800 Lancaster Avenue19085VillanovaPAUSA" ]	[ "Логические исследования Logical Investigations 2019. Т. 25. № 1. С. 100-119 2019" ]	The present article is a brief informal survey of computability logic (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic.In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems understood in their most general, interactive sense are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool.CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics.	10.21146/2074-1472-2019-25-1-100-119	[ "https://logicalinvestigations.ru/article/download/525/545?lang=en" ]	61,153,474	1902.05172	277c08b953d90e30007a707a537d240c3b685cff	Computability logic: Giving Caesar what belongs to Caesar 2019 Giorgi Japaridze [email protected] Villanova University and Institute of Philosophy Russian Academy of Sciences 800 Lancaster Avenue19085VillanovaPAUSA Giorgi Japaridze Villanova University and Institute of Philosophy Russian Academy of Sciences 800 Lancaster Avenue19085VillanovaPAUSA Computability logic: Giving Caesar what belongs to Caesar Логические исследования Logical Investigations 2019. Т. 25. № 1. С. 100-119 2019 251201910.21146/2074-1472-2019-25-1-100-119Computability logicgame semanticsconstructive logicintuitionistic logiclinear logicinteractive computability For citation: Japaridze "Computability logic: Giving Caesar what belongs to Caesar", The present article is a brief informal survey of computability logic (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic.In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems understood in their most general, interactive sense are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool.CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics. Computability logic versus classical logic Not to be confused with the generic term "computational logic", "computability logic" (CoL) is the proper name of an approach and ongoing ambitious project initiated by myself back in 2003 [Japaridze, 2003]. I characterize it as a "formal theory of computability in the same sense as classical logic is a formal theory of truth". To see what this means, let us compare the two logics. • In classical logic, the central semantical concept is truth; formulas represent statements; and the main utility of the logic is that it provides a systematic answer to the questions "What is (always) true?" or "Does truth of P (always) follow from truth of Q?". 1 • In computability logic, the central semantical concept is computability; formulas represent computational problems; and the main utility of the logic is that it provides a systematic answer to the questions "What is (always) computable?" or "Does computability of P (always) follow from computability of Q?". As we see, the second bulleted item is identical to the first one, only with "truth" replaced by "computability" everywhere and, correspondingly, "statements" by "computational problems" (for computability is the desired property of computational problems just like truth is the desired property of statements). In positive cases computability logic additionally provides a systematic answer to not only questions in the style "what...", but also "how...", such as "How to (always) compute P ?", or "How to (always) obtain an algorithm for P from an algorithm for Q?". With potential applications in mind, such questions are of course more interesting than their "what" style counterparts. Things are naturally set up so that statements of classical logic turn out to be special cases of computational problems, and classical truth a special case of computability. Eventually this makes classical logic a conservative fragment of CoL: the language of CoL is a proper extension of that of classical logic, but if we limit the former to the latter, CoL validates nothing more and nothing less than what classical logic does. Computability logic versus intuitionistic and linear logics Similarly, intuitionistic and linear logics can also be viewed as fragments of CoL, albeit "not quite" conservative ones, as CoL validates certain principles not provable in those logics, even if there are more similarities than differences. To me this fact indicates that those two logics are incomplete and do not fully correspond to their underlying philosophies and intuitions. Let me take the liberty to philosophize a little bit here. I believe the right way to build a new logic is to: (I) Start with the philosophy and intuitions that we want to capture call this informal semantics. (II) Then elaborate a formal semantics that adequately corresponds to the informal semantics. (III) And only after that ask what should be provable and what not in a proof system for the resulting logic, construct such a system and verify its soundness and completeness. This is the way classical logic evolved, culminating in Gödel's completeness theorem for first order logic. CoL, too, follows the same pattern. On the other hand, I would say that intuitionistic and linear logics jumped from informal semantics directly into proof systems, skipping the formal semantics phase. Take Heyting's intuitionistic logic for instance. Its construction started by looking at proof systems for classical logic and removing the postulates that appeared to be wrong from the informal intuitionistic point of view, such as the law of excluded middle. Similarly, linear logic was obtained from Gentzen's sequent calculus for classical logic as a result of deleting certain rules obviously incompatible with the resource philosophy of linear logic, such as contraction. Yes, in both cases the underlying philosophical and intuitive considerations were sufficient to clearly see that the expelled principles were indeed wrong. But where is the guarantee that, together with the law of excluded middle or contraction, some innocent, deeply hidden principles did not vanish as well? Idiomatically speaking, where is the guarantee that such a revision of classical logic did not throw out the baby with the bathwater? And, indeed, I dare to argue that this is exactly what happened. In the case of intuitionistic logic, among such "babies" is (¬P → A ∨ B) ∧ (¬Q → C ∨ D) ∧ ¬(P ∧ Q) → (¬P → A) ∨ (¬P → B) ∨ (¬Q → C) ∨ (¬Q → D). (1) And an example of an innocent victim of rudely rewriting classical logic into linear logic is (A ∧ B) ∨ (C ∧ D) → (A ∨ C) ∧ (B ∨ D),(2) with its connectives understood in the multiplicative sense. I call the latter Blass's principle as Andreas Blass [Blass, 1992] was the first to study it as an example of a game-semantically vaild principle underivable in linear logic. Of course some, mostly retroactive, attempts have been made to create formal semantics matching the proof systems of intuitionistic or linear logics. But the reasonable way to go is to match a proof system with a convincing formal semantics rather than vice versa. It is always possible to come up with some formal semantics that matches the target proof system, but the whole question is how adequately and convincingly that semantics captures the philosophy and intuitions underlying the logic. When constructing a deductive system, we ask what should be provable in it and what not. An answer to this question stems from the underlying semantics and only semantics, formal or informal: those things should be provable that are semantically valid. Some popular approaches to intuitionistic logic have attempted to explain everything in terms of proofs. For instance, you can see the meaning of A∨B explained by saying that this formula should be considered "good" (true? provable?) if either A or B can be proven. But the whole point is that we are just trying to understand what should be provable and what not. Trying to justify provability in terms of provability creates a vicious circle. Why is taking a shortcut from the earlier described stage (I) directly to stage (III) wrong? Because it is hardly possible to convincingly argue directly that a given proof system corresponds to a given informal semantics. On the other hand, adequacy (soundness and completeness) of a proof system with respect to a formal semantics can be proven mathematically, as both, unlike informal semantics, are mathematical objects. Now you can ask here: "OK, but where is then the guarantee that the formal semantics adequately captures the informal semantics and thus the original motivations and philosophy underlying the logic?". Of course, there is no guarantee, as this cannot be proven mathematically. But it is easier to argue that the two match each other (when they really do) because both are semantics. Comparing apples with apples is easier than comparing them with oranges. I have been pushing forward the above points since long ago. While having heard the angry "How dare you!" many times from sympathizers of intuitionistic or linear logics, I am still waiting to see some more convincing attempts to refute them. Summarizing much of what has been said in this section, my favorite excerpt from [Japaridze, 2009], not without sarcasm, notes: The reason for the failure of the principle of excluded middle in CoL is not that this principle ... is not included in its axioms. Rather, the failure of this principle is exactly the reason why it, or anything entailing it, would not be among the axioms of a sound system for CoL. Computational problems as games Anyway, what is computability? Before trying to answer or even ask this question, one should first understand what a computational problem is, for computability is a property of computational problems. So, what is a computational problem? According to Church, a computational problem is nothing but a function (to be computed). That is, the task of systematically generating the values of that function at different arguments. The tradition of seeing computational problems as functions has since firmly established in theoretical computer science. Such an approach, however, as acknowledged by Turing [Turing, 1936] himself, is too narrow. Most tasks performed by computers are interactive, far from being as simple as just receiving an input and generating an output. For instance, take a look at the work of a network server. It is in fact an infinite process, with signals moving back and forth between it and its environment in a not quite synchronized or regulated fashion, affecting not only current events but some future events as well. Such tasks are not always reducible to functions, at least reducible in some "nice" way. We need something more here, a more general concept to be able to adequately model complex tasks performed by computers. Such "something more" for us are games: a computational problem is a game between a machine, denoted , and its environment, denoted ⊥. Then computability is understood as existence of a machine which always wins the game, i.e., wins it no matter how the environment acts. In this presentation I am not giving you any formal definitions, including definitions of our concepts of games or game-playing. But such definitions, of course, do exist. Even though often it is us who act in the role of ⊥, we are fans of rather than ⊥. That is because (machine) is a tool, and its losing the game would mean failing to perform the task it was supposed to perform for us. The behavior (game-playing strategy) of , as the word "machine" suggests, should be algorithmic as it is a mechanical device. On the other hand, there are no restrictions on the behavior of ⊥, as the latter represents a capricious user, the blind forces of nature or the devil himself (and you can't ask the devil to only follow algorithmic strategies). Games can be visualized as trees in the style of Figure 1. Vertices of such a tree represent positions in the game, and edges their labels, that is represent legal moves, prefixed with or ⊥ to indicates which player can make the move. On the other hand, the label or ⊥ of a vertex indicates which player is considered to be the winner if the game ends in the corresponding position. The game can end anywhere, it does not have to continue to the "end": after all, some branches can be infinite and thus there will be nothing that could be understood as the "end". So, if the machine made the move α in the game of Figure 1, the environment responded with γ and no further moves were made, the machine loses as the corresponding vertex of the tree is ⊥-labeled. ⊥ α ⊥β ⊥γ ⊥β ⊥γ α ⊥ α β γ ⊥ ¢ ¢ ¢ ¢ β f f f f γ ⊥ ¢ ¢ ¢ ¢ β γ f f f f α ⊥ α ⊥ ⊥ ⊥ Figure 1: A game of depth 3 Games in logic have been studied by many authors, but our understanding of games is apparently unique in that it does not impose any regulations on the order in which the players should or could move, and permits positions where both players have legal moves. For instance, the root position of the game of Figure 1, as we see, allows either player to move. In that position, a move (if any) will be made by the player which can or want to act faster. It turns out that, in the sort of games we consider, the relative speed of either player does not matter. Namely, it never hurts a player to postpone making moves and let the adversary go first whenever possible. Such games are said to be static, and they are defined by imposing a certain technical yet simple condition on games. Striving to keep this presentation non-technical, I will not discuss that condition here. Suffice it to say that all "pure" (speedindependent) interactive problems turn out to be static, and the class of static games is closed under all game operations studied in CoL. The game of Figure 1 is static, in which the machine has a winning strategy. An interactive algorithm that guarantees the machine a win reads as follows: Regardless of what the adversary is doing or has done, go ahead and make move α; make β as your second (and last) move if and when you see that the adversary has made move γ, no matter whether this happened before or after your first move. It is left as an exercise for the reader to see that , following this interactive algorithm (strategy), wins no matter what and how fast ⊥ does. Computational problems in the traditional sense, i.e. functions, are static games of depth 2 of the kind seen in Figure 2. Figure 2: The successor function as a game Input ⊥0 ⊥1 ⊥2 h h h h h h h h h h h h ... Output ⊥ 0 ¢ ¢ ¢ ¢ 1 2 f f f f 3 d d d d ... ⊥ ⊥ ⊥ ⊥ 0 ¢ ¢ ¢ ¢ 1 2 f f f f 3 d d d d ... ⊥ ⊥ ⊥ ⊥ 0 ¢ ¢ ¢ ¢ 1 2 f f f f 3 d d d d ... ⊥ ⊥ ⊥ In such a game, the upper level edges represent possible inputs provided by the environment, so they are ⊥-labeled. The lower level edges represent possible outputs generated by the machine, so they are -labeled. The root is -labeled because it corresponds to the situation where nothing happened, namely, no input was provided by the environment. The machine has nothing to answer for in this case, so it wins. The middle level nodes are ⊥-labeled because they correspond to situations where there was an input but the machine failed to generate an output, so the machine loses. Each group of the bottom level nodes has exactly one -labeled node, because a function has exactly one (correct) value at each argument. It is not hard to see that the particular game of Figure 2 represents the successor function x + 1: if the input is 0, the machine, in order to win, should generate the output 1, if the input is 1, the output should be 2, etc. Now CoL rhetorically asks why limit ourselves only to trees of the kind seen in Figure 2. First of all, we may want to allow branches to be longer than 2, or even infinite to be able to model long or infinite tasks performed by computing machines. And why not allow all sorts of distributions of and ⊥ in nodes or on edges? For instance, consider the task of computing the function 3/x. It would be natural to make the node to which the input 0 takes us not ⊥-labeled, but -labeled. Because the function is not defined at 0, so the machine cannot be held responsible for failing to generate an output on such an input. It makes sense to generalize computational problems not only in the direction of increasing their depths, but also decreasing. Games of depth 0 are said to be elementary. These are games with no legal moves (the game "tree" is just its root), and thus games where one of the players automatically wins by doing nothing. We understand true atomic sentences of classical logic such as 2 × 2 = 4 or as the elementary game automatically won by the machine, and false sentences such as 2 × 2 = 5 or ⊥ as the elementary game lost by the machine. Note the two different yet related meanings of the symbols and ⊥ in CoL: depending on the context, such a symbol stands either for the corresponding elementary game, or the player which wins that game. Thus, classical propositions for us are nothing but elementary games. This generalizes to predicates in the standard way. In classical logic, predicates can be thought of as "propositions that (may) depend on variables". Similarly, we allow "games that (may) depend on variables", with predicates being nothing but elementary sorts of such games. As a result, classical logic becomes a special case of CoL CoL where only elementary games are allowed. Choice operators Logical operators in CoL stand for operations on games. There is a whole zoo of them, with (at least) four sorts of conjunction and disjunction as well as universal and existential quantifiers, a bunch of so called recurrence (repetition) operations and corresponding implication-style and negation-style operations, and more. In this short presentation we shall only look at the following subset of the logical operators studied in CoL: [¬, ∧, ∨, →, ∀, ∃, , , , , • \| , • \| , • -, •¬ .] Using the classical notation for the first six of these is no accident. They are conservative generalizations of their classical counterparts from elementary games to all games. Conservative in the sense that, when applied to elementary games (propositions, predicates) only, their extensional meanings and logical behavior turn out to be exactly classical. This is how classical logic naturally becomes a special (elementary) fragment of CoL. We start with the choice connectives (conjunction) and (disjunction). The way they combine two games A and B to get the new game A B or A B is depicted in Figure 3. A B ⊥0 d d d d ⊥1 A B A B ⊥ 0 d d d d 1 A B Figure 3: Choice conjunction and disjunction As we see, A B is the game where the first legal move is (only) by the environment. Such a move should be either 0 or 1. If move 0 is made, the game "turns into" A, in the sense that it continues and the winner is determined according to the rules of A. Similarly for B in the case of move 1. Intuitively, making move 0 or 1 means choosing between the left disjunct and the right disjunct. Making such a choice is not only a privilege of the environment, but also an obligation: as seen in the picture, the root of A B is -labeled, meaning that the environment loses if it fails to make an initial move/choice. A B is fully symmetric/dual to A B: in it, it is the machine rather than the environment who makes the initial choice and who loses if no choice is made. For simplicity, let us agree that the universe of discourse is always {0, 1, 2, · · · }. If so, the choice universal quantification xA(x) (note that is larger than ) can be understood as the infinite choice conjunction A(0) A(1) A(2) · · · , and the choice existential quantification xA(x) as the infinite disjunction A(0) A(1) A(2) · · · . So, now a choice is made not just between 0 or 1, but among 0, 1, 2, · · · , as shown in Figure 4. xA(x) ⊥0 d d d d ⊥2 ⊥1 . . . Having these operators in the language, we may now conveniently express standard computational problems (and beyond) without drawing trees. So, for instance, the problem of computing the successor function depicted in Figure 2 can be simply written as x y(y = x + 1). In this game, the first move for instance 2 is by the environment. Intuitively, this can be seen as asking the machine the question "What is the successor of 2?". The game continues as y(y = 2 + 1). The next move say 3 is by the machine, which amounts to saying that 3 is the successor of 2. The game is now brought down ("continues as") 3 = 2 + 1. This is an elementary game with no further moves, and the machine has won because 3 = 2 + 1 is true. Had the machine made the move 4 instead of 3, or no move at all, it would have lost. A(0) A(1) A(2) xA(x) ⊥ 0 d d d d 2 1 . . . A(0) A(1) A(2) Rather similarly, the problem of deciding a predicate p is expressed by x ¬p(x) p(x) . Negation Negation ¬ is an operation which flips the roles of the two players, turning 's wins and legal moves into ⊥'s wins and legal moves, and vice versa. For instance, if Chess is the game of chess from the point of view of the white player, then ¬Chess is the same game as seen by the black player. Figure 5 illustrates how applying ¬ to a game A generates the exact "negative image" of A, with and ⊥ interchanged both in the nodes and on the arcs of the game tree. A ⊥0 d d d d ⊥1 ⊥ 0 ¢ ¢ ¢ ¢ f f f f 1 ⊥ ⊥ 0 ¢ ¢ ¢ ¢ f f f f 1 ⊥ ¬A ⊥ 0 d d d d 1 ⊥0 ¢ ¢ ¢ ¢ f f f f ⊥1 ⊥ ⊥0 ¢ ¢ ¢ ¢ f f f f ⊥1 ⊥ Figure 5: Negation Obviously if A is a true proposition, i.e., an elementary game automatically won by the machine, then ¬A remains an elementary game but now lost by the machine; in other words, ¬A is a false proposition. This is exactly what was meant when promising that the meaning of ¬, or any other operator for which we use classical notation, is exactly classical when limited to elementary games. It can be easily seen that the games ¬¬A and A are identical: switching the roles twice brings each player to its original status. Similarly, it can be seen that ¬ interacts with choice operations in the kind old DeMorgan fashion. E.g., ¬(A B) = ¬A ¬B. Looking back at Figure 5, notice that the game A shown there is nothing but ( ⊥) (⊥ ), and ¬A is its DeMorgan dual (⊥ ) ( ⊥). Parallel connectives The operations ∧ and ∨ are called parallel conjunction and parallel disjunction. Unlike their choice counterparts A B and A B, in A ∧ B or A ∨ B no choice between A and B is made by either player. Rather, the play proceeds in parallel in both components. To win in A ∧ B, the machine should win in both A and B, while for winning A ∨ B winning in just one of the two components is sufficient. Consider, for instance, Chess ∧ Chess. This is in fact a play on two boards, where plays white on both boards. Perhaps it plays against two adversaries: Peter and Paul, though, for , they together form just what it calls the (one) environment. In order to win, needs to defeat Peter on the left board and Paul on the right board. The first move in this compound game is definitely by , as the opening move is by the white player on both boards. But, after makes its first move, say against Peter, the situation changes. Now both and its environment naturally have legal moves. Namely, has a legal move against Paul, while Peter (and thus the environment from 's point of view) also has a legal move in response to 's initial move. It would be unnatural here to impose some regulations regarding which player can go next. This is why CoL's understanding of games does not insist that in each position only either or ⊥ (but not both) should be allowed to move. To appreciate the difference between the choice and the parallel sorts of connectives, let us compare the two games ¬Chess Chess and ¬Chess ∨ Chess. We assume that draw outcomes are ruled out in Chess, and the player who fails to make a move on his turn is considered to have lost. Imagine I am playing in the role of , and the world champion Kasparov in the role of ⊥. In ¬Chess Chess, I have a choice between playing on the left board (¬Chess) or on the right board (Chess). That is, I get to decide whether I want to play black or play white. After such a choice is made, I have to defeat Kasparov on the chosen board, while the other board is discarded. Obviously I stand no chance to win, regardless of whether I choose to play black or white. On the other hand, I can easily beat Kasparov in ¬Chess ∨ Chess. This is a parallel play on two boards. At the beginning, both Kasparov and I have legal moves: Kasparov on the left board where he is playing white, and I on the right board. Rather than hurrying to make an opening move, I wait to let Kasparov move first. If he, too, chooses to do nothing, then I win due to being the winner on the left board. Now suppose Kasparov makes his opening move on the left board. Can you guess how I should respond? Yes, by making the exact same move on the right board. I wait again. Whatever move Kasparov makes on the right board in response, I copy that move back on the left board. And so on. By using this copy-cat strategy, I am in fact letting Kasparov play against himself. Eventually, both he and I are guaranteed to win on one board and lose on the other. Since this is a disjunction, having won in one of the disjuncts makes me the winner in the overall game. In general, the law of excluded middle "¬A OR A" is invalid in CoL with OR understood as but valid when OR is understood as ∨: one can prove that, while the above seen copy-cat strategy wins all games of the form ¬A ∨ A, for some A no machine can win ¬A A against a sufficiently smart adversary. Putting things where they belong What is meant by "Giving Caesar what belongs to Caesar" (... and God what belongs to God) in the title of this article? The twentieth century has witnessed endless and fruitless fights between the classically-minded and the constructivistically-minded regarding whether the law of excluded middle should be accepted or rejected. It is obvious that the two schools of thought were talking about two very different meanings of disjunction. Yet, for some strange reason, they chose the same symbol ∨ for both, and then started arguing with each other. Not quite serious I would say. CoL neutralizes this and similar controversies by putting things where they belong. And, as pointed out in Section 2., it does so semantically, not because it allows or forbids them among the postulates of some purportedly "right" deductive system. Give the classically minded what belongs to the classically-minded (∨), and the constructivists what belongs to the constructivists ( )! • Yes, classical logic is right: ¬A ∨ A is indeed valid. • Yes, intuitionistic logic is right: ¬A A is indeed invalid. No subject for arguing! The classical tautology (¬A ∧ ¬A) ∨ A fails in CoL unless A is stipulated to be elementary. Observe that, at least, the copy-cat trick used earlier in our winning strategy for ¬Chess∨Chess no longer works for the "similar" (¬Chess∧ ¬Chess) ∨ Chess. I can try to copy Kasparov's moves in Chess within both conjuncts of ¬Chess ∧ ¬Chess and vice versa. However, Kasparov may start acting in different ways in these two conjuncts, and then, at best, I will be able to synchronize only one of them with Chess. It is then possible that eventually I lose in Chess and in the unsynchronized conjunct of ¬Chess ∧ ¬Chess, which makes me lose in the overall game (¬Chess∧¬Chess)∨Chess. Anyway, classical logic accepts the principle (¬A ∧ ¬A) ∨ A and linear logic rejects it (with ∧, ∨ seen as multiplicatives). Which one is "right"? The formal language of pure CoL has two sorts of nonlogical atoms: elementary and general. Elementary atoms are meant to be interpreted as elementary games, and general atoms as any games, elementary or not. We use the lowercase p, q, · · · for elementary atoms and the uppercase P, Q, · · · for general atoms. And, again, Caesar is being given what belongs to Caesar and God what belongs to God. The semantics of CoL classifies: • (¬p ∧ ¬p) ∨ p as valid. Yes, classical logic is right! • (¬P ∧ ¬P ) ∨ P as invalid. Yes, linear logic is right! (As for the earlier discussed law of excluded middle, both ¬P ∨ P and ¬p ∨ p are valid and both ¬P P and ¬p p are invalid.) From CoL's perspective, classical logic differs from intuitionistic logic in its understanding of logical constants (operators), and differs from linear logic in its understanding of logical variables (nonlogical atoms). Reduction The implication operation → is defined in the standard way by A → B = def ¬A ∨ B. The intuition associated with this operation is that of a reduction of the consequent to the antecedent. Since A is negated here and thus the roles of the two players are interchanged in it, A can be seen by as an environment-provided resource rather than a task. Namely, can observe how the environment is playing in A and use that information in its play in B. The task of is to win B as long as the environment wins A; in other words, to solve problem B as long as the environment is (correctly) solving problem A. To get a feel of → as a reduction operation, consider the game x y y = Father(x) ∧ x y y = Mother(x) → x y y = PaternalGrandmother(x) , where Father(x) is the function "x's father", and similarly for Mother(x) and PaternalGrandmother(x). Here, the task is facing is telling the name of an arbitrary person's paternal grandmother while the environment (correctly) tells the name of an arbitrary person's father and the name of an arbitrary person's mother. In other words, this is the problem of reducing the paternal grandmotherhood problem to the fatherhood and motherhood problems. Winning this game is easy and does not require any knowledge of anyone's relative's names. Here is a strategy for : Wait till ⊥ makes a move a in the consequent (if not, wins automatically). Intuitively, such a move amounts to asking the question "Who is a's paternal grandmother?". Make the same move a in the first conjunct of the antecedent, i.e., ask the counterquestion "Who is a's father?". ⊥ will have to answer correctly, or else it loses. Let us say ⊥'s answer/move is b. Make the same move b in the second conjunct of the antecedent, thus asking ⊥ to tell who b's mother is. ⊥, again, will have to provide the correct answer, let us say c. Now, by making the same move c in the consequent, i.e., answering "c" to ⊥'s original question regarding a's paternal grandmother, wins: c is indeed a's paternal grandmother (unless the environment lied in the antecedent about a's father or b's mother, but in that case, as already noted, is no longer responsible for anything). Blind quantifiers The operations ∀ and ∃, called blind quantifiers, conservatively generalize their classical counterparts, just like ¬, ∧, ∨, → do. Unlike the choice quantifiers, there are no moves associated with ∀ or its dual ∃. Playing ∀xA(x) or ∃xA(x) means playing A(x) "blindly", without knowing the value of x as the latter is not specified by either player. In order to win ∀xA(x) (resp. ∃xA(x)), needs to play A(x) in such a way that it wins for all (resp. at least one) possible values of x. An alternative intuitive characterization of ∀xA(x) and ∃xA(x) would be that, in these games, a third party chooses a value for x but never shows it to either player. In order to win ∀xA(x) (resp. ∃xA(x)), (resp. ⊥) needs to play A(x) in a way that guarantees success regardless of what that chosen value might have been. Let us compare the games x Even(x) Odd(x) and ∀x Even(x) Odd(x) . x Even(x) Odd(x) , which is a game of depth 2, is easy to win: wait till the adversary selects a value m for x; if m is even, respond by choosing the left disjunct of Even(m) Odd(m), otherwise respond by choosing the right disjunct, and rest your case. On the other hand, ∀x Even(x) Odd(x) is a game of depth 1, and it is impossible to win. Here the value of x is not specified by the adversary or whoever for that matter, yet you should do the impossible task of choosing between Even(x) and Odd(x) so that all of the elementary games/propositions Even(0), Even(1), Even(2), · · · (if you chose Even(x)) or Odd(0), Odd(1), Odd(2), · · · (if you chose Odd(x)) are won/true. This should not suggest than all ∀-games are unwinnable. Consider ∀x Even(x) Odd(x) → y Even(x + y) Odd(x + y) . Here, given a number chosen by the environment for y, let us say 5, in order to tell whether x + 5 is even or odd it is not necessary to know the actual value of x. Rather, just knowing whether x is even or odd is sufficient. And, luckily, this piece of information on x will have to be provided by the environment as mandated by Even(x) Odd(x) in the antecedent. If the environment claims that x is even, then chooses Odd(x + 5) and wins; otherwise, it chooses Even(x + 5). ∀ can be seen to be stronger than , in the sense that the semantics of CoL validates the principle ∀xA(x) → xA(x) but not its contrapositive. This means that xA(x) is reducible to ∀xA(x) but not vice versa. Symmetrically, ∃ is weaker than . Speaking philosophically, choice quantifiers are constructive versions of their blind counterparts. While not as popular as the law of excluded middle, ∃x∀y p(x) ∨ ¬p(y) is another example of a valid principle of classical logic which, however, is not valid in any constructive sense, and not provable in intuitionistic logic. Again giving Caesar what belongs to Caesar, CoL unsurprisingly establishes: • Both ∃x∀y p(x)∨¬p(y) and ∃x∀y P (x)∨¬P (y) are valid. Yes, classical logic is right! • Both x y p(x) ∨ ¬p(y) and x y P (x) ∨ ¬P (y) are invalid. Yes, intuitionistic logic is right! On the other hand, the valid principle ∀y∃x p(x) ∨ ¬p(y) of classical logic is commonly recognized to be valid in every reasonable constructive sense, and is provable in intuitionistic logic. As expected, CoL validates this principle with both (blind and choice) sorts of quantifiers and both (elementary and general) sorts of atoms. Recurrences Out of several types of so called recurrence operations studied within the framework of CoL, here we shall only take a look at branching recurrence • \| . Its dual corecurrence operation • \| can simply be understood as ¬ • \| ¬. When applied to a game G, • \| turns it into a game playing which means repeatedly playing G. When G is seen as a resource (e.g., when it is in the antecedent of an implication), • \| generates multiple "copies" of G, thus making G a reusable/recyclable resource. In classical logic, this sort of an operation would be meaningless, because classical logic is resource-blind, seeing no difference between one and many copies of G. In the resource-conscious CoL, however, recurrence operations are not only meaningful, but also necessary to achieve a satisfactory level of expressiveness and realize CoL's potential and ambitions. Hardly any computer program is used only once; rather, it is run over and over again. Loops within such programs also assume multiple repetitions of the same subroutine. In general, the tasks performed in real life by computers, robots or humans are typically recurring ones or involve recurring subtasks. Let me use our old friend Chess to explain the meaning of • \| . A play of • \| Chess starts as an ordinary play of Chess. At any time, however, the environment may decide to split the current position into two identical ones, thus creating two runs of Chess out of one that have a common past but possibly diverging futures. From that point on, the play continues on two boards. At any time, the environment can again create two identical copies of the thencurrent position on either board, and the play correspondingly continues on three boards. The environment can keep splitting positions in this fashion, creating more and more sessions of Chess to be played in parallel. Eventually, will be considered the winner if it wins in all of those sessions. • \| Chess is similar, with the difference that now splitting positions is 's privilege, and wins if it wins in at least one of the multiple sessions of Chess. Brimplication The implication-style operation • -, called brimplication ("br" for "branching"), is defined by A • -B = def • \| A → B. Intuitively A • -B, just like A → B , is a problem of reducing B to A. The difference between the two reduction operations is that, while in A → B the machine has a single copy of A available as an environment-provided informational resource for solving B, in A • -B the resource A as well as any game/position it has evolved to can be duplicated and reused any number of times. As a result, A • -B is easier for to win than A → B because, as a resource, the antecedent of A • -B is stronger (very much so) than the antecedent of A → B. While being the most basic sort of reduction allowing us to naturally define •or other reduction-style operations, → is a stricter and thus less general operation of reduction than • -. In fact, according to Thesis 1 below, •is the most general sort of reduction. We say that a problem/game B is brimplicatively reducible to a problem A iff there is a machine with a winning strategy for A • -B. Thesis 1. Brimplicative reducibility is an adequate mathematical counterpart of our intuition of reducibility in the weakest and hence the most general algorithmic sense possible. Namely, for all games/problems A and B, we have: (I): Whenever B is brimplicatively reducible A, B is also algorithmically reducible to A according to everyone's reasonable intuition. (II): Whenever B is algorithmically reducible to A according to everyone's reasonable intuition, B is also brimplicatively reducible to A. This is pretty much in the same sense as, by Church's celebrated thesis, a function f is Turing-machine computable iff f is algorithmically computable according to everyone's reasonable intuition. It should be also mentioned that, unsurprisingly, brimplicative reducibility turns out to be a conservative generalization of Turing reducibility, commonly accepted in theoretical computer science as the most general relation of reducibility between the traditional, non-interactive sorts of problems. On intuitionistic logic once again According to Kolmogorov's [Kolmogorov, 1932] well known thesis, intuitionistic logic is a logic of problems. This thesis was stated by Kolmogorov in rather abstract, philosophical terms. No past attempts to find a strict and adequate mathematical explication of it have fully succeeded. The following theorem tells a partial success story ("partial" because it is limited to only positive propositional fragment of intuitionistic logic): Theorem 1. [Japaridze [Japaridze, 2007b]; Mezhirov and Vereshchagin [Mezhirov, Vereshchagin, 2010]] The positive (negation-free) propositional fragment of Heyting's intuitionistic calculus is sound and complete with respect to the semantics of CoL, with intuitionistic implication understood as • -, conjunction as and disjunction as . As for the intuitionistic operators not mentioned in the above theorem, CoL sees the intuitionistic universal quantifier as , existential quantifier as , and negation as what it calls brefutation •¬ , defined by •¬ A = def A • -⊥. 2 So, formula (1) from Section 2. should in fact have been written as ( •¬ P • -A B) ( •¬ Q • -C D) •¬ (P Q) • - ( •¬ P • -A) ( •¬ P • -B) ( •¬ Q • -C) ( •¬ Q • -D).(3) This formula, as noted earlier, is valid in CoL but unprovable in Heyting's calculus, making the latter incomplete with respect to the semantics of CoL. At the same time, Heyting's calculus in its full first order language has been shown [Japaridze, 2007a] to be sound with respect to CoL's semantics. So, intuitionistic logic at least, Heyting's formal version of it is a fragment of CoL but, unlike classical logic, "not quite" a conservative one. Nevertheless, since (3) is the shortest formula known to separate Heyting's calculus from the corresponding fragment of CoL, one can say that Heyting's calculus is quite close to being complete. Conclusion Computability logic (CoL) is a formal theory of computability in the same sense as classical logic is a formal theory of truth. Its formulas represent computational problems, logical operators stand for operations on such problems, and validity means being "always computable". Computational problems, in turn, are understood in their most general interactive sense and, mathematically, are defined as games played by a machine against its environment. This article was a brief, informal and incomplete survey of CoL. The latter is not a subject that can be duly introduced within a 1-hour presentation and, in order to well understand it, one will have to use additional sources. Out of the numerous publications devoted to CoL, the most recommended reading for a beginner are the first ten sections of [Japaridze, 2009]. An even more comprehensive and the most up-to-date survey of CoL can be found online in [Japaridze, 2019]. There was no discussion of related literature in this article. Such discussions can be found elsewhere, including the already mentioned [Japaridze, 2009] or [Japaridze, 2019]. I just want to point out here that the main precursors of CoL are Lorenzen's [Lorenzen, 1961] dialogue semantics for intuitionistic logic, Hintikka's [Hintikka, 1973] game-theoretic semantics for classical logic and Blass's [Blass, 1992] game semantics for linear logic, the latter being the closest one. The language of CoL is much more expressive than the fragment surveyed in the present article. Important topics not covered here also include the proof theory of CoL. And, of course, actual and potential applications of CoL outside logic itself. Such applications include theory of (interactive) computation, knowledgebase systems, systems for planning and action, declarative programming languages, constructive applied theories, and more. So far the most manifestly realized extra-logical utility of CoL has been using it as a logical basis for applied theories [Japaridze, 2010]- [Japaridze, 2016c], with such theories offering substantial advantages over their classicallogic-based counterparts. CoL-based number theory, termed clarithmetic, will be the subject of a forthcoming paper expected to appear in the next issue of this journal. I want to close this article by pointing out that, despite having been evolving for 15 years already, CoL, due to its ambitiousness, still remains at an early stage of development, with more open questions than answered ones. A researcher who decides to join the project will find enough interesting material to be occupied with for many years to come. Students are especially encouraged to try. Figure 4 : 4Choice quantifiers Of course, one is a special case of the other. As we remember from Section 3., the meaning of the logical constant ⊥ in CoL is standard: this is an always-false proposition, i.e., the elementary game automatically lost by the machine. A game semantics for linear logic. A Blass ; Blass, Annals of Pure and Applied Logic. 56Blass, 1992 -Blass, A. "A game semantics for linear logic", Annals of Pure and Applied Logic, 1992, Vol. 56, pp. 183-220. J Hintikka ; -Hintikka, Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Clarendon PressHintikka, 1973 -Hintikka, J. Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Clarendon Press, 1973. Introduction to computability logic. G Japaridze ; Japaridze, Annals of Pure and Applied Logic. 123Japaridze, 2003 -Japaridze, G. "Introduction to computability logic", Annals of Pure and Applied Logic, 2003, Vol. 123, pp. 1-99. Introduction to computability logic. Japaridze, G Japaridze, Acta Cybernetica. 181Japaridze, 2007a -Japaridze, G. "Introduction to computability logic", Acta Cyber- netica, 2007, Vol. 18, No. 1, pp. 77-113. The intuitionistic fragment of computability logic at the propositional level. Japaridze, G Japaridze, Annals of Pure and Applied Logic. 1431Japaridze, 2007b -Japaridze, G. "The intuitionistic fragment of computability logic at the propositional level", Annals of Pure and Applied Logic, 2007, Vol. 143, No. 1, pp. 187-227. In the beginning was game semantics. G Japaridze ; Japaridze, Games: Unifying Logic, Language, and Philosophy. O. MajerSpringerJaparidze, 2009 -Japaridze, G. "In the beginning was game semantics", in: Games: Unifying Logic, Language, and Philosophy, eds. by O. Majer, A.-V. Pietarinen and T. Tulenheimo. Springer, 2009, pp. 249-350. Towards applied theories based on computability logic. Japaridze, G Japaridze, Journal of Symbolic Logic. 75Japaridze, 2010 -Japaridze, G. "Towards applied theories based on computability logic", Journal of Symbolic Logic, 2010, Vol. 75, pp. 565-601. Introduction to clarithmetic I. Japaridze, 209Japaridze, 2011 -Japaridze, G. "Introduction to clarithmetic I", nformation and Com- putation, 2011, Vol. 209, pp. 1312-1354. Introduction to clarithmetic III. G Japaridze ; Japaridze, Annals of Pure and Applied Logic. 165Japaridze, 2014 -Japaridze, G. "Introduction to clarithmetic III", Annals of Pure and Applied Logic, 2014, Vol. 165, pp. 241-252. Introduction to clarithmetic II. Japaridze, G Japaridze, Information and Computation. 247Japaridze, 2016a -Japaridze, G. "Introduction to clarithmetic II", Information and Computation, 2016, Vol. 247, pp. 290-312. Build your own clarithmetic I: Setup and completeness. Japaridze, G Japaridze, Logical Methods in Computer Science. 12Paper 8Japaridze, 2016b -Japaridze, G. "Build your own clarithmetic I: Setup and complete- ness", Logical Methods in Computer Science, 2016, Vol. 12, Issue 3, Paper 8, pp. 1-59. Computability logic: Giving Caesar what belongs to Caesar 119. Computability logic: Giving Caesar what belongs to Caesar 119 Build your own clarithmetic II: Soundness. Japaridze, G Japaridze, Logical Methods in Computer Science. 12Paper 12Japaridze, 2016c -Japaridze, G. "Build your own clarithmetic II: Soundness", Logical Methods in Computer Science, 2016, Vol. 12, Issue 3, Paper 12, pp. 1-62. Computability Logic Homepage", An Online Survey of Computability Logic. G Japaridze ; -Japaridze, www.csc.villanova.edu/$\sim$japaridz/CL/, accessed on 21.02.2019Japaridze, 2019 -Japaridze, G. "Computability Logic Homepage", An Online Survey of Computability Logic. 2019. [www.csc.villanova.edu/$\sim$japaridz/CL/, accessed on 21.02.2019] Zur Deutung der intuitionistischen Logik. Kolmogorov, A N Kolmogorov, Mathematische Zeitschrift. 35Kolmogorov, 1932 -Kolmogorov, A. N. "Zur Deutung der intuitionistischen Logik", Mathematische Zeitschrift, 1932, Vol. 35, pp. 58-65. Ein dialogisches Konstruktivitätskriterium. P Lorenzen ; -Lorenzen, Infinitistic Methods. PWN, Proc. Symp. WarsawLorenzen, 1961 -Lorenzen, P. "Ein dialogisches Konstruktivitätskriterium", in: Infin- itistic Methods. PWN, Proc. Symp. Foundations of Mathematics., Warsaw, 1961, pp. 193-200. On abstract resource semantics and computability logic. Mezhirov, Vereshchagin, I Mezhirov, N Vereshchagin, Journal of Computer and Systems Sciences. 76Mezhirov, Vereshchagin, 2010 -Mezhirov, I., Vereshchagin, N. "On abstract resource semantics and computability logic", Journal of Computer and Systems Sciences, 2010, Vol. 76, pp. 356-372. On Computable numbers with an application to the entsheidungsproblem. A Turing ; -Turing, Proceedings of the. theLondon Mathematical Society2Turing, 1936 -Turing, A. 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[ "Orbits of Hamiltonian Paths and Cycles in Complete Graphs", "Orbits of Hamiltonian Paths and Cycles in Complete Graphs" ]	[ "Samuel Herman [email protected] \nDivision of Natural Sciences\nNew College of Florida Sarasota\n34243FLUnited States\n", "Eirini Poimenidou [email protected] \nDivision of Natural Sciences\nNew College of Florida Sarasota\n34243FLUnited States\n" ]	[ "Division of Natural Sciences\nNew College of Florida Sarasota\n34243FLUnited States", "Division of Natural Sciences\nNew College of Florida Sarasota\n34243FLUnited States" ]	[]	We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings representing subgraphs. These orbits are enumerated using Burnside's lemma. The technique used also provides an alternative proof of the formulae found by S. W. Golomb and L. R. Welch which give the number of distinct n-gons on fixed, regularly spaced vertices up to rotation and optionally reflection.	null	[ "https://arxiv.org/pdf/1905.04785v3.pdf" ]	152,282,341	1905.04785	4e47fb64c5b84b77d7bd38f533a257ec7c3b5664	Orbits of Hamiltonian Paths and Cycles in Complete Graphs 27 Dec 2019 Samuel Herman [email protected] Division of Natural Sciences New College of Florida Sarasota 34243FLUnited States Eirini Poimenidou [email protected] Division of Natural Sciences New College of Florida Sarasota 34243FLUnited States Orbits of Hamiltonian Paths and Cycles in Complete Graphs 27 Dec 2019 We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings representing subgraphs. These orbits are enumerated using Burnside's lemma. The technique used also provides an alternative proof of the formulae found by S. W. Golomb and L. R. Welch which give the number of distinct n-gons on fixed, regularly spaced vertices up to rotation and optionally reflection. Introduction All graphs in this paper are considered as their geometric realizations, which are defined as follows: if a graph G has n vertices, its geometric realization is the figure obtained by first associating its vertices with n regularly spaced points on a circle, then representing its edges as line segments between said points. For example, the geometric realizations associated to the complete graphs K n for 3 ≤ n ≤ 6 are Recall that a Hamiltonian path is a path in a graph which visits every vertex exactly once, and a Hamiltonian cycle is a Hamiltonian path which is a cycle. Any Hamiltonian path or cycle in a graph induces a subgraph whose vertex set is the same as the original graph, but whose edges consist of those traversed in the path or cycle; e.g.: The investigation in this paper begins with a natural observation regarding the shapes of subgraphs induced by Hamiltonian paths in complete graphs. To see this observation for yourself, consider the set of subgraphs of K 4 induced by Hamiltonian paths which have an endpoint at the top left vertex: Notice that these subgraphs form one of just three distinct shapes-that is, any subgraph of K 4 induced by a Hamiltonian path is obtainable as a rotation or reflection of one of the following three graphs: The analogous observation in the case of K 5 yields eight of these shapes: Furthermore, the analogous observations regarding subgraphs induced by Hamiltonian cycles in K 4 and K 5 yield two and four of these shapes, respectively: The natural inclination at this point is to ask whether there are formulae for enumerating the distinct shapes formed by the subgraphs induced by Hamiltonian paths or cycles in K n . As it turns out, there are! However, before we may show you, we must first make this question more precise. Definition 1. Let P n and C n denote the sets of subgraphs of the complete graph K n which are induced by Hamiltonian paths or cycles, respectively. Define the following equivalence relations on P n and C n : 1. Two subgraphs G 1 , G 2 are said to be similar, denoted by G 1 ≡ S G 2 , if they are obtainable from one another by a rotation or reflection. 2. Two subgraphs G 1 , G 2 are said to be equivalent, denoted by G 1 ≡ E G 2 , if they are obtainable from one another by a rotation (but not a reflection). Example 2. . This definition allows us to state our problem as one of enumerating the equivalence classes of either P n or C n under either ≡ S or ≡ E , that is, we seek the sizes of the sets P n / ≡ S , P n / ≡ E , C n / ≡ S , and C n / ≡ E . The sizes of these sets are given by \|P n / ≡ S \| = 1 4 (n − 1)! + ( n 2 + 1)(n − 2)!!], if n is even; (n − 1)!!], if n is odd. , \|P n / ≡ E \| = 1 2 (n − 1)! + (n − 2)!!], if n is even; 0, if n is odd. , \|C n / ≡ S \| = 1 4n 2   d\|n φ n d 2 n d d d! + n!! n(n+6) 4 , if n is even; n 2 (n − 1)!!, if n is odd.   , \|C n / ≡ E \| = 1 2n 2   d\|n φ n d 2 n d d d! + n 2 n!!, if n is even; 0, if n is odd.   , where n!! denotes the product of n with every natural number less than n of the same parity as n, i.e., n!! = n(n − 2) · · · (2), if n is even; n(n − 2) · · · (3)(1), if n is odd. These formulae are proved in Theorems 6, 7, 8, and 9 respectively. Further, illustrations of P n / ≡ S and C n / ≡ S for 3 ≤ n ≤ 6 may be found in Figures 4 and 5, respectively. Finally, Table 1 gives the size of each of these sets for 3 ≤ n ≤ 10. 3 1 1 1 1 4 3 4 2 2 5 8 12 4 4 6 38 64 12 14 7 192 360 39 54 8 1320 2544 202 332 9 10176 20160 1219 2246 10 91296 181632 9468 18264 n \|P n / ≡ S \| \|P n / ≡ E \| \|C n / ≡ S \| \|C n / ≡ E \| Setup We obtain these answers by converting the original problem into one of enumerating the orbits of a specific group action. These orbits are enumerated by way of Burnside's lemma, which is a standard tool in the theory of finite group actions. Burnside's lemma. Consider a group G acting on a set A. For each g ∈ G, let fix(g) denote the set of elements of A which are fixed by g, i.e., fix(g) = {a ∈ A \| g · a = a }. Let A/G denote the set of orbits of this action. Then the number of orbits under the action of G on A is given by \|A/G\| = 1 \|G\| g∈G \|fix(g)\|. We first define a set of strings which will represent the elements of P n or C n . Definition 3. Fix a labelling of the vertices of K n with the setn = {0, 1, . . . , n − 1}, and let X n denote the set of n-length strings which are permutations of the elements ofn, i.e., X n = {x 0 x 1 · · · x n−1 \| x i ∈n and i = j ⇒ x i = x j } . Note that X n has n! elements. Next, we associate each string in X n with its interpretation as a graph in either P n or C n as follows. 1. (X n −→ P n ) Associate each string (x 0 x 1 · · · x n−1 ) ∈ X n with the subgraph of K n induced by the Hamiltonian path which traverses the vertices of K n in the order indicated by the string, i.e., the association is of the form (x 0 x 1 · · · x n−1 ) −→ x 0 → x 1 → · · · → x n−1 . This interpretation is illustrated by Figure 1. Notice that both a string and its reversal are mapped to the same subgraph in P n . Figure 1: Interpretations of strings in X n as graphs in P n . Figure 2: Interpretations of strings in X n as graphs in C n . 2. (X n −→ C n ) Associate each string (x 0 x 1 · · · x n−1 ) ∈ X n with the subgraph of K n induced by the Hamiltonian cycle which traverses the vertices of K n in the order indicated by the string, i.e., the association is of the form (x 0 x 1 · · · x n−1 ) −→ x 0 → x 1 → · · · → x n−1 → x 0 . This interpretation is illustrated by Figure 2. Notice that all cyclic permutations of a string as well as each of their reversals are mapped to the same subgraph in C n . Next, for each choice of P n or C n and ≡ S or ≡ E , we define a group to act on X n such that the orbits of this action will coincide with the desired equivalence classes. This group will be a direct product where the first coordinate acts purely on strings (i.e., an action of the first coordinate may only send strings to strings which have the same interpretation), while the second coordinate acts on a string's interpretation as a graph. Definition 4. 1. Define the following two groups which correspond to considering either P n or C n : S(P, n) = v \| v 2 = 1 and S(C, n) = v, c \| c n = v 2 = 1, vcv = c −1 . Note that S(P, n) is isomorphic to the cyclic group of order 2, and S(C, n) is isomorphic to the dihedral group of order 2n. 2. Define the following two groups which correspond to considering either ≡ S or ≡ E : G(≡ S , n) = r, s \| r n = s 2 = 1, srs = r −1 and G(≡ E , n) = r \| r n = 1 . Note that G(S, n) is isomorphic to the dihedral group of order 2n, and G(E, n) is isomorphic to the cyclic group of order n. 3. Finally, given a choice of α = P n , C n and a choice of β =≡ S , ≡ E , the acting group with respect to these choices is given by Γ(n, α, β) = S(α, n) × G(β, n). For example, the acting group for P n under ≡ S is Γ(n, P, ≡ S ) = S(P, n) × G(≡ S , n). Definition 5. The elements of Γ(n, α, β) act on strings in X n as follows: (c, 1) · (x 0 x 1 · · · x n−1 ) = (x 1 · · · x n−1 x 0 ), (v, 1) · (x 0 x 1 · · · x n−1 ) = (x n−1 · · · x 1 x 0 ), and (1, r) · (x 0 x 1 · · · x n−1 ) = (x 0 + 1) (x 1 + 1) · · · (x n−1 + 1), (1, s) · (x 0 x 1 · · · x n−1 ) = (−x 0 )(−x 1 ) · · · (−x n−1 ), where (x i + 1) denotes the sum (x i + 1) taken modulo n, and (−x i ) denotes the (additive) inverse of x i modulo n. The correct geometric interpretations of the actions of the second component are illustrated in Figure 3. This action has the following important properties: 1. Two strings in X n have the same interpretation as graphs in P n or C n if and only if they are contained in the same orbit under the action of S(P, n) or S(C, n) on X n , respectively. 2. If two strings in X n are contained in the same orbit under the action of G(≡ S , n) or G(≡ E , n), then their interpretations as graphs are similar or equivalent, respectively. Considering these properties in the context of Burnside's lemma yields the observation that the equivalence classes of P n or C n under ≡ S or ≡ E correspond bijectively to the orbits of X n under the appropriate acting group. In particular, we have the following: \|P n / ≡ S \| = \|X n /Γ(n, P, ≡ S )\| = 1 4n g∈Γ(n,P,≡ S ) \|fix(g)\|, \|P n / ≡ E \| = \|X n /Γ(n, P, ≡ E )\| = 1 2n g∈Γ(n,P,≡ E ) \|fix(g)\|, \|C n / ≡ S \| = \|X n /Γ(n, C, ≡ S )\| = 1 4n 2 g∈Γ(n,C,≡ S ) \|fix(g)\|, \|C n / ≡ E \| = \|X n /Γ(n, C, ≡ E )\| = 1 2n 2 g∈Γ(n,C,≡ E ) \|fix(g)\|. The path cases We begin by considering the cases involving P n . This is because these cases turn out to be considerably simpler than those involving C n , and thus they provide a suitable starting point for our investigation. We first enumerate the classes of P n / ≡ S , and the size of P n / ≡ E will follow as corollary. Theorem 6. Let n ≥ 3 be an integer. Then the number of equivalence classes of P n under ≡ S is given by \|P n / ≡ S \| = 1 4 (n − 1)! + ( n 2 + 1)(n − 2)!!], if n is even; (n − 1)!!], if n is odd. Proof. Notice that each element of Γ(n, P, ≡ S ) may be expressed in exactly one of the forms (1, r k ), (1, sr k ), (v, r k ), (v, sr k ) for some integer 0 ≤ k ≤ n − 1. Considering this fact in the context of Burnside's lemma yields the observation that \|X n /Γ(n, P, ≡ S )\| = 1 4n (A 1 + A 2 + A 3 + A 4 ), where A 1 = n−1 k=0 \|fix(1, r k )\|, A 2 = n−1 k=0 \|fix(1, sr k )\|, A 3 = n−1 k=0 \|fix(v, r k )\|, A 4 = n−1 k=0 \|fix(v, sr k )\|. We now evaluate each of these sums. 1. Clearly (1, r k ) will fix (x 0 x 1 · · · x n−1 ) only when k = 0. Hence A 1 = \|fix(1, 1)\| = n!. 2. For each 0 ≤ k ≤ n − 1, if the action of (1, sr k ) fixes the string (x 0 x 1 · · · x n−1 ), then we must have x i ≡ n −x i − k for all 0 ≤ i ≤ n − 1, and so 2x i + k ≡ n 0. But since there is some x j such that x j = 0, it follows that k = 0 and so sr k = s, which will clearly fix no strings. Thus A 2 = 0. 3. Notice that (v, r k ) will fix (x 0 x 2 · · · x n−1 ) if and only if x i ≡ n x −(i+1) + k and x −(i+1) ≡ n x i + k for all 0 ≤ i ≤ n − 1. This implies that x i ≡ n x i + 2k and thus that 2k ≡ n n. Hence n must be even and, since (v, 1) will clearly fix no strings, we have k = n/2. Now, x i ≡ n x −(i+1) +n/2 implies that x i −x −(i+1) ≡P ind = {0, n − 1} , {1, n − 2} , . . . , n 2 − 1, n 2 P lab = 0, n 2 , 1, n 2 + 1 , . . . , n 2 − 1, n − 1 ; we obtain 2 n/2 strings, for 2 n/2 · (n/2)! = n!! fixed strings in total; that is, we have A 3 = n!!, if n is even; 0, if n is odd. 4. Notice that (v, sr k ) will fix (x 0 x 1 · · · x n−1 ) if and only if x i ≡ n −(x −(i+1) + k) for all 0 ≤ i ≤ n − 1. First note that if k is even, then there is some entry x j such that x j ≡ n −k/2. Hence x j ≡ n −k/2 ≡ n −x −(j+1) − k, which implies that x j ≡ n x −(j+1) , and thus that n must be odd. We must consider the following cases. (a) If n is even and k is odd, then for each of n/2 possible values of k and each of the (n/2)! bijections between the sets of pairs of compatible indices and pairs of compatible labels: P ind = {0, n − 1} , {1, n − 2} , . . . , n 2 − 1, n 2 P (k) lab = n − k − 1 2 , n − k + 1 2 , . . . , {n − 1, n − k + 1} ; we have 2 n/2 fixed strings, for a total of (n/2)n!! fixed strings for this case. (b) If n is odd and k is even, we have 2xn−1 2 ≡ n −k, and thus xn−1 2 = n − k 2 . Hence our set of pairs of compatible indices is a pairing of the setn − n−1 2 , and our set of pairs of compatible labels is a pairing of the setn − n − k 2 . So for each even value of k and each of the ( n−1 2 )! bijections between fixed strings, again yielding (n − 1)!! fixed strings in total for this case. Hence over all n possible values of k we have a total of n(n − 1)!! fixed odd-length strings. P ind = {0, n − 1} , {1, n − 2} , . . . , n − 3 2 , n + 1 2 P (k) lab = {0, n − k} , . . . , n − k − 1 2 , n − k + 1 2 , {n − k + 1, n − 1} , . . . , n − 1 − k 2 , n + 1 − k 2 , Thus we obtain A 4 = ( n 2 )n!!, if n is even; n(n − 1)!!, if n is odd. Having evaluated each of these sums, the desired theorem now follows. Since Γ(n, P, ≡ E ) is a subgroup of Γ(n, P, ≡ S ), the number of equivalence classes of P n under ≡ E follows as an easy corollary. Theorem 7. Let n ≥ 3 be a integer. Then the number of equivalence classes of P n under ≡ E is given by \|P n / ≡ E \| = 1 2 (n − 1)! + (n − 2)!!], if n is even; 0, if n is odd. The cycle cases We now turn to the more difficult problem of enumerating equivalence classes of C n . As before, we begin by enumerating C n / ≡ S , and the size of C n / ≡ E will follow as corollary. Theorem 8. Let n ≥ 3 be an integer. Then the number of equivalence classes of C n under ≡ S is given by \|C n / ≡ S \| = 1 4n 2   d\|n φ n d 2 n d d d! + n!! n(n+6) 4 , if n is even; n 2 (n − 1)!!, if n is odd. \|fix(c m , sr k )\|, B 3 = n−1 k,m=0 \|fix(c m v, r k )\|, B 4 = n−1 k,m=0 \|fix(c m v, sr k )\|. As before, we proceed to evaluate each of these sums. 1. Notice that (c m , r k ) fixes (x 0 x 1 . . . x n−1 ) if and only if x i ≡ n x i+m +k for all 0 ≤ i ≤ n−1, and so x i ≡ n x i+ℓm + ℓk for all 0 ≤ i ≤ n − 1 and all ℓ ≥ 0. In particular, for ℓ = n gcd(n,m) , we must have that n \| ℓm and so x i = x i+ℓm . Hence x i ≡ n x i + ℓk and ℓk ≡ n 0; that is, n \| ℓk and thus gcd(n, m) \| k. Consequently, we have gcd(n, m) \| gcd(n, k). Similarly, for ℓ = n gcd(n,k) , we have n \| ℓm, implying that gcd(n, k) \| m and hence gcd(n, k) \| gcd(n, m). We conclude that gcd(n, k) = gcd(n, m) = d for some divisor d of n, and therefore Now, fix some particular k, m, d with d = gcd(n, k) = gcd(n, m). We seek to determine the size of fix(c m , r k ). Both of r k and c m have order n/d, and hence, if (c m , r k ) fixes (x 0 x 1 . . . x n−1 ), then x i ≡ n x i+ℓm + ℓk for all 0 ≤ i ≤ n − 1 and 0 ≤ ℓ ≤ n d − 1. Hence, each choice of label x i determines the labels of all positions of the form x i+ℓm for 0 ≤ ℓ ≤ n d − 1. Note that, for t ∈ {0, 1, . . . , n − 1} with gcd(t, n) = d and α ∈ {0, . . . , d − 1}, the elements of the set F (t) α = α + ℓt \| 0 ≤ ℓ ≤ n d − 1 are all congruent to α modulo d yet are all distinct modulo n [3]. Setting t = k, m, we see that the set {0, 1, . . . , n − 1} may be partitioned in two different ways via Π m and Π k , where Π m = 0, m, . . . , ( n d − 1)m , . . . , d − 1, d − 1 + m, . . . , 1 + d − ( n d − 1)m , and Π k = 0, k, . . . , ( n d − 1)k , . . . , d − 1, d − 1 + k, . . . , 1 + d − ( n d − 1)k . Consequently, the label of x i determines all labels with indices in the set F Combining this with (1), we obtain B 1 = n−1 k,m=1 \|fix(c m , r k )\| = d\|n gcd(k,n)=d gcd(m,n)=d \|fix(c m , r k ), = d\|n n d d d! gcd(k,n)=d gcd(m,n)=d 1, = d\|n φ n d · φ n d · n d d · d! , where φ denotes Euler's totient function. 2. Notice that (c m , sr k ) fixes (x 0 x 1 . . . x n−1 ) if and only if −x i+m − k ≡ n x i for all 0 ≤ i ≤ n − 1. Thus we have x i+m + x i ≡ n −k, and so x i+2m + x i+m ≡ n −k, which implies that x i ≡ n x i+m for all 0 ≤ i ≤ n − 1, and hence m = 0 or m = n/2. But if m = 0, then from the evaluation of A 2 above, no strings will be fixed, and so we must have m = n/2 and n must be even. Hence we have that x i + x i+ n 2 ≡ n −k. Note that k cannot be even, since, as n is even, −k would also be even, and so we would have x i = −k/2 = x i+ n 2 , which cannot be the case since c n 2 fixes no points. Thus k must be odd, and so for all n/2 odd choices of k we have (n/2)! bijections P ind = 0, n 2 , 1, n 2 + 1 , . . . , n 2 − 2, n − 2 , n 2 − 1, n − 1 P (k) lab = {0, −k} , {1, −(k + 1)} , . . . , k − 1 2 , k + 1 2 , each of which affords 2 (n/2) fixed strings. Therefore we obtain B 2 = n 2 n!!, if n is even; 0, if n is odd. 3. Notice that (c m v, r k ) fixes (x 0 x 1 . . . x n−1 ) if and only if x −(i+1)+m + k ≡ n x i for all 0 ≤ i ≤ n − 1,n 2 − 1, n 2 − 1 − k ; each of which, as before, yields 2 n 2 fixed strings, for a total of n!! fixed strings for each value of m. (a) If n is odd, then it is tedious but not difficult to see that an analogous argument to the evaluation of A 4 in the proof of Theorem 6 applies for all n values of m, yielding a total of n 2 (n − 1)!! fixed strings. Hence we obtain B 3 = (b) If n, m are both even, then it is again not difficult to see that an analogous argument to the evaluation of A 4 in the proof of Theorem 6 applies for all n/2 even values of m. That is, each even choice of m allows for n/2 odd values of k, each of which affords n!! fixed strings. (c) If n is even and m is odd, then there are exactly two indices 0 ≤ a, b ≤ n − 1 such that 2a ≡ n 2b ≡ n m + 1. It follows that m − a − 1 ≡ n a and m − b − 1 ≡ n b, and so −x a − k ≡ n x a and −x b − k ≡ n x b . Hence 2x a ≡ n 2x b ≡ n −k, and so k must be even. Now, for particular values of m, a, b, there are n/2 possible even values of k. Each choice of k fixes the values of {x a , x b }, which may be ordered in two ways; and, by the same methods as before, both choices of ordering afford (n − 2)!! fixed strings. Hence, in this case we have (n/2) · (n/2) · 2 · (n − 2)!! = (n/2)n!! fixed strings in total. Thus we obtain B 4 = ( n 2 + 1) n 2 n!!, if n is even; n 2 (n − 1)!!, if n is odd. Having evaluated each of these sums, the desired theorem now follows. As before, since Γ(n, C, ≡ E ) is a subgroup of Γ(n, C, ≡ S ), the number of equivalence classes of C n under ≡ E follows as an easy corollary.   Finally, it is worth noting a small corollary to the above theorems. Since φ(p) = (p − 1) for any prime p, we have the following. Here we note some interesting connections which the authors noticed over the course of writing this paper. After completing the enumeration of P n / ≡ S , we discovered that there are exactly as many of them as there are tone rows in n-tone music-the enumeration of which may be found in a paper of Reiner [2]. The corresponding OEIS sequence is sequence A099030-which, as has been noted, is identical to sequence A089066. Further, for reasons which should be clear, there are exactly as many classes in C n / ≡ S as there are classes of similar n-gons (that is, classes of n-gons which are equivalent up to rotations and reflections). These classes-as well as the analogous case of n-gons equivalent up to rotations only-were enumerated in a 1960 paper of Golomb and Welch [1]. As such, this paper provides an alternative proof of their result. The corresponding OEIS sequences are A000940 and A000939, respectively. It should also be noted that the evaluation of B 1 in the proof of Theorem 8 is in large part an adaptation of an argument of Moser [3]. In particular, the sum of Euler φ terms which makes an appearance in this paper as well as in the paper of Golomb and Welch [1] is the same as that which appears in the case of a = 1 in Moser's paper [3]. This connection is (as far as the authors are aware) not yet noted anywhere. Figure 3 : 3Geometric interpretation of the action of G(S, n) on strings in X n . n n/ 2 . 2Hence, for each of the (n/2)! bijections between the sets of compatible pairs of indices {i, −(i + 1)} and compatible pairs of labels x i , x −(i+1) : , yielding (n − 1)!! fixed strings in total for this case.(c) If n and k are both odd, as above we have 2xnset of pairs of compatible indices is again a pairing of the setn− n−1 2 , and our set of pairs of compatible labels is a pairing of the setn − n−k 2 . So, as above, for each odd value of k and each of the ( n−1 2 )! bijections between P ind = {0, n − 1} , {1, n − 2} , . . Notice that each element of Γ(n, C, ≡ S ) may be expressed in exactly one of the forms(c m , r k ), (c m , sr k ), (c m v, r k ), (c m v, sr k )for some integers 0 ≤ k, m ≤ n − 1. Considering this fact in the context of Burnside's lemma yields the observation that \|X n /Γ(n, C, ≡ S )\| = 1 4n 2 (B 1 + B 2 + B 3 + B 4 Figure 4 : 4Representatives from each class of P n / ≡ S for 3 ≤ n ≤ 6. Figure 5 : 5Representatives from each class of C n / ≡ S for 3 ≤ n ≤ 6. c m , r k )\|. In fact, we see that the labels of the initial substring (x 0 x 1 . . . x d−1 ) completely determine the labelling of the rest of the string.Hence, for each of the d! bijections between Π m and Π k , each of the (n/d) d possible sets of choices of labelling i → x i ∈ F (k) i determines a unique fixed string. That is, for each particular valid k, m, d, we have \|fix(c m , r k ) which is the case if and only if x −(i+1) + k ≡ n x i−m for all x i . There are three cases. (a) If n is odd, then for all values of m there exists a unique 0 ≤ a ≤ n − 1 such that −(a + 1) ≡ n a − m, and hence x −(a+1) = x a−m and consequently k = 0. But, since all other entries of the string are moved, it follows that no odd-length strings will be fixed by (c m v, r k ). (b) If n is even and m is odd, then there exist exactly two indices 0 ≤ a, b ≤ n − 1 such that −(a + 1) ≡ n a − m and −(b + 1) ≡ n b − m, and hence x −(a+1) = x a−m and x −(b+1) = x b−m . Then, just as above, we have k = 0 and thus no strings will be fixed. (c) If both n and m are even, then, since there must be some x i = 0, each of n/2 choices of m will fully determine the value of k. Hence, in the same manner as before, for each m we consider the (n/2)! bijections between P (m) ind = {0, n − 1 + m} , {1, n − 2 + m} , . . = {0, −k} , {1, 1 − k} , . . . , Notice that (c m v, sr k ) fixes (x 0 x 1 . . . x n−1 ) if and only if −(x −(i+1)+m + k) ≡ n x i for all 0 ≤ i ≤ n − 1. There are three cases. Theorem 9 . 9Let n ≥ 3 be an integer. Then the number of equivalence classes of C n under ≡ E is given by\|C n / ≡ E \| !!, if n is even; 0,if n is odd. Corollary 10 . 10Let p > 2 be prime. Then\|C p / ≡ S \| = 1 4p (p − 1) 2 + p(p − 1)!! + (p − 1)! , and \|C p / ≡ E \| = 1 2p [(p − 1) 2 + (p − 1)!]. Table 1 : 1Table of values for 3 ≤ n ≤ 10. AcknowledgementsThe authors would like to thank Chris Kottke for providing valuable feedback on several early versions this paper. We would also like to thank the anonymous reviewer for his or her constructive suggestions, many of which were incorporated into this final version. Finally, the first author would like to thank Nika Sigua for his role in the initial discovery of the problems discussed in this paper. On the enumeration of polygons. S W Golomb, L R Welch, Amer. Math. Monthly. 67S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly 67 (1960), 349-353. Enumeration in music theory. D L Reiner, Amer. Math. Monthly. 92D. L. Reiner, Enumeration in music theory, Amer. Math. Monthly 92 (1985), 51-54. A (modest) generalization of the theorems of Wilson and Fermat. W O J Moser, Canad. Math. Bull. 33W. O. J. Moser, A (modest) generalization of the theorems of Wilson and Fermat, Canad. Math. Bull. 33 (1990), 253-256. Mathematics Subject Classification: Primary 05C30; Secondary 05E18. Keywords: Hamiltonian Path, equivalence class, group action, Burnside's lemma. Mathematics Subject Classification: Primary 05C30; Secondary 05E18. Keywords: Hamiltonian Path, equivalence class, group action, Burnside's lemma.	[]
[ "Evidence of Coulomb interaction induced Lifshitz transition and robust hybrid Weyl semimetal in T d MoTe 2", "Evidence of Coulomb interaction induced Lifshitz transition and robust hybrid Weyl semimetal in T d MoTe 2" ]	[ "N Xu \nInstitute of Advanced Studies\nWuhan University\n430072WuhanChina\n", "Z W Wang \nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n", "A Magrez \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n", "P Bugnon \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n", "H Berger \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n", "C E Matt \nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n\nLaboratory for Solid State Physics\nETH Zürich\nCH-8093ZürichSwitzerland\n", "V N Strocov \nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n", "N C Plumb \nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n", "M Radovic \nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n", "E Pomjakushina \nLaboratory for Developments and Methods\nPaul Scherrer Institut\nCH-5232VilligenSwitzerland\n", "K Conder \nLaboratory for Developments and Methods\nPaul Scherrer Institut\nCH-5232VilligenSwitzerland\n", "J H Dil \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n\nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n", "J Mesot \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n\nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n\nLaboratory for Solid State Physics\nETH Zürich\nCH-8093ZürichSwitzerland\n", "R Yu \nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n", "H Ding [email protected] \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\nBeijingChina\n", "M Shi \nSwiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland\n" ]	[ "Institute of Advanced Studies\nWuhan University\n430072WuhanChina", "School of Physics and Technology\nWuhan University\n430072WuhanChina", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Laboratory for Solid State Physics\nETH Zürich\nCH-8093ZürichSwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Laboratory for Developments and Methods\nPaul Scherrer Institut\nCH-5232VilligenSwitzerland", "Laboratory for Developments and Methods\nPaul Scherrer Institut\nCH-5232VilligenSwitzerland", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland", "Laboratory for Solid State Physics\nETH Zürich\nCH-8093ZürichSwitzerland", "School of Physics and Technology\nWuhan University\n430072WuhanChina", "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina", "Collaborative Innovation Center of Quantum Matter\nBeijingChina", "Swiss Light Source\nPaul Scherrer Institut\nCH-5232Villigen PSISwitzerland" ]	[]	Using soft x-ray angle-resolved photoemission spectroscopy we probed the bulk electronic structure of T d MoTe 2 . We found that on-site Coulomb interaction leads to a Lifshitz transition, which is essential for a precise description of the electronic structure. A hybrid Weyl semimetal state with a pair of energy bands touching at both type-I and type-II Weyl nodes is indicated by comparing the experimental data with theoretical calculations. Unveiling the importance of Coulomb interaction opens up a new route to comprehend the unique properties of MoTe 2 , and is significant for understanding the interplay between correlation effects, strong spin-orbit coupling and superconductivity in this van der Waals material.	10.1103/physrevlett.121.136401	[ "https://arxiv.org/pdf/1808.08816v1.pdf" ]	52,973,548	1808.08816	36dc57d4196295edfe992255870e68c2e4ba7d1b	Evidence of Coulomb interaction induced Lifshitz transition and robust hybrid Weyl semimetal in T d MoTe 2 N Xu Institute of Advanced Studies Wuhan University 430072WuhanChina Z W Wang School of Physics and Technology Wuhan University 430072WuhanChina A Magrez Institute of Physics École Polytechnique Fédérale de Lausanne CH-1015LausanneSwitzerland P Bugnon Institute of Physics École Polytechnique Fédérale de Lausanne CH-1015LausanneSwitzerland H Berger Institute of Physics École Polytechnique Fédérale de Lausanne CH-1015LausanneSwitzerland C E Matt Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland Laboratory for Solid State Physics ETH Zürich CH-8093ZürichSwitzerland V N Strocov Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland N C Plumb Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland M Radovic Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland E Pomjakushina Laboratory for Developments and Methods Paul Scherrer Institut CH-5232VilligenSwitzerland K Conder Laboratory for Developments and Methods Paul Scherrer Institut CH-5232VilligenSwitzerland J H Dil Institute of Physics École Polytechnique Fédérale de Lausanne CH-1015LausanneSwitzerland Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland J Mesot Institute of Physics École Polytechnique Fédérale de Lausanne CH-1015LausanneSwitzerland Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland Laboratory for Solid State Physics ETH Zürich CH-8093ZürichSwitzerland R Yu School of Physics and Technology Wuhan University 430072WuhanChina H Ding [email protected] Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina Collaborative Innovation Center of Quantum Matter BeijingChina M Shi Swiss Light Source Paul Scherrer Institut CH-5232Villigen PSISwitzerland Evidence of Coulomb interaction induced Lifshitz transition and robust hybrid Weyl semimetal in T d MoTe 2 Page 1 Using soft x-ray angle-resolved photoemission spectroscopy we probed the bulk electronic structure of T d MoTe 2 . We found that on-site Coulomb interaction leads to a Lifshitz transition, which is essential for a precise description of the electronic structure. A hybrid Weyl semimetal state with a pair of energy bands touching at both type-I and type-II Weyl nodes is indicated by comparing the experimental data with theoretical calculations. Unveiling the importance of Coulomb interaction opens up a new route to comprehend the unique properties of MoTe 2 , and is significant for understanding the interplay between correlation effects, strong spin-orbit coupling and superconductivity in this van der Waals material. Page 2 The T d phase of MoTe 2 belongs to the materials family of two-dimensional transition metal dichalcogenides (2D TMDs), which exhibit a wide variety of novel physical properties and have thus been of increasing interest recently [1][2]. These compounds are formed by layers with strong covalent intra-layer bonds, but adjoining layers are coupled by weak van der Waals' forces. This remarkable anisotropy makes it possible to exfoliate crystals down to a single layer with potential applications for nanoscale electronics [3]. Distinct from its counterpart 2H MoTe 2 and other TMDs in hexagonal structure, T d MoTe 2 (in the manuscript only this structural polytype is discussed), forms an orthorhombic lattice with a non-centrosymmetric unit cell containing two MoTe 2 layers (Fig. 1a). Furthermore, it shows no semiconducting properties but instead behaves as a semimetal with large and unsaturated magnetoresistance [4]. It has been intensively studied as a potential candidate for a quantum spin Hall insulator in the 2D limit [2]. Recently, density functional theory (DFT) calculations [5][6][7] predicted bulk MoTe 2 to be a promising candidate for a type-II Weyl semimetal (WSM). This prediction is supported by some spectroscopic evidences [8][9][10]. In contrast to the iso-structural WTe 2 , MoTe 2 undergoes a structural transition from T d to monoclinic 1T' above 240 K, and an additionally the structural transition from 2H to 1T' can be driven by laser illumination or electrostatic doping [11][12][13][14]. Furthermore, among all the T d phase TMDs, a superconducting transition at ambient pressure has only been observed in MoTe 2 [15]. These exotic properties in this 2D material make MoTe 2 as a promising material candidate for nano-electronics and topological quantum device applications. Although many novel features have been revealed in MoTe 2 , the underlying bulk electronic structure which plays a decisive role in the transport and topological properties, has still not been directly determined from experimental investigations. Due to the large lattice constant along the c axis (Fig. 1a), the Brillouin zone (BZ) is tiny along k z (π/c = 0.226 Å -1 ) (Fig. 1b) and requires an experimental probe having high k z resolution to resolve the three-dimensional electronic structure. Here, using soft X-ray angle-resolved photoemission spectroscopy (SX-ARPES) with increased k z resolution [16], we present a comprehensive study of the bulk electronic structure of MoTe 2 . The determined Fermi surface (FS) and band structure clearly show a periodicity along the k z direction, indicating coherent hopping of electrons between the layers similar to WTe 2 [17]. In contrast to previous studies (reproduced in Fig. 1c) Page 3 [6][7], we demonstrate that the inclusion of on-site Coulomb interactions in the DFT calculations is essential for the correct description of the band structure and FS topology. This is especially clear for the (N+1)th band (black lines in Fig. 1c-e) around the Y(T) point, which is expected to form type-II Weyl nodes with the Nth band (brown lines in Fig. 1c-d) according to DFT calculations [6][7]. Our SX-ARPES and DFT+U results further suggest that correlation effects can lead to a stable hybrid Weyl semimetal state, where both type-I and type-II Weyl nodes are formed by the same pair of bands near the chemical potential [18]. High quality 1T' single-crystals were produced by chemical vapor transport (CVT). A sealed ampule containing high purity Mo, Te and Iodine, used as transport agent, was placed in a horizontal furnace with the reaction zone at 1000°C and growth zone at 940°C. Large and high quality 1T'-MoTe 2 single crystals were obtained after a few weeks. The chemical stoichiometry was measured by x-ray fluorescence with a 30 micrometer spot size. The T d -phase is obtained when cooling the 1T'-MoTe 2 below 240 K. Clean (001) surfaces of T d -MoTe 2 were prepared for SX-ARPES measurements by cleaving the samples in situ in a vacuum better than 5 × 10 -11 Torr and at a temperature lower than 20 K. Bulk-sensitive SX-ARPES measurements were performed at the Advanced Resonant Spectroscopies beam line (ADRESS) at SLS [19] using a hemispherical electrostatic spectrometer manufactured by the Specs GmbH. observed electronic states. However, the periodicity in k z is 4π/c which is twice of that in the bulk BZ along the k z direction (2π/c). The constant energy map at binding energy (E B ) of 0.5 eV (Fig. 2b) shows also a 4π/c periodicity in the k z direction. This phenomenon is due to matrix element effects in the non-symmorphic crystal symmetry which causes the measured ARPES intensity to depend on the BZ number [22][23][24]. The 2π/c periodicity can be revealed by measuring at k x = 0 and k y = 2π/b, and the observed valence band dispersion along Γ-Z-Γ can be well reproduced by the DFT calculation with spin-orbit coupling and onsite Coulomb interaction (U eff = 2.4 eV) taken into account (Fig. 2c). To further explore the bulk electronic structure of MoTe 2 , we acquired SX-ARPES data in the k z = 0 plane with photon energy hν = 600 eV. The determined bulk FSs consist of a pair of banana-shaped hole-like pockets (α) slightly off the Γ-Y symmetry line and a pair of electron-like pockets (β) located farther from the center in the form of smaller crescent-shape facing the α pockets (Fig. 3a). The DFT (green lines) and DFT+U (red lines) calculations produce similar band dispersions along the Γ-X direction, and both of them are consistent with the experimental results (Fig. 3d). On the other hand, DFT and DFT+U methods give qualitatively different band dispersions along the Γ-Y (Fig. 3e), leading to a change of Fermi surfaces topology near the Y point (Fig. 3c,e). In the DFT calculations (green lines in Fig. 3e), two electron-like bands with tiny spin splitting (γ and δ) cross E F around the Y(T) point, forming almost 2D FSs with no dispersive feature along the Y-T direction. However, both the γ and δ electron pockets are absent in the ARPES spectra ( Fig. 3a and e). The acquired SX-ARPES spectra in a large number of BZ along the Y-T direction show no band within 0.5 eV below E F (Fig. 3e), thus excluding the existence of the γ and δ pockets. We have also collected SX-ARPES data in a large momentum space, over several Brillouin zones along both in-plane and out-of-plane directions, and found no sign of the electron-like pockets around the Y(T) point. We noticed that the band bottoms of the β and γ bands occur at almost the same energy positions in DFT calculations (Fig. 3d-e). The absence of the γ and δ electron pockets cannot result from any possible doping, because the β pockets are clearly observed in Fig. 3a and d. To obtain insight into the qualitative difference between the experimentally determined band structure and the one from DFT calculations, we performed electronic structure calculations on MoTe 2 using the DFT+U method to include Page 5 on-site correlations. The most pronounced effect of U eff is that the electron-like bands near the Y(T) point are pushed above E F , resulting in the absence of the γ and δ pockets (Fig. 3b, e), which is consistent with our ARPES results. In normal DFT an unrealistic lattice expansion of 5% is needed to push the states above the Fermi level at the Y(T) point, simultaneously completely altering the dispersion close to the Γ-point (Fig. S1 in Supplementary Material), thus ruling out a lattice effect as a possible explanation for our observations. Further theoretical analysis shows that, for the γ pocket, a Lifshitz transition takes place at a critical value of U eff ∼ 1.8 eV (Fig. 1d). We further demonstrate that band structure calculations using the The on-site Coulomb interaction in MoTe 2 , which was rarely taken into account in previous theoretical consideration, has a striking influence on its topological properties, as summarized in Fig. 4a-d. In the DFT+U calculation with U eff = 2.4 eV and lattice parameters from Ref. [7], the band minimum of the (N+1)th band near the Y(T) point is shifted to above E F . The predicted eight type-II Weyl nodes (open circles in Fig. 4b) in the non-interacting calculations [6], which are located at two different energies in the k z = 0 plane and are formed by the (N+1)th and Nth bands, transform into four Weyl nodes of a single type (W1) near E F in the k z = 0 plane (solid circles in Fig. 4c). Furthermore, the Weyl cone of W1 has a type-I like dispersion, i.e. the Fermi velocities of the crossing bands have opposite sign in any momentum plane that cuts the Weyl cones and contains Weyl points. As shown in Fig. 4e,g,h, the band dispersions passing through W1 from the DFT+U calculation show an overall consistency with the ARPES results along the k x , k y and k z directions. We note that W1 is very close to the critical point between type-I and type-II Weyl nodes ( Fig. Page 6 4e,f), which can, mathematically, be considered as an analogue of a black hole horizon [25]. Interestingly, the Nth and (N+1)th bands from the DFT+U calculation touch again and form eight type-II Weyl nodes (W2) in the k z = ±0.08 Å -1 plane (open circles in Fig. 4c). The band structure around W2 corresponds to type-II Weyl cone dispersions with a node at 12 meV below E F (Fig. 4i-j). The good agreement between the ARPES data and the band structure from the DFT+U calculation ( Fig. 4i-l) strongly suggests that MoTe 2 contains type-II Weyl nodes off the k z = 0 plane, along with type-I Weyl nodes in the k z = 0 plane. The combined ARPES and DFT+U results suggest a novel topological phase exists in MoTe 2 , -i.e. a hybrid Weyl semimetal phase induced by electron correlation in which the same conduction and valence bands form both type-I and type-II Weyl nodes near E F (Fig. 4d). Such a hybrid WSM is predicted to show a unique Landau-level structure and quantum oscillations [18]. We also note that the Weyl nodes with opposite chirality are well separated, which makes the correlated hybrid Weyl semimetal state robust against lattice perturbations. We further theoretically examine the topological properties of MoTe 2 using another set of lattice parameters that were experimentally determined at low temperature. A previous DFT calculations suggests that a 0.3% smaller lattice constant a could induce a topological phase transition from eight type-II Weyl nodes at two different binding energies (open circles in Fig. 4b) to four type-II Weyl nodes at a single energy (open circles in Fig. 4a) [7]. In contrast, our DFT+U calculation shows that the topological properties of Ref. [6] and lattice 2 from Ref. [7]). c, The same as b, but calculated using the DFT method. d, ARPES spectrum near E F along the -X direction. The overlaid solid lines are the energy bands calculated by using the DFT+U (green) and DFT (red) methods, respectively. e, Same as d, but along the -Y direction. f, ARPES spectrum along T-Y direction, taken with photon energies covering a k z range of over 26π/c. Data were collected using circular-polarized light with an overall energy/angular resolution in the range of 50-80 meV/0.1° at T ~ 10 K. Electronic structure calculations were performed based on DFT method. The electron-electron interaction between Mo 4d electrons is simulated through Dudarev's method by setting effective Coulomb interaction U eff = (U-J) in the DFT+U scheme, where the Hund's coupling J = 0.4 eV is taken as the typical value for Mo-related compounds [20-21]. Slightly changing of U-J value is not changing the conclusion of this work. A tight-binding model based on the maximally localized Wannier functions method has been constructed in order to investigate the Weyl points of T d MoTe 2 . Figure 2a 2ashows the ARPES spectra in the k x -k z plane, acquired from the cleaved (001) surface of MoTe 2 with the photon-energy in the range of 350-650 eV. The periodic variation of the Fermi momentum in k z confirms the bulk origin of the Page 4 hybrid-functional method (HSE06) or van der Waals density functional method cannot explain the experimentally observed bulk FS configuration changes. The band dispersions and FS topology around the Γ point from different functionals would deviate from the SX-ARPES results if the Liftshiz transition had occurred near the Y point (Figs. S2-S3 in Supplementary Material). The good consistency between the SX-ARPES spectra and the band structure calculated using the DFT+U method with U eff = 2.4 eV strongly indicates that the on-site Coulomb interaction plays a crucial role in influencing both the electronic structure near the chemical potential and the FS topology. The U eff value of 2.4 eV is also the typical value for other Mo-related compounds in previous studies[20][21]. FiguresFigure 1 .Figure 2 . 12MoTe 2 are insensitive to the 0.3% variation of the lattices constant, i.e. the number and types of Weyl nodes are unchanged, and the locations of Weyl nodes are almost at the same binding energies. The hybrid Weyl semimetal state occurs for U eff larger than 2 eV, which is found to reproduce the experimentally observed Lifshitz transition of FS near the Y(T) point, and it persists for values of U eff up to 3 eV. This covers the common U eff values usually used in Mo and other 4d electron systems. In summary, we demonstrated that electron correlation plays a crucial role in the description of the band structure and FS in T d MoTe 2 . Our theoretical analysis shows that a Lifshitz transition occurs upon increasing the on-site U eff over a threshold of ∼ 1.8 eV. The bulk band structure measured by SX-ARPES measurements agrees well with the DFT+U calculations using U eff = 2.4 eV. Our combined results indicate that a Page 7 novel topological phase, the hybrid Weyl semimetal state, could emerge from the van der Waals material MoTe 2 . The hybrid Weyl semimetal state induced by electron correlation is robust against disorder and lattice perturbations. MoTe 2 with correlated 4d electrons provides a versatile platform for studying the fundamental physics of novel topological phases and the interplay between spin-obit coupling, Coulomb interaction and superconductivity. Correlation induced Lifshitz transition in MoTe 2 . a, Crystal structure of the orthorhombic (T d ) phase of MoTe 2 . b, Bulk Brillouin zone of MoTe 2 . c-e, Calculated band dispersions using the DFT and DFT+U methods with U eff = 1.8 eV and 3 eV, respectively. The crossings of Nth band (brown line) and (N+1)th band (black line) form type-II Weyl nodes in DFT calculations [Bulk electronic structure of MoTe 2 . a, Photoemission intensity of the bulk electronic states as a function of energy relative to the Fermi level in the k x -k z plane, obtained from ARPES measurements in the photon-energy range of 350 -650 eV. b,Constant energy map at 0.5 eV below E F in the k x -k z plane. c, ARPES intensity plot along the -Z direction, taken with photon energies covering a k z range over 10 BZs.The DFT+U calculated bands are overlaid for a direct comparison.Page 12 Figure 3 . 3Effect of Coulomb interaction on electronic structure of MoTe 2 . a, FS intensity map in the k x -k y plane with k z = 0, acquired with hν = 600 eV in the ARPES measurements. b, Corresponding spectral functions from bulk band calculations using the DFT+U method. To illustrate the effect of a slight variation of the lattice parameters we plot corresponding FSs for two sets of lattice constants (lattice 1 from Figure 4 . 4Evolution of topological properties of MoTe 2 induced by Coulomb interaction. a-b, Illustration of Type-II Weyl nodes in MoTe 2 obtained from the DFT calculations using the lattice constants from Ref. [6] and [7], respectively. c, Weyl nodes in MoTe 2 obtained from DFT+U calculations. Solid and open circles indicate type-I and type-II Weyl nodes, respectively. The blue and red colours represent different chirality. d, Band structure passing through both W1 and W2 from DFT+U calculations, which shows the hybrid Weyl semimetal state in MoTe 2 . The labeled points on the horizontal axis are A = (-0.36, 0.13, 0.08), B = (-0.18, 0.13, 0.08) and C = (-0.18, -0.01, -0.08) in the k-space with the unit of Å -1 . e, The ARPES spectrum passing through W1 and along the k x direction. 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[ "UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION", "UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION", "UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION", "UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION" ]	[ "Michael J Curran ", "Michael J Curran " ]	[]	[]	We establish upper bounds for the joint moments of the 2k th power of the Riemann zeta function with the 2h th power of its derivative for 0 ≤ h ≤ 1 and 1 ≤ k ≤ 2. These bounds are expected to be sharp based upon predictions from random matrix theory.	10.4064/aa220127-11-4	[ "https://arxiv.org/pdf/2106.00165v1.pdf" ]	235,265,915	2106.00165	3849e29cd0758274f8b99515a69a99715a726528	UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION 1 Jun 2021 Michael J Curran UPPER BOUNDS FOR FRACTIONAL JOINT MOMENTS OF THE RIEMANN ZETA FUNCTION 1 Jun 2021 We establish upper bounds for the joint moments of the 2k th power of the Riemann zeta function with the 2h th power of its derivative for 0 ≤ h ≤ 1 and 1 ≤ k ≤ 2. These bounds are expected to be sharp based upon predictions from random matrix theory. INTRODUCTION In the past two decades, conjectural connections between the zeros of the Riemann zeta function ζ(s) and eigenvalues of random unitary matrices have led to many interesting developments in understanding the moments of the zeta function. In the recent random matrix theory literature, there has been a fair bit of interest in understanding the joint moments of the characteristic polynomial of a random unitary matrix with its derivative. In this paper, the primary objects are the joint moments of ζ(s), given by I T (k, h) = 2T T \|ζ( 1 2 + it)\| 2k−2h ζ ′ ( 1 2 + it) 2h dt, as well as the joint moments of the Hardy Z function J T (k, h) = 2T T \|Z(t)\| 2k−2h \|Z ′ (t)\| 2h dt, where Z(t) = π −it/2 Γ 1 4 + it 2 Γ 1 4 + it 2 ζ( 1 2 + it). Note in particular that \|Z(t)\| = \|ζ( 1 2 + it)\|, and that Z(t) is real valued for t ∈ R. The work of Keating and Snaith [22,23], Hughes [20], and Hall [19] has led to the conjecture that whenever k > − 1 2 and − 1 2 < h ≤ k + 1 2 I T (k, h) ∼ C ζ (k, h)T (log T ) k 2 +2h , J T (h, k) ∼ C Z (k, h)T (log T ) k 2 +2h(1) for a certain constants C ζ (k, h), C Z (k, h) as T → ∞. There are conjectured values for the constants C Z (k, h) for general real h, k, but values for C ζ (k, h) are only conjectured for integral h, k. In both cases, the constants split as a product of an arithmetic factor and a random matrix factor. The arithmetic factor is a well understood product over primes. The random matrix factor has many different expressions including combinatorial sums [13,14,20], a multiple contour integral in the case h = k [12], and a determinant of Bessel functions [2,12]. For h, k not necessarily equal, the random matrix factor can be solved for finite N and is related to the solution of a Painlevé V type differential equation [4]. Furthermore, the limit as N → ∞ is related to the solution of a certain Painlevé III equation [1,2,4,15]. Previously the asymptotics (1) were known for h, k ∈ {0, 1, 2} with h ≤ k due to Ingham [16] and Conrey [8], and upper bounds of the right order were only known for half integer valued h, k ≤ 2 due to work of Conrey [8] and Conrey Ghosh [11]. The aim of this paper is to establish upper bounds for I T (k, h) and J T (k, h) of the right order in a larger range of h and k. Theorem 1. Let 1 ≤ k ≤ 2 and 0 ≤ h ≤ 1. Then for large T I T (k, h) ≪ T (log T ) k 2 +2h , and the same bound holds for J T (k, h). The proof we give is based on the work of Heap, Radziwiłł and Soundararajan [18] which in turn is based on the method introduced in Radziwiłł and Soundararajan [26]. The general principle in these works is that if one can compute the 2k th moment of a given L-function twisted by an arbitrary Dirichlet polynomial then one can find upper bounds of the right order for all of its lower order moments. In particular, this approach is used to prove Theorem 1 in the case h = 0. We combine the ideas of the paper [18] with twisted joint moment calculations to deduce Theorem 1 in the case of h = 1 and then deduce the result from Hölder's inequality-the bounds we obtain are of the right order since the exponent of log T in (1) is linear in h. We are forced to take k ∈ [1, 2] because 2k − 2h is only nonnegative when k ≥ 1 at the boundary case h = 1. It is likely that one could establish sharp bounds on I T (k, h) and J T (k, h) in the full range k > − 1 2 and − 1 2 < h ≤ k + 1 2 assuming the Riemann hypothesis. ACKNOWLEDGEMENTS The author would like to thank his supervisor Jonathan P. Keating for introducing him to this problem and for his encouragement. OUTLINE OF THE PROOF We will deduce Theorem 1 from the following. Proposition 1. Let T be large and 1 ≤ k ≤ 2. Then 2T T \|ζ( 1 2 + it)\| 2k−2 \|ζ ′ ( 1 2 + it)\| 2 dt ≪ T (log T ) k 2 +2 , and the same bound holds when ζ( 1 2 + it) is replaced by Z(t). Proof of Theorem 1. Recall theorem 1 of [18] gives for 0 ≤ k ≤ 2 2T T \|ζ( 1 2 + it)\| 2k dt ≪ T (log T ) k 2 . Therefore by Hölder's inequality with p = 1 h and q = 1 1−h , this estimate and Theorem 1 give I T (k, h) ≤ 2T T \|ζ( 1 2 + it)\| 2k−2 \|ζ ′ ( 1 2 + it)\| 2 dt h 2T T \|ζ( 1 2 + it)\| 2k dt 1−h ≪ T (log T ) k 2 +2h . The case of the joint moments of Z(t) is similar since \|Z(t)\| = \|ζ( 1 2 + it)\|. To prove Proposition 1, we will approximate the logarithm of ζ(s) by a truncated sum over primes p≤X p −s . Following the works [17,25,27], we will break up this sum into increments which have progressively smaller variance. This in turn allows us to work with a Dirichlet polynomial of length T θ for some small but fixed θ > 0, which is long enough to give a good enough approximation of ζ(s). We follow the notation introduced in [18]. Denote by log j the j-fold iterated logarithm, and take ℓ to be the largest integer so that log ℓ T ≥ 10 4 . Now define a sequence T j for 1 ≤ j ≤ ℓ by T 1 = e 2 and T j = exp log T (log j T ) 2 for 2 ≤ j ≤ ℓ, and for 2 ≤ j ≤ ℓ and s ∈ C set P j (s) = T j−1 ≤p<T j 1 p s , and P j = T j−1 ≤p<T j 1 p . The hope is then that on average log ζ(s) will be controlled by the sum of the increments P j (s), where P j is the variance of the j th increment on the half line. By Merten's second estimate, note that P j ∼ 2 log j T − 2 log j+1 T. Next define for 2 ≤ j ≤ ℓ the truncated Taylor expansion N j (s; α) = T j−1 ≤p<T j Ω(n)≤500P j α Ω(n) g(n) n s where g is the multiplicative function given by g( p m ) = 1/m! on prime powers. So for most t ∈ [T, 2T ] we expect 2≤j≤ℓ N j ( 1 2 + it; α) to behave similarly to ζ( 1 2 + it) α . Now each N j is a Dirichlet polynomial of length at most T 500P j j so 2≤j≤ℓ N j ( 1 2 + it; α) is a Dirichlet polynomial of length at most T 1/10 , which is amenable to analysis. We will deduce Proposition 1 in two steps. First we bound the integrand by a product of integral powers of ζ and ζ ′ with short Dirichlet polynomials. Proposition 2. For 1 ≤ k ≤ 2 and s = 1 2 + it with t ∈ R \|ζ(s)\| 2k−2 \|ζ ′ (s)\| 2 ≤ 2k\|ζ(s)\| 2 \|ζ ′ (s)\| 2 2≤j≤ℓ \|N j (s; k − 2)\| 2 + (4 − 2k)\|ζ ′ (s)\| 2 2≤j≤ℓ \|N j (s; k − 1)\| 2 + 2≤v≤ℓ 2k\|ζ(s)\| 2 \|ζ ′ (s)\| 2 2≤j<v \|N j (s; k − 2)\| 2 + (4 − 2k)\|ζ ′ (s)\| 2 2≤j≤ℓ \|N j (s; k − 1)\| 2 P v (s) 50P v 2⌈50Pv ⌉ . The same bound holds when ζ(s) is replaced by Z(t). The proof of Proposition 2 is almost identical to the proof of proposition 1 in [18], so it is omitted. The only difference is that one uses the conjugate exponents p = 1 k−1 and q = 1 2−k , and then one multiplies the resulting inequality by \|ζ ′ (s)\| 2 or \|Z ′ (t)\| 2 . This reduces the proof of Proposition 1 to the calculation of two types of twisted moments. Proposition 3. For 1 ≤ k ≤ 2 2T T \|ζ ′ ( 1 2 + it)\| 2 2≤j≤ℓ \|N j ( 1 2 + it; k − 1)\| 2 dt ≪ T (log T ) k 2 +2(2) and for 2 ≤ j ≤ ℓ and 0 ≤ r ≤ 2⌈50P v ⌉ 2T T \|ζ ′ ( 1 2 + it)\| 2 2≤j<v \|N j ( 1 2 + it; k − 1)\| 2 \|P v ( 1 2 + it)\| 2r dt (3) ≪ T (log T ) 3 (log T v−1 ) k 2 −1 (2 r r!P r v exp(P v )) , and the same bounds hold when ζ( 1 2 + it) is replaced by Z(t). Proposition 4. For 1 ≤ k ≤ 2 2T T \|ζ( 1 2 + it)\| 2 \|ζ ′ ( 1 2 + it)\| 2 2≤j≤ℓ \|N j ( 1 2 + it; k − 2)\| 2 dt ≪ T (log T ) k 2 +2(4) and for 2 ≤ v ≤ ℓ and 0 ≤ r ≤ 2⌈50P v ⌉ 2T T \|ζ( 1 2 + it)\| 2 \|ζ ′ ( 1 2 + it)\| 2 2≤j<v \|N j ( 1 2 + it; k − 2)\| 2 \|P v ( 1 2 + it)\| 2r dt (5) ≪ T (log T ) 6 (log T v−1 ) k 2 −4 (18 r r!P r v exp(P v )) , and the same bounds hold when ζ( 1 2 + it) is replaced by Z(t). We will derive estimates for general twisted joint moments of ζ in the following section, and then use these estimates to prove Propositions 3 and 4 in the final section. Before we undertake this, let us see how these estimates imply Proposition 1. Proof of Proposition 1. Our estimates give 2T T \|ζ( 1 2 + it)\| 2k−2 \|ζ ′ ( 1 2 + it)\| 2 dt ≪ T (log T ) k 2 +2 + 2≤v≤ℓ T (log T v−1 ) k 2 +2 log T log T v−1 3 2 ⌈50Pv⌉ ⌈50P v ⌉!P ⌈50Pv ⌉ v exp(P v ) (50P v ) 2⌈50Pv ⌉ + log T log T v−1 6 18 ⌈50Pv⌉ ⌈50P v ⌉!P ⌈50Pv⌉ v exp(P v ) (50P v ) 2⌈50Pv ⌉ ≪ T (log T ) k 2 +2 , where the final bound follows by the same reasoning as [18]. The conclusion for the Z function is the same. TWISTED MOMENT FORMULAE We will derive the necessary twisted joint moment formulae from formulae for twisted moments of ζ(s) with small shifts off of the critical line. Fortunately there are many known formulae for computing twisted moments of ζ due to connections with the proportion of zeros of ζ lying on the critical line [7,24]. Then following work of Young [28], we can differentiate these formulae with respect to the shifts to obtained the desired twisted joint moments. The formula in [28] is valid for Dirichlet polynomials of length T 1/2−ε , and we note that work of Bettin Chandee and Radziwiłł [6] provides asymptotics for the twisted second moment without shifts for any Dirichlet polynomial of length at most T 17/33−ε . The twisted fourth moment formula we use was first proven by Hughes and Young [21] for Dirichlet polynomials of length at most T 1/11−ε , which was later increased to T 1/4−ε by Bettin Bui Li and Radziwiłł [5]. Following these works, we will bound the desired twisted moments by introducing a smooth cutoff. Going forward, we fix a smooth φ : R → R such that supp φ ⊂ [3/4, 9/4] and φ(t) = 1 for all t ∈ [0, 1]. Lemma 1. Given a Dirichlet polynomial A(s) = h≤T θ a h h s with θ < 1/2, if F (z 1 , z 2 ) = h,k≤T θ a h a k [h, k] (h, k) z 1 +z 2 h z 1 k z 2 , then I (1) (T ) := R \|ζ ′ ( 1 2 + it)\| 2 \|A( 1 2 + it)\| 2 φ(t/T )dt ≪ T (log T ) 3 max \|z j \|=3 j / log T \|F (z 1 , z 2 )\|, and the same bound holds when ζ( 1 2 + it) is replaced by Z(t). Proof. Let α, β ∈ C have modulus less than 1/ log T . Then by [28] we may write I T (α, β) := 2T T ζ( 1 2 + α + it)ζ( 1 2 + β + it)\|A( 1 2 + it)\| 2 dt = h,k≤T θ a h a k [h, k] R (h, k) α+β h α k β ζ(1 + α + β)+ t 2π −α−β (h, k) −α−β h −α k −β ζ(1 − α − β) φ(t/T )dt + O(T 1−δ ) for some δ > 0, which is holomorphic in α, β sufficiently small. We may express the main term as a multiple contour integral around α and β: by lemma 2.5.1 of [10] and a shift of contours we find I T (α, β) = − 1 (2πi) 2 \|z 2 −β\|=9/ log T \|z 1 −α\|=3/ log T F (z 1 , −z 2 ) ζ(1 + z 1 − z 2 )(z 1 − z 2 ) 2 (z 1 − α)(z 1 + β)(z 2 − α)(z 2 + β) ×   R t 2π z 1 −z 2 −β−α 2 φ(t/T )dt   dz 1 dz 2 + O(T 1−δ ) = − 1 (2πi) 2 \|z 2 \|=9/ log T \|z 1 \|=3/ log T F (z 1 , −z 2 ) ζ(1 + z 1 − z 2 )(z 1 − z 2 ) 2 (z 1 − α)(z 1 + β)(z 2 − α)(z 2 + β) ×   R t 2π z 1 −z 2 −β−α 2 φ(t/T )dt   dz 1 dz 2 + O(T 1−δ ). Note we do not cross any poles when shifting contours since \|α\|, \|β\| < 1/ log T . Now since I T (α, β) is holomorphic with respect to small α and β, as in [28] the derivatives of I T (α, β) with respect to α and β can be obtained via Cauchy's theorem as contour integrals along circles of radii ≍ 1/ log T . Since the error term holds uniformly on these contours, we conclude I T (α, β) := 2T T ζ ′ ( 1 2 + α + it)ζ ′ ( 1 2 + β + it)\|A( 1 2 + it)\| 2 dt = d dα d dβ − 1 (2πi) 2 \|z 2 \|=9/ log T \|z 1 \|=3/ log T F (z 1 , −z 2 ) ζ(1 + z 1 − z 2 )(z 1 − z 2 ) 2 (z 1 − α)(z 1 + β)(z 2 − α)(z 2 + β) ×   R t 2π z 1 −z 2 −β−α 2 φ(t/T )dt   dz 1 dz 2 + O(T 1−δ ). To compute I (1) (T ), we evaluate these derivatives and then set α = β = 0, obtaining I (1) (T ) = 1 (2πi) 2 \|z 2 \|=9/ log T \|z 1 \|=3/ log T F (z 1 , −z 2 )ζ(1 + z 1 − z 2 )(z 1 − z 2 ) 2 R z 1 + z 2 + z 1 z 2 2 log t 2π × z 1 + z 2 − z 1 z 2 2 log t 2π t 2π z 1 −z 2 2 φ(t/T )dt dz 1 z 4 1 dz 2 z 4 2 + O(T 1−δ ). Finally, since \|z j \| = 3 j / log T and supp φ ⊂ [3/4, 9/4], notice that ζ(1 + z 1 − z 2 ) ≪ log T, (z 1 − z 2 ) 2 ≪ (log T ) −2 , and R z 1 + z 2 + z 1 z 2 2 log t 2π z 1 + z 2 − z 1 z 2 2 log t 2π t 2π z 1 −z 2 2 φ(t/T )dt ≪ T (log T ) −2 , so the claim now follows. The case for twisted moments of Z is similar. The main difference is that applying lemma 2.5.1 of [10] gives up to a power savings the simpler formula − 1 (2πi) 2 \|z 2 \|=9/ log T \|z 1 \|=3/ log T F (z 1 , −z 2 ) ζ(1 + z 1 − z 2 )(z 1 − z 2 ) 2 (z 1 − α)(z 1 + β)(z 2 − α)(z 2 + β) ×   R t 2π z 1 −z 2 2 φ(t/T )dt   dz 1 dz 2 . Then differentiating with respect to α and β and setting the shifts to zero we obtain 1 (2πi) 2 \|z 2 \|=9/ log T \|z 1 \|=3/ log T F (z 1 , −z 2 )ζ(1 + z 1 − z 2 )(z 2 1 − z 2 2 ) 2   R t 2π z 1 −z 2 2 φ(t/T )dt   dz 1 z 4 1 dz 2 z 4 2 , which satisfies the same bound. Lemma 2. Given a Dirichlet polynomial A(s) = h≤T θ a h h s with θ < 1/4, if G(z 1 , z 2 , z 3 , z 4 ) = h,k≤T θ a h a k [h, k] B z 1 ,z 2 ,z 3 ,z 4 h (h, k) B z 3 ,z 4 ,z 1 ,z 2 k (h, k) , where B z 1 ,z 2 ,z 3 ,z 4 (n) = p m n j≥0 σ z 1 ,z 2 (p j+m )σ z 3 ,z 4 (p j ) p j j≥0 σ z 1 ,z 2 (p j )σ z 3 ,z 4 (p j ) p j −1 and σ z 1 ,z 2 (n) = ab=n a −z 1 b −z 2 , then I (2) (T ) := R \|ζ( 1 2 +it)\| 2 \|ζ ′ ( 1 2 +it)\| 2 \|A( 1 2 +it)\| 2 φ(t/T )dt ≪ T (log T ) 6 max \|z j \|=3 j / log T \|G(z 1 , z 2 , z 3 , z 4 )\|. The same bound holds when ζ( 1 2 + it) is replaced by Z(t). Proof. This is similar to the proof of Lemma 1. Using the twisted 4th moment formula with shifts in [5] and lemma 2.5.1 of [10], we can write up to a power savings in T I T (α, β) = R ζ( 1 2 + α + it)ζ( 1 2 + it)ζ( 1 2 + β − it)ζ( 1 2 − it)\|A( 1 2 + it)\| 2 φ(t/T )dt = 1 4(2πi) 4 \|z j \|=3 j / log T A(z 1 , z 2 , −z 3 , −z 4 )G(z 1 , z 2 , −z 3 , −z 4 )∆(z 1 , z 2 , z 3 , z 4 ) 2 ×   R t 2π z 1 +z 2 −z 3 −z 4 −α−β 2 φ(t/T )dt   4 m=1 dz m z 2 m (z m − α)(z m + β) , where ∆(z 1 , z 2 , z 3 , z 4 ) = 1≤j<k≤4 (z k − z j ) is the Vandermonde determinant and A(z 1 , z 2 , z 3 , z 4 ) = ζ(1 + z 1 + z 3 )ζ(1 + z 1 + z 4 )ζ(1 + z 2 + z 3 )ζ(1 + z 2 + z 4 ) ζ(2 + z 1 + z 2 + z 3 + z 4 ) . Now differentiating with respect to α and β and then setting α = β = 0 gives after some algebraic manipulation I (2) (T ) = 1 4(2πi) 4 \|z j \|=3 j / log T A(z 1 , z 2 , −z 3 , −z 4 )G(z 1 , z 2 , −z 3 , −z 4 )∆(z 1 , z 2 , z 3 , z 4 ) 2 × R z 2 1 z 2 2 z 2 3 z 2 4 log t 2π 2 − (z 1 z 2 z 3 + z 1 z 2 z 4 + z 1 z 3 z 4 + z 2 z 3 z 4 ) 2 × t 2π z 1 +z 2 −z 3 −z 4 2 φ(t/T )dt 4 m=1 dz m z 6 m . Now to deduce the claim, notice that A(z 1 , z 2 , −z 3 , −z 4 ) ≪ (log T ) 4 , ∆(z 1 , z 2 , z 3 , z 4 ) 2 ≪ (log T ) −12 , R z 2 1 z 2 2 z 2 3 z 2 4 log t 2π 2 − (z 1 z 2 z 3 + z 1 z 2 z 4 + z 1 z 3 z 4 + z 2 z 3 z 4 ) 2 × t 2π z 1 +z 2 −z 3 −z 4 2 φ(t/T )dt ≪ T (log T ) −6 for \|z j \| = 3 j / log T and t ∈ [3T /4, 9T /4]. As in the previous proof, the analysis for the Z function is simpler, and the same bound holds. PROOF OF PROPOSITIONS 3 AND 4 The proofs of Propositions 3 and 4 are straightforward modifications of the proof of proposition 3 in [18]. In fact we will see Proposition 4 is an immediate consequence of Lemma 2 and a bound for G(z 1 , z 2 , z 3 , z 4 ) proven in [18]. This will then conclude the proof of Theorem 1. Proof of Proposition 3. We will apply Lemma 1 to the Dirichlet polynomials 2≤j≤ℓ N j (s; k − 1) and 2≤j<v N j ( 1 2 + it; k − 1) P v ( 1 2 + it) r . By multiplicativity, it suffices to bound the sums Now we handle the sums (7). Write p\|mn⇒T j−1 ≤p<T j Ω(n)=Ω(n)=r r! 2 g(n)g(m) [n, m] ≤ r! 2 r j=0 p\|d⇒T j−1 ≤p<T j Ω(d)=j 1 d p\|n⇒T j−1 ≤p<T j Ω(n)=r−j g(nd) n 2 . Now using that g(nd) ≤ g(n), we may further bound this by r! 2 r j=0 1 j! P j v 1 (r − j)! P r−j v = r!P r v r j=0 r j P r−j v (r − j)! ≤ 2 r r!P r v exp(P v ). The claim now readily follows by Lemma 1. Proof of Proposition 3. This is a direct consequence of Lemma 2 and proposition 3 of [18], where it is shown in the first case that max \|z j \|=3 j / log T \|G(z 1 , z 2 , z 3 , z 4 )\| ≪ T (log T ) k 2 −4 , and in the second case that max \|z j \|=3 j / log T \|G(z 1 , z 2 , z 3 , z 4 )\| ≪ (log T v−1 ) k 2 −4 (18 r r!P r v exp(P v )) . p\|m,n⇒T j−1 ≤p<T j Ω(n),Ω(n)≤500P j (k − 1) Ω(n)+Ω(m) g(n)g(m)[n, m] · (m, n) z 1 +z 2 m z 1 n z 2andIn both cases, we will estimate (m, n) z 1 +z 2 m z 1 n z 2 ≪ 1, which holds under the assumptions \|z j \| ≤ 9/ log T and m, n ≤ T 1/10 .First we handle(6). Using Rankin's trick, we may drop the condition Ω(n), Ω(n) ≤ 500P j incurring an error term of at mostTherefore by Lemma 1 and Merten's third estimate we conclude the integral in(2)is T Assiotis, J P Keating, J Warren, arXiv:2005.13961On the joint moments of the characteristic polynomials of random unitary matrices. T. Assiotis, J. P. Keating, J. 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[ "Solar System Experiments and the Interpretation of Saa's Model of Gravity with Propagating Torsion as a Theory with Variable Plank \"Constant\"", "Solar System Experiments and the Interpretation of Saa's Model of Gravity with Propagating Torsion as a Theory with Variable Plank \"Constant\"" ]	[ "P Fiziev [email protected] ", "S Yazadjiev ", "\nDepartment of Theoretical Physics\nFaculty of Physics\nDepartment of Theoretical Physics\nFaculty of Physics\nBogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research\nSofia University\n5 James Bourchier Boulevard141980, 1164Dubna, Moscow Region, SofiaRussia, Bulgaria\n", "\nSofia Univer-sity\n5 James Bourchier Boulevard1164SofiaBulgaria\n" ]	[ "Department of Theoretical Physics\nFaculty of Physics\nDepartment of Theoretical Physics\nFaculty of Physics\nBogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research\nSofia University\n5 James Bourchier Boulevard141980, 1164Dubna, Moscow Region, SofiaRussia, Bulgaria", "Sofia Univer-sity\n5 James Bourchier Boulevard1164SofiaBulgaria" ]	[]	It is shown that the recently proposed interpretation of the transposed equi-affine theory of gravity as a theory with variable Plank "constant" is inconsistent with basic solar system gravitational experiments. * 1 We use the Schouten's normalization conventions[7]which differs from the original ones in[1]-[5].	10.1142/s0217732399000560	[ "https://arxiv.org/pdf/gr-qc/9807025v2.pdf" ]	10,001,466	gr-qc/9807025	aecaecdcfe6808d3f6283d74553a6d791e61f9c3	Solar System Experiments and the Interpretation of Saa's Model of Gravity with Propagating Torsion as a Theory with Variable Plank "Constant" Jul 1998 January 2, 2018 P Fiziev [email protected] S Yazadjiev Department of Theoretical Physics Faculty of Physics Department of Theoretical Physics Faculty of Physics Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research Sofia University 5 James Bourchier Boulevard141980, 1164Dubna, Moscow Region, SofiaRussia, Bulgaria Sofia Univer-sity 5 James Bourchier Boulevard1164SofiaBulgaria Solar System Experiments and the Interpretation of Saa's Model of Gravity with Propagating Torsion as a Theory with Variable Plank "Constant" Jul 1998 January 2, 2018arXiv:gr-qc/9807025v2 19 It is shown that the recently proposed interpretation of the transposed equi-affine theory of gravity as a theory with variable Plank "constant" is inconsistent with basic solar system gravitational experiments. * 1 We use the Schouten's normalization conventions[7]which differs from the original ones in[1]-[5]. Recently a new model of gravity involving propagating torsion was proposed by A. Saa [1]- [5]. In this model a special type of Einstein-Cartan geometry is considered in which the usual volume element √ −g d 4 x is replaced with new one: e −3Θ √ −g d 4 x -covariantly constant with respect to the transposed affine connection ∇ T , hence the name transposed-equi-affine theory of gravity [6]. As a result the torsion vector S α = S αβ α turns to be potential: S α = ∂ α Θ, Θ being its scalar potential 1 . Because of the exponential factor e −3Θ in the volume element Saa's model has a very important feature: it leads to a consistent application of the minimal coupling principle both in the action principle and in the equations of motion for all matter fields. These equations are of autoparallel type and may be derived via the standard action principle for a nonstandard action integral: A tot = A G + A M F = 1 c L G e −3Θ √ −gd 4 x + 1 c L M F e −3Θ √ −gd 4 x,(1) where L G = − c 2 2κ R is the lagrangian of the geometrical fields: the metric g αβ , and the torsion S αβ γ , R being the Cartan scalar curvature, c being the speed of light, and L M F is the usual lagrangian of the corresponding matter fields: scalar fields φ(x), spinor fields ψ(x), electromagnetic fields A α (x), Yang-Mills fields A α (x), e.t.c. But this property not held for the equations of motion of classical particles and fluids which turn to be of geodesic type [6]. Most probably this inconsistency leads to the negative result obtained in [8]: Saa's model is inconsistent with the basic solar system gravitational experiments. Then having in mind to preserve the good features of Saa's model and in the same time somehow to avoid this problem we are forced to try some further modifications of the model. The simplest one is to use the Saa's modification of the volume element only in the action integrals like (1) and the usual volume element in all other physical, or geometrical formulae [6]. This leads to the action for a classical spinles particle in a form: A m = −mc e −3Θ ds(2) where m is the rest mass of the particle and ds = g αβ dx α dx β is the usual fourdimensional interval. The corresponding action integral for spinless fluid (See for details [6]) is: A µ = 1 c L µ e −3Θ √ −gd 4 x = − 1 c (µc 2 + µΠ) e −3Θ √ −gd 4 x,(3) µ(x) being fluid's density, Π being the elastic potential energy of the fluid. This situation calls for a new curious interpretation of the torsion potential Θ as a quantity which describes the space-time variations of the Plank "constant" according to the lawh (x) =h ∞ e 3Θ(x) ,(4) h ∞ being the Plank constant in vacuum far from matter. Indeed, according to the first principles we actually need lagrangians and action integrals to write down the quantum transition amplitude in a form of Feynman path integral on the histories of all fields and particles. In the variant of the theory under consideration it has the form: D g αβ (x), S αβ γ (x), φ(x), ψ(x), A α (x), A α (x), ...; x(t), ... exp 1 h ∞ d 4 xe −3Θ(x) (L G + L M F ) − mc e −3Θ ds .(5) Now it is obvious that the very Plank constanth may be included in the factor e 3Θ(x) , but more important is the observation that we must do this, because the presence of this uniform factor in the formula (5) means that we actually introduce a local Plank "constant" at each point of the space-time. Indeed, if the geometric field Θ(x) changes slowly in a cosmic scales, then in the framework of the small domain of the laboratory we will see an effective "constant": h(x) ≈ h ∞ e 3Θ(x laboratory ) = const =h. In presence of spinless matter only an Einstein-Cartan geometry with semisymmetric torsion tensor S αβ γ = S [α δ γ β] appears and the following equations for geometrical fields are obtained G µν + ∇ µ ∇ ν Θ − g µν ✷Θ = κ c 2 ((ε + p)u µ u ν − pg µν ) ∇ σ S σ = ✷Θ = − 2κ c 2 (ε + 3p)(6) where G µν is Einstein tensor with respect to the affine connection ∇ µ , κ being the Einstein gravitational constant, ε, p and u µ are the energy density, pressure and four velocity of the relativistic perfect fluid [6]. Using the standard variational principle for the action (3) one can obtain the equations of motion for the perfect fluid: (ε + p)u β ∇ β u α = δ β α − u α u β ∇ β p + F α(7) where F α = −2(ε + p) δ β α − u α u β ∇ β Θ is the torsion force, as defined in [6]. This nonzero value of the torsion force shows that in the present model with variable Plank "constant" (VPC model) the matter equations of motion are not of autoparallel, nor of geodesic type in contrast to all equations for matter fields which are of autoparallel type. This inconsistency of the model is not enough to reject it immediately as far as the very requirement for all dynamical equations in theory to be of the same type is not founded on a well established principle, nevertheless it seems to be necessary for validity of the corresponding generalization of the equivalence principle in spaces with torsion [9]. The main purpose of this letter is to investigate the consistency of the VPC model with basic solar-system experimental facts. To do this we have to consider the motion of a test particle in presence of a metric and torsion fields. The standard variation of the action (2) yields the equations of motion we need, but it's more convenient to investigate directly the corresponding Hamilton-Jacoby equation: g µν ∂ µ S∂ ν S = mce −3Θ 2 .(8) The conform transformation g µν → * g µν = e −6Θ g µν yields the effective metric * g µν and the following form of the equation (8) * g µν ∂ µ S∂ ν S = m 2 c 2(9) which is well known from general relativity. Thus we may consider the motion of a test particles precisely as in general relativity working with the metric * g µν . Therefore the simplest way to compare the predictions of the VPC model with the experimental facts is to consider post-Newtonian expansion of the metric * g µν in vacuum in vicinity of a star like the Sun. The asymptotically flat, static and spherically symmetric general solution of the equations (6) for geometric fields in vacuum is known [11], [12]. In isotropic coordinates it's given by a two-parameter -(r 0 , k) family of solutions ds 2 = 1 − r 0 r 1 + r 0 r 2 ρ(k) (c dt) 2 − 1 − r 2 0 r 2 2 1 − r 0 r 1 + r 0 r 2 ρ(k) (3k−1) dr 2 + r 2 dΩ 2 , (10) Θ = k 2 ν (11) where ρ(k) = 3 k − 1 2 2 + 1 4 . In the VPC model under consideration the whole geometry (metric and torsion) causes a gravitational force (of pure geometrical nature). The parameter k presents the ratio of the torsion part of this force and its metric part. In the case when k = 0 we have the usual torsionless Schwarzshild's solution and r g ≡ 4r 0 is the standard gravitational radius. From equations (11) we obtain the effective metric * g µν and the effective fourinterval d * s 2 = 1 − r 0 r 1 + r 0 r 2 ρ(k) (1−3k) (c dt) 2 − 1 − r 2 0 r 2 2 1 − r 0 r 1 + r 0 r −2 ρ(k) dr 2 + r 2 dΩ 2 . (12) The asymptotic expansion of the metric in (12) at r → ∞ gives d * s 2 ≈ 1 − 4r 0 (1 − 3k) ρ(k)r + 8r 2 0 (1 − 3k) 2 ρ(k) 2 r 2 (c dt) 2 − 1 + 4r 0 ρ(k)r dr 2 + r 2 dΩ 2 .(13) In the asymptotic region r → ∞ we must have Newtonian gravity. Consequently the mass "seen" by the test particles is M = 2r 0 (1 − 3k) ρ(k) .(14) Therefore we may represent the effective four-interval in the asymptotic form d * s 2 ≈ 1 − 2M r + 2M 2 r (c dt) 2 − 1 + 1 1 − 3k 2M r dr 2 + r 2 dΩ 2 .(15) From the above expression it immediately follows that two of post-Newtonian parameters corresponding to the effective metric * g µν are * β= 1, * γ = 1 1 − 3k .(16) As it's well known, solar system gravitational experiments set tight constrains on post-Newtonian parameters [14]: \| * β −1 \|< 1 * 10 −3 , \| * γ −1 \|< 2 * 10 −3 .(17) Therefore, to avoid contradictions with the basic experimental facts we must have 3k 1 − 3k < 2 * 10 −3 .(18) In order to specify the theoretically possible values of k we must investigate a model of a star. As a simplest basic model we may consider a static spherically symmetric star. Putting the metric in the standard form ds 2 = e ν (c dt) 2 − e λ dr 2 − r 2 (dθ 2 + sin 2 (θ)dϕ 2 ) we obtain from the general field equations (6), (7) the following complete system of ordinary differential equations for the star's fluid equilibrium ξ ′ + 2 r ξ + 2ξ − λ ′ 2 ξ − 3S r ξ = −(ε + 3p)e λ S ′ r + 2 r S r + 2ξ − λ ′ 2 S r − 3S 2 r = −(ε + 3p)e λ e λ − 1 + r 2 (2ξ − λ ′ ) = −3rS r + 2(ε + 2p)r 2 e λ ξ ′ − λ ′ 2 ξ + ξ 2 − λ ′ r = 3S ′ r − 3 2 λ ′ S r + 3S 2 r − 2(ε + 2p)e λ p ′ = −(ε + p)(ξ − 3S r ) p = p(ε)(19) where ξ = 1 2 ν ′ , S r = Θ ′ , p = p(ε) is the matter state equation, ε and p are the energy density and the pressure. The prime denotes differentiation with respect to r. The regular at the center of the star (r = 0) solution corresponds to the initial conditions [13]: ξ(0) = 0, S r (0) = 0. As we see the first two equations of the system (19) coincide. Then by virtue of the same initial conditions for ξ and S r we obtain equal solutions ξ = S r . Hence, in VPC model the only possible value of the parameter k is k = 1. This means that in this model the torsion part of gravitational force equals to the metric one in magnitude. As a consequence it is impossible to fulfill the condition (18). Moreover, if r 0 > 0 the value k = 1 leads to a negative mass of the star (See equation (14)). This result shows that the interpretation of the Saa's model as a theory with variable Plank's constant is inconsistent with the well known solar system gravitational experiments. AcknowledgmentsThis work has been partially supported by the Sofia University Foundation for Scientific Researches, Contract No. 245/98, and by the Bulgarian National Foundation for Scientific Researches, Contract F610/98. One of us (PF) is grateful to the leadership of the Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia for hospitality and working conditions during his stay there in the summer of 1998. . A Saa, Gen. Rel. and Grav. 29205A. Saa, Gen. Rel. and Grav. 29, 205, (1997). . A Saa, Mod. Phys. Lett. A8. 2565A. Saa, Mod. Phys. Lett. A8, 2565, (1993). . A Saa, Mod. Phys. Lett. A9. 971A. Saa, Mod. Phys. Lett. A9, 971, (1994). . A Saa, Class. Quant. Grav. 12A. Saa, Class. Quant. Grav. 12, L85, (1995). . A Saa, J. Geom. and Phys. 15A. Saa, J. Geom. and Phys. 15, 102, (1995). Spinless matter in Transposed-Equi-Affine theory of gravity, Eprint gr-qc/9712004, to appear in the journal General Relativity and Gravitation. P Fiziev, P. Fiziev, Spinless matter in Transposed-Equi-Affine theory of gravity, E- print gr-qc/9712004, to appear in the journal General Relativity and Gravi- tation. J A Schouten, Tensor Analysis for Physicists. OxfordClarendon PressJ. A. Schouten, Tensor Analysis for Physicists, (Clarendon Press, Oxford, 1954); . Ricci-Calculus, SpringerBerlinRicci-Calculus (Springer, Berlin, 1954). Solar-system experiments and Saa's Model of gravity with Propagating Torsion. P Fiziev, S Yazadjiev, gr-qc/9806063E-printP. Fiziev, S. Yazadjiev, Solar-system experiments and Saa's Model of gravity with Propagating Torsion, E-print gr-qc/9806063. P Fiziev, the Proceedings of the 2-nd Bulgarian Workshop "New Trends in Quantum Field Theory. A. Ganchev, R.Kerner, I.TodorovRazlog; Sofia, 248Heron Press Science SeriesOn the Action Principle in Affine Flat Spaces with TorsionP. Fiziev, On the Action Principle in Affine Flat Spaces with Torsion, in the Proceedings of the 2-nd Bulgarian Workshop "New Trends in Quantum Field Theory", Razlog, 1995, editors A. Ganchev, R.Kerner, I.Todorov, Heron Press Science Series, Sofia, 248, (1996); . E-Print, gr-qc/9712002E-print: gr-qc/9712002. . C Brans, R H Dicke, Phys. Rev. 124925C. Brans, R. H. Dicke, Phys. Rev. 124, 925 (1961). . C Brans, Phys. Rev. 1252194C. Brans, Phys. Rev. 125, 2194 (1961). . B C Xanthopoulos, T , Zannias Phys. Rev. 402564B. C. Xanthopoulos, T. Zannias Phys. Rev. D40, 2564 (1989). Yazadjiev Neutron star in presence of torsiondilaton field. T Boyadjiev, P Fiziev, S , gr-qc/9803084E-printT. Boyadjiev, P. Fiziev, S. Yazadjiev Neutron star in presence of torsion- dilaton field, E-print gr-qc/9803084. . C M Will, Theory and Experiment in Gravitational Physics. Cambridge University PressC. M. Will, Theory and Experiment in Gravitational Physics, (Cambridge University Press, 1993).	[]
[ "Forces on an exterior algebra bundle", "Forces on an exterior algebra bundle" ]	[ "Jason Hanson " ]	[]	[]	In a previous article, the exterior algebra bundle over spacetime was used as a common geometric framework for obtaining the Dirac and Einstein equations, with other forces incorporated using minimal coupling. Here the fundamental forces that are allowed within this framework are explicitly enumerated.	null	[ "https://arxiv.org/pdf/2202.00527v1.pdf" ]	246,442,159	2202.00527	8c2b48265f97dae6f0e9a035dd70445bc9c45e24	Forces on an exterior algebra bundle 30 Jan 2022 February 2, 2022 Jason Hanson Forces on an exterior algebra bundle 30 Jan 2022 February 2, 2022 In a previous article, the exterior algebra bundle over spacetime was used as a common geometric framework for obtaining the Dirac and Einstein equations, with other forces incorporated using minimal coupling. Here the fundamental forces that are allowed within this framework are explicitly enumerated. Introduction In the article [3], we used the exterior algebra bundle * M over a spacetime manifold M as a geometric framework for incorporating both the Dirac and Einstein equations. We also indicated briefly how other forces can be introduced using minimal coupling, namely by using a connection on the exterior algebra bundle of the form ∇ α = ∂ α +Γ α + θ α . HereΓ α is the torsion-free metric-compatible connection on M extended to * M, and θ α is a collection of four 16 × 16 matrices that encode the forces. These matrices must satisfy certain constraints in order for the variational principle to yield the Dirac equation γ α ∇ α ψ = mψ. Such a collection θ = {θ 0 , θ 1 , θ 2 , θ 3 } is a tensor, which we call a force tensor -although it represents a potential rather than a force. In this article, we describe the distinct types of forces that are allowed through minimal coupling. This is done by considering the action of the Lorentz group on the space of all possible force tensors V . Under this action, V is a representation of the Lorentz group. And as such, it can be decomposed into irreducible subrepresentations. Each irreducible subrepresentation corresponds to a distinct force. Article summary. In the remainder of this section, we review the relevant constructions used in the geometric framework introduced in [3], as well as the necessary constraints on a force tensor θ. In section 2, we use geodesic normal coordinates to find a basis for the space of all possible force tensors, and in section 3 we write down the irreducible subspaces. Each irreducible subspace is identified with a type of fundamental force field on spacetime M: scalar, vector, anti-symmetric tensor, symmetric tensor, and Fierz tensor. In section 4, we use the curvature of the connection ∇ α to obtain field equations for each of the fundamental forces in a flat spacetime. Geometric framework Suppose M is a Lorentz manifold with metric g. Let x = (x α ), with α = 0, 1, 2, 3, be local coordinates for M, and let e α . = ∂ α = ∂/∂x α denote the corresponding basis vectors of the tangent bundle of M at x. In this basis, g αβ denotes the components of g, and g αβ denotes the components of g −1 . We assume that g has signature (+, −, −, −, −). Exterior algebra bundle. The exterior algebra bundle * M is formed by taking the complex exterior algebra of each fiber of the tangent bundle. We may choose e ∅ , e 0 , e 1 , e 2 , e 3 , e 01 , e 02 , e 03 , e 12 , e 13 , e 23 , e 012 , e 013 , e 023 , e 123 , e 0123 as basis vectors for the fibers of * M. Here e I . = e α 1 ∧ e α 2 ∧ · · · ∧ e α k for the multi-index I = α 1 α 2 · · · α k . We denote the length of I by \|I\|; i.e., \|I\| = k. A field ψ is a section M → * M, and we write ψ = ψ I e I , where I = ∅, 0, 1, . . . , 0123. We useψ I to denote the complex conjugate of ψ I . Gamma matrices. In addition to the exterior product e α ∧ ψ, we have the interior product ι α ψ, which is the linear operator defined by the rules (i) ι α e β = g αβ e ∅ and (ii) ι α (e I ∧ e J ) = (ι α e I ) ∧ e J + (−1) \|I\| e I ∧ (ι α e J ). The gamma matrix γ α is then defined to be the linear operator γ α ψ . = e α ψ − ι α ψ. We find that γ α γ β + γ β γ α = −2g αβ . As usual, we set γ α . = g αβ γ β . Extended metric. The spacetime metric g is extended to a Hermitian metricĝ on * M by settingĝ(ψ, φ) =ψ Iĝ IJ ψ J . Hereĝ IJ = 0 if \|I\| = \|J\|, andĝ IJ = det(g α i β j ) if I = α 1 · · · α k and J = β 1 · · · β k with k = \|I\| = \|J\|. One shows that γ † αĝ +ĝγ α = 0. Extended group and algebra actions. A local transformation A on the tangent bundle of M extends to a transformationÂ on * M via the rule Â e α 1 ···α k . = (Ae α 1 ) ∧ · · · ∧ (Ae α k ). In particular, a change of basis B for the tangent bundle of M, e ′ α = B −1 e α , extends to a change of basisB on * M. If G is a Lie group acting locally on the tangent bundle, then we can extend the action of G to * M by using this rule. On the other hand, the Lie algebra g of G extends to a Lie algebra action via the rulê ae α 1 ···α k = k i=1 e α 1 ∧ · · · ∧ e α i−1 ∧ (ae α i ) ∧ e α i+1 ∧ · · · ∧ e α k(1) for all a ∈ g. Note thatâe ∅ = 0. Allowed forces We use minimal coupling to model forces. That is, we assume there is a connection ∇ on * M, where ∇ α ψ = ∂ α ψ + C α ψ for some collection of four 16 × 16 matrices C α . However, we need to ensure that we obtain the Dirac equation Dψ = mψ, where Dψ . = γ α ∇ α ψ, by varying the Lagrangian density L K = ( 1 2 ψ †ĝ Dψ + c.c. − mψ †ĝ ψ) ω, ω . = − det(g),(2) with respect toψ. As observed in [3], the conditions C † αĝ +ĝC α = ∂ αĝ and [γ α , C β ] = ∂ β γ α + Γ α βǫ γ ǫ ,(3) where Γ α βǫ are the Christoffel symbols for the metric connection on M, are sufficient to guarantee this. The metric connection on M extends to a connection on * M via the Leibniz rule. Moreover, the extended metric connection matricesΓ α satisfy both conditions in equation (3). So to incorporate forces other than gravity, we look for connection matrices of the form C α =Γ α + θ α , with θ † αĝ +ĝθ α = 0 and [γ α , θ β ] = 0.(4) We call any collection θ α of 16 × 16 matrices that satisfy (4) a force tensor. We write θ to denote the collection {θ 0 , θ 1 , θ 2 , θ 3 }. By general principles, under a local change of basis B for the tangent bundle of M, the connection matrices C α transform as C ′ α = (B −1 ) β α (−∂ βB +BC β )B −1 . The extended metric connectionΓ α satisfies this equation, so a force tensor must satisfy the transformation rule θ ′ α = (B −1 ) β αB θ βB −1 . (5) 2 The space of force tensors The set V (g) of all force tensors is a real vector space. We will determine a basis in the case of the metric g = η on Minkowski space M, where η . = diag(1, −1, −1, −1). For a multi-index I, let Θ I denote the 16 × 16 matrix that commutes with gamma matrices and such that Θ I e ∅ = e I . We will see (Fact 1(c) below) that this completely determines Θ I . It should be pointed out that even though Θ α e ∅ = e α and γ α e ∅ = e α , necessarily Θ α = γ α . The above theta matrices can be used to construct a basis of the space of force tensors V (η). Specifically, we let Θ β I denote the collection of matrices with (Θ β I ) α = 0 if α = β, and (Θ β I ) α = Θ I if α = β. I.e., (Θ β I ) α = δ β α Θ I . We will show that V (η) and β = 0, 1, 2, 3. In particular, V (η) has dimension 64. The space V (η) Fact 1. Suppose I = α 1 · · · α k is a unique multi-index. That is, it contains no duplicate indices. (a)η is diagonal, andη II = η α 1 α 1 · · · η α k α k . (b) Set γ I . = γ α 1 · · · γ α k if I = ∅, and γ ∅ . = I. Then γ I e ∅ = e I . (c) Θ I e J = γ J e I for all unique multi-indices J. (d) Θ † Iη = s Iη Θ I , where s I = −1 if \|I\| = 1, 2, and s I = 1 if \|I\| = 0, 3, 4. Here, I denotes the 16 × 16 identity matrix. It should be noted that if I is not unique, then Θ I = 0. Moreover, the conclusion of (b) is false in this case. The identity Θ I e J = γ J e I trivially holds if I is not unique, but otherwise does not hold if J is not unique. Fact 2. The space CΘ of all 16 ×16 matrices that commute with gamma matrices is a sixteen-dimensional complex vector space spanned by the matrices Θ I for all unique multi-indices I. Moreover, CΘ is a matrix ring with unit Θ ∅ = I, and forms a Clifford algebra representation: Θ α Θ β +Θ β Θ α = −2η αβ . Item (d) of Fact 1 is equivalent to the statement: Θ † Iη +ηΘ I = 0 for \|I\| = 1, 2, and (iΘ J ) †η +η(iΘ J ) = 0 for \|J\| = 0, 3, 4. Thus the subspace of CΘ consisting of matrices N with N †η + ηN = 0 is a sixteen-dimensional real vector space with basis Θ I for \|I\| = 1, 2, and iΘ J with \|J\| = 0, 3, 4. It follows that the space V (η) of force tensors on Minkowski space indeed has a basis given by equation (6). Proof. All statements in both facts are straightforward, with the exception of Fact 1(d), for which we sketch an ad hoc argument. First verify that γ † Iη = s Iη γ I using the identities γ † αη = −ηγ α and γ α γ β = −γ β γ α if α = β. Set N = z I Θ I , where z I = 1 for \|I\| = 1, 2 and z I = i for \|I\| = 0, 3, 4. Compute that e † K (N †η +ηN)e ∅ =η KI (z I + s KzI ) = 0 for any unique K. Now compute e † K (N †η +ηN)e J = s J e † K γ † J (N †η +ηN)e ∅ for any unique J, which must be zero by the previous computation. The vector space CΘ has a secondary complex structure J , given by the matrix J . = Θ 0123 . That is, J 2 = −I. The two complex structures combine to give a perplex structure iJ : (iJ ) 2 = I. Observe that J induces a complex structure on two subspaces of V (η). The first is the subspace spanned by force tensors of the form iΘ α I where \|I\| = 0, 4. And the second is the subspace spanned by Θ α I with \|I\| = 2. On the other hand, the subspace spanned by Θ α I and iΘ β J with \|I\| = 1 and \|J\| = 3 has a perplex structure induced by iJ . Theta matrix identities The gamma and theta matrices both satisfy the Clifford algebra relation, so the usual gamma matrix algebra and trace identities will also be satisfied by the theta matrices. In fact, the gamma and theta matrices are similar in the sense that Θ α = Sγ α S −1 . Indeed, S can be taken to be diagonal with S I I = 1 if \|I\| = 0, 1, 4, and S I I = −1 if \|I\| = 2, 3. We state some useful identities specific to theta matrices. Here, ǫ αβρσ is the totally anti-symmetric Levi-Civita symbol. Θ I J = (−1) \|I\| J θ I J Θ αβρ = −ǫ αβρ σ Θ σ J Θ αβ = 1 2 ǫ αβ ρσ Θ ρσ Θ αβρσ = ǫ αβρσ J tr (Θ I ) = 16 δ I∅ tr (Θ α Θ ρσ ) = 0 tr (J Θ α Θ ρσ ) = 0 tr (Θ αβ Θ ρσ ) = 16(η ασ η βρ − η αρ η βσ ) tr (J Θ αβ Θ ρσ ) = −16 ǫ αβρσ Θ α Θ α = −4 I Θ α Θ β = −Θ αβ − η αβ I Θ α Θ αβ = Θ βα Θ α = 3Θ β η ρσ Θ αρ Θ σβ = 2Θ αβ + 3η αβ I η ρσ Θ αρ Θ µν Θ σβ = −Θ αβ Θ µν − Θ µν Θ αβ − η αβ Θ µν [Θ αβ , Θ ρσ ] = 2η βρ Θ ασ + 2η ασ Θ βρ − 2η αρ Θ βσ − 2η βσ Θ αρ Irreducible force tensors A Lorentz transformation Λ defines a change of basis for the tangent bundle of M. We can extend this to a change of basisΛ for * M. In this way the Lorentz group O(g) acts on the space of force tensors V (g). Indeed, from equation (5), (Λ · θ) α . = (Λ −1 ) β αΛ θ βΛ −1 for any force tensor θ. The corresponding Lorentz algebra action is then (L · θ) α = −L β α θ β +Lθ α − θ αL = [L, θ α ] − L β α θ β(7) for L ∈ so(g). We say that a force tensor θ is irreducible if its orbit under the O(g) action, or equivalently under the so(g) action, lies in an irreducible real representation of O(g). The irreducible sub-representations of V (g) can be computed using standard Lie algebra techniques, such as in [1]. In this section, we will present the results for the case g = η. Let us first indicate explicitly how so(η) affects each of the basic force tensors in equation (6). The Lorentz algebra so(η) is a real six-dimensional vector space consisting of 4 × 4 matrices L that satisfy L T η + ηL = 0. From equation (7), the action of L ∈ so(η) on the force tensor Θ β I is given by (L · Θ β I ) α = [L, (Θ β I ) α ] − L ν α (Θ β I ) ν = δ β α [L, Θ I ] − L β α Θ I .(8) In [3] it is shown thatΛγ αΛ −1 = Λ β α γ β for any Λ ∈ O(η). It follows that [L, γ α ] = L β α γ β . And with this, one shows that [L, Θ I ] commutes with gamma matrices. So that its value is determined by its effect on e ∅ , which can be computed using equation (1). As an example, we compute L · Θ β βρ . First, [L, Θ βρ ]e ∅ =LΘ βρ e ∅ =Le βρ = L µ β e µρ + L µ ρ e βµ = (L µ β Θ µρ + L µ ρ Θ βµ )e ∅ (recall thatLe ∅ = 0). Whence [L, Θ βρ ] = L µ β Θ µρ + L µ ρ Θ βµ . By equation (8), we then have (L · Θ β βρ ) α = L µ α Θ µρ + L µ ρ Θ αµ − L β α Θ βρ = L µ ρ Θ αµ = (L µ ρ Θ β βµ ) α . That is, L · Θ β βρ = L µ ρ Θ β βµ , so that Θ β βρ transforms like e ρ . In general, one shows that Θ β βρ 1 ···ρ k transforms like e ρ 1 ···ρ k . Irreducible summands: overview Viewing the space of force tensors V (g) as a Lie algebra representation of so(g), we decompose it into irreducible summands. Schematically, the decomposition is V (g) = V r (g) ⊕ iV c (g) V r (g) = 1 ⊕ 4 ⊕ 4 ′ ⊕ 6 ⊕ 9 ⊕ 16 V c (g) = 1 ′ ⊕ 4 ′′ ⊕ 4 ′′′ ⊕ 6 ′ ⊕ 9 ′(9) where each summand has the indicated dimension. All summands of the same dimension are isomorphic. The dimension four summands are isomorphic to ( 1 2 , 1 2 ) in the classification of representations of the Lorentz group. The dimension six, nine, and sixteen summands are isomorphic to (1, 0) ⊕ (0, 1), (1, 1), and ( 3 2 , 1 2 ) ⊕ ( 1 2 , 3 2 ), respectively. While the summands 6 and 16 are reducible over the complex numbers, they are irreducible over the reals. In the case when g = η, we can be more specific about the summands in (9). One computes that the complex structure matrix J commutes with the extended Lorentz Lie algebra action. So that if we define J to act component-wise on Θ α I , that is (J Θ α I ) β = δ α β J Θ I , then L · (J Θ α I ) = J (L · Θ α I ). And we have 1 ′ = J 1, 6 ′ = J 6, 4 ′ = J 4, 4 ′′′ = J 4 ′′ , 9 ′ = J 9 as representations of so(η). Whence V (η) = (1 + iJ )1 ⊕ (1 + J )4 ⊕ i(1 + J )4 ′′ ⊕ (1 + iJ )6 ⊕ (1 + iJ )9 ⊕ 16.(10) Each of the summands will be described explicitly in the remainder of this section. One-dimensional force tensors The summand 1 is the one-dimensional subspace spanned by the single force tensor u . = Θ α α . Indeed, L · u = 0 for all L ∈ so(η). Any force tensor in (1 + iJ )1 can thus be written in the form U = φ(cos ζ + i sin ζ J )u for a real scalar field φ, and constant angle ζ. The individual component matrices of U are U α = φ(cos ζ + i sin ζ J )Θ α .(11) Remark. For all compound summands in V (η), such as (1 + iJ )1, there will be an angular parameter ζ. It gives a free parameter of the model, and although we use the same symbol, its value is not the same across distinct compound summands. Real four-dimensional force tensors The four-dimensional summand 4 in (9) has a basis given by the force tensors v α . = Θ β βα . As we have seen, these force tensors transform exactly like the basis vectors e α of M under Lorentz transformations. Any force tensor in (1 + J )4 can therefore be written in the form F = B α (cos ζ + sin ζ J )v α = B α e ζJ v α for some 4-vector field B α and fixed angle ζ. Here e ζJ is the matrix exponential e ζJ = cos ζ I + sin ζ J . The components of F can be written as F α = B β e ζJ Θ αβ .(12) Imaginary four-dimensional force tensors The four-dimensional summand i4 ′′ in (9) has a basis consisting of the force tensors w α . = iη αβ Θ β ∅ . One computes that so(η) also acts on the force tensors w α in exactly the same way as on the vectors e α . Consequently, any force tensor in i(1 + J )4 ′′ is of the form M = A β e ζJ w β for some 4-vector field A α . In other words, the component matrices of M are M α = iA α e ζJ .(13) Six-dimensional force tensors The summand 6 in (9) is spanned by the force tensors v αβ = J w αβ , for α < β, where w αβ . = Θ ǫ ǫαβ . The action of so(η) on w αβ , and consequently on v αβ as well, is the same as on e αβ . Hence any force tensor from (1 + iJ )6 can be written as H = W αβ (cos ζ + i sin ζ J )v αβ , for some anti-symmetric tensor field W αβ . That is, the component matrices of H are H α = −iW ρσ (sin ζ + i cos ζ J )Θ αρσ .(14) 3.6 Nine-dimensional force tensors Let 10 be the ten-dimensional real vector space spanned by the force tensors u αβ . = 1 2 η αǫ Θ ǫ β + η βǫ Θ ǫ α for α ≤ β. Necessarily, 10 is a subspace of V (η). We have u αβ = u βα , and in fact we may identify 10 with the second symmetric power S 2 (M) of M. Indeed, one computes that the so(η) action is given by L · u αβ = L ǫ α u ǫβ + L ǫ β u αǫ for any L ∈ so(η). Therefore, a force tensor on 10 is of the form S αβ u αβ for some symmetric tensor field S αβ on M. Observe that η αβ u αβ = Θ α α = u, so that 1 is a subspace of 10. Moreover, the map π : 10 → 1 given by π(S αβ u αβ ) = 1 4 S α α u, where S α α . = η αβ S αβ , is a projection. The space 9 is the complement of 1 in 10. That is, a force tensor on 9 is of the form S αβ u αβ , where S αβ is now symmetric and contraction-free. I.e., S α α = 0. From the above argument, a force tensor in (1 + iJ )9 can be written as N = S αβ (cos ζ + i sin ζ J )u αβ , which has component matrices N α = η αβ S βǫ (cos ζ + i sin ζ J )Θ ǫ(15) for S αβ a symmetric and contraction-free real field on M. Sixteen-dimensional force tensors Consider the 24-dimensional subspace 24 of V (η) spanned by the force tensors u αρσ . = η αβ Θ β ρσ with ρ < σ. One verifies that L · u αρσ = L ǫ α u ǫρσ + L ǫ ρ u αǫσ + L ǫ σ u αρǫ for any L ∈ so(η). Consequently as u αρσ = −u ασρ , 24 can be identified with the tensor product M ⊗ 2 (M) of Minkowski space and its second exterior power. Thus any force tensor in 24 can be written as F αρσ u αρσ for some tensor field F αρσ on M with F αρσ = −F ασρ . Because η αρ u αρσ = Θ β βσ = v σ , we see that 4 is a subspace of 24. And consequently, so is J 4. In fact, we have that ǫ αρστ u αρσ = 2η τ β J v β . The projections π 1 : 24 → 4 and π 2 : 24 → J 4 are given by π 1 (F αρσ u αρσ ) = 1 4 η αρ F αρσ v σ and π 2 (F αρσ u αρσ ) = 1 3 η µν ǫ ναρσ F αρσ J v µ . One computes that π 1 • π 2 = 0, so that the subspaces 4 and J 4 are orthogonal. Thus (1+J )4 forms an 8-dimensional subspace of 24. The space 16 is the complement of (1 + J )4 in 24. It follows that any force tensor in 16 is of the form X = F αρσ u αρσ where F αρσ satisfies F αρσ = −F ασρ , η αρ F αρσ = 0, ǫ αρστ F αρσ = 0.(16) Tensors F αρσ that satisfy (16) are called (self-dual) Fierz tensors, and are used in spin-2 field theory, see [2]. The component matrices of X are X α = η αβ F βρσ Θ ρσ .(17) Fundamental forces Each distinct irreducible summand in (10) corresponds to a distinct fundamental force. That is, there are six fundamental forces (not including gravity) allowed by this model. In this section, for each fundamental force we find the potential energy under the assumption L V . = 1 2 ω tr (Ω αβ Ω αβ ) + c.c., where Ω αβ = ∂ α C β − ∂ β C α + C α C β − C β C α is the curvature matrix of the total connection C α =Γ α + θ α , with θ a force tensor. We also give the field equations obtained by varying the Lagrangian density L = L K − τ L V(18) with respect to the field components. Here L K is as in equation (2), and τ is a constant. Again we assume that g = η, so that C α = θ α . Scalar field The potential energy of the force tensor U in equation (11), associated with the summand (1 + iJ )1, is found to be L V = −96 cos 2ζ [(∂ α φ)(∂ α φ) + 8 cos 2ζ φ 4 ].(19) Variation of the Lagrangian density (18) with respect to the real scalar field φ then gives the field equation ∂ α ∂ α φ − 16 cos 2ζ φ 3 = 1 192τ cos 2ζ ψ †η γ α (cos ζ + i sin ζ J )Θ α ψ.(20) Like ζ, τ is a free parameter of the model, and is not constant across the distinct compound summands in (10). Vector fields There are two compound summands of dimension four in (10). Each will give rise to a different 4-vector field. Vector field of the first type The potential energy of the force tensor F associated to the summand (1 + J )4 given in equation (12) is L V = − 384(B α B α ) 2 cos 4ζ − 256(B α B α ∂ β B β − B α B β ∂ α B β ) cos 3ζ − 32[2(∂ α B β )(∂ α B β ) + (∂ α B α ) 2 ] cos 2ζ − 32ǫ αβρσ (∂ α B β )(∂ ρ B σ ) sin 2ζ.(21) The last summand is a divergence, and has no effect on the field variation. The field equation for the real four-vector field B α is then ∂ β ∂ β B α + 1 2 ∂ α ∂ β B β + 6 cos 3ζ cos 2ζ (B β ∂ α B β − B α ∂ β B β ) − 12 cos 4ζ cos 2ζ (B β B β )B α = 1 128τ cos 2ζ ψ †η γ β e ζJ Θ βα ψ.(22) Vector field of the second type On the other hand, for the force tensor M on i(1 + J )4 ′′ in equation (13), we have the potential energy L V = −32 cos 2ζ [(∂ α A β )(∂ α A β ) − (∂ α A β )(∂ β A α )](23) for the real 4-vector field A α . This leads to the field equation ∂ β ∂ β A α − ∂ α ∂ β A β = i 64τ cos 2ζ ψ †η γ α e ζJ ψ.(24) Antisymmetric tensor field In this case, the potential energy term of the Lagrangian for the force tensor H on (1 + iJ )6 in equation (14) is L V = 1024 cos 2 2ζ [W αβ W βρ W ρσ W σα − (W αβ W αβ ) 2 ] + 64 cos 2ζ [(∂ α W ρσ )(∂ α W ρσ ) + 2(∂ β W αβ )(∂ σ W ασ )].(25) The field equation for the real antisymmetric tensor field W αβ is then ∂ µ ∂ µ W αβ − ∂ α ∂ σ W βσ + ∂ β ∂ σ W ασ + 16 cos 2ζ [W αρ W ρσ W σβ + (W ρσ W ρσ )W αβ ] = i 128τ cos 2ζ ψ †η γ ν (sin ζ + i cos ζ J )Θ ναβ ψ. (26) Symmetric contraction-free tensor field The potential energy of the force tensor N on the summand (1 + iJ )9, equation (15), is given by L V = 64 cos 2 2ζ [S αβ S βρ S ρσ S σα − (S αβ S αβ ) 2 ] − 32 cos 2ζ [(∂ α S βρ )(∂ α S βρ ) − (∂ α S βρ )(∂ β S αρ )]. To compute the variation of the total Lagrangian with respect to S αβ , we need to take into account the constraint S ρ ρ = η ρσ S ρσ = 0. The result is ∂ µ ∂ µ S αβ − 1 2 ∂ α ∂ µ S µβ − 1 2 ∂ β ∂ µ S µα + 1 4 η αβ ∂ µ ∂ ν S µν + 4 cos 2ζ [S αµ S µν S νβ − (S µν S µν )S αβ − 1 4 η αβ S ν µ S σ ν S µ σ ] = 1 128τ cos 2ζ ψ †η (cos ζ + i sin ζ J )(γ α Θ β + γ β Θ α − 1 2 η αβ γ µ Θ µ )ψ(28) with S αβ a real symmetric contraction-free tensor field. Fierz tensor field Here the potential energy of the term of the force tensor X in equation (17) is given by L V = −32F αβρσ F αβρσ ,(29) where F αβρσ . = ∂ α F βρσ − ∂ β F αρσ + 4(F αρ ν F βνσ − F βρ ν F ανσ ) and F αρσ is a Fierz tensor: a tensor that satisfies equation (16). Constrained variation of the total Lagrangian yields ξ αρσ − 1 3 η αρ ξ β βσ + 1 3 η ασ ξ β βρ − 1 6 ǫ αρσω ǫ λµνω ξ λµν = χ αρσ − 1 3 η αρ χ β βσ + 1 3 η ασ χ β βρ − 1 6 ǫ αρσω ǫ λµνω χ λµν where ξ αρσ = 1 2 (ξ αρσ −ξ ασρ ) and ξ αρσ . = 8(2F βν ρ ∂ β F ασν + F βν ρ ∂ α F βνσ + F αρ ν ∂ β F β σν ) − 32(F βρµ F α µν F β νσ − F βρµ F βµν F ανσ ) + ∂ β ∂ β F αρσ − ∂ β ∂ α F β ρσ χ αρσ . = 1 128τ ψ †η γ α Θ ρσ ψ. J = ∅, 012, 013, 023, 123, 0123 Representation Theory: A First Course. William Fulton, Joe Harris, Graduate Texts in Mathematics. SpringerWilliam Fulton and Joe Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, Springer, New York, 1991. Spin-2 field theory in curved spacetime in the Fierz representation. M Novello, R P Neves, Class. Quantum Grav. 195335M. Novello and R. P. Neves, Spin-2 field theory in curved spacetime in the Fierz representation, Class. Quantum Grav. 19, 5335 (2002). Coupling the Dirac and Einstein equations through geometry. Hanson Jason, Found. Phys. 521jason hanson, Coupling the Dirac and Einstein equations through geom- etry, Found. Phys. 52, 1 (2022).	[]
[ "COMPARING INVARIANTS OF LEGENDRIAN KNOTS", "COMPARING INVARIANTS OF LEGENDRIAN KNOTS" ]	[ "Marco Golla " ]	[]	[]	We prove the equivalence of the invariants EH(L) and L − (±L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vértesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement S 3 \ K and the knot Floer homology of (S 3 , K) and define intermediate Legendrian invariants.	10.4171/qt/66	[ "https://arxiv.org/pdf/1404.1203v1.pdf" ]	119,144,694	1404.1203	00a6edb0c70bdfe856a701aff0d61cb1b6443e1c	COMPARING INVARIANTS OF LEGENDRIAN KNOTS Marco Golla COMPARING INVARIANTS OF LEGENDRIAN KNOTS We prove the equivalence of the invariants EH(L) and L − (±L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vértesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement S 3 \ K and the knot Floer homology of (S 3 , K) and define intermediate Legendrian invariants. Introduction In recent years, many Floer-theoretic invariants for Legendrian knots have been introduced: in 2008, Ozsváth, Szabó and Thurston [OSzT] used grid diagrams to define two invariants λ ± (L) and λ ± (L) of oriented Legendrian knots L in (S 3 , ξ st ), taking values in a combinatorial version of knot Floer homology. Shortly afterwards, Lisca, Ozsváth, Stipsicz and Szabó [LOSSz] used open books to construct two other invariants of oriented nullhomologous Legendrian knots L, called L(L) and L − (L), taking values in the original version knot Floer homology. In 2006, Juhász defined a version of Heegaard Floer homology for manifolds with "marked" boundary, which he called sutured Floer homology [Ju]. Honda, Kazez and Matić soon constructed invariants for contact manifolds with convex boundary, taking values in a sutured Floer cohomology group [HKM1]: the key feature of their invariant (and of sutured Floer homology) is its behaviour with respects to gluing manifolds along their (compatible) boundaries [HKM2]. In this context, to every Legendrian knot L in a contact three-manifold one can associate a contact manifold with convex boundary, and therefore a contact invariant EH(L) living in some sutured Floer homology group. Some natural questions arise at this point: is there any relation between EH(L) and the L invariants? If so, what is this relation exactly? Late in 2008, a first answer to these questions was given by Stipsicz and Vértesi, who explained how EH(L) determines L(L) [SV]; recently, Baldwin, Vela-Vick and Vértesi were able to prove the equivalence of the combinatorial invariants λ and the LOSS invariants L [BVV]. Our main result is the following (−L means L with the reversed orientation). Theorem 1.1. For two oriented, topologically isotopic Legendrian knots L 0 , L 1 in (S 3 , ξ st ), the following are equivalent: The same result has been obtained, in greater generality, by Etnyre, Vela-Vick and Zarev [EVZ]. In fact, using the same techniques together with a generalisation of [LOT,Theorem 11.35], one can prove the generalisation of Theorem 1.1 to Legendrian knots in arbitrary contact 3-manifolds (Y, ξ) such that c(Y, ξ) = 0, and it is always the case that EH(L) determines L − (±L). Organisation. This paper is organised as follows: we first review the setting we're working in, giving a brief introduction to sutured Floer homology in Section 2 and the EH invariants in Section 3. Then we analyse in some detail the groups and the maps we are dealing with, in Section 4. In Section 5 the relation between various sutured Floer homology associated to a knot complement and HF K − are explained; this will lead to the proof of the equivalence of the two invariants EH and L − in the last section. for suggesting me the problem, for many helpful discussions, and for his support. I want to thank Paolo Lisca, Olga Plamenevskaya, András Stipsicz and David Shea Vela-Vick for interesting conversations, and the referees for helpful comments and suggestions. Part of this work has been done while I was visiting the Simons Center for Geometry and Physics: I acknowledge their support. The author has been supported by the ERC grant LTDBUD. 2. Sutured Floer homology and gluing maps 2.1. Sutured manifolds. The definition of balanced sutured manifold is due to Juhász [Ju]. Definition 2.1. A balanced sutured manifold, is a pair (M, Γ) where M is an oriented 3-manifold with nonempty boundary ∂M , and Γ is a family of oriented curves in ∂M that satifies: • Γ intersects each component of ∂M ; • Γ disconnects ∂M into R + and R − , with ±Γ = ∂R ± (as oriented manifolds); • χ(R + ) = χ(R − ). Remark 2.2. The condition χ(R + ) = χ(R − ) is called the balancing condition. Since this is the only kind of sutured manifolds we're dealing with, we prefer to just drop the adjective 'balanced'. Example 2.3. Any M oriented 3-manifold with S 2 -boundary, can be turned into a sutured manifold (M, {γ}) by choosing any simple closed curve γ in ∂M . We'll often write M = Y (1), where Y = M ∪ ∂ D 3 is the "simplest" closed 3-manifold containing M . For every integer f , we have a sutured manifold S 3 K,f given by pairs (S 3 \ N (K), {γ f , −γ f }), where γ f is an oriented curve on the boundary torus ∂N (K) of an open small neighbourhood N (K) of K. The slope of γ f is λ S + f · µ, and −γ f is a parallel push-off of γ f , with the opposite orientation. Here λ S denotes the Seifert longitude of K. We'll use the shorthand Γ f for {γ f , −γ f }. Example 2.4. To any Legendrian knot L ⊂ (Y, ξ) in an arbitrary 3-manifold Y one can associate in a natural way a sutured manifold, that we'll denote with Y L , constructed as follows: there's a standard open Legendrian neighbourhood ν(L) for L, whose complement has convex boundary. The dividing set Γ L on the boundary consists of two parallel oppositely oriented curves parallel to the contact framing of L. The manifold Y L is then defined as the pair (Y \ ν(L), Γ L ). In the case we're mainly interested in, where Y = S 3 and L is of topological type K, we have S 3 L = S 3 K,tb(L) . More generally, the same identification {framings} ↔ Z can be made canonical whenever K is nullhomologous in Y and H 2 (Y ) = 0, and we then have Y L = Y K,tb(L) . We'll often use Y L also to denote the contact manifold with convex boundary (Y \ ν(L), ξ\| Y \ν(L) ), without creating any confusion. There's a decomposition/classification theorem for sutured manifolds, completely analogous to the Heegaard decomposition/Reidemeister-Singer theorem for closed three-manifolds. Consider a compact surface Σ with boundary together with two collections of pairwise disjoint simple closed curves α, β ⊂ Σ, such that each collection is a linearly independent set in H 1 (Σ; Z); suppose moreover that \|α\| = \|β\|. We can build a balanced sutured manifold out of this data as follows: take Σ × [0, 1], glue a 2-handle on Σ × {0} for each α-curve, and a 2-handle on Σ × {1} for each β-curve, and let M be the manifold obtained after smoothing corners; declare Γ = ∂Σ × {1/2}. The pair (M, Γ) is a balanced sutured manifold, and (Σ, α, β) is called a (sutured) Heegaard diagram of (M, Γ). Theorem 2.5 ( [Ju]). Every balanced sutured manifold admits a Heegaard diagram, and every two such diagrams become diffeomorphic after a finite number of isotopies of the curves, handleslides and stabilisations taking place in the interior of the Heegaard surface. 2.2. The Floer homology packages. This is meant to be just a recollection of facts about the Floer homology theories we'll be working with. The standard references for the material in this subsection are [OSz3,OSz4,Li] for the Heegaard Floer part, and [Ju] for the sutured Floer part. In order to avoid sign issues, we'll work with F = F 2 coefficients. Consider a pointed Heegaard diagram H = (Σ g , α, β, z) representing a three-manifold Y , and form two Heegaard Floer complexes CF (Y ) and CF − (Y ): the underlying modules are freely generated over F and F[U ] by g-tuples of intersection points in i,j (α i ∩ β j ), so that there's exactly one point on each curve in α ∪ β. The differentials ∂, ∂ − are harder to define, and count certain pseudoholomorphic discs in a symmetric product Sym g (Σ g ), or maps from Riemann surfaces with boundary in Σ g × R × [0, 1], with the appropriate boundary conditions. The homology groups HF (Y ) = H * ( CF (Y ), ∂) and HF − (Y ) = H * (CF − (Y ), ∂ − ) so defined are called Heegaard Floer homologies of Y , and are independent of the (many) choices made along the way [OSz3]. Sutured Floer homology is a variant of this construction for sutured manifolds (M, Γ). The starting point is a sutured Heegaard diagram H = (Σ, α, β) for (M, Γ). We form a complex SF C(M, Γ) in the same way, generated over F by d-tuples of intersection points as above, where d = \|α\| = \|β\|. The differential ∂ is defined by counting pseudo-holomorphic discs in Sym d (Σ) or maps from Riemann surfaces to Σ × R × [0, 1], again with the appropriate boundary conditions. The homology SF H(M, Γ) = H * (SF C(M, Γ), ∂) is called the sutured Floer homology of (M, Γ), and is shown to be independent of all the choices made [Ju]. It naturally corresponds to a 'hat' theory. Proposition 2.6 ( [Ju]). For a closed 3-manifold Y , HF (Y ) = SF H(Y (1)). For a knot K in a closed 3-manifold Y , HF K(Y, K) = SF H(Y K,m ), where m is the meridian for K in Y . 2.3. Floer-theoretic contact invariants. The first contact invariant to be defined in Heegaard Floer homology was Ozsváth and Szabó's c [OSz5]. We sketch here the construction of the contact class EH [HKM1], and we will relate it to c below. Definition 2.7. A partial open book is a triple (S, P, h) where S is a compact open surface, P is a proper subsurface of S which is a union of 1-handles attached to S \ P and h : P → S is an embedding that pointwise fixes a neighborhood of ∂P ∩ ∂S. We can build a contact manifold with convex boundary out of these data in a fashion similar to the usual open books: instead of considering a mapping torus, though, we glue two asymmetric halves, quotienting the disjoint union S × [0, 1/2] P × [1/2, 1] by the relations (x, t) ∼ (x, t ) for x ∈ ∂S, (y, 1/2) ∼ (y, 1/2), (h(y), 1/2) ∼ (y, 1) for y ∈ P . The contact structure is uniquely determined if we require -as we do -tightness and prescribed sutures on each half S × [0, 1/2]/∼ and P × [1/2, 1]/∼ (see [Ho] for details). Moreover, to any contact manifold with convex boundary we can associate a partial open book, unique up to Giroux stabilisations. We can build a balanced diagram out of a partial open book. The Heegaard surface Σ is obtained by gluing P to −S along the common boundary. Definition 2.8. A basis for (S, P ) is a set a = {a 1 , . . . , a k } of arcs properly embedded in (P, ∂P ∩ ∂S) whose homology classes generate H 1 (P, ∂P ∩ ∂S). Given a basis as above, we produce a set b = {b 1 , . . . , b k } of curves using a Hamiltonian vector field on P : we require that under this perturbation the endpoints of a i move in the direction of ∂P , and that each a i intersects b i in a single point x i , and is disjoint from all the other b j 's. Finally define the two sets of attaching curves: α = {α i } and β = {β i }, where α i = a i ∪ −a i and β i = h(b i ) ∪ −b i : the sutured manifold associated to (Σ, α, β) is (M, Γ). We call x(S, P, h) the generator {x 1 , . . . , x k } in SF C(Σ, β, α) supported inside P . The type of invariants that we're going to deal with are either invariants of (complements of) Legendrian knots or invariants coming from contact structures on closed manifolds: this allows us to consider only sutured manifolds with sphere/torus boundary and one/two sutures, as described in Examples 2.3 and 2.4. Consider a closed contact manifold (Y, ξ), and let B ⊂ Y be a small, closed Darboux ball with convex boundary. Then consider the manifold (Y (1), ξ(1)) where Y (1) is obtained from Y by removing the interior of B, and ξ(1) is ξ\| Y (1) . Proposition 2.11 ( [HKM1]). There is an isomorphism of graded complexes from HF (Y ) to SF H(Y (1)) that maps the Ozsváth-Szabó contact invariant c(Y, ξ) to the Honda-Kazez-Matić class EH(Y (1), ξ(1)). Suppose now that L ⊂ Y is a Legendrian knot with respect to a contact structure ξ: the contact manifold Y L defined in Example 2.4 determines a contact invariant EH(Y L ) ∈ SF H(−Y L ). We'll denote this invariant by EH(L), considering it as an invariant of the Legendrian isotopy class of L rather than of its complement. 2.4. Gluing maps. In their paper [HKM2], Honda, Kazez and Matić define maps associated to the gluing of a contact manifold to another one along some of the boundary components, and show that these maps preserve their EH invariant. Consider two sutured manifolds (M, Γ) ⊂ (M , Γ ), where M is embedded in Int(M ); let ξ be a contact structure on N := M \ Int(M ) such that ∂N is ξ-convex and has dividing curves Γ ∪ Γ . For simplicity, and since this will be the only case we need, we'll restrict to the case when each connected component of N intersects ∂M (i.e. gluing N to M doesn't kill any boundary component). Theorem 2.12. The contact structure ξ on N induces a gluing map Φ ξ , that is a linear map Φ ξ : SF H(−M, −Γ) → SF H(−M , −Γ ). If ξ M is a contact structure on M such that ∂M is ξ M -convex with dividing curves Γ, then Φ ξ (EH(M, ξ M )) = EH(M , ξ M ∪ ξ). This theorem has interesting consequences, even in simple cases: Corollary 2.13. If (M, Γ) embeds in a Stein fillable contact manifold (Y, ξ), and ∂M is ξ-convex, divided by Γ, then EH(M, ξ\| M ) is not trivial. There's also a naturality statement, concerning the composition of two gluing maps: suppose that we have three sutured manifolds (M, Γ) ⊂ (M , Γ ) ⊂ (M , Γ ) as at the beginning of the section, and suppose that ξ and ξ are contact structures on M \ Int(M ) and M \ Int(M ) respectively, that induce sutures Γ, Γ and Γ on ∂M , ∂M and ∂M respectively. Theorem 2.14. If ξ and ξ are as above, then Φ ξ∪ξ = Φ ξ • Φ ξ . Much of our interest will be devoted to stabilisations of Legendrian knots and associated maps, whose discussion will occupy Subsection 3.3: we give a brief summary of the contact side of their story here. Let's start with a definition, due to Honda [Ho]: Definition 2.15. Let η be a tight contact structure on T 2 ×I with two dividing curves on each boundary component: call γ i , −γ i the homology class of the two dividing curves on T 2 × {i}, and let s i ∈ Q ∪ {∞} be their slope. (T 2 × I, η) is a basic slice if it is of the form above, and also satisfies the following three conditions: • {γ 0 , γ 1 } is a basis for H 1 (T 2 ); • ξ is minimally twisting, i.e. if T t = T × {t} is convex, the slope of the dividing curves on T t belongs to [s 0 , s 1 ] (where we assume that if s 0 > s 1 the interval [s 0 , s 1 ] is [−∞, s 1 ] ∪ [s 0 , ∞]); Honda proved the following: Proposition 2.16 ( [Ho]). For every integer t there exist exactly two basic slices (T 2 × I, ξ j ) (for j = 1, 2) with boundary slopes t/1 and (t − 1)/1. The sutured complement of a stabilisation L of L is gotten by attaching one of the two basic slices to Y L , where the trivialization of T 2 is given by identifying the slopes 0/1 and t/1 with a meridian µ and the contact framing c for L, respectively. These two different layers correspond to the positive and negative stabilisation of L, once we've chosen an orientation for the knot; reversing the orientation swaps the labelling signs. Since we'll be considering oriented Legendrian knots, we can label the two slices with a sign. Definition 2.17. We call stabilisation maps the gluing maps associated to the attachment of a stabilisation basic slice: these will be denoted with σ ± . Remark 2.18. As it happens for the Stipsicz-Vértesi map [SV], these basic slice attachments correspond to single bypass attachments, too. 3. A few facts on SF H(S 3 K,n ) and σ ± Given a topological knot K in S 3 , denote with S 3 m (K) the manifold obtained by (topological) m-surgery along K, and let K be the dual knot in S 3 m (K), that is the core of the solid torus we glue back in. Notice that an orientation on K induces an orientation of K, by imposing that the intersection of the meridian µ K of K on the boundary of the knot complement has intersection number +1 with the meridian µ K of K on the same surface. Fix a contact structure ξ on S 3 and a Legendrian representative L of K: we'll write t for tb(L). Since t measures the difference between the contact and the Seifert framings of L, S 3 t (K) K,∞ and S 3 L are sutured diffeomorphic: in particular, EH(L) lives in SF H(−S 3 t (K) K,∞ ) = HF K(−S 3 t (K), K), the identification depending on the choice of an orientation for K (or K). We will often write CF K(Y, K) to denote any chain complex computing HF K(Y, K) that comes from a Heegaard diagram, even though the complex itself depends on the choice of the diagram. 3.1. Gradings and concordance invariants. The groups HF K(S 3 , K) and HF K(−S 3 m (K), K) come with a grading, that we call the Alexander grading. A Seifert surface F ⊂ S 3 for K gives a relative homology class [F, ∂F ] ∈ H 2 (S 3 \ N (K), ∂N (K)) = H 2 (S 3 m (K) \ N ( K), ∂N ( K)) . Given a generator x ∈ CF K(S 3 , K), there's an induced relative Spin c structure s(x) in Spin c (S 3 , K) [HP,Equation 2], and the Alexander grading of x is defined as A(x) = 1 2 c 1 (s(x)) − P D([µ K ]), [F, ∂F ] , where P D denotes Poincaré duality. Likewise, given a generator x ∈ CF K(−S 3 m (K), K), there's an induced relative Spin c structure s(x) ∈ Spin c (S 3 m (K), K), and we can define A(x) as (3.1) A(x) = 1 2 c 1 (s(x)) − P D([µ K ]), [F, ∂F ] . We now turn to recalling the definition of τ (K), due to Ozsváth and Szabó [OSz1]. Recall that the Alexander grading induces a filtration on the knot Floer chain complex ( CF K(S 3 , K), ∂), where the differential ∂ ignores the presence of the second basepoint, that is H * ( CF K(S 3 , K), ∂) = HF (S 3 ). In particular, every sublevel CF K(S 3 , K) A≤s is preserved by ∂, and we can take its homology. Definition 3.1. τ (K) is the smallest integer s such that the inclusion of the s-th filtration sublevel induces a nontrivial map H * ( CF K(S 3 , K) A≤s , ∂) −→ HF (S 3 ) = F. This invariant turns out to provide a powerful lower bound for the slice genus of K, in the sense that \|τ (K)\| ≤ g * (K) [OSz1]. One of the properties it enjoys, and that we'll need, is that τ (K) = −τ (K) for every K. Modules. We now turn our attention back to HF K(−S 3 t (K), K) SF H(−S 3 K,t ). Recall that this is a F-vector space on which the A defines a grading. The group CF K(S 3 , K) is a graded vector space that comes with two differentials, ∂ K and ∂, such that the complex ( CF K(S 3 , K), ∂) has homology HF (S 3 ) = F, while the complex ( CF K(S 3 , K), ∂ K ) is the associated graded object with respect to the Alexander filtration. By definition HF K(S 3 , K) is the homology of this latter complex; as such, it inherits an Alexander grading that we call A. Let's call d = dim HF K(S 3 , K), and fix a basis B = {η i , η j \| 0 ≤ i < d} of CF K(S 3 , K) such that the set {η top i , (η j ) top } of the highest nontrivial Alexander-homogeneous components of the η i 's and η j 's is still a basis for CF K(S 3 , K), and the following relations hold (see [LOT,Section 11.5]): ∂η 0 = 0 ∂ K η 0 = 0 ∂η 2i−1 = η 2i ∂ K η i = 0 ∂η 2j−1 = η 2j ∂ K η 2j−1 = η 2j . Observe that the set of homology classes of the η i 's is a basis for HF K( S 3 , K) = H * ( CF K(S 3 , K), ∂ K ). We'll write A(η) for A(η top ). Finally, call δ(i) = A(η 2i−1 ) − A(η 2i ); let's remark that by definition A(η 0 ) = τ := τ (K). Theorem 3.2 ([LOT]). The homology group HF K(−S 3 m (K), K) is an F- vector space with basis {d i,j , d * i,j , u \| 1 ≤ i ≤ k, 1 ≤ j ≤ δ(i), 1 ≤ ≤ \|2τ −m\|}, where the generators satisfy A(d i,j ) = A(η 2i ) − (j − 1) − (m − 1)/2 = −A(d * i,j ) and A(u ) = τ − ( − 1) − (m − 1)/2. Generators with a * are to be thought of as symmetric to the generators without it, and each family {d i,j } j can be interpreted as representing the arrow η 2i−1 ∂ → η 2i (notice that i varies among positive integers), counted with a multiplicity equalling its length (i.e. the distance it covers in Alexander grading). Remark 3.3. Not any basis of HF K(−S 3 m (K), K) with the same degree properties works for our purposes: we're actually choosing a basis that's compatible with stabilisation maps, as we're going to see in Theorem 3.7. Definition 3.4. Call S + the subspace of HF K(−S 3 m (K), K) generated by {d i,j }, and S − the one generated by {d * i,j }: the subspace S = S + ⊕ S − is the stable complex, and elements of S are called stable elements. The subspace spanned by {u } is called the unstable complex and will be denoted with U m (although the subscript will often be dropped), so that HF K (−S 3 m (K), K) decomposes as S + ⊕ U m ⊕ S − . It's worth remarking that the decomposition given in the definition above does depend on our choice of the basis: the three stable subspaces S ± and S are independent on this choice, but the unstable complex isn't; see also Remark 3.9 below. There's a good and handy pictorial description when \|m\| is sufficiently large; we'll be mostly dealing with negative values of m, so let's call m = −m 0. Consider a direct sum C = m i=1 C i of m copies of C = CF K(S 3 , K), and (temporarily) denote by x i the copy of the element x ∈ C in C i . Endow C with a shifted Alexander grading: Figure 3.1. We represent here the top (on the right) and bottom (on the left) parts of HF K(S 3 m (K), K) for m 0. Each vertical tile is a copy of CF K(S 3 , K), and the arrows show the direction of the differentials. A(x i ) = A(x) − (i − 1) − (m − 1)/2 for i ≤ m /2 −A(x) − (i − 1) − (m − 1)/2 for i > m /2 . . . . . . Ã for each homogeneous x in CF K(S 3 , K). We picture this situation by considering each copy of C as a vertical tile of 2g(K) + 1 boxes -each corresponding to a value for the Alexander grading, possibly containing no generators at all, or more than one generator -and stacking the m copies of C in staircase fashion, with C 1 as the top block and C m as the bottom block. Notice that, by our grading convention, the copies in the bottom part of the picture are turned upside down: for example, if x max ∈ C has maximal Alexander degree A(x) = g(K), then x max 1 lies in the top box of C 1 , while x max m lies in the bottom box of C m . Likewise, an element x τ ∈ C has Alexander degree A(x) = τ , then x τ 1 lies in the (g(K) − τ + 1)-th box from the top in C 1 , and x τ m lies in the (g(K) − τ + 1)-th box from the bottom in C m . Our construction is a variant of Hedden's construction: while in general our chain complex for HF K(S 3 m (K), K) differs in from his complex in the region with intermediate Alexander grading, the resulting homologies nevertheless agree. The situation is depicted in Figure 3.1: in this concrete example we have g(K) = 2 and τ (K) = −1; accordingly, there are 2g(K) + 1 = 5 boxes in each vertical column and x τ 1 lies in the fourth box from the top in C 1 . Now define a differential ∂ on C in the following way: ∂ :        (η 0 ) i → 0 for small and large i (η 2j−1 ) i → (η 2j ) i+δ(j) → 0 for small i (η 2j−1 ) i → (η 2j ) i−δ(j) → 0 for large i (η 2j−1 ) i → (η 2j ) i → 0 for every i . We extend the differential to be any map ∂ such that the level {A = j} is a subcomplex for every j, whose homology is F for intermediate values of j (this is possible since {A = j} has odd rank for every intermediate value of j). We're now going to analyse what happens on the top and bottom part of the complex (i.e. when i is small or large, in what follows), when we take the homology. Pairs (η 2j−1 ) i , (η 2j ) i cancel out in homology. The element (η 2j ) i is a cycle for each i, j, and it's a boundary only when j > 0 and either i > δ(j) or i < m − δ(j): so there are 2δ(j) surviving copies of η 2j , in degrees A(η 2j ) − k − (m − 1)/2 and −A(η 2j ) + k + (m − 1)/2 for k = 0, . . . , δ(j) − 1. We can declare d i,j = [(η top 2j ) i ] and d * i,j = [(η top 2j ) m −i ]. The element (η 0 ) i is a cycle for every i, and it's never canceled out, so it survives when taking homology. Given our grading convention, for small values of i, A((η 0 ) i ) = A(η 0 )−(i−1)−(m−1)/2 = τ (K)−(i−1)−(m−1)/2, and in particular we have a nonvanishing class [(η top 0 ) i ] = u i in degrees τ (K)−(m− 1)/2, τ (K)−(m−1)/2−1, . . . On the other hand, when i is large, [(η 0 ) i ] lies in degree −τ (K) − (i − 1) − (m − 1)/2, and we get a nonvanishing class [(η top 0 ) i ] = u 2τ (K)+i+(m−1)/2 in degrees −τ (K) + (m − 1)/2, −τ (K) + (m − 1)/2 + 1, . . . We also have a string of F summands in between, giving us a strip of unstable elements of length 2τ (K) − m, as in Theorem 3.2. 3.3. Stabilisation maps. We're going to study the action of the two stabilisation maps σ ± of Definition 2.17 on the sutured Floer homology groups SF H(−S 3 L ). It's worth stressing that these maps do not depend on the particular Legendrian representative, but only on its Thurston-Bennequin number: in fact, the topological type of L determines the complement S 3 \ ν(L) and tb(L) determines the sutures on ∂ν(L), hence the sutured manifold S 3 L depends only on these data. A gluing map Φ ξ : SF H(M, Γ) → SF H(M , Γ ) only depends on the contact structure ξ on the layer and not on the contact structure on (M, Γ) (in fact, no such contact structure is required in the definition of Φ ξ ). Notice that if L is a Legendrian knot in S 3 with tb(L) = n, then, as a sutured manifold, S 3 L is just S 3 K,n . Moreover, if L is a stabilisation of L, then S 3 L is isomorphic to S 3 K,n−1 as a sutured manifold. Recall that we have two families (indexed by the integer n) of stabilisation maps, σ ± : SF H(−S 3 K,n ) → SF H(−S 3 K,n−1 ), corresponding to the gluing of the negative and positive stabilisation layer: if the knot K is oriented, these maps can be labelled as σ − or σ + . With a slight abuse of notation, we're going to ignore the dependence of these maps on the framing. Remark 3.5. Notice that orientation reversal of L or K isn't seen by the sutured groups nor by EH(L), but it swaps the rôles of σ − and σ + . Remark 3.6. Let's recall that for an oriented Legendrian knot L of topological type K in S 3 the Bennequin inequality holds: tb(L) + r(L) ≤ 2g(K) − 1. In [Pl], Plamenevskaya proved a sharper result: (3.2) tb(L) + r(L) ≤ 2τ (K) − 1. This last form of the Bennequin inequality, together with Theorem 3.2, tells us that, whenever we're considering knots in the standard S 3 , the unstable complex is never trivial in SF H(−S 3 K,n ): more precisely we're always (strictly) below the threshold 2τ := 2τ (K), so that 2τ − m is always positive; in particular, the dimension of the unstable complex is always positive and increases under stabilisations. We'll state the theorem in its full generality anyway, even though this remark tells us we need just half of it when working in (S 3 , ξ st ). The following theorem is proved in [Go,Section 3.4]. Theorem 3.7. The maps σ − , σ + : SF H(−S 3 K,n ) → SF H(−S 3 K,n−1 ) act as follows: σ − :    d i,j → d i,j u → u d * i,j → d * i,j+1 , σ + :    d i,j → d i,j+1 u → u +1 d * i,j → d * i,j for n ≤ 2τ ; σ − :        d i,j → d i,j u → u u n−2τ → 0 d * i,j → d * i,j+1 , σ + :        d i,j → d i,j+1 u → u −1 u 1 → 0 d * i,j → d * i,j for n > 2τ. Notice that we're implicitly choosing an appropriate isomorphism between the group SF H(−S 3 K,n ) and the vector space generated by the d i,j 's and the u i 's (see Theorem 3.2). There's an interpretation of the maps σ ± : SF H(−S 3 K,n ) → SF H(−S 3 K,n−1 ) in terms of Figure 3.1, when n 0: fix a chain complex C computing HF K(S 3 , K) and call ( C n , ∂) and ( C n−1 , ∂) the two complexes defined in the previous section, computing SF H(−S 3 K,n ) and SF H(−S 3 K,n−1 ) starting from C. We have two "obvious" chain maps s ± : C n → C n−1 : s − sends x i ∈ C n to x i ∈ C n−1 , while s + sends x i ∈ C n to x i+1 ∈ C n−1 . The maps s ± induce the two stabilisation maps σ ± at the homology level. s − is the inclusion C n → C n−1 that misses the leftmost vertical tile (that is, the copy C 1−n of C that's in lowest Alexander degree), while s + is the inclusion that misses the rightmost vertical tile (the copy C 1 of C that lies in highest Alexander degree). As a corollary (of the proof), we obtain a graded version of the result: Corollary 3.8. The maps σ ± are Alexander-homogeneous of degree ∓ 1 /2. Remark 3.9. Notice that the maps σ − preserve S + and eventually kill S − , whereas the maps σ + have the opposite behaviour. Moreover, σ − and σ + are injective on the unstable complex for n ≤ 2τ , while they eventually kill it for n > 2τ . Namely, for n ≤ 2τ , the subcomplex S ± = m>0 ker σ m ± = ker σ N ± for some large N (depending on K, but not on the slope n: any N > 2g(K) works), do not depend on the basis we've chosen. For n < 2τ , though, the unstable subspace does depend on this choice: this reflects the fact that it is a section for the projection map SF H(−S 3 K,n ) → SF H(−S 3 K,n )/(S + + S − ). On the other hand, for m > 2τ the situation is reversed: the unstable complex is the intersection of the kernels of σ N ± , and S ± is a section of the projection map ker σ N ∓ → (ker σ N ∓ )/(ker σ N − ∩ ker σ N + ). The action of σ ± on the unstable complex is just by degree shift, as in Theorem 3.7. 4. An apparently new Legendrian invariant 4.1. Some remarks on EH(L). Given an oriented Legendrian knot L, we define L m,n to be the Legendrian knot obtained from L via m negative and n positive stabilisations. The main character of the subsection will be an unoriented Legendrian knot L in the 3-sphere S 3 , equipped with some contact structure ξ. Proposition 4.1. If tb(L) ≤ 2τ (K), the pair {EH(L 0,n ), EH(L n,0 )} deter- mines EH(L). Strictly speaking, since L is not oriented, EH(L 0,n ), EH(L n,0 ) are not individually defined, but the pair {EH(L 0,n ), EH(L n,0 )} is, as the unordered pair {σ n − (EH(L)), σ n + (EH(L))} for either orientation of L. Proof. Since σ − preserves S − and σ + preserves S + , knowing the pair we know what the stable part of EH(L) is. Let's consider now the unstable component of EH(L): since EH(L) is represented by a single generator in the chain complex, it is Alexander-homogeneous; moreover, since the stable and unstable complexes are generated by homogeneous elements, both the stable and unstable components of EH(L) are Alexander-homogeneous. We now state a proposition that will turn out to be useful later, and we will prove it below. Now, if ξ is overtwisted, EH(L) is stable, so we're done. On the other hand, if ξ = ξ st , the unstable component of EH(L) is nonvanishing, and -when fixing either orientation -has Alexander degree 2Ã(EH(L)) = A(EH(L n,0 )) +Ã(EH(L 0,n )), and this suffices to determine it. Remark 4.3. Proposition 4.2 is a analogue to Theorem 1.2 in [LOSSz], which tells us that L − (L) is mapped to c(ξ) by setting U = 1 in the complex HF K − (−S 3 , K). See also Proposition 4.14 below. Proof of Proposition 4.2. We're first going to prove that if EH(L) is stable, ξ is overtwisted, via the following lemma (which will turn out to be useful also later). Let ψ ∞ denote the gluing map associated to the gluing of the standard neighbourhood of a Legendrian knot (i.e. the difference T ∞ = Y (1) \ Int(Y L )). Proof. Consider the Legendrian unknot L ⊂ (S 3 , ξ st ) with tb(K 0 ) = −1, and stabilise it once (with either sign) to get L . By gluing T ∞ to either S 3 L or S 3 L we obtain the contact structure ξ st on S 3 . Observe now that S 3 L is obtained from S 3 L by a stabilisation basic slice: it follows in particular that the union T of this basic slice and T ∞ is a tight solid torus. Honda's classification of tight contact structures of solid tori tells us that T is isotopic to T ∞ . Now the associativity of gluing maps (Theorem 2.14) tells us that, as T is isotopic (as a contact manifold) to T ∞ , ψ ∞ • σ ± = ψ ∞ . Suppose that x is stable, then there exists a positive integer N such that (σ − • σ + ) N (x) = 0, and therefore ψ ∞ (x) = ψ ∞ ((σ − • σ + ) N (x)) = 0. Suppose now that x is not stable. Then (σ − • σ + ) N (x) = 0 for all N . Notice that σ − • σ + carries homogenous elements to homogenous elements, and has degree 0. By Theorem 3.7, there is a sufficiently large integer N such that the image of x under (σ − • σ + ) N lies in the middle part of the complex. More precisely, it lies in a homogenous component of dimension 1, and in particular x N = (σ − • σ + ) N (x) is the generator of the unstable complex in its Alexanderdegree summand. We claim that ψ ∞ doesn't kill x N . Now take a knot L that is Legendrian with respect to the standard contact structure, and consider EH(L ). From the first part of the proof, we know that, for all k, ≥ 0, (σ k − • σ + )(EH(L )) ψ∞ −→ c(ξ st ) . But there are positive integers k, , m such that (σ k − •σ + )(EH(L )) and x N +m have the same Alexander degree, and are both nonzero. Since they live in the same 1-dimensional summand, they're equal, and in particular ψ ∞ (x N +m ) = ψ ∞ (EH(L )) = c(ξ st ) = 0. We can now conclude the proof of Proposition 4.2: recall that Eliashberg [El] proved that the only tight contact structure on S 3 is the standard one, and in particular a contact structure ξ on S 3 is overtwisted if and only if c(ξ) = 0. By the lemma above, though, c(ξ) = 0 if and only if EH(L) is stable. We can pin down the Alexander grading of EH(L) using an argument analogous to the one that Ozsváth and Stipsicz use for L − (L) [OS]. where x L is a generator representing L − (L) in some Heegaard diagram for S 3 . Let's consider the following set up: let (Σ, α, β, γ, z, w) be a triple Heegaard diagram, where (Σ, α, β, z, w) is obtained from an open book compatible with L as in [LOSSz], so that L − (L) is represented by a generator x in Notice that (Σ, β, γ, z, w) represents an unknot in # g−1 (S 1 × S 2 ), therefore we can choose a generator Θ representing the top-dimensional class in HF K(Σ, β, γ, z, w). Ozsváth and Szabó proved in [OSz2, Section 2] that, whenever we have a triangular domain ψ ∈ π 2 (x, y, Θ), then (4.1) s(y) − s(x) = (n w (ψ) − n z (ψ))P D(µ). We exhibit in Figure 4.1 a Whitney triangle ψ in π 2 (x, y, Θ) with n w (ψ) = 1, n z (ψ) = 0 connecting the generator x in (Σ, β, α, z, w) representing L(L) and the generator y for in (Σ, γ, α, D) representing EH(L), where D is a disc on γ 0 that touches the two regions of Σ \ (α ∪ γ) containing z and w. Notice that x and y live in the cohomology groups of CF L − (Σ, α, β, z, w) and SF C(Σ, α, γ, D) = CF K(Σ, α, γ, z, w), so we need to be careful when using Equation 4.1. More precisely, we want to consider a map SF C(Σ, β, α) → SF C(Σ, γ, α) (we omit basepoint for the sake of clarity), that is dual to a map SF C(Σ, α, γ) → SF C(Σ, α, β) so we should be looking at triangles in the triple Heegaard diagram (Σ, α, γ, β) instead of (Σ, α, β, γ). In particular, the grading shifts are reversed: for every triangular domain D in (Σ, α, β, γ) we associate the domain −D in (Σ, α, γ, β), so that n z and n w change signs. Since c 1 (s + α) = c 1 (s) + 2α for every s ∈ Spin c (Y ) and every α ∈ H 2 (Y ), it follows from the computations in [OS,Section 4] that We now prove that the hypothesis tb(L) ≤ 2τ (K) − 1 above is necessary: Proposition 4.6. For every non-loose unknot L in S 3 , EH(L) is nonvanishing and purely unstable. Proof. When K is the unknot, the stable complex of S 3 K,n is trivial for all values of n. Also, τ (K) = 0. According to Eliashberg and Fraser [EF], L has non-negative Thurston-Bennequin number tb(L) ≥ 0 = 2τ (K), and admits a tight Legendrian surgery (Y, ξ). Since L is topologically unknotted, Y is a lens space, and any tight contact structure on a lens space is Stein fillable: in particular c(Y, ξ) = 0. Then Lemma 2.13 applies, showing that also EH(L) = 0. We conclude the section by giving an alternative proof of the following fact, due to Etnyre and Van Horn-Morris, and Hedden [EV,He2]. If K ⊂ S 3 is a fibred knot, then it's the binding of an open book for S 3 , and any fibre is a minimal genus Seifert surface for K: call ξ K the contact structure on S 3 supported by this open book. Theorem 4.7. ξ K is tight if and only if τ (K) = g(K). Proof. K sits in ξ K as a transverse knot, and sl(K) = 2g(K) − 1. Let's consider a ξ K -Legendrian approximation L of K such that tb(L) 0. Vela-Vick proved that L(L) = 0 [Ve], therefore EH(L) = 0 [SV]. Since K is fibred, HF K(S 3 , K; g(K)) is 1-dimensional [OSz5]: using Proposition 4.5 above, together with Theorem 3.2 we see that EH(L) is the only nonzero element in the top degree component of SF H(−S 3 L ). If τ (K) = g(K), then EH(L) is also the generator in top degree of the unstable complex, and in particular 0 = ψ ∞ (EH(L)) = c(ξ K ). If τ (K) < g(K), on the other hand, the unstable complex is supported in degree strictly less than A(EH(L)), so 0 = ψ ∞ (EH(L)) = c(ξ K ). Thus, applying Eliashberg's classification result [El] as above, ξ K is tight if and only if c(ξ K ) = 0 if and only if τ (K) = g(K). Remark 4.9. Since we're taking a direct limit, what counts is just what happens for sufficiently large indices. In particular, we just need to know what happens for n ≥ n 0 := −2τ (K) + 1: this also fits in the picture of contact topology, since this is the only interval where EH(L) can live for a Legendrian L in (S 3 , ξ st ). What happens for other indices is that, with respect to the maps ψ m,n , the only component that survives is S: this is going to be more precise below, even though we discuss just the interval n ≥ n 0 . The group As defined, SF H − −− → is just a graded F-vector space: using the other (i.e. the positive) stabilisation map σ + , we can endow it with an F[U ]-module structure. One way to do it is to identify the projective limit with the quotient of the disjoint union A n by the relations x i ∼ x j whenever there exists N such that ψ i,N (x i ) = ψ j,N (x j ) and defining U · [x] = [σ + (x)]: since σ − and σ + commute, the map is well defined. Notice that the map σ + has now Alexander degree −1 (due to the degree shift introduced), and so does the map U · on HF K − (S 3 , K). Alternatively, we can see the map induced by σ + in a more abstract (and universal) way, considering the following diagram: SF H − −− → (−S 3 , K) SF H − −− → (−S 3 , K) U · O O A m ιm•σ + : : ψm,n / / ιm 8 8 A n ιn•σ + d d ιn f f . Ignoring the dashed arrow, the diagram commutes, since σ − and σ + commute, and by the universal property of the direct limit (and of the arrows ι n !), there's a uniquely defined map U ·, that is the dashed arrow. Remark 4.10. We have a dual direct system A + defined using σ + rather than σ − , and changing the sign of the degree shift. Reversing the orientation of K induces, as expected, an isomorphism of F-vector spaces SF H − −− → (−S 3 , K) SF H − −− → (−S 3 , −K): this follows from the fact that the two direct systems A − and A + are isomorphic. Moreover, the universal isomorphism commutes with the U -action, and this U -equivariance gives the isomorphism in the category of F[U ]-modules. This symmetry can also be seen as a choice for the labelling of positive vs negative stabilisation, which is in fact equivalent to the choice of an orientation. Before diving into the proof, recall Ozsváth and Szabó's description of HF K − (see for example [OSz6], especially Figures 1 and 2). The complex is a direct sum of countably many copies of HF K, each thought of as U k · HF K for k ∈ N: this gives the complex the F[U ]-structure; we think of each copy drawn as a vertical tile of Alexander-homogeneous components, and that all copies stacked in the plane like a staircase parallel to the x = y diagonal; the differential comes from the complex ( HF K, ∂) computing HF (S 3 ), and it can be depicted as a set of arrows pointing horizontally, each coming from a vertical arrow in ( HF K, ∂) and corresponding to a domain crossing the auxiliary basepoint w. There's a quite striking similarity between the first chunks of this complex and the first chunks of the complexes computing A n 's, and this similarity is both the inspiration and the key of the proof of the theorem. Proof. We'll split the proof in two steps: first we'll prove the isomorphism of the two as graded F-vector spaces, and then as F[U ]-modules. As usual, we'll call g = g(K) and τ = τ (K). Step 1. We want to prove there are maps j n : A n → H := HF K − (−S 3 , K) such that (H, {A n }, {j n }) satisfy the universal property for the direct limit of A − : C H φ O O A m φm ; ; ψm,n / / jm = = A n φn b b jǹ`. We need to define the maps j n first, and then we need to prove that for every commutative diagram with maps φ n to a module C there is a unique (dashed) map φ making the full diagram commute. The maps j n are easily defined: thanks to the previous description, HF K − (−S 3 , K) is the direct sum of a copy of S − ⊂ A n and a copy of F[U ], with A(U k ) = τ −k; imagining a superposition between the two pictures for the complexes computing A n and H yields to the claim that j n would like to be a fixed (i.e. not depending on n) graded isomorphism on S − , zero on S + and the degree 0, injective map U n → F[U ]: the commutativity of the lower triangle of the diagram is clear by the description of the maps σ ± . Now we can consider the full diagram, and show that φ is uniquely defined by (φ n ) n≥n 0 : consider an element x m = a m + s m ∈ A m , with s m ∈ S + and a m ∈ S − ⊕ U m , and consider the diagram for n = m + d m: since the lower triangle is commutative, we have that φ m (x m ) = φ n (σ d − (x m )) = φ n (σ d − (a m )) = φ m (a m ) , so φ m (S + ) = 0: this implies that the map φ m factors through j m . Now, define φ by φ\| S − = φ m \| S − for some m and φ\| F[U ]/(U m ) = φ m •j −1 m : notice how φ is well defined (since σ − is an isomorphism on S − and the injection of degree + 1 /2 on the unstable complex), and makes the diagram commute. Since j m is injective on S − ⊕ U m and F[U ] is the direct limit of F[U ]/(U k ), this is the only way we can define φ, and this concludes the first part of the proof. Remark 4.12. It's worth remarking explicitly what we've proven: we've shown that the inclusion map ι n : HF K(−S 3 −n (K),K) → SF H − −− → (−S 3 , K) is injective on S − ⊕ U , and that S + = ker ι n for each n ≥ n 0 . Moreover, for n sufficiently large, the map ι n is an isomorphism between truncations of A n and HF K − (−S 3 , K) that forgets of all elements of low Alexander degree. Step 2. We now need to prove that the two F[U ]-module structure correspond under some map: we just need to show that the universal map Φ in the diagram H SF H − −− → (−S 3 , K) Φ O O A m jm 8 8 ψm,n / / ιm 8 8 A n jn f f ιn f f is U -equivariant, since the universal property for (H, {A n }, {ι n }) already implies that it's an F-isomorphism. For x ∈ A n , the map Φ sends ι n (x) to the class [x] = j n (x). We have a good way to picture Φ when the framing is large: in this case, we just superpose the picture of the complex described in Section 3.2 above with Ozváth and Szabó's description, and identify generators pointwise. But we're working with the projective limit SF H − −− → , which is not the disjoint union A n , but rather its quotient by the relation x ∼ ψ m,n (x). Up to changing the choice of n and x, we can suppose that Theorem 3.2 above applies: in this case, the map σ + is just an injection of A n on the bottom of A n+1 , which, in Ozsváth and Szabó's picture corresponds to shifting each copy U k · HF K(−S 3 , K) to the next one, U k+1 · HF K(−S 3 , K), hence proving the U -equivariance of Φ. Remark 4.15. EH(L) is an unoriented invariant, i.e. doesn't see orientation reversal, whereas the sign of the stabilisation does (see Remark 3.5), so one apparently can find a contradiction in Proposition 4.14. What happens is that when we reverse the orientation of L, we also reverse the orientation of K and we swap the rôles the two maps σ − and σ + play. The two resulting groups, associated to A − and A + are -as already noticed -isomorphic, but in the first one σ − acts trivially and σ + acts as U (as seen in the proof of Proposition 4.14.(i,ii)), while in the second one we'd have to write: Notice how the theorem above is formally identical to our Proposition 4.14: it's therefore natural to compare the two invariants EH − − → and L − . We postpone the proof of the main theorem to the last subsection, and draw some conclusions from the theorem, first. It's now worth stressing and making precise what we've announced in the introduction, that EH(L) (but not EH − − → (L)!) contains at least as much information as L − (L) and L − (−L) together. We can prove the following refinement of Theorem 1.1: Theorem 5.3. For two oriented Legendrian knots L 0 , L 1 in (S 3 , ξ) of topological type K, with tb(L 0 ), tb(L 1 ) ≤ 2τ (K), the following are equivalent: In general, withouth any restriction on the Thurston-Bennequin numbers of L 0 and L 1 , (i) implies (ii). Proof. EH(L i ) determines both L − (L i ) and L − (L i ) by Theorem 5.2, so (ii) follows from (i). Let's now suppose that the constraint on the Thurston-Bennequin invariants holds. As already observed (see Remarks 3.5 and 4.15), EH − − → is an oriented invariant of Legendrian knots with the following property: EH − − → (L 1 ) = EH − − → (L 2 ) if and only if the components of EH(L 1 ) and EH(L 2 ) along S − and U agree. In particular, if L − (L 0 ) = L − (L 1 ) the components of EH(L i ) along S − and along U are equal; if L − (−L 0 ) = L − (−L 1 ), then the S + components of EH(L 1 ) and EH(L 2 ) agree, too, thus showing that EH(L 1 ) = EH(L 2 ). 5.1. Triangle counts. The proof of Theorem 5.2 relies on bypass attachments on contact sutured knot complements and the induced gluing maps, henceforth called simply bypass maps. There is another description of a sutured manifold with torus boundary and annular R + we're going to need: an arc diagram H a is a quintuple (Σ, α, β a , β c , D), where Σ is a closed surface, α and β c are sets of nondisconnecting, simple closed curves in Σ, D is a closed disc disjoint from α ∪ β c and β a is an arc properly embedded Σ \ (Int(D) ∪ β c ). We further ask that \|α\| = g = g(Σ), and \|β c \| = g − 1. We will often drop D from the notation and write β for β c ∪ {β a }, for sake of brevity. We build a sutured manifold (M, Γ) with torus boundary and two parallel sutures out of H a as follows: the set of α-curves determines how to attach g upside-down 2-handles on Σ×{0} ⊂ Σ×[0, 1]; we attach a 0-handle (a ball) to fill up the remaining component of the lower boundary; the set β c of β-curves determines the attaching circles of 2-handles on Σ × {1}. We define M to be the manifold obtained by smoothing corners after these handle attachments; notice that D is an embedded disc in ∂M , and β a is an embedded arc in ∂M . Let R + be a small regular neighbourhood of D ∪ β a and Γ be its boundary. We can now consider the chain complex SF C(H a ) as usual, by taking gtuples of intersection points of α-curves and β-curves and arcs, so that no two points lie on the same curve or arc, and the differential counts holomorphic discs whose associated domains do not touch the disc D. It is clear that SF C(H a ) is isomorphic as a chain complex to the complex associated to a doubly-pointed Heegaard diagram representing the dual knot K inside S 3 γ (K); in particular, it is also chain homotopic to a complex computing SF H(M, Γ). Remark 5.4. The construction above is related to Zarev's bordered sutured manifolds and their bordered sutured diagrams [Za], and in fact generalises to sutured manifolds with connected R + . What we called arc diagrams are in fact similar to bordered sutured diagrams (but not to what he calls arc diagrams). In order to obtain the bypass maps we need to count holomorphic triangles in triple arc diagrams. At the level of arc diagrams, attaching a bypass to (M, Γ) corresponds to choosing another arc γ a on Σ, which intersects β a transversely in a single point θ a . Every γ-curve is a small perturbation of a β-curve in H β , and therefore there is a preferred choice among the two intersection points (see [OSz3] and Section 5.1.2 below), giving an element Θ. We then have: Theorem 5.5 ( [Ra]). The bypass map is induced by the triangle count map F (· ⊗ Θ) associated to the triple diagram described above. Somewhat confusingly, the rôles of α-and β-curves are reversed when talking about contact invariants, since we're looking at elements in SF H(−M, −Γ) rather than in SF H(M, Γ): we will be very explicit and careful about the issue of triangle counts in this setting, as we discuss below. Recall that, given a Legendrian knot L in any contact 3-manifold, EH(L) is the class of a generator x EH ∈ SF C(Σ, β, α), where H = (Σ, α, β) is an arc diagram representing S 3 L (notice the order of α and β). 5.1.1. α-slides. If we want to do a handle-slide among the α-curves in H, say changing α = {α i } to α = {α i }, what we are doing is replacing the second set of curves in a (doubly-pointed) Heegaard diagram. A triangle count in (Σ, β, α, α ) gives a map CF (β, α) ⊗ CF (α, α ) ⊗ CF (α , β) → F, which in turn gives a map F αα : CF (β, α) ⊗ CF (α, α ) → CF (β, α ). (Here we've been dropping Σ from the notation, and we'll do it again later.) In all cases we're going to meet, the top-dimensional generator in HF (α, α ) will be represented by a single generator, that we call Θ αα , and the map we'll be looking at is Ψ αα : F αα (· ⊗ Θ αα ). Consider a holomorphic triangle ψ connecting x to y giving a nontrivial contribution to the Ψ αα ; that is, the Maslov index of ψ is 0 and the moduli space contains an odd number of points. The boundary of the domain D(ψ) associated to ψ has the following behaviour along its boundary: A1. ∂∂ α D(ψ) = x − Θ αα ; A2. ∂∂ α D(ψ) = Θ αα − y; A3. ∂∂ β D(ψ) = y − x. This amounts to saying that if we travel along ∂D(ψ) following the orientation induced by D(ψ) we cyclicly run along curves in the order β, α , α. β-slides. On the contrary, if we're doing some triangle count that changes the β-curves or arcs instead (as we will see below), we are going to face the opposite behaviour. More precisely, consider a set of curves β . A triangle count in (Σ, β, α, β ) gives a map CF (β, α) ⊗ CF (α, β ) ⊗ CF (β , β) → F, which in turn gives a map F βαβ : CF (β, α) ⊗ CF (β , β) → CF (β , α). In all cases we're going to meet, the top-dimensional generator in HF (β , β) will be represented by a single generator, that we call Θ β β , and the map we'll be looking at is Ψ ββ : F αα (· ⊗ Θ ββ ). Notice that Θ β β represents the bottom-dimensional generator of HF (β, β ). Therefore, if we call ψ a triangle as above, giving a nontrivial summand y in F ββ (x) connecting generators x and y, we get the following conditions on D(ψ): B1. ∂∂ α D(ψ) = x − y; B2. ∂∂ β D(ψ) = Θ β β − x; B3. ∂∂ β D(ψ) = y − Θ β β . That is to say that moving along ∂D(ψ) following the orientation induced by D(ψ) we meet the curves β, β , α. 5.2. Proof of Theorem 5.2. The idea underlying the proof is to find explicit representatives for the two contact invariants EH − − → (L) and L − (L) that live in suitable Heegaard diagram, and compare them. The proof will be divided in three steps: (1) We construct an open book (S 3 , ξ, L ) for a single negative stabilisation L of L, together with an associated arc diagram H sut and an associated doubly-pointed Heegaard diagram H knot , representing SF H(−S 3 L ) and HF K − (−S 3 , L ) respectively. (2) We consider a large negative stabilisation L stab of L . Stabilisations corresponds to bypass attachments on H sut : we compute the associated triangle counts, obtaining a generator in a diagram H stab , representing EH(L stab ). Moreover, H stab has a handle that is very similar to the winding region (see Figure 5.1). (3) Finally, we handle-slide a single α-curve and compare H stab with H knot using a refinement of a result of Hedden [He1]. Figure 5.1. The winding region: the picture represent a handle, with the top and the bottom sides of the rectangle identified according to arrows. The horizontal curve (in red) is a α-curve, the vertical curve (in blue) is a β-curve, representing the meridian for the knot in S 3 , whereas the curve that winds along the handle (in green) is the γ-curve representing the given framing on the boundary of S 3 \ ν(K): basepoints are placed such that (Σ, α, β, z, w) represents (S 3 , K), while (Σ, α, γ, z , w) represents (S 3 n (K),K). Proof. Step 1. Recall the definition of L − : given L ⊂ (S 3 , ξ), using an idea of Giroux ([Gi], see also [Et]) we can construct an open book (F, φ) with L sitting on one of the pages (identified with F , so that L ⊂ F ) as a homologically nontrivial curve. We then choose a basis for F (in the sense of Definition 2.8) with only one arc, say a 1 , intersecting L. We can construct a doubly-pointed Heegaard diagram as we did for the EH-diagram, the only thing to take care of being placing the two basepoints (see [LOSSz]). A representative for L − is now given by the only intersection point entirely supported in F ⊂ Σ. The following lemma is implicitly used by Stipsicz and Vértesi [SV]. Lemma 5.6. The partial open book (S, P, h) := (F, F \ ν(L), φ\| P ) represents the manifold (S 3 L , ξ\| S 3 L ), where ν(L) is a small neighbourhood of L in S 3 . Proof. The contact sutured manifold (M, Γ) associated to (S, P, h) embeds in S 3 as the complement of a small neighbourhood of L, since we can embed the two halves of M inside the two halves of S 3 given by (F, φ), respecting the foliation: this shows that (M, Γ) is contactomorphic to S 3 L . Remark 5.7. We can read off a sutured Heegaard diagram associated to (S, P, h) directly from the doubly-pointed Heegaard diagram for (S 3 , L): we just need to remove the basepoints, together with a (small, open) neighbourhood of L in the Heegaard surface, and erase the two curves corresponding to a 1 and b 1 . The remaining a i 's form a basis for the (S, P, h), so the EH invariant is already on the picture. If we also want to have an arc diagram for S 3 L , we can to do the following. We start with the doubly-pointed Heegaard diagram, and replace the curve β 1 with a curve λ parallel to L. Then we add a disc D along it this curve, that is disjoint from α 1 and lies in the two regions that are occupied by the basepoints. Finally, we just forget about the basepoints and let β c 1 be the arc with endpoints in D that runs along λ. Notice that this arc arc intersects a single α-curve (namely, α 1 ) exactly once. Notice that in this case the chain complexes associated to the sutured Heegaard diagram and the arc diagram are trivially isomorphic (as chain complexes), since they have exactly the same generators and count precisely the same curves (since the arc intersects only α 1 , and in a single point). L is isotopic to L inside F , except that it runs once along the handle [On]. Let's see what happens at the level of arc diagrams: recall that the invariant EH(L ) is represented by a chain x EH in the arc diagram H a = (Σ, β a , β c , α) coming from the open book (F , φ ) together with the embedding L ⊂ F . In particular we have that Σ = F ∪ −F and D ∪ β a ⊂ F ⊂ Σ. Call g + 1 the genus of Σ; the α-curves are obtained after choosing a basis {a 0 , a 1 , . . . , a g } for F . We choose this basis so that a 0 is the co-core of the handle H ⊂ F , and is the only arc intersecting L inside the page, and a 1 is the only other arc intersecting the curve c above (this is always possible). Finally, we let β 0 = β a the arc that runs parallel to L inside F , α 0 = a 0 ∪ −a 0 , which is the only curve that intersects β 0 , α 1 = a 1 ∪ −a 1 , and we number the remaining curves so that α i and β i intersect once inside F . Recall that x EH is the generator consisting of all the intersection points x i = α i ∩ β i inside F . Step 2. We now want to attach bypasses to the sutured knot complement S 3 L and compute the associated gluing maps, as indicated in Theorem 5.5. When we stabilise L we attach a bypass to the sutured knot complement, and the framing of the sutures decreases by 1. We're going to obtain an arc diagram (Σ, γ a , γ c , α) for a stabilisation L of L by attaching a bypass to the sutured knot complement S 3 L (see [SV]): the Heegaard surface Σ and the curves α are the same as in H a ; also, γ c = β c . The curve γ 0 is obtained by juxtaposing β 0 and µ as in Figure 5.3, where µ is the meridian of L ⊂ S 3 . Notice that µ can be obtained by taking a 0 = α 0 ∩ F and letting µ = a 0 ∪ −φ (a 0 ); in other words, µ is the curve β 0 in the doubly-pointed Heegaard diagram of representing HF K − (−S 3 , L). Observe also that the arc γ 0 intersects β 0 transversely in a single point, θ 0 . We're now ready to compute the action of the bypass map on x EH ; we're going to denote the bypass maps induced by negative stabilisations σ − . In order to be able to do a triangle count, we need to perturb the β-curves to obtain curves γ 1 , . . . , γ g . We choose the perturbations so that γ i has the following two properties: • it intersects β i transversely in two points, both inside F and separated along β i ∩ F by α i (see Figure 5.4); • it intersects α i ∩ F transversely in a single point y i . The two intersection points of β i and γ i are connected by a bigon B inside F . We label them θ i and θ i so that this a B connects θ i to θ i . Notice that this is the opposite of the usual convention for triangle counts (see 5.1.2 above). We let Θ = {θ i }. We're going to do a triangle count in (Σ, β, α, γ, D); in the notation of 5.1.2 above, the map σ − is induced by F − = F βαγ (· ⊗ Θ). Let F − (x EH ) = n k=1 y k , where all summands are distinct (such a representation exists and is unique up to permutations, since we're working with coefficients in F). In this picture, we represent F together with a small neighbourhood of c inside −F ; the dashed curve represents the meridian µ for L ⊂ S 3 , and γ 0 is obtained from β 0 through a right-handed Dehn twist along µ. We omit the curves β 1 and γ 1 to avoid cluttering the picture. Lemma 5.8. For each k = 1, . . . , n, the intersection point of y k along α i for i > 0 is y i . Proof. Let ψ be a holomorphic triangle contributing to the summand y j in σ − (x) = y k . Let's consider F ⊂ Σ in a neighbourhood of α i containing also β i and γ i . The arcs a i = α i ∩ F for i ≥ 0 don't disconnect F by construction; moreover, the arc β 0 is entirely contained in F and does not meet any α i for i > 0, while γ 0 ∩ F is made of two arcs that run along β except near α 0 . It follows that the two unbounded regions to the left and right of Figure 5.4 are in fact the same region, which touches the disc D. Therefore, the multiplicity of D(ψ) on this regions is 0. Since the multiplicity at the left of x i has to vanish, the corner of D(ψ) at x i is acute and is contained in the small triangle, shaded in the picture. Since the multiplicity at the right of y i , too, vanishes, there has to be a corner at y i , to, and in particular the α i -component of y j has to be y i . Moreover the domain D(ψ) has to be a small triangle in the pictured region. We now look at the intersection points of α 0 ∩ γ 0 . Let's call y 0 the first intersection point of γ 0 and α 0 we meet when we travel along γ 0 starting from D and going in the direction of θ 0 . Lemma 5.9. For each k = 1, . . . , n, the intersection point of y k along α 0 is y 0 . Proof. The remaining intersection point of y k lies on α 0 and β 0 , since all other curves already have an intersection point on them. Notice also that there's a small triangle connecting θ 0 , x 0 and y 0 , so that y = {y 0 , . . . , y g } does in fact appear in the sum. See Figure 5.3 Consider now a holomorphic triangle ψ and its domain D = D(ψ). ∂ β D has to be the the short arc connecting x 0 and y 0 in the handle H, since the complement of this arc touches the base-disc D. Consider a small pushoff a of α 0 disjoint from this arc and from α 0 itself. Observe that ∂D is nullhomologous and ∂D ∩ a = ∂ γ D ∩ a. Therefore, a has to have trivial algebraic intersection with the γ-boundary of D. Suppose that D connects x 0 with another intersection of γ 0 with α 0 : its γ-boundary ∂ γ D is homologous to a linear combination of α 0 and the meridian µ, where µ appears with nonzero multiplicity. In particular, this contradicts the fact that a intersects ∂ γ D trivially, since \|a ∩ µ\| = 1. In particular, all y k s are equal, therefore EH(L ) = σ − (EH(L )) = [y]. We want to iterate the procedure, and stabilise L . The bypass we need to attach only modifies γ 0 by juxtaposition with µ, and in particular Lemma 5.8 holds in this case as well. Notice also that the only thing we used in proving Lemma 5.9 is that x 0 and y 0 were the first intersection points of α 0 with the arcs β 0 and γ 0 respectively, so -up to notational modifications -Lemma 5.9 holds for iterations of bypass attachments. In particular, we've computed the action of σ n − on EH(L ) for every n ≥ 0. Step 3. We now slide α 1 over α 0 to obtain α 1 . Recall that a i = α i ∩ F intersects the curve c that we used to stabilise the open book (F, φ) only if i = 0, 1. In particular, α 1 is disjoint from µ and the only α-curve that intersects µ is α 0 . Call H f inal the Heegaard diagram (Σ, β, α , D). We're going to compute the action of the map HS induced by this handleslide on the contact invariant. Let (Σ, β, α, α ) be the triple Heegaard diagram associated to the handleslide, and x EH be the contact invariant as computed in the previous step and y c be the intersection point in H f inal that is closest to x EH (see below for a more precise description). Lemma 5.10. The handleslide map HS sends x EH to y c . Proof. As above, let HS(x EH ) = y k where all summands are distinct. On all curves other than α 0 and α 1 the same argument as in Lemma 5.8 applies with no modification (see 5.1.1 for the orientation issues): Figure 5.5 represents what happens locally around x i and is obtained from Figure 5.4 through a rotation by 180 degrees. Figure 5.6. This picture represents a neighborhood of the sliding region between α 0 and α 1 in F In fact, the same argument applies to the triple β 1 , α 1 , α 1 : looking at Figure 5.6, we see that for every y k in the sums the intersection point on α 1 is the intersection of α 1 and β 1 on F . First of all, there is a small triangle T 1 connecting x 1 to y 1 inside F . To prove that there can be no other domain, we observe that the multiplicity has to vanish in the corner at x 1 across from T 1 , since this region touches the disc D. On the other hand, this is enough for the proof of Lemma 5.8 to work. Finally, we take care of the intersection point of β 0 ∩ α 0 , that is the first intersection point when moving from the D along β 0 , traversing the handle H first. This is similar to the proof of Lemma 5.9 above, and it follows from the same homological considerations. Observe that a neighbourhood of the meridian µ L of L ⊂ S 3 in the diagram looks like half of the winding region, as in Figure 5.7. Call x 0 the intersection point of µ L with α 0 , and number the intersection points of α 0 with β 0 as Figure 5.7. The neighbourhood of µ L in H f inal . The twisting is all on one side of µ L (the vertical curve). The intersection points on α 0 (the horizontal curve) are labelled x 0 , x 1 , . . . from right to left. We also put the basepoints z and w (in gray) to represent the knot K ⊂ S 3 . x 1 , x 2 , . . . according to the order in which we meet them when travelling along α 0 (so that x 0 comes first). An easy adaptation of the proof of Theorem 4.1 in [He1] shows the following: Proposition 5.11. All generators in H f inal with sufficiently large Alexander degree have an intersection point in the winding region. Moreover, the map Φ : SF C A≥N (−S 3 L stab ) → CF K − A≥N (S 3 , K) defined by Φ({x n } ∪ x) = U n · ({x 0 } ∪ x) induces an isomorphism of chain complexes when N is sufficiently large and N = N + tb(L stab )+1 2 . In particular, the generator y 0 we've shown to represent EH(L stab ) is of the form {x 1 }∪x, where the generator {x 0 }∪x ∈ CF K − (S 3 , K) represents L − (L). It follows that under map induced at the chain level by Φ maps EH(L stab ) to L − (L), therefore concluding the proof of Theorem 5.2. (i) EH(L 0 ) = EH(L 1 ); (ii) L − (L 0 ) = L − (L 1 ) and L − (−L 0 ) = L − (−L 1 ). The chain x(S, P, h) ∈ SF C(Σ, β, α) is a cycle, and its class in SF H(−M, −Γ) is an invariant of the contact manifold (M, ξ) defined by the partial open book (S, P, h). Definition 2.10. EH(M, ξ) is the class [x(S, P, h)] ∈ SF H(−M, −Γ) for some partial open book (S, P, h) supporting (M, ξ). Proposition 4.2. ξ is overtwisted if and only if EH(L) is stable. Lemma 4.4. A homogeneous element x ∈ SF H(−S 3 K,n ) is stable if and only if ψ ∞ (x) = 0. Proposition 4. 5 . 5Identifying SF H(−S 3 L ) = HF K(S 3 tb(L) (K), K) as in Proposition 2.6, EH(L) is homogenous of Alexander degree −r(L)/2. Proof. In [OS, Theorem 4.1], Ozsváth and Stipsicz compute the Alexander degree of L − (L) by a combinatorial argument on an open book compatible with L. They obtain that A(L − (L)) = 1 2 c 1 (s(x L )) − P D([µ L ]), [F, ∂F ] = tb(L) − r(L) + 1 2 , Figure 4 . 1 . 41The triple Heegaard diagram used in the proof of Proposition 4.5 CF K − (Σ, β, α, z, w). Now define γ to be obtained from β by replacing β 0 with L ⊂ F as sitting inside the page of the open book, and positioned with respect to z, w as inFigure 4.1. c 1 (s(y)), [F, ∂F ] = c 1 (s(x)), [F, ∂F ] −2 = 2A(L − (L))+1−2 = tb(L)−r(L). If we now plug this in Equation 3.1 and we use P D([µ K ]), [F, ∂F ] = tb(L), we get A(EH(L)) = c 1 (s(y)) − P D([µ K ]), [F, ∂F ] 2 = − r(L) 2 , since P D([µ K ]), [F, ∂F ] = tb(L) by construction of S 3 L . SF H − −− → . Let's step back for a second, and consider an oriented topological knot K in S 3 .Given a graded vector spaceV = d V d , we denote with V {s} a graded vector space with graded components (V {s}) d = V d−s . Consider the family of graded F-vector spaces A n := HF K(−S 3 −n (K),K){(1 − n)/2}, indexed by integers (notice the − signs in the definition of A n ); for each n we have a degree 0 map σ − : A n → A n+1 , the (negative) stabilisation map, induced by the negative basic slice attachment; these data can be conveniently summarized in a direct system A − := ((A n ), (ψ m,n ) n≥m ), where the map ψ m,n : A m → A n is σ n−m − . Definition 4.8. Let SF H − −− → (−S 3 , K) to be the direct limit lim − → A σ , and call ι n the universal map ι n : A n → SF H − −− → (−S 3 , K). Theorem 4 . 11 . 411The groups SF H − −− → (−S 3 , K) and HF K − (−S 3 , K) are isomorphic as F[U ]-modules. 4. 3 . 3EH − − → invariants. Suppose now we have an oriented Legendrian knot L in (S 3 , ξ), of topological type K: by construction, we have a naturally defined oriented contact class in SF H − −− → (−S 3 , K). Definition 4.13. Define the class EH − − → (L) ∈ SF H − −− → (−S 3 , K) as [EH(L)], in the identification SF H − −− → (−S 3 , K) = A n / ∼. We can immediately read off some facts about this new invariant, that follow straight away from the definition: Proposition 4.14. Consider an oriented Legendrian L in (S 3 , ξ) of topological type K; then: (i) for a negative stabilisation L of L, EH − − → (L ) = EH − − → (L); (ii) for a positive stabilisation L of L, EH − − → (L ) = U · EH − − → (L); (iii) EH − − → (L) is an element of U -torsion if and only if ξ is overtwisted. (iv) EH − − → (L) sits in Alexander grading tb(L) − r(l) ) L is a negative stabilisation of L, so EH(L ) = σ − (EH(L)), and EH − − → (L ) = [EH(L )] = [σ − (EH(L))] = [EH(L)] = EH − − → (L). (ii) L is a positive stabilisation of −L, so EH(L ) = σ + (EH(L)), and EH − − → (L ) = [EH(L )] = [σ + (EH(L))] = U · [EH(L)] = U · EH − − → (L). ( iii ) iiiBy definition, an element [x] of SF H − −− → (−S 3 , K) vanishes if and only if σ k − (x) = 0 for some k, and is of U -torsion if and only if [σ h + (x)] = 0 for some h: in particular, since σ − and σ + commute, [x] is of U -torsion if and only if (σ − • σ + ) (x) = 0 for some . If tb(L) > 2τ (K) (and therefore ξ is overtwisted), we know that SF H(−S 3 L ) = ker(σ − • σ + ) , so in particular EH(L) is U -torsion. On the other hand, if tb(L) < 2τ (L), Lemma 4.2 tells us that (σ − • σ + ) (EH(L)) vanishes if and only if ξ is overtwisted. (iv) EH(L) lives in the group HF K(−S 3 tb(L) (K), K), and by Proposition 4.5, its Alexander degree is −r(L)/2. Therefore, it lives in degree tb(L)−r(L)+1 2 in A −tb(L) and in SF H − −− → (−S 3 , K). EH − − → (L ) = [EH(L )] = [σ − (EH(L))] = U · [EH(L)], EH − − → (L ) = [EH(L )] = [σ + (EH(L)] = [EH(L))].4.4. Transverse invariants. Let's just recall the classical theorem relating transverse and Legendrian knots: it will be the key fact throughout this subsection. Theorem 4 . 416 ([EH]). Two transverse knots are transverse isotopic if and only if any two of their Legendrian approximations are Legendrian isotopic up to negative stabilisations.As it happens for L − , also EH − − → descends to a transverse isotopy invariant of transverse knots:Definition 4.17. Given a transverse knot T in (S 3 , ξ) of topological type K, we can define EH − − → (T ) = EH − − → (L) for a Legendrian approximation L of T .The transverse element is well-defined, in light of Proposition 4.14 and Theorem 4.16. A stronger statement holds, the natural counterpart of Proposition 4.1, that reveals a transverse nature of EH:Theorem 4.18. Suppose L, L are two oriented Legendrian knots in S 3 that have the same classical invariants. Suppose also that both the transverse pushoffs of L, L and the ones of −L, −L are transversely isotopic. Then EH(L) = EH(L ). Proof. Since the pushoffs of L and L (respectively, of −L and −L ) are transverse isotopic, EH − − → (L) = EH − − → (L ) (resp. EH − − → (−L) = EH − − → (−L )). By Remark 4.12, and by the behaviour of σ ± on the unstable complex, we can reconstruct all three components (that is, along S ± and U ) of EH(L) from EH − − → (L) and EH − − → (−L), and this concludes the proof. 5. EH − − → vs L − Fix an oriented Legendrian knot L in (S 3 , ξ), of topological type K: the LOSS invariant L − (L) is an element of HF K − (−S 3 , K), which has just been proven isomorphic to SF H − −− → (−S 3 , K), where EH − − → (L) lives. Let's also recall the following theorem: Theorem 5.1. [LOSSz, Theorems 1.2 and 1.6] For L as before: (i) for a negative stabilisation L of L, L − (L ) = L − (L); (ii) for a positive stabilisation L of L, L − (L ) = U · L − (L); (iii) L − (L) is an element of U -torsion if and only if ξ is overtwisted. (iv) L − (L) sits in Alexander degree tb(L)−r(L)+1 2 . Theorem 5 . 2 . 52Given L as before, there's an isomorphism of bigraded F[U ]modules SF H − −− → (−S 3 , K) → HF K − (−S 3 , K) taking EH − − → (L) to L − (L). ( i ) iEH(L 0 ) = EH(L 1 ); (ii) L − (L 0 ) = L − (L 1 ) and L − (−L 0 ) = L − (−L 1 ). Figure 5 . 2 . 52On the left we have a 1-handle of the page of an open book for (Y, ξ, L), where the arrow represents L. On the right we have the page with the additional handle H (shaded), the curve c along which we perform a positive Dehn twist; the arrow represents L (which otherwise agrees with L). We now want to know what happens to this picture when we stabilise L negatively to get L . If L sits on a page F of the open book (F, φ), L sits on a page of the open book (F , φ ) = (F ∪ H, φ • δ c ), where H is a 1-handle attached to the boundary of F as in Figure 5.2 and δ is a positive Dehn twist along the curve c, dashed in the figure. Figure 5 . 3 . 53Figure 5.3. In this picture, we represent F together with a small neighbourhood of c inside −F ; the dashed curve represents the meridian µ for L ⊂ S 3 , and γ 0 is obtained from β 0 through a right-handed Dehn twist along µ. We omit the curves β 1 and γ 1 to avoid cluttering the picture. Figure 5 . 4 . 54This picture represents a neighborhood of α i in F , for some positive i. Figure 5 . 5 . 55This picture represents a neighborhood of α i in F , for some positive i. Acknowledgments. I'm very grateful to my supervisor, Jake Rasmussen, On the equivalence of Legendrian and transverse invariants in knot Floer homology. J Baldwin, D Vela-Vick, V Vértesi, Geom. Topol. 172J. Baldwin, D. Vela-Vick, V. Vértesi: On the equivalence of Legendrian and transverse invariants in knot Floer homology, Geom. Topol. 17 (2013), no. 2, 925-974. Contact 3-manifolds twenty years since J. Martinet's work. Y Eliashberg, Ann. Inst. Fourier (Grenoble). 421-2Y. Eliashberg: Contact 3-manifolds twenty years since J. 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[ "Multi-Season Analysis Reveals the Spatial Structure of Disease Spread", "Multi-Season Analysis Reveals the Spatial Structure of Disease Spread" ]	[ "Inbar Seroussi ", "Nir Levy ", "Elad Yom-Tov ", "Microsoft ", "Herzeliya ", "Israel " ]	[]	[]	Understanding the dynamics of infectious disease spread in a heterogeneous population is an important factor in designing control strategies. Here, we develop a novel tensor-driven multi-compartment version of the classic Susceptible-Infected-Recovered (SIR) model and apply it to Internet data to reveal information about the complex spatial structure of disease spread. The model is used to analyze state-level Google search data from the US pertaining to two viruses, Respiratory Syncytial Virus (RSV), and West Nile Virus (WNV). We fit the data with correlations of R 2 = 0.70, and 0.52 for RSV and WNV, respectively. Although no prior assumptions on spatial structure are made, human movement patterns in the US explain 27-30% of the estimated inter-state transmission rates. The transmission rates within states are correlated with known demographic indicators, such as population density and average age. Finally, we show that the patterns of disease load for subsequent seasons can be predicted using the model parameters estimated for previous seasons and as few as 7 weeks of data from the current season. Our results are applicable to other countries and similar viruses, allowing the identification of disease spread parameters and prediction of disease load for seasonal viruses earlier in season.	10.1016/j.physa.2020.124425	[ "https://arxiv.org/pdf/1902.04073v1.pdf" ]	60,441,173	1902.04073	3fa6233903b6b9594bf777b1f409c7dd82deb143	Multi-Season Analysis Reveals the Spatial Structure of Disease Spread Inbar Seroussi Nir Levy Elad Yom-Tov Microsoft Herzeliya Israel Multi-Season Analysis Reveals the Spatial Structure of Disease Spread Understanding the dynamics of infectious disease spread in a heterogeneous population is an important factor in designing control strategies. Here, we develop a novel tensor-driven multi-compartment version of the classic Susceptible-Infected-Recovered (SIR) model and apply it to Internet data to reveal information about the complex spatial structure of disease spread. The model is used to analyze state-level Google search data from the US pertaining to two viruses, Respiratory Syncytial Virus (RSV), and West Nile Virus (WNV). We fit the data with correlations of R 2 = 0.70, and 0.52 for RSV and WNV, respectively. Although no prior assumptions on spatial structure are made, human movement patterns in the US explain 27-30% of the estimated inter-state transmission rates. The transmission rates within states are correlated with known demographic indicators, such as population density and average age. Finally, we show that the patterns of disease load for subsequent seasons can be predicted using the model parameters estimated for previous seasons and as few as 7 weeks of data from the current season. Our results are applicable to other countries and similar viruses, allowing the identification of disease spread parameters and prediction of disease load for seasonal viruses earlier in season. Introduction The spread of infectious diseases through a population of susceptible individuals varies in time, space, and according to the characteristics of the susceptible individuals and the disease [1]. Modeling this spread can provide information on the spreading mechanism of the infection and can assist in designing strategies for control of the disease. One common method to model infections is compartmental models [2,3], which in their basic form assume a single population divided into 3 sub-groups of susceptible, infected, or recovered (SIR) individuals. This model describes a global spreading mechanism where each infected individual can infect any susceptible individual. Another common model describes local spreading [4], where infected individuals only spread infection to a limited subset of susceptible individuals. In reality, many epidemics spread through a combination of two or more spreading mechanisms; hybrid spreading [5]. Such spreading can be modeled by multiple compartments encompassing different demographic characteristics [4], various disease strains [6,7,8,9,10], and cross-immunity in an age structured model [11,12,13]. However, fitting data from a given epidemic to these models remains a challenge owing to a dearth of data of sufficiently high spatial and temporal resolution. This problem is exacerbated the higher the resolution of the model, since the number of model parameters grows with the number of compartments of the model. Thus, in many cases researchers introduce simplifying assumptions in order to reduce the dimensionality of the models [14,15,16]. In the absence of actual data, the spreading topology can generally be analyzed by numerical simulation of models using synthetic data, for example, of the patterns of human movements at the desired scale [16,17]. It can also be analyzed through the use of theoretical tools to detect critical phenomena in networks [18,19,5]. Another approach to overcome a lack of epidemiological data is to use proxy data, such as mobile phone data [20] or Internet data [21]. Internet data, including search engine queries, social media postings [22,23], or Wikipedia visits can serve as a proxy for such information and provide data at high spatial and temporal resolutions from very large populations. One demonstration of the value of these data is the recent evaluation of the childhood influenza vaccine campaign in England [23], which showed that vaccination of primary school-age children significantly reduces influenza rates as estimated from Twitter posts and Bing queries. In this study, we focus on modeling the spatial and temporal spread of infectious disease using a tensor-driven multi-compartment version of the classic Susceptible-Infected-Recovered (SIR) model. We illustrate the ability of the model to capture disease parameters by fitting it to Internet data on two common viruses, Respiratory Syncytial Virus (RSV) and West Nile Virus (WNV) in the United States (US). Our results demonstrate that the spatial and temporal dynamics of these viruses can be captured by this model with almost no a-priori assumptions on the spreading topology and interaction. This allow us to characterize the spread of the virus of future seasons. The inferred model parameters are shown to be correlated with population indices such as human movement dynamics, and with demographic and environmental factors which are likely modulating transmission. The model The proposed model is a generalization of the single population SIR model [2], that describes the average evolution of an infectious disease in a single population. In the SIR model, disease spread is represented by a system of ordinary differential equations for the susceptible S, infected I, and recovered R individuals. It accounts for only one pathogen and assumes equal probability of being infected by any individuals regardless of his position i.e. mean field interaction. The model equations are: dS dt = −βSI, dI dt = βSI − γI, dR dt = γI,(1) where β > 0 is the infection rate and γ > 0 is the recovery rate. This model, Eq. (1), can be generalized to account for multiple viruses and spatial effects due to groups of sub-populations and virus strains, by transforming to a multidimensional representation in which S, I, R, β, and γ are tensors (Eq. (2)). These tensors act as time-dependent state representation of the entire system, reflecting its various attributes or dimensions. That is, each two-dimensional projection represents different sub-groups e.g. viruses, spatial regions. Here, the system describes the evolution of an infectious sub-population interacting with other sub-populations under the influence of multiple viruses. The dynamics for the systems state tensors evolution is represented by the following system of ordinary differential equations: dS dt = −Iβ S dI dt = Sβ I − γ I dR dt = γ I,(2) where boldface represents a tensor. We refer to this model as a multi-compartment SIR model (mcSIR). This model, and its reduction to account for multiple viruses and virus strains in one population, is presented by Levy et al. [6]. Thus, the multi-compartment model is represented by a multi-dimensional set of equations. This representation of sub population and viruses can also be generalized to other epidemic models, such as the SIRS and the SIS [1]. In this work, we focus on the spreading mechanism in a group of N subpopulations (different spatial regions) analyzed separately for each virus. Within each sub-population the interaction between individual is well mixed i.e., of mean field type. In this case, S, I, and R take the form of square matrices with dimension N . The specific elements in these matrices, e.g. S ij (t), are defined as the fraction of susceptible individuals traveling from region i to j at time t, where i = j, represents the fraction of susceptible individuals in sub-population i present at time t. The off-diagonal elements of the matrix β account for the infection rates between sub-populations and the diagonal elements represent the infection rates within each sub-population. Note that, β is not a symmetric matrix, as the probability to move from i to j is not assumed to be the same as moving from j to i. The matrix γ, under the assumption of distinct sub-populations, is defined as a diagonal matrix. Each element, γ ii > 0, is the average recovery rate for individuals belonging to the ith sub-populations. For our purposes, we also assume that the number of infected people in each region is the sum of all the infected people traveling to this region, and that the number of susceptible people traveling is negligible compared to the sizes of the susceptible sub-populations in the region. In this case, the model is reduced to a traditional sub-population model [4]. The advantage of using a sub-population model is that in contrast to situations where populations might be well mixed (a common assumption of models without spatial structure) local disease spread is taken into account. This means that most of the population is not exposed to the infection immediately upon its introduction. Rather, the disease may have to pass through many intermediate individuals before reaching all members of the population. This is mainly due to the fact that at a larger scale, individuals that are close together in space are likely to come into contact with each other more frequently than individuals that are more distant. In addition, this locally structured populations may exhibit clique behavior. Moreover, in many cases one can find a spatial epidemic wave pattern depending on the virus type and location [24,21,25]. This assumption, which to some extent holds true even today when people move longer distances over shorter times, may smooth-out the clique behavior. Understanding the spatial nature of disease transmission has important consequences for control measures aimed at a disease per location. In order to validate the use of sub-population models presented in Eq. (2) to describe the observed Internet data, we tested the model on seasonal data derived from search engine queries about two commonly spread viruses in the US, RSV and WNV. In this case, the sub-populations are the 50 states. We chose these two viruses to fit to an SIR model since there is no vaccine or medicine for prevention or treatment. Moreover the symptoms are usually mild meaning people are less likely to visit a doctor or have the case reported to the national health care authorities. On the other hand, Internet search data in these cases may provide real time high resolution estimate for the disease activity. Methods Data sources We used two Internet data sources in our analysis: Google Trends[26] and Twitter. Google Trends was used to determine the relative number (per state in the US) of queries made about each of the two viruses as described below. Data was extracted at a weekly resolution. Twitter was used to estimate human mobility patterns as described below. Respiratory Syncytial Virus (RSV) Data about RSV were extracted for the years 2013 to 2018 i.e., 5 seasons. A season starts at the beginning of October every year. The data comprised of the number of queries for the term "RSV". A single term was sufficient in this case because of the observation [21] that this number is in high correlation with the number of people reported as infected in each state, as published by the Centers for Disease Control and Prevention (CDC). Our data included all US states excluding Hawaii. West Nile Virus (WNV) Data about WNV were extracted for the years 2014 to 2018 i.e., 4 seasons. A season starts at the beginning of April every year. Preliminary analysis of Google Trends data showed that the number of queries for the term "West Nile" from each state during 2017 was highly correlated with the yearly total number of infected cases reported by the CDC, reaching R 2 = 0.58 (p-value=1.4 · 10 −5 ). Thus, we used the number of people querying for "West Nile" from 2014 to 2018 as a proxy for WNV disease load. Our data included all contiguous US states, excluding Hawaii and Alaska. Data on human mobility patterns in the US We collected all messages from Twitter having a GPS location in the US from October 1, 2015 through April 31, 2016. For each message we extracted an anonymized user identifier, the time of the message, and the location from where it was made. These data comprised of approximately 50 million messages. The exact GPS location of each message was mapped as its encompassing US state. From these data we created a matrix of mobility patterns, where the (i, j)-th cell comprised of the number of people whose location in one tweet was state i and in their following tweet was state j. This matrix was normalized by dividing the number of people moving from state i to state j by the total number of people who moved from state i to any other state. Demographic information We extracted from the US census [27] variables that may affect the spreading rate of a virus within a population. These data, extracted per state, included: Parameter matching Fitting is carried out in two steps. In the first step, a dictionary of functions is generated. Each function is the infection rate function from an instantiation of a solution to the single population SIR model (Eq. (1)) chosen from a wide range of parameter values. Specifically, we used γ ∈ [10 −4 , 0.1] with 50 steps and β ∈ [10 −4 , 1] with 200 steps. Each function entry is intended to be proportional to the number of infected individuals per state over time sampled at a weekly resolution. Out of the dictionary, one function is chosen, such that it has the highest correlation with the data from each state over a period of one season. In the second step, we fit the data to the multi-compartment SIR (mcSIR) model presented in Eq. (2). The parameters found in the previous step are used as an initialization to the mcSIR fitting process. Specifically, the on-diagonal elements of the matrix β were initialized to the values found from the single SIR model. The off-diagonal values of β were initialized to uniformly distributed random values between 0 and the average β found from the fit to the single population SIR. We perform a search in the parameter space for the solution of the mcSIR model which maximizes the highest geometric mean across states of the correlations between the mcSIR estimated infection rates and data acquired from Google Trends calculated over four seasons. The use of multiple seasons provides us with more data to increase robustness to noise, under the assumption that the change in the infection rate from season to season and the recovery rate over the years can be accounted for by a single season-dependence multiplicative scaling factor D. Another assumption made in the fitting of the mcSIR model is the use of a scalar recovery rate γ for all seasons and all states. This assumption was made because of the observation from the single SIR fit that the recovery rates are similar for all states and seasons. The gradient descent optimization process was limited to 15 iterations. We repeated the optimization process 100 times, each time with random initialization on the off-diagonal terms of the β matrix. The run with highest correlation between the data and the infected rate estimated by the mcSIR model was chosen. The single population SIR and mcSIR models are solved numerically. All analyses were performed using Matlab R2017b [28]. Results The algorithm reached an average correlation (across states) R 2 of 0.70 for RSV and R 2 of 0.52 for WNV between the Google Trends data and the mcSIR model. correlation distribution is centered around 0.8. Although, the data for WNV (Figure 1b) was noisier, there was still a significant number of states with a correlation higher than 0.6. We hypothesize that WNV data is noisier than RSV due to the lower incidence of the former. Infection rate within states After fitting the mcSIR model to the data, the infection rate matrix β and the recovery rate γ of each virus could be examined. We note that the best value found for the recovery rate is γ = 0.02 for both RSV and WNV. The values along the main diagonal of the matrix β represent the infection rate within each sub-population normalized by the sub-population size [4]. Google Trends data is normalized to the maximal number of queries in each state in a given season. This, the infection rate within each state require similar normalization. Since the number of queries in a state are correlated with its population size, in order to estimate the infection rate in each state we multiply the diagonal elements of β by the state's population size. Then, in order to explain the estimated infection rate values, we built a rank regression model where the independent variables were the known demographic variables of each state (see Methods) and the estimated infection rates in each state. In the case of RSV, the model fit is R 2 = 0.42 (p-value= 9.9 · 10 −7 ). Statistically significant variables are population density (slope: 0.64) and average age (slope: −0.35). The model for WNV reached a fit of R 2 = 0.49 (p-value= 2.92 · 10 −7 ). Statistically significant variables are population density (slope: 0.64), poverty (slope: 0.31), and average age (slope: −0.30). Infection rates of viruses increase in areas of denser population [29] and so both viruses have a positive correlation with population density. For both viruses there is a negative correlation with average age meaning that younger people are more susceptible. RSV is known to be more severe in infants [30]. WNV is commonly spread by mosquitoes, which may explain the positive correlation with poverty, as mosquitoes could be more prevalent in such areas [31]. Inter-state infection rates The off-diagonal elements of the infection rate matrix β represent the rate of infection between states. That is, these elements represent the rate at which susceptible people in one state are likely to be infected by people from another state. Thus, we hypothesized that these elements should be correlated with human mobility patterns among states. To estimate the correlation between the state-level mobility patterns estimated from Twitter data and the values of β, we normalized β by the average number of infected people estimated by the mcSIR model in each state during a particular season i.e., the infection rate matrix β is multiplied by a diagonal matrix of the average number of people transmitting the disease in a season in each state. This quantity is defined as the average transmission of the infection between different states. The Spearman correlation between the movement patterns, as estimated from Twitter, and the estimate normalized infection rates for RSV was ρ = 0.30 (p-value < 10 −10 ) and for WNV was ρ = 0.27 (p-value < 10 −10 ). Thus, approximately a significant portion of of the variance in transmission rates of the examined viruses is explained by human mobility patterns. Embedding the transmission rate matrix on a map of the US provides a clearer understanding of disease spread. Figure 2 shows the transmission rate matrix averaged over the first 25% of the season for the RSV virus in 2016. The color of each state represents the week of peak infection as estimated by the model. Thus, the model predicts Florida would be the first state where the disease peaks. This also identifies Florida as the source of the virus. This is in line with our knowledge of the virus spread in the US [30]. The color of the arrows in Figure 2 represents the strength of the interaction between each two states given by the elements of the transmission matrix β. Since the transmission matrix is correlated with the movements of people, the matrix elements (arrows on the map) represent the beginning of spatial virus spread in the US predicted by the model. Characterization of the disease's spread in the following seasons Our results suggest that the parameters of disease spread change little from year to year. This can be observed by looking at the season dependent scaling factors, D (defined in the Methods). This factor approaches a value of 1 when there is seasonal invariance. For RSV the average value of D, over 4 seasons was 0.93±0.06, and for WNV the average value over 3 seasons was 0.92±0.08. Thus, we sought to predict infection rates by applying the model with the estimated parameters from previous seasons, corrected using the first few weeks of data from the current season. This allow us to predict the infection rates of the entire season and the temporal location of the infection peak in each state. The latter is simply the extremal point of the model Eq. (2). Here, we used the estimates of β and γ from the previews seasons, together with increasing amounts of data from the 2018 season to predict the seasonal dynamics for the entire 2018 season. Specifically, we used the parameters from the previews seasons as initial parameters for the model and adjusted them using the first N weeks of data from the 2018 season, by running a few iterations of the optimization algorithm. Figures 3a and 3c show the correlation between the actual and predicted infection rates given increasing amounts of data for the current season (increasing N ), for RSV and WNV, respectively. Figures 3b and 3d show the error in prediction of peak infection for RSV and WNV, respectively. Figure 3 shows that within 7 weeks from the beginning of the season it is possible to predict the progression of the entire season with reasonably high accuracy, both for RSV and WNV. Discussion Multi-compartment epidemic models provide a useful tool in the understanding of disease spread. However, limits on availability of epidemiological data has made the testing and validation of such models complicated. Here, we demonstrated the utility of matching proxy data from search queries to theoretical models. This matching allows us to confirm the spatial and temporal dynamics of disease spread for two sample viruses, RSV and WNV. Additionally, this matching provides information on the parameters of disease spread. It reveals and emphasizes the importance of accounting for the spatial structural complexity of the spreading mechanism. These parameters were shown to be correlated with the demographics of people in each state, and the mobility of people across states. In addition, we show that learning the parameters of the disease spread from proxy data over multiple seasons generalizes relatively well to the next season. Our work has drawbacks, especially, in the use of proxy data, rather than epidemiological data. Still, although less accurate, the proxy data are derived at higher spatial resolution and from a much larger, and arguably a more representative, population. The correlation between the parameters we find and the demographics of people provide an indication for the validity of this data. Therefore, this approach can be used to estimate factors influencing disease intervention and control especially in cases where epidemiological data is lacking or insufficient providing more information about the spreading mechanism. The modeling approach we proposed makes minimal assumptions on the structure of the population, interventions, or behavior. Our approach can be used to track and model other viruses and sub-populations. Future work will focus on modeling sub-populations (for example, different age groups) within a geographic area, as well as the effects of interventions such as vaccination. 1 . 1Population size (number of people) 2. Density per square mile of land area. 3. Children in the following age groups: 0-4, 5-11,12-14, 14-17 (%) 4. Percentage of people in each of 6 main racial groups 5. Average income per household and per family (Dollars) 6. Poverty rate in adults and child under 18 (%) 7. Senior people (65 years and over) in the population (Unemployment rate (%). Figure 1 Figure 1 : 11shows the distribution of the model fit (R 2 ) between the Google Trends data and the mcSIR estimated infection rate in each state over 4 seasons for RSV (200 state and season pairs), and 3 seasons for WNV (147 state and season pairs). For both viruses, the majority of states show high correlation between mcSIR model and the Google Trends data. 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[ "Effect of current injection into thin-film Josephson junctions", "Effect of current injection into thin-film Josephson junctions" ]	[ "V G Kogan \nDepartment of Energy\nAmes Laboratory\n50011AmesIowaUS, USA\n", "R G Mints \nThe Raymond and Beverly Sackler School of Physics and Astronomy\nTel Aviv University\n69978Tel AvivIsrael\n" ]	[ "Department of Energy\nAmes Laboratory\n50011AmesIowaUS, USA", "The Raymond and Beverly Sackler School of Physics and Astronomy\nTel Aviv University\n69978Tel AvivIsrael" ]	[]	New thin-film Josephson junctions have recently been tested in which the current injected into one of the junction banks governs Josephson phenomena. One thus can continuously manage the phase distribution at the junction by changing the injected current. A method of calculating the distribution of injected currents is proposed for a half-infinite thin-film strip with source-sink points at arbitrary positions at the film edges. The strip width W is assumed small relative to Λ = 2λ 2 /d, λ is the bulk London penetration depth of the film material, d is the film thickness.	10.1103/physrevb.90.184504	[ "https://arxiv.org/pdf/1410.1088v1.pdf" ]	16,985,089	1410.1088	587447da45306f23b67a1cf996b55874c97817d8	Effect of current injection into thin-film Josephson junctions V G Kogan Department of Energy Ames Laboratory 50011AmesIowaUS, USA R G Mints The Raymond and Beverly Sackler School of Physics and Astronomy Tel Aviv University 69978Tel AvivIsrael Effect of current injection into thin-film Josephson junctions numbers: 7455+v Ec7478-w8525Cp New thin-film Josephson junctions have recently been tested in which the current injected into one of the junction banks governs Josephson phenomena. One thus can continuously manage the phase distribution at the junction by changing the injected current. A method of calculating the distribution of injected currents is proposed for a half-infinite thin-film strip with source-sink points at arbitrary positions at the film edges. The strip width W is assumed small relative to Λ = 2λ 2 /d, λ is the bulk London penetration depth of the film material, d is the film thickness. New thin-film Josephson junctions have recently been tested in which the current injected into one of the junction banks governs Josephson phenomena. One thus can continuously manage the phase distribution at the junction by changing the injected current. A method of calculating the distribution of injected currents is proposed for a half-infinite thin-film strip with source-sink points at arbitrary positions at the film edges. The strip width W is assumed small relative to Λ = 2λ 2 /d, λ is the bulk London penetration depth of the film material, d is the film thickness. I. INTRODUCTION In recent years, the physics of Josephson phenomena enjoyed a number of important developments. Introduction of π and 0-π junctions, 1 various ways to have a different from π phase shift, 2,3 effect of vortices in the junction vicinity, [4][5][6] to name a few. Striking improvements in managing Josephson phenomena came after injection of currents into one of the thin-film banks was introduced that allowed for continuous control of the phase difference on the junction [7][8][9] and, in particular, to imitate the 0-π behavior. This development necessitates evaluation of the injected supercurrent distribution in one of the junction banks since this determines the distribution of the superconducting phase. In one of common realizations, the junction is formed by two "half-infinite" thin-film strips with overlapped edges. Extra current injectors are attached to the edges of one of the films, shown schematically in Fig. 1. The injected current affects the phase distribution in the thinfilm bank where it flows and thus the phase difference on the junction. We show in this communication that for sufficiently thin films with the size W smaller than the Pearl length Λ = 2λ 2 /d the problem of the injected currents can be solved under very general assumptions, so that the design of junctions with needed properties becomes possible. II. STREAM FUNCTION Consider a half-infinite thin-film strip of a width W Λ = 2λ 2 /d where λ is the London penetration depth of the film material and d is the film thickness. Choose x along the strip, 0 < x < ∞, and y across so that 0 < y < W , Fig. 1b. Let the injection points be at (x 1 , y 1 ) and (x 2 , y 2 ) at the film edge. The London equation integrated over the film thickness reads: h z + 2πΛ c curl z g = 0 .(1) Here, g is the sheet current density and h is the self-field of the current g. The Biot-Savart integral for h z in terms of g shows that h z is of the order g/c, whereas the second term on the left-hand side of Eq. (1) is of the order gΛ/cW g/c. Hence, in narrow strips with W Λ, the self-field can be disregarded. Introducing the scalar stream function S via g = curl[S(x, y)ẑ], we obtain instead of Eq. (1): ∇ 2 S = 0 .(2) Physically, this simplification comes about since in narrow films the major contribution to the system energy is the kinetic energy of supercurrents, while their magnetic energy can be disregarded. The boundary condition of zero current component normal to edges, e.g., g y = −∂ x S = 0 at the edge y = 0, translates to S = constant along the edges. This constant, however, is not necessarily the same everywhere, in particular, it should experience a finite jump at injection points. Consider a point contact at the edge as illustrated at Fig. 1, take its position as the origin of polar coordinates, and integrate the r component of the current g along the small half-circle centered at the injection point. The total injected current is: I = π 0 g r r dφ = π 0 ∂S ∂φ dφ = S(π) − S(0) . (3) Hence, on two sides of the injection point the stream function experiences a jump equal to the total injected current. Clearly, at the current sink S-jump has the opposite sign. Thus, we can choose S = 0 everywhere at the edges of the half-infinite strip, except the segment between the injection and sink points, where S = I. To solve the Laplace equation (2) we first employ the conformal mapping of the half-strip to a half-plane: 6,10 u + iv = −i cosh π(x + iy) .(4) It is seen that the half-plane u > 0 is transformed to the half-strip of a width 1 (hereafter we use W as a unit length). Explicitly, this transformation reads: u = sinh πx sin πy , v = − cosh πx cos πy . (5) Hence, we have to solve the Laplace equation on a halfplane u > 0 subject to boundary conditions S = I at the edge u = 0 in the interval v 1 < v < v 2 with v 1 = − cosh πx 1 cos πy 1 , v 2 = − cosh πx 2 cos πy 2 ,(6) and S = 0 otherwise. To proceed, we first write the "step-function" S(0, v) at the edge u = 0 as a Fourier integral: S(0, v) = ∞ −∞ dk 2π S(0, k)e ikv ,(7)S(0, k) = I v2 v1 dv e −ikv = I ik e −ikv1 − e −ikv2 .(8) Since S(0, v) is a linear superposition of plane waves e ikv , we first consider the solution of the Laplace equation (∂ 2 u + ∂ 2 v )s(u, v) = 0 subject to the boundary condition s(0, v) = e ikv . Separating variables we obtain s(u, v) = e ikv e −\|k\|u . Hence, the solution for the actual boundary condition is: S(u, v) = ∞ −∞ dk 2π S(0, k)e ikv−\|k\|u .(9) Substituting here S(0, k) of Eq. (8) one obtains: S(u, v) = I π tan −1 v − v 1 u − tan −1 v − v 2 u .(10) It is seen that at u → 0, S = I if v 1 < v < v 2 and S = 0 othewise, as it should be. Thus, we have the steam function at the half-plane u > 0 for arbitrary positions v 1 and v 2 of the current contacts at the edge u = 0. We now can go back to the (x, y) plane and specify the injection positions. A. Injectors at the edge x = 0 Let the injector and the sink be at (0, y 1 ) and (0, y 2 ). We obtain: S(x, y) I/π = tan −1 cos πy 1 − cosh πx cos πy sinh πx sin πy − tan −1 cos πy 2 − cosh πx cos πy sinh πx sin πy . The lines of the current g are given by g × dr = 0 or by ∂ x S dx + ∂ y S dy = dS = 0, in other words, by contours of S = const. An example is shown in Fig. 2. S(x, y) I/π = tan −1 cosh πx 1 − cosh πx cos πy sinh πx sin πy − tan −1 cosh πy 2 − cosh πx cos πy sinh πx sin πy . An example is shown in Fig. 3. It is worth noting that the same method can be employed for currents injected to thin-film samples of any polygonal shape. According to the Schwartz-Christoffel theorem any polygon can be mapped onto a halfplane. The general solution (10) on the (u, v) plane will hold. Therefore, knowing the function which realizes the needed transformation, one obtains S(x, y). The only physical precondition for this method to work is the requirement of a small sample size on the scale of Pearl length 2λ 2 /d, that allows one to reduce the problem to the Laplace equation for the stream function S. The method can be applied for more than two injection points or to extended injections, which require though different boundary conditions imposed on S. III. PHASE We now note that the sheet current is expressed either in terms of the gauge invariant phase ϕ or via the stream function S: g = − cφ 0 4π 2 Λ ∇ϕ = curl Sz This relation written in components shows that S(r) and (cφ 0 /4π 2 Λ)ϕ(r) are the real and imaginary parts of an analytic function. It is easy to construct the phase on the (u, v) plane since −i ln(u + iv) = tan −1 (v/u) − i ln u 2 + v 2 . Hence, the phase corresponding to the stream function of Eq. (10) obeys: cφ 0 4π 2 Λ ϕ = − I π ln u 2 + (v − v 1 ) 2 u 2 + (v − v 2 ) 2 ,(13)or ϕ = − I I 0 ln u 2 + (v − v 1 ) 2 u 2 + (v − v 2 ) 2 , I 0 = cφ 0 4πΛ .(14) The characteristic current I 0 depends on the Pearl length. Thus, the phase is proportional to the reduced injected current j = I/I 0 and a factor ϕ/j depending on the film geometry and injection positions. Substituting here u(x, y) and v(x, y) one obtains the phase as a function of (x, y). Examples of the phase near the edge x = 0 are shown in Fig. 4. Clearly, one can make the phase "jump" steeper by putting injection contacts closer. It is worth noting that the phase change due to injected currents can be used, e.g., to imitate properties of 0-π junctions, or in fact to have any phase shift by choosing properly the injected current. IV. JOSEPHSON CRITICAL CURRENT The total Josephson current through a rectangular patch (the shaded region in Fig. 1a) with the size ∆x along the x axis is J(I/I 0 ) J c0 = ∆x 0 dx 1 0 dy sin[ϕ(x, y, I/I 0 ) + ϕ 0 ] ,(15) where J c0 is the critical Josephson current density (which in the following is set equal to 1) and ϕ 0 is an overall phase imposed by the transport current through the junction. To find the critical current, we maximize this relative to ϕ 0 to obtain J c = √ A 2 + B 2 where A = A. Symmetric injection Consider J c (j) for the rectangular junction of the width ∆x = 0.1 W , similar to the experimental set up. 11 The injection contacts are at the edge x = 0 and y 1 = 0.3 and y 2 = 0.7, i.e. they are symmetric relative to the strip middle y = 1/2. The current distribution for this case is given in Fig. 2. J c (j) evaluated with the help of Eq. (16) is shown in Fig. 5. We note that J c (0) is proportional to the junction width ∆x, since the Josephson critical current density is constant in the absence of injected currents. Fig. 6 shows J c (j) in the same junction with contacts at y 1 = 0.48 and y 2 = 0.52 so that they are separated by ∆y = 0.04, ten times closer than in the previous example. Comparing these plots we see that the first zero of J c at j ≈ 0.5 in the first graph whereas it is at j ≈ 5 in the second. We then conclude that zeros roughly scale as the inverse of the contact separation, 1/∆y. We also observe that maxima of J c seem to be independent of contacts separation. It is shown below that these properties of J c (j) can be traced back to general expressions (15) and (16) for narrow junctions, ∆x 1, and small contacts separations ∆y 1. To this end, we note that being a solution of the Laplace equation for a half-infinite strip, the phase ϕ(x, y) changes considerably on distances of the order W = 1 from the edge x = 0 where the contacts are placed. Hence, for narrow junctions with ∆x 1, one can set in the first approximation: ϕ(x, y) ≈ ϕ(0, y) = −j ln (v − v 1 ) 2 (v − v 2 ) 2 = −j ln (cos πy − cos πy 1 ) 2 (cos πy − cos πy 2 ) 2 . Here, y 2,1 = (1 ± ∆y)/2, and for ∆y 1 one can expand the last expression in powers of ∆y: ϕ(0, y) = 2πj∆y cos πy + O(∆y) 3 .(18) Since cos πy is odd relative to the strip middle, and so is sin ϕ(0, y), we have A = 0, and J c = \|B\|, see Eq. (16): asymmetric injections, we note that J c (j) of all of them have the property that their minima do not reach zeros. J c = V. DISCUSSION In summary, we have shown that the current distribution in thin film samples small on the scale of the Pearl length Λ = 2λ 2 /d can be found by solving the Laplace equation for the stream function under boundary conditions specified for injection sources at arbitrary points at sample edges. When this film constitutes one of the Josephson junction banks, the contribution of the phase associated with injected currents to the junction phase difference is proportional to the injected current j = 4πΛI/cφ 0 . Hence, the thinner the film (or the larger the Pearl Λ) the smaller injected currents are needed for the same effect upon the junction properties. The critical Josephson current J c (j), for certain (symmetric) infection geometries, has zeros, the position of which scales as the inverse distance between the injection points. If j n is one of these zeros, application of a field H parallel to the junction plane results in a pattern J c (H, j n ) with zero at H = 0 instead of a standard maximum, the property seen in experiments. 7,8 PACS numbers: 74.55.+v Ec, 74.78.-w, 85.25.Cp FIG. 1 . 1(a) A sketch of two semi-infinite thin-film strips forming the Josephson junction in the overlapping shaded region. (b) The top film; arrows show positions of injection point contacts. (c) The stream function S at the film edge. FIG. 2 . 2The current distribution for the injection points (0, 0.3) and (0, 0.7). B. The injector at y = 0 and the sink at x = 0 This situation corresponds to positions (x 1 , 0) and (0, y 1 ): FIG. 3. The current distribution for the asymmetric injection points (0.2, 0) and (0, 0.7). FIG. 4 . 4The phase ϕ(x0, y) for j = 1 at fixed distances x0 from the edge x = 0 as a function of the transverse coordinate y for symmetric contacts at y1 = 0.3 and y2 = 0.7. The solid line is for x0 = 0.05, the dashed line is for x0 = 0.2. cos[ϕ(x, y; j)] . FIG. 5 .FIG. 6 . 56Jc(j) for the junction width ∆x = 0.1 and symmetric contacts at the edge x = 0. y1 = 0.3 and y2 = 0.7 so that the distance between contacts is ∆y = 0.4. Jc(I/I0) for the junction width ∆x = 0.1 and symmetric contacts at the edge x = 0 and y1 = 0.48 and y2 = 0.52 so that the distance between contacts is ∆y = 0.04. FIG. 7 . 7Jc(j) corresponding to the current distribution of Fig. 3 for the injection points (0.2, 0) and (0, 0.7). The authors are grateful to E. Goldobin for sharing experimental information and many helpful discussions. The Ames Laboratory is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358.Clearly, when no current is injected, J c (0) = ∆x as is seen in Figs. 5 and 6. The integral over y can be done:where p F q is the generalized hypergeometric function and η = 2πj∆y. This function is expressed in terms of Bessel and Struve functions so that it oscillates when η changes:It is worth noting that J c of Eqs.(19)and(20)depends on the injected current j and the contacts separation ∆y only via the product η = 2πj∆y. In other words, the curves J c (j, ∆y) for different contact separations ∆y can be rescaled to a universal curve J c (η). In particular, the zeros of J c (j) and the positions of its maxima should scale as 1/∆y. Unlike their positions, the absolute value of its n-th maximum is independent of the separation ∆y.The two first roots of J c (η) found numerically are 1. Let us now consider the magnetic field H applied parallel to the junction plane (x, y). In general, the critical current should depend on both H and j, J c = J c (j, H). In the absence of injected currents, J c (0, H) has a standard shape with maximum at H = 0. The presence of zeros of J c (j) for symmetric injection in zero field, has an important consequence. If the injected current j n is such that J c (j n ) = 0, application of the magnetic field will result in the pattern J c (j n , H) such that J c (j n , 0) = 0, i.e., the curve J c (j n , H) will have zero at H = 0 instead of the standard maximum. This situation is similar to the famous case of "0-π" junction, 1 however, here the injected currents cause a necessary phase shift. Precisely this situation has been seen in experiment 7,8 with symmetric injectors.B. Asymmetric injectionIf the injection contacts are arranged asymmetrically as, e.g., atFig. 3, the minima of J c (j) do not reach zeros as shown inFig. 7. Without discussing a variety of . L N Bulaevskii, V V Kuzii, A A Sobyanin, Sol. St. Com. 251053L. N. Bulaevskii, V. V. Kuzii and A. A. Sobyanin, Sol. St. Com, 25, 1053 (1978). . C C Tsuei, J R Kirtley, Rev. Mod. Phys. 72969C. C. Tsuei and J.R. Kirtley, Rev. Mod. Phys. 72, 969 (2000). . R G Mints, Phys. Rev. 573221R. G. Mints, Phys. Rev. B57, R3221 (1998). . T Golod, A Rydh, V M Krasnov, Phys. Rev. Lett. 104227003T. Golod, A. Rydh, and V. M. Krasnov, Phys. Rev. Lett.104, 227003 (2010). . V G Kogan, R G Mints, Phys. Rev. 8914516V. G. Kogan, R. G. Mints, Phys. Rev. B89, 014516 (2014). . V G Kogan, R G Mints, Phys Superc, 50258V. G. Kogan, R. G. Mints, Phys. C Superc. 502, 58 (2014). . A V Ustinov, Appl. Phys. Lett. 803153A. V. Ustinov, Appl. Phys. Lett. 80, 3153 (2002). . E Goldobin, A Sterck, T Gaber, D Koelle, R Kleiner, Phys. Rev. Lett. 9257005E. Goldobin, A. Sterck, T. Gaber, D. Koelle, and R. Kleiner, Phys. Rev. Lett.92, 057005 (2004). . A Dewes, T Gaber, D Koelle, R Kleiner, E Goldobin, Phys. Rev. Lett. 101247001A. Dewes, T. Gaber, D. Koelle, R. Kleiner, and E. Goldobin, Phys. Rev. Lett.101, 247001 (2008). . P M Morse, H , Feshbach Methods of Theoretical Physics. McGraw-HillP. M. Morse and H. Feshbach Methods of Theoretical Physics, McGraw-Hill, 1953. . E Goldobin, private communicationE. Goldobin, private communication.	[]
[ "EXACT CONTROLLABILITY OF A LINEAR KORTEWEG-DE VRIES EQUATION BY THE FLATNESS APPROACH", "EXACT CONTROLLABILITY OF A LINEAR KORTEWEG-DE VRIES EQUATION BY THE FLATNESS APPROACH" ]	[ "Philippe Martin ", "Ivonne Rivas ", "Lionel Rosier ", "Pierre Rouchon " ]	[]	[]	We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.2010 Mathematics Subject Classification: 37L50, 93B05	10.1137/18m1181390	[ "https://arxiv.org/pdf/1804.06209v1.pdf" ]	58,944,693	1804.06209	1794e1e0528840b9e58a1c53c03fc6cbb6b2b28e	EXACT CONTROLLABILITY OF A LINEAR KORTEWEG-DE VRIES EQUATION BY THE FLATNESS APPROACH Philippe Martin Ivonne Rivas Lionel Rosier Pierre Rouchon EXACT CONTROLLABILITY OF A LINEAR KORTEWEG-DE VRIES EQUATION BY THE FLATNESS APPROACH Korteweg-de Vries equationexact controllabilitycontrollability to the trajectoriesflat- ness approachGevrey classsmoothing effect We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.2010 Mathematics Subject Classification: 37L50, 93B05 INTRODUCTION The Korteweg-de Vries (KdV) equation is a well-known dispersive equation that may serve as a model for the propagation of gravity waves on the surface of a canal or a lake. It reads ∂ t y + ∂ 3 x y + y∂ x y + ∂ x y = 0, (1.1) where t is time, x is the horizontal spatial coordinate, and y = y(x,t) stands for the deviation of the fluid surface from rest position. As usual, ∂ t y = ∂ y/∂t, ∂ x y = ∂ y/∂ x, ∂ 3 x y = ∂ 3 y/∂ x 3 , etc. When the equation is considered on a bounded interval (0, L), it has to be supplemented with three boundary conditions, for instance y(0,t) = u(t), y(L,t) = v(t), ∂ x y(L,t) = w(t), (1.2) and an initial condition y(x, 0) = y 0 (x). (1. 3) The controllability of the Korteweg-de Vries equation with various boundary controls has been considered by many authors since several decades (see e.g. the surveys [18,2]). The exact controllability in the energy space L 2 (0, L) was derived by Rosier in [15] (resp. by Glass and Guerrero in [6]) with w as the only control input (resp. with v as the only control input). On the other hand, if we take u as the only control input, the exact controllability fails in the energy space [17], because of the smoothing effect. Nevertheless, both the null-controllability and the controllability to the trajectories hold with the left Dirichlet boundary control, see [17] and [5]. The aim of the present paper is to go a step further by investigating the exact controllability in a "narrow" space with the left Dirichlet boundary control. Due to the smoothing effect, the space in which the exact controllability can hold is a space of analytic functions. For the sake of simplicity, we will focus on a linear KdV equation (removing the nonlinear term y∂ x y). Performing a scaling in time and space, there is no loss of generality in assuming that L = 1. 1 By a translation, we can also assume that x ∈ (−1, 0). The first-order derivative term will assume the form a∂ x y where a ∈ R + is some constant. The case a = 1 corresponds to the linearized KdV equation ∂ t y + ∂ 3 x y + ∂ x y = 0, (1.4) while the case a = 0 corresponds to the "simplified" linearized KdV equation ∂ t y + ∂ 3 x y = 0, (1.5) which is often considered when investigating the Cauchy problem on the line R (instead of a bounded interval) by doing the change of unknownỹ(x,t) = y(x + t,t). The paper will be concerned with the control properties of the system: ∂ t y + ∂ 3 x y + a∂ x y = 0, x ∈ (−1, 0), t ∈ (0, T ), (1.6) y(0,t) = ∂ x y(0,t) = 0, t ∈ (0, T ), (1.7) y(−1,t) = u(t), t ∈ (0, T ), (1.8) y(x, 0) = y 0 (x), x ∈ (−1, 0), (1.9) where y 0 = y 0 (x) is the initial data and u = u(t) is the control input. We shall address the following issues: 1. (Null controllability) Given any y 0 ∈ L 2 (−1, 0), can we find a control u such that the solution y of (1.6)-(1.9) satisfies y(., T ) = 0? 2. (Reachable states) Given any y 1 ∈ R (a subspace of L 2 (−1, 0) defined thereafter), can we find a control u such that the solution y of (1.6)-(1.9) with y 0 = 0 satisfies y(., T ) = y 1 ? We shall investigate both issues by the flatness approach and derive an exact controllability in R by combining our results. The null controllability of (1.6)-(1.9) was established in [17] (see also [5]) by using a Carleman estimate. The control input u was found in a Sobolev space (e.g. u ∈ H 1 2 −ε (0, T ) for all ε > 0 if y 0 ∈ L 2 (−1, 0), see [5]). Here, we shall improve this result by designing a control input in a Gevrey class. Furthermore, the trajectory and the control will be given explicitly as the sums of series parameterized by the flat output. To state our result, we need introduce a few notations. A function u ∈ C ∞ ([t 1 ,t 2 ]) is said to be Gevrey of order s ≥ 0 on [t 1 ,t 2 ] if there exist some constant C, R ≥ 0 such that \|∂ n t u(t)\| ≤ C (n!) s R n ∀n ∈ N, ∀t ∈ [t 1 ,t 2 ]. The set of functions Gevrey of order s on [t 1 , t 2 ] is denoted by G s ([t 1 ,t 2 ]). A function y ∈ C ∞ ([x 1 , x 2 ] × [t 1 ,t 2 ] ) is said to be Gevrey of order s 1 in x and s 2 in t on [x 1 , x 2 ] × [t 1 ,t 2 ] if there exist some constants C, R 1 , R 2 > 0 such that \|∂ n 1 x ∂ n 2 t y(x,t)\| ≤ C (n 1 !) s 1 (n 2 !) s 2 R n 1 1 R n 2 2 ∀n 1 , n 2 ∈ N, ∀(x,t) ∈ [x 1 , x 2 ] × [t 1 ,t 2 ]. The set of functions Gevrey of order s 1 in x and s 2 in t on [x 1 , x 2 ] × [t 1 ,t 2 ] is denoted by G s 1 ,s 2 ([x 1 , x 2 ] × [t 1 ,t 2 ]). The first main result in this paper is a null controllability result with a control input in a Gevrey class. Theorem 1.1. Let y 0 ∈ L 2 (−1, 0), T > 0, and s ∈ [ 3 2 , 3) . Then there exists a control input u ∈ G s ([0, T ]) such that the solution y of (1.6)-(1.9) satisfies y(., T ) = 0. Furthermore, it holds that y ∈ C([0, T ], L 2 (−1, 0)) ∩ G s The second issue investigated in this paper is the problem of the reachable states. For the heat equation, an important step in the characterization of the reachable states was given in [11] with the aid of the flatness approach. It was proven there that reachable states can be extended as holomorphic functions on some square of the complex plane, and conversely that holomorphic functions defined on a ball centered at the origin and with a sufficiently large radius give by restriction to the real line reachable states. See also [4] for an improvement of this result as far as the domain of analyticity of the reachable states is concerned. To the best knowledge of the authors, the determination of the reachable states for (1.6)-(1.9) has not been addressed so far. From the controllability to the trajectories established in [17,5], we know only that any function y 1 = y 1 (x) that can be written as y 1 (x) =ȳ(x, T ) for some trajectoryȳ of (1.6)-(1.9) associated with some y 0 ∈ L 2 (−1, 0) and u = 0 is reachable. But such a function is in G x + a∂ x ) n y 1 (−1) = 0 for all n ≥ 1, according to Proposition 2.1 (see below). Proceeding as in [11], we shall obtain a class of reachable states that are less regular than those for the controllability to the trajectories (namely y 1 ∈ G 1 ([−1, 0]) for which no boundary condition has to be imposed at x = −1. To state our second main result, we need to introduce again some notations. For z 0 ∈ C and R > 0, we denote by D(z 0 , R) the open disk D(z 0 , R) := {z ∈ C; \|z − z 0 \| < R}, and by H(D(z 0 , R)) the set of holomorphic (i.e. complex analytic) functions on D(z 0 , R). Introduce the operator Py := ∂ 3 x y + a∂ x y, so that (1.6) can be written ∂ t y + Py = 0. (1.11) Since ∂ t and P commute, it follows from (1.11) that for all n ∈ N * ∂ n t y + (−1) n−1 P n y = 0 (1.12) where P n = P • P n−1 and P 0 = Id. We are in a position to define the set of reachable states: for any R > 1, let 1,0] , and (P n y)(0) = ∂ x (P n y)(0) = 0 ∀n ≥ 0}. R R := {y ∈ C 0 ([−1, 0]); ∃z ∈ H(D(0, R)), y = z \|[− The following result is the second main result in this paper. (1) As for the heat equation, it is likely that any reachable state for the linear Kortewegde Vries equation can be extended as an holomorphic function on some open set in C. (2) The reachable states corresponding to the controllability to the trajectories are in G 1 2 ([−1, 0]), so that they can be extended as functions in H(C). By contrast, the reachable functions in Theorem 1.2 need not be holomorphic on the whole set C: they can have poles outside D(0, R). 1 It is conjectured that it belongs to G = {y ∈ C([−1, 0]); ∃(a n ) n∈N ∈ R N , ∞ ∑ n=0 \|a n \|r 3n < ∞ ∀r ∈ (0, R) and y(x) = ∞ ∑ n=0 a n x 3n+2 ∀x ∈ [−1, 0]}. Note that y(−1) needs not be 0 for y ∈ R R . Examples of functions in R R include (i) the polynomial functions of the form y(x) = ∑ N n=0 a n x 3n+2 ; (ii) the entire function y(x) = e x + je jx + j 2 e j 2 x where j := e i 2π 3 . Note that y is real-valued and y(−1) > 0 (see Fig. 1). Combining Theorem 1.1 and Theorem 1.2, we obtain the following result which implies the exact controllability of (1.6)-(1.9) in R R for R > R 0 . Corollary 1.1. Let a ∈ R + , T > 0, R > R 0 , y 0 ∈ L 2 (−1, 0) and y 1 ∈ R R . Then there exists u ∈ G 3 ([0, T ]) such that the solution of (1.6)-(1.9) satisfies y(., T ) = y 1 . Since system (1.6)-(1.9) is linear, it is sufficient to pick u = u 1 + u 2 where u 1 is the control given by Theorem 1.1 for y 0 and u 2 is the control given by Theorem 1.2 for y 1 . The paper is outlined as follows. Section 2 is devoted to the null controllability of the linear KdV equation. The flatness property is established in Proposition 2.1. The smoothing effect for the linear KdV equation from L 2 (−1, 0) to G 1 2 ([−1, 0]) is derived in Proposition 2.2. The section ends with the proof of Theorem 1.1. Section 3 is concerned with the study of the reachable states. The flatness property is extended to the limit case s = 3 in Proposition 3.1. Theorem 1.2 then follows from Proposition 3.1 and some version of Borel theorem borrowed from [11]. NULL CONTROLLABILITY BY THE FLATNESS APPROACH In this section, we are concerned with the null controllability of (1.6)-(1.9). 2.1. Flatness property. Our first task is to establish the flatness property, namely the fact that the solution of (1.6)-(1.8) can be parameterized by the "flat ouput" ∂ 2 x y(0, .). More precisely, we consider the ill-posed system ∂ t y + ∂ 3 x y + a∂ x y = 0, x ∈ (−1, 0), t ∈ (0, T ), (2.1) y(0,t) = ∂ x y(0,t) = 0, t ∈ (0, T ), (2.2) ∂ 2 x y(0,t) = z(t), t ∈ (0, T ),(2.3) and we prove that it admits a solution y ∈ G s 3 ,s ([−1, 0] × [0, T ]) whenever z ∈ G s ([0, T ]) and 1 ≤ s < 3. The trajectory y and the control input u can be written as y(x,t) = ∑ i≥0 g i (x)z (i) (t), (2.4) u(t) = y(−1,t) = ∑ i≥0 g i (−1)z (i) (t) (2.5) where the generating functions g i , i ≥ 0, are defined as in [10]. More precisely, the function g 0 is defined as the solution of the Cauchy problem g 0 (x) + ag 0 (x) = 0, x ∈ (−1, 0), (2.6) g 0 (0) = g 0 (0) = 0, (2.7) g 0 (0) = 1 (2.8) (where = d/dx), while the function g i for i ≥ 1 is defined inductively as the solution of the Cauchy problem g i (x) + ag i (x) = −g i−1 (x), x ∈ (−1, 0), (2.9) g i (0) = g i (0) = g i (0) = 0. (2.10) It is well known that g i for i ≥ 1 can be expressed in terms of g 0 and g i−1 as g i (x) = − x 0 g 0 (x − ξ )g i−1 (ξ ) dξ . (2.11) Remark 2.1. (1) If a = 0, then it follows from direct integrations of (2.6)-(2.8) and (2.9)-(2.10) that g i (x) = (−1) i x 3i+2 (3i + 2)! , x ∈ [−1, 0], i ≥ 0. (2.12) (2) If a > 0, then the solution of (2.6)-(2.8) reads g 0 (x) = 1 a (1 − cos( √ ax)). (2.13) To ensure the convergence of the series in (2.4), we first have to establish some estimates for g i L ∞ (−1,0) . Lemma 2.1. Let a ∈ R + . Then for all i ≥ 0 \|g i (x)\| ≤ \|x\| 3i+2 (3i + 2)! ∀x ∈ [−1, 0]. (2.14) Proof. If a = 0, then (2.14) is a direct consequence of (2.12). Assume now that a > 0 and let us prove (2.14) by induction on i. It follows from (2.13) that (2.15) so that (2.14) is true for i = 0. Assume now that (2.14) is true for some i − 1 ≥ 0. Then, integrating by parts twice in (2.11) and using (2.7), we see that 0 ≤ g 0 (x) ≤ x 2 2 , ∀x ∈ [−1, 0],g i (x) = − g 0 (x − ξ ) ξ 0 g i−1 (ζ )dζ x 0 =0 − x 0 g 0 (x − ξ ) ξ 0 g i−1 (ζ )dζ dξ = − g 0 (x − ξ ) ξ 0 ζ 0 g i−1 (σ )dσ dζ x 0 =0 − x 0 g 0 (x − ξ ) cos √ a(x−ξ ) ξ 0 ζ 0 g i−1 (σ )dσ dζ dξ . It follows that \|g i (x)\| ≤ x 0 ξ 0 ζ 0 \|g i−1 (σ )\|dσ dζ dξ ≤ x 0 ξ 0 ζ 0 σ 3i−1 (3i − 1)! dσ dζ dξ = \|x\| 3i+2 (3i + 2)! , as desired. We are now in a position to solve system (2.1)-(2.3). Proposition 2.1. Let s ∈ [1, 3), z ∈ G s ([0, T ]), and y = y(x,t) be as in (2.4). Then y ∈ G s 3 ,s ([−1, 0]×[0, T ]) and it solves (2.1)-(2.3). Proof. We need to estimate the behavior of the constants in the equivalence of norms in W n,p ([−1, 0]) as n → ∞. For n ∈ N, p ∈ [1, ∞], and f ∈ W n,p (−1, 0), we denote f p = f L p (−1,0) and f n,p = n ∑ i=0 ∂ i x f p . The following result will be used several times. Its proof is given in appendix, for the sake of completeness. Lemma 2.2. Let p ∈ [1, ∞] and P = ∂ 3 x +a∂ x , where a ∈ R + . Then there exists a constant K = K(p, a) > 0 such that for all n ∈ N, (1 + 1 a ) −1 (1 + a) −n n ∑ i=0 P i f p ≤ f 3n,p ≤ K n n ∑ i=0 P i f p , ∀ f ∈ W 3n,p (−1, 0). (2.16) We follow closely [8]. Pick any z ∈ G s ([0, T ]) for some s ∈ [0, 3). We can find some numbers M > 0 and R < 1 such that \|z (i) (t)\| ≤ M (i!) s R i , ∀i ∈ N, ∀t ∈ [0, T ]. Pick any m, n ∈ N. Then ∂ m t P n (g i (x)z (i) (t)) = z (i+m) (t)(−1) n g i−n (x) if i − n ≥ 0, 0 if i − n < 0. (2.17) Assume that i ≥ n. Setting j = i − n and N = n + m, so that j + N = i + m, we have that ∂ m t P n g i (x)z (i) (t) ≤ M (i + m)! s R i+m 1 (3(i − n) + 2)! ≤ M ( j + N)! s R j+N 1 (3 j + 2)! · Let S := ∑ i≥n \|∂ m t P n (g i (x)z (i) (t))\|. Using the classical estimate ( j + N)! ≤ 2 j+N j! N! and the equivalence (3 j)! ∼ 3 3 j+ 1 2 (2π j) −1 ( j!) 3 which follows at once from Stirling formula, we obtain that S ≤ M ∑ j≥0 ( j + N)! s R j+N 1 (3 j + 2)! ≤ M ∑ j≥0 (2 j+N j! N!) s R j+N 2π( j + 1) (3 j + 2)(3 j + 1)3 3 j+ 1 2 ( j!) 3 ≤ M (N!) s ( R 2 s ) N for some positive constants M , M". Indeed, since s < 3, we have that ∑ j≥0 (2 j j! ) s R j 2π( j + 1) (3 j + 2)(3 j + 1)3 3 j+ 1 2 ( j!) 3 < +∞. Using again the fact that N! = (n + m)! ≤ 2 n+m n! m!, we arrive to S ≤ M n! s m! s R n 2 R m 1 with R 1 = R 2 = R/2 s . Let K be as in Lemma 2.2 for p = ∞. Then we have ∑ i≥n ∂ m t (g i (x)z (i) (t)) 3n,∞ ≤ K n ∑ i≥n ∑ 0≤ j≤n ∂ m t P j (g i (x)z (i) (t)) ∞ ≤ M K n ∑ 0≤ j≤n j! s m! s R j 2 R m 1 ≤ M n! s R n 2 m! s R m 1 for some R 2 > 0 and some M > 0. This shows that the series of derivatives ∂ m t ∂ l x g i (x)z (i) (t) is uniformly convergent on [−1, 0] × [0, T ] for all m, l ∈ N, so that y ∈ C ∞ ([−1, 0] × [0, T ]) and it satisfies for l ≤ 3n \|∂ m t ∂ l x y(x,t)\| ≤ M n! s R n 2 m! s R m 1 ∀x ∈ [−1, 0], ∀t ∈ [0, T ] for some constant M > 0. Note that n! s ∼ 2πn(3n)! 3 3n+ 1 2 s 3 · It follows that if l ∈ {3n − 2, 3n − 1, 3n}, then n! s /R n 2 ≤ C l! s 3 /R l 2 for some C > 0 and R 2 > 0. This yields \|∂ m t ∂ l x y(x,t)\| ≤ C M m! s R m 1 l! s 3 R l 2 , ∀x ∈ [−1, 0], ∀t ∈ [0, T ], as desired. The fact that y solves (2.1)-(2.3) is obvious. 2.2. Smoothing effect. We now turn our attention to the smoothing effect. We show that any solution y of the initial value problem (1.6)-(1.9) with u ≡ 0 and y 0 ∈ L 2 (−1, 0) is a function Gevrey of order 1/2 in x and 3/2 in t for t > 0. Proposition 2.2. Let y 0 ∈ L 2 (−1, 0) and u(t) = 0 for t ∈ R + . Then the solution y of (1.6)-(1.9) satisfies y ∈ G 1 2 , 3 2 ([−1, 0] × [ε, T ]) for all 0 < ε < T < ∞. More precisely, there exist some positive constant K, R 1 , R 2 such that \|∂ n t ∂ p x y(x,t)\| ≤ Kt − 3n+p+3 2 n! 3 2 R n 1 p! 1 2 R p 2 ∀p, n ∈ N, ∀t ∈ (0, T ], ∀x ∈ [−1, 0]. (2.18) Proof. Using (2.18) on intervals of length one, we can, without loss of generality, assume that T = 1. Let us introduce the operator Ay = −Py = −∂ 3 x y − a∂ x y with domain D(A) = {y ∈ H 3 (−1, 0); y(−1) = y(0) = ∂ x y(0) = 0} ⊂ L 2 (−1, 0). Then it follows from [15] that A generates a semigroup of contractions in L 2 (−1, 0), and tht a global Kato smoothing effect holds. More precisely, if y = e tA y 0 is the mild solution issuing from y 0 at t = 0, then we have for all T > 0 y(T ) L 2 ≤ y 0 L 2 (2.19) T 0 ∂ x y(.,t) 2 L 2 dt ≤ 1 3 (aT + 1) y 0 2 L 2 , (2.20) where f L 2 = ( 0 −1 \| f (x)\| 2 dx) 1 2 . For simplicity, we denote f H p = (∑ p i=0 ∂ i x f 2 L 2 ) 1 2 for p ∈ N.y(.,t) H 1 ≤ C √ t y 0 L 2 ∀t ∈ (0, 1]. (2.21) To prove Claim 1, we follow closely [14]. Pick any y 0 ∈ X 3 = D(A), we have by a classical property from semigroup theory that y ∈ C([0, T ], D(A)), and that z(.,t) = Ay(.,t) satisfies z(.,t) = e tA z(., 0). It follows by (2.19) that z(.,t) L 2 ≤ z(., 0) L 2 ∀t ∈ (0, 1]. Summing with (2.19), this yields y(.,t) L 2 + Py(.,t) L 2 ≤ y(.,t) L 2 + Py 0 L 2 , t ∈ (0, 1]. Using Lemma 3.2, this yields y 0 H 3 ≤ C 3 y 0 H 3 , t ∈ (0, 1] (2.22) for some C 3 = C 3 (a) > 0. Using interpolation, and noticing that x 1 = [X 0 , X 3 ] 1 3 , we infer the existence of some constant C 1 = C 1 (a) > 0 such that y(.,t) H 1 ≤ C 1 y 0 H 1 ∀y 0 ∈ X 1 , ∀t ∈ (0, 1]. (2.23) This yields for 0 < s < t ≤ 1 y(.,t) 2 H 1 ≤ C 2 1 y(., s) 2 H 1 , which gives upon integration over (0,t) for t ∈ (0, 1] t y(.,t) 2 To prove Claim 2, we pick again y 0 ∈ D(A) and set z(.,t) = Ay(.,t). We infer from (2.21) applied to z(.,t) that H 1 ≤ C 2 1 t 0 y(., s) 2 H 1 ds ≤ C 2 1 3 ((a + 3)T + 1) y 0 2 L 2 whereAy(.,t) H 1 = z(.,t) H 1 ≤ C √ t z(., 0) H 1 = C √ t Ay 0 . Combined with (2.21), this gives y(.,t) H 1 + (∂ 3 x + a∂ x )y(.,t) H 1 ≤ C √ t ( y 0 L 2 + Ay 0 L 2 ). (2.25) We know from Lemma 3.2 that for z ∈ H 3 (−1, 0) z H 3 ≤ C( z L 2 + Pz L 2 ). It follows that for z ∈ X 4 (C denoting a positive constant that may change from line to line, and that do not depend on t and on y 0 ) z H 4 ≤ C( z H 3 + ∂ 4 x z L 2 ) ≤ C z L 2 + Pz L 2 + ∂ x (∂ 3 x z + a∂ x z) L 2 + a ∂ 2 x z L 2 ≤ C( z H 1 + Az H 1 ). Combined with (2.25), this gives y(.,t) H 4 ≤ C √ t y 0 H 3 ∀t ∈ (0, 1] (2.26) for all y 0 ∈ X 4 , and also for all y 0 ∈ X 3 = D(A) by density. Interpolating between (2.21) and (2.26), we obtain (2.24). The proof of Claim 2 is achieved. Using Claim 2 inductively and spitting [0,t] into [0,t/3] ∪ [t/3, 2t/3] ∪ [2t/3,t], we infer that a∂ x y(.,t) L 2 ≤ Ca √ t y 0 L 2 t ∈ (0, 1], (2.27) ∂ 3 x y(.,t) L 2 ≤ C t 3 ∂ 2 x y(., 2t 3 ) L 2 ≤   C t 3   2 ∂ x y(., t 3 ) L 2 ≤   C t 3   3 y 0 L 2 (2.28) Combining (2.27) and (2.28), we infer the existence of a constant C = C (a) > 0 (say C ≥ 1, for simplicity), such that Ay(t) L 2 ≤ C t 3 2 y 0 L 2 , for y 0 ∈ L 2 (−1, −), t ∈ (0, 1]. (2.29) For y 0 ∈ D(A n−1 ), z(t) = A n−1 y(t) satisfies z(.,t) = e tA (A n−1 y 0 ) and thus A n y(.,t) L 2 = Az(.,t) L 2 C t 3 2 A n−1 y 0 . For y 0 ∈ L 2 (−1, 0) and t ∈ (0, 1], splitting [0,t] into [0, t n ] ∪ [ t n , 2t n ] ∪ · · · ∪ [ n−1 n t,t], we obtain A n y(.,t) L 2 ≤ C ( t n ) 3 2 A n−1 y( n − 1 n t) ≤ · · · ≤ C ( t n ) 3 2 n y 0 = C n t 3n 2 n 3n 2 y 0 L 2 · (2.30) If p ∈ N is given, we pick n ∈ N such that 3n − 3 ≤ p ≤ 3n − 1. Then, by Sobolev embedding, we have that ∂ p x y(.,t) L ∞ ≤ C y(.,t) H p+1 ≤ C y(.,t) H 3n ≤ C ( y(.,t) L 2 + Py(.,t) L 2 + · · · + P n y(.,t) L 2 ) ≤ C 1 + C t 3 2 + · · · + C n t 3n 2 n 3n 2 y 0 L 2 ≤ C C n (n + 1)n 3n 2 t 3n 2 y 0 L 2 · Since (n + 1) (3n) 3n 2 3 3n 2 ≤ (1 + p 3 ) 3 p+1 2 (p + 3) p+3 2 ≤ C p 3 4 ( e 3 ) p 2 ((p + 1)!) 1 2 we see that there are some constants C > 0 and R > 0 such that \|∂ p x y(x,t)\| ≤ C R p t p+3 2 (p!) 1 2 , p ∈ N, t ∈ (0, 1], x ∈ [0, 1]. From (2.30), we have that y ∈ C((0, 1], D(A n )) for all n ≥ 0 and hence that y ∈ C ∞ ([0, 1] × (0, 1]). Finally, for all n ≥ 0 and p ≥ 0, we have that ∂ n t ∂ p x y = (−1) n P n ∂ p x y = (−1) n (∂ 3 x + a∂ x ) n ∂ p x y = (−1) n n ∑ q=0 n q a n−q ∂ n+2q+p x y and hence, assuming R < 1, \|∂ n t ∂ p x y(x,t)\| ≤ C n ∑ q=0 n q a n−q (n + 2q + p)! 1 2 R n+2q+p t n+2q+p+3 2 ≤ C (n + 1)(2a) n (3n + p)! 1 2 R 3n+p t 3n+p+3 2 ≤ C (n + 1)(2a) n 2 3n+p 2 (3n!) 1 2 p! 1 2 R 3n+p t 3n+p+3 2 ≤ K t 3n+p+3 2 n! 3 2 R n 1 p! 1 2 R p 2 for some K,C 1 ,C 2 ∈ (0, +∞) and for all t ∈ (0, 1] and all x ∈ [0, 1]. The proof of Proposition 2.2 is complete. It is actually expected that for y 0 ∈ L 2 (−1, 0) and u ≡ 0, we have that y ∈ G 1 3 ,1 ([−1, 0] × [ε, T ]) ∀ 0 < ε < T < ∞. Proving such a property seems to be challenging. The smoothing effect from L 2 to G 1/3 is much easy to establish on R for data with compact support. The proof of the following result is given in appendix. ∂ t y + ∂ 3 x y = 0, t > 0, x ∈ R, (2.31) y(x, 0) = y 0 (x), x ∈ R. (2.32) Then y ∈ G ∂ tȳ + ∂ 3 xȳ + a∂ xȳ = 0, x ∈ (−1, 0), t ∈ (0, T ), (2.33) y(0,t) = ∂ xȳ (0,t) =ȳ(−1,t) = 0, t ∈ (0, T ), (2.34) y(x, 0) = y 0 (x), x ∈ (−1, 0). (2.35) It follows from Proposition 2.2 thatȳ ∈ G 1 2 , 3 2 ([−1, 0] × [ε, T ]) for any ε ∈ (0, T ). In particular, ∂ 2 xȳ (0, .) ∈ G 3 2 ([ε, T ]) for any ε ∈ (0, T ). Pick any τ ∈ (0, T ) and let z(t) = φ s t − τ T − τ ∂ 2 xȳ (0,t), where φ s is the "step function" φ s (ρ) =        1 if ρ ≤ 0, 0 if ρ ≥ 1, e − M (1−ρ) σ e − M ρ σ +e − M (1−ρ) σ if ρ ∈ (0, 1), with M > 0 and σ := (s − 1) −1 . As φ s is Gevrey of order s (see e.g. [12]) and s ≥ 3/2, we infer that z ∈ G s ([ε, T ]) for all ε ∈ (0, T ). Let y(x,t) = y 0 (x) if x ∈ [−1, 0], t = 0, ∑ i≥0 g i (x)z (i) (t) if x ∈ [−1, 0], t ∈ (0, T ]. Then, by Proposition 2.1, y ∈ G ∂ p x y(0,t) = ∂ p xȳ (0,t), ∀t ∈ (0, T ), ∀p ∈ {0, 1, 2}, so that y(x,t) =ȳ(x,t) ∀(x,t) ∈ [−1, 0] × (0, τ), by Holmgren theorem. We infer that y ∈ C([0, T ], L 2 (−1, 0)) and that it solves (1.6)-(1.9) if we define u as in (2.5). Note that u(t) = 0 for 0 < t < τ and that u ∈ G s ([0, T ]). Finally y(., T ) = 0, for z (i) (T ) = 0 for all i ≥ 0. The proof of Theorem 1.1 is complete. REACHABLE STATES 3.1. The limit case s = 3 in the flatness property. The following result extends the flatness property depicted in Proposition 2.1 to the limit case s = 3. Proposition 3.1. Assume that z ∈ G 3 ([0, T ]) with \|z ( j) (t)\| ≤ M (3 j)! R 3 j ∀ j ≥ 0, ∀t ∈ [0, T ] (3.1) where R > 1, and let y = y(x,t) be as in ( Proof. We follow closely [11]. Pick any m, n ∈ N. By (2.17), we can assume that i ≥ n. Setting j = 3i − 3n and N = 3n + 3m, so that j + N = 3i + 3m, we have that ∂ m t P n g i (x)z (i) (t) ≤ M (3i + 3m)! R 3i+3m 1 (3(i − n) + 2)! ≤ M ( j + N)! R j+N 1 ( j + 2)! · Let S := ∑ i≥n \|∂ m t P n (g i (x)z (i) (t))\|. If N ≤ 2, then S ≤ M ∑ j≥0 R −( j+N) < ∞ for R > 1. Assume from now on that N ≥ 2. Then S ≤ M ∑ j≥0 ( j + 3) · · · ( j + N) R j+N ≤ M ∑ k≥0 ∑ kN≤ j<(k+1)N ( j + 3) · · · ( j + N) R j+N ≤ M ∑ k≥0 N (k + 2)N N−2 R (k+1)N ≤ MN N−1 ∑ k≥0 k + 2 R k+1 N · Pick any σ ∈ (0, 1) and let a := sup k≥0 k+2 (R 1−σ ) k+1 < ∞. We infer from [11, Proof of Proposition 3.1] that ∑ k≥0 k + 2 R k+1 N ≤ a N R Nσ − 1 , so that S ≤ MN N−1 a N R Nσ − 1 ≤ M ae R σ N N! N 3 2 for some constant M > 0, by using Stirling formula. Next, we have that N! = (3n + 3m)! ≤ 2 3n+3m (3n)!(3m)! and using again the estimate (3m)! m! 3 ∼ 3 3m √ 3 2πm , we infer that 3 2 for some positive constants M , M , R 1 and R 2 . There is no loss of generality in assuming that R 2 < 1. Let K be as in Lemma 2.2 for p = ∞. Then we have S ≤ M ae R σ 3m+3n 2 3n+3m (3n)! 3 3m m + 1 (m!) 3 1 (m + n + 1) 3 2 ≤ M (3n)! R 3n 2 (m!) 3 R m 1 1 (n + 1)∑ i≥n ∂ m t (g i (x)z (i) (t)) 3n,∞ ≤ K n ∑ i≥n ∑ 0≤ j≤n ∂ m t P j (g i (x)z (i) (t)) ∞ ≤ M K n ∑ 0≤ j≤n (m!) 3 R m 1 (3 j)! R 3 j 2 1 ( j + 1) 3 2 ≤ M K n (3n)! R 3n 2 (m!) 3 R m 1 ∑ j≥0 1 ( j + 1) 3 2 · This shows that the series of derivatives ∂ m t ∂ l x g i (x)z (i) (t) is uniformly convergent on [−1, 0] × [0, T ] for all m, l ∈ N, so that y ∈ C ∞ ([−1, 0] × [0, T ]), and that the function y satisfies for l ≤ 3n \|∂ m t ∂ l x y(x,t)\| ≤ M K n (3n)! R 3n 2 (m!) 3 R m 1 ∀x ∈ [−1, 0], ∀t ∈ [0, T ]. for some constant M > 0. Finally, if l ∈ {3n − 2, 3n − 1, 3n}, then (3n)!(K/R 3 2 ) n ≤ C l!/R l 2 for some C > 0, R 2 > 0. This yields \|∂ m t ∂ l x y(x,t)\| ≤ C M (m!) 3 R m 1 l! R l 2 , ∀x ∈ [−1, 0], ∀t ∈ [0, T ]. 3.2. Proof of Theorem 1.2. Pick any R > R 0 = e (3e) −1 (1 + a) 1 3 and any y 1 ∈ R R . Our first task is to write y 1 in the form y 1 (x) = ∑ i≥0 b i g i (x), x ∈ [−1, 0]. (3.2) Note that if (3.2) holds with a convergence in W n,∞ (−1, 0) for all n ≥ 0, then ∂ 2 x P n y 1 (0) = ∑ i≥0 b i ∂ 2 x P n g i (0) = ∑ i≥n b i ∂ 2 x g i−n (0) = b n . Set b n := ∂ 2 x P n y 1 (0), ∀n ≥ 0. Since y 1 ∈ R R with R > R 0 , there exists for any r ∈ (R 0 , R) a constant C = C(r) > 0 such that \|∂ n x y 1 (x)\| ≤ C n! r n , ∀x ∈ [−r, 0], ∀n ∈ N. Using Lemma 2.2, we infer that \|b n \| = \|P n ∂ 2 x y 1 (0)\| ≤ (1 + 1 a )(1 + a) n ∂ 2 x y 1 3n,∞ ≤ C (1 + a) n (3n + 2)! r 3n+2 for some C > 0 and all n ≥ 0. We need the following version of Borel Theorem, which is a particular case of [11,Proposition 3.6] (with a p = [3p(3p − 1)(3p − 2)] −1 for p ≥ 1). Proposition 3.2. Let (d q ) q≥0 be a sequence of real numbers such that \|d q \| ≤ CH q (3q)! ∀q ≥ 0 for some H > 0 and C > 0. Then for allH > e e −1 H, there exists a function f ∈ C ∞ (R) such that f (q) (0) = d q ∀q ≥ 0, \| f (q) (x)\| ≤ CH q (3q)! ∀q ≥ 0, ∀x ∈ R. Since r > R 0 , we can pick two numbers H ∈ (0, e −e −1 ) and C > 0 such that \|b n \| ≤ C H n (3n)! ∀n ≥ 0. By Proposition 3.2, there exists a function f ∈ G 3 ([0, T ]) and a number R > 1 such that f (i) (T ) = b i ∀i ≥ 0, (3.3) \| f (i) (t)\| ≤ C (3i)! R 3i ∀i ≥ 0, ∀t ∈ [0, T ]. (3.4) Pick any τ ∈ (0, T ) and let g(t) = 1 − φ 2 t − τ T − τ for t ∈ [0, T ]. Note that g ∈ G 2 ([0, T ]) and that g(T ) = 1, g (i) (T ) = 0 for all i ≥ 1. Setting z(t) = g(t) f (t) ∀t ∈ [0, T ], we have that z ∈ G 3 ([0, T ]) and that 6). Furthermore, we have by (3.5) that z (i) (T ) = b i ∀i ≥ 0, (3.5) z (i) (0) = 0 ∀i ≥ 0, (3.6) \|z (i) (t)\| ≤ C (3i)! R 3i ∀i ≥ 0, ∀t ∈ [0, T ]y(x, T ) = ∑ i≥0 g i (x)z (i) (T ) = ∑ i≥0 b i g i (x), x ∈ [−1, 0]. From the proof of Proposition 3.1, we know that for all l, m ∈ N, the sequence of partial sums of the series ∑ i≥0 ∂ m t ∂ l x g i (x)z (i) (t) converges uniformly on [−1, 0] × [0, T ] to ∂ m t ∂ l x y, and hence for all n ≥ 0 P n y(0, T ) = ∑ i≥0 b i P n g i (0) = ∑ i≥n b i g i−n (0) = 0, ∂ x P n y(0, T ) = ∑ i≥0 b i ∂ x P n g i (0) = ∑ i≥n b i ∂ x g i−n (0) = 0, ∂ 2 x P n y(0, T ) = ∑ i≥0 b i ∂ 2 x P n g i (0) = ∑ i≥n b i ∂ 2 x g i−n (0) = b n = ∂ 2 x P n y 1 (0). To conclude that y(x, T ) = y 1 (x), ∀x ∈ [−1, 0], it is sufficient to prove the following CLAIM 3. If h ∈ G 1 ([−1, 0]) is such that P n h(0) = ∂ x P n h(0) = ∂ 2 x P n h(0) = 0 for all n ∈ N, then h ≡ 0. Indeed, we notice that P 0 = id and that P n = ∂ 3n x + · · · , ∂ x P n = ∂ 3n+1 x + · · · , and ∂ 2 x P n = ∂ 3n+2 x + · · · , where · · · stands for less order derivatives. Then we obtain by induction that ∂ 3n x h(0) = ∂ 3n+1 x h(0) = ∂ 3n+2 x h(0) = 0 for all n ≥ 0, so that h ≡ 0. This completes the proof of Claim 3 and of Theorem 1.2. APPENDIX 3.3. Proof of Lemma 2.2. We first need to prove two simple lemmas. We still use the notation P = ∂ 3 x + a∂ x . Lemma 3.1. Let a ∈ R + and p ∈ [1, ∞]. Then for all n ∈ N, we have P n f p ≤ (1 + a) n f 3n,p ∀ f ∈ W 3n,p (−1, 0). (3.8) Proof. The proof is by induction on n. For n = 0, the result is obvious. If it is true at rank n − 1, then P n−1 (P f ) p ≤ (1 + a) n−1 P f 3n−3,p ≤ (1 + a) n−1 f 3n−3,p + a f 3n−3,p ≤ (1 + a) n f 3n,p . Lemma 3.2. Let a ∈ R + and p ∈ [1, ∞]. Then there exists a constant C 1 = C 1 (p, a) > 0 such that f 3,p ≤ C 1 ( f p + P f p ) ∀ f ∈ W 3,p (−1, 0). (3.9) Proof. The seminorm \|\|\| f \|\|\| := f p + P f p is clearly a norm in W 3,p (−1, 0). Let us check that W 3,p (−1, 0), endowed with the norm \|\|\| · \|\|\|, is a Banach space. Pick any Cauchy sequence ( f n ) n≥0 for \|\|\| · \|\|\| . Then \|\|\| f m − f n \|\|\| = f m − f n p + P f m − P f n p → 0, as m, n → +∞. Since L p (−1, 0) is a Banach space, there exist f , g ∈ L p (−1, 0) such that f n − f p + P f n − g p → 0, as n → ∞. (3.10) Since f n → f in D (−1, 0), P f n → P f in D (−1, 0) as well, and P f = g. Thus f = g(x)dx − a f ∈ L p (−1, 0), and f ∈ W 2,p (−1, 0). This yields f ∈ W 1,p (−1, 0) and f = g − a f ∈ L p (−1, 0), and hence f ∈ W 3,p (−1, 0). Note that (3.10) can be written \|\|\| f n − f \|\|\| → 0. This proves that (W 3,p (−1, 0), \|\|\| · \|\|\|) is a Banach space. Now, applying the Banach theorem to the identity map from the Banach space (W 3,p (−1, 0), · 3,p ) to the Banach space (W 3,p (−1, 0), \|\|\| · \|\|\|), which is linear, continuous, and bijective, we infer that its inverse is continuous; that is, (3.9) holds. Let us prove Lemma 2.2. We proceed by induction on n. Both inequalities in (2.16) are obvious for n = 0. Assume now that both inequalities in (2.16) are satisfied up to the rank n − 1. Let us first prove the left inequality in (2.16) at the rank n. Pick any f ∈ W 3n,p (−1, 0). Then, by the induction hypothesis and Lemma 3.1, we obtain But it follows from Lemma 3.2 that ∂ 3n−3 x f 3,p ≤ C 1 ( ∂ 3n−3 x f p + P∂ 3n−3 x f p ) ≤ C 1 ( ∂ 3n−3 x f p + ∂ 3n−3 x P f p ) ≤ C 1 ( f 3n−3,p + P f 3n−3,p ) ≤ C 1 K n−1 n−1 ∑ i=0 P i f p + n ∑ i=1 P i f p ≤ 2C 1 K n−1 n ∑ i=0 P i f p . Combined with (3.11), this yields f 3n,p ≤ (1 + 2C 1 )K n−1 n ∑ i=0 P i f p . It is sufficient to pick K := 1 + 2C 1 . 3.4. Proof of Proposition 2.3. Let Ai denote the Airy function defined as the inverse Fourier transform of ξ → exp(iξ 3 /3). Then it is well known (see e.g. [7]) that Ai is an entire (i.e. complex analytic on C) function satisfying Ai (x) = xAi(x), ∀x ∈ C. (3.12) To prove that y ∈ G As it was noticed in [10, Remark 2.7], we have s i i! ≤ i ∏ j=1 (r + js) ≤ s i (i + 1)! ∀s ∈ N, ∀r ∈ [0, s]. Theorem 1. 2 . 1 3 > 1 . 211Let a ∈ R + , T > 0, and R > R 0 := e (3e) −1 (1 + a) Pick any y 1 ∈ R R . Then there exists a control input u ∈ G 3 ([0, T ]) such that the solution y of (1.6)-(1.9) with y 0 = 0 satisfies y(., T ) = y 1 . Furthermore, y ∈ G 1,3 ([−1, 0] × [0, T ]). ( 3 ) 3The set R R takes a very simple form when a = 0. Indeed, in that caseR R = {y ∈ C( Fig. 1 . 1The function y(x) = e x + je jx + j 2 e j 2 x . For p ∈ {0, 1, 2, 3, 4}, we introduce the Banach spacesX 0 = L 2 (−1, 0), X 1 = H 1 0 (−1, 0), X 2 = {y ∈ H 2 (−1, 0); y(−1) = y(0) = ∂ x y(0) = 0}, X 3 = D(A), and X 4 = {y ∈ H 4 (−1, 0), y(−1) = y(0) = ∂ x y(0) = ∂ 3 x y(−1) = ∂ 3 x y(0) = 0}, X p being endowed with the norm · H p for p = 0, ..., 4. CLAIM 1 There is a constant C = C(a) > 0 such that Proposition 2. 3 . 3Let y 0 ∈ L 2 (R) be such that y 0 (x) = 0 for a.e. x ∈ R \ [−L, L] for some L > 0. Let y = y(x,t) denote the solution of the Cauchy problem 1 3 1,1 ([−l, l] × [ε, T ]) for all l > 0 and all 0 < ε < T .2.3. Proof of Theorem 1.1. Pick any y 0 ∈ L 2 (−1, 0), T > 0, and s ∈ [ 3 2 , 3). Letȳ denote the solution of the free evolution for the KdV system: 0] × [ε, T ]) for all ε ∈ (0, T ), and it satisfies (2.1)-(2.3). Furthermore, 2.4). Then y ∈ G 1,3 ([−1, 0] × [0, T ]) and it solves (2.1)-(2.3). C > 0. (The fact that the constant R > 0 in (3.7) is the same as in (3.4) is proved as in [11, Lemma 3.7].) Let y be as in (2.4). Then by Proposition 3.1 we know that y ∈ G 1,3 ([−1, 0] × [0, T ]) and that it solves (2.1)-(2.3). Let u(t) = y(−1,t) for t ∈ [0, T ]. Then u ∈ G 3 ([0, T ]) and y solves (1.6)-(1.9) with y 0 = 0, by (3. 1 3 1,1 ([−l, l] × [ε, T ]), we need first check that the Airy function is itself Gevrey of order 1/3. CLAIM 4. Ai ∈ G 1 3 ([−l, l]) for any l > 0. Let us prove Claim 4. Differentiating k + 1 times in (3.12) and letting x = 0 results in Ai (3+k) (0) = (Ai ) (k+1) (0) = (k + 1)Ai (k) (0) ∀k ∈ N. 3 j), Ai (3k+2) (0) = 0. ,s ([−1, 0] × [ε, T ]) ∀ε ∈ (0, T ). (1.10) ACKNOWLEDGEMENTSThis work was done when the second author (IR) was visiting CAS, MINES ParisTech. The third author (LR) was supported by the ANR project Finite4SoS (ANR-15-CE23-0007).Thus there is some constant C 1 > 0 such that Ai (3k+q) (0) ≤ C 1 3 k k! ∀k ≥ 0, ∀q ∈ {0, 1, 2}.From Stirling formula we have (3k)! ∼ √ 3 2πk (3 k k!) 3 so thatfor any R ∈ (0, 1) and some constants C 2 ,C 3 > 0. The following result comes from[9,13].Lemma 3.3. Let s ∈ (0, 1) and let (a n ) n≥0 be a sequence such that \|a n \| ≤ C n! s R n ∀n ≥ 0for some constants C, R > 0. Then the function f (x) = ∑ n≥0 a n x n n! is Gevrey of order s on [−l, l] for all l > 0, withIt follows from (3.13) and Lemma 3.3 that the Airy function is Gevrey of order 1/3 on each intervalAix − s (3t)1 3y 0 (s) ds.It is clear that y is of class C ∞ on R x × (0, +∞) t . Pick any l > 0 and any 0 < ε < T . Since y solves (2.31), we have for all x ∈ [−l, l], t ∈ [ε, T ] and p, q ∈ N thatfor some constants K, K , R 1 , R 2 > 0 which depend on l, L, ε and T . The proof of Proposition 2.3 is complete. Neumann boundary controllability of the Korteweg-de Vries equation on a bounded domain. M A Caicedo, R De A. Capistrano, B.-Y Filho, Zhang, SIAM J. Control Optim. 556M. A. Caicedo, R. de A. Capistrano Filho, B.-Y. Zhang, Neumann boundary controllability of the Korteweg-de Vries equation on a bounded domain, SIAM J. Control Optim. 55 (2017), no. 6, 3503-3532. Control of a Korteweg-de Vries equation: a tutorial. E Cerpa, Math. Control Relat. Fields. 41E. Cerpa, Control of a Korteweg-de Vries equation: a tutorial, Math. Control Relat. Fields 4 (2014), no. 1, 45-99. Boundary controllability of the Korteweg-de Vries equation on a bounded domain. E Cerpa, I Rivas, B.-Y Zhang, SIAM J. Control Optim. 514E. Cerpa, I. Rivas, B.-Y. 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Rouchon, Controllability of some evolution equations by the flatness approach, in press. A F Pazoto, L Rosier, Stabilization of a Boussinesq system of KdV-KdV type. 57A. F. Pazoto, L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems & Control Letters 57 (2008), 595-601. Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. L Rosier, ESAIM Control Optim. Calc. Var. 2L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33-55. A fundamental solution supported in a strip for a dispersive equation. L Rosier, Special issue in memory of Jacques-Louis Lions. 21L. Rosier, A fundamental solution supported in a strip for a dispersive equation. Special issue in memory of Jacques-Louis Lions, Comput. Appl. Math. 21 (2002), no. 1, 355-367. Control of the surface of a fluid by a wavemaker. L Rosier, ESAIM Control Optim. Calc. Var. 103L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var. 10 (3) (2004), 346-380. Control and stabilization of the Korteweg-de Vries equation: recent progresses. L Rosier, B.-Y Zhang, J. Syst. Sci. Complex. 224L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: recent progresses. J. Syst. Sci. Complex. 22 (2009), no. 4, 647-682. CALLE 13 NO. 100-00, A.A. 25360, CALI, COLOMBIA E-mail address: ivonne.rivasl@correounivalle. Centre Automatique, Mines Systèmes (cas), Psl Paristech, Boulevard Research University, 60, Universidad Del Saint-Michel ; Fr, Ciudadela Valle, Universitaria Meléndez, address: [email protected] CENTRE AUTOMATIQUE ET SYSTÈMES (CAS). CAS) AND CENTRE DE ROBOTIQUE, MINES PARISTECH, PSL RESEARCH UNIVERSITY, 60 BOULEVARD SAINT-MICHEL, 75272 PARIS CEDEX 06, FRANCE E-mail; MINES PARISTECH, PSL RESEARCH UNIVERSITY, 60 BOULEVARD SAINT-MICHEL75272 PARIS CEDEX 06, FRANCE E-mail address: philippe.martin@mines-paristech. 75272 PARIS CEDEX 06, FRANCE E-mail address: [email protected] AUTOMATIQUE ET SYSTÈMES (CAS), MINES PARISTECH, PSL RESEARCH UNIVERSITY, 60 BOULEVARD SAINT-MICHEL, 75272 PARIS CEDEX 06, FRANCE E-mail address: [email protected] UNIVERSIDAD DEL VALLE, CIUDADELA UNIVERSITARIA MELÉNDEZ, CALLE 13 NO. 100-00, A.A. 25360, CALI, COLOMBIA E-mail address: [email protected] CENTRE AUTOMATIQUE ET SYSTÈMES (CAS) AND CENTRE DE ROBOTIQUE, MINES PARISTECH, PSL RESEARCH UNIVERSITY, 60 BOULEVARD SAINT-MICHEL, 75272 PARIS CEDEX 06, FRANCE E-mail address: [email protected] CENTRE AUTOMATIQUE ET SYSTÈMES (CAS), MINES PARISTECH, PSL RESEARCH UNIVERSITY, 60 BOULEVARD SAINT-MICHEL, 75272 PARIS CEDEX 06, FRANCE E-mail address: [email protected]	[]
[ "Microstates of black holes in expanding universe from interacting branes", "Microstates of black holes in expanding universe from interacting branes" ]	[ "Shotaro Shiba [email protected] \nTheory Center\nHigh Energy Accelerator Research Organization (KEK)\n1-1 Oho305-0801TsukubaIbarakiJapan\n" ]	[ "Theory Center\nHigh Energy Accelerator Research Organization (KEK)\n1-1 Oho305-0801TsukubaIbarakiJapan" ]	[]	Thermodynamics of the near extremal black p-branes can be described by collective motions of gravitationally interacting branes. This proposal is called the p-soup model. In this paper, we check this proposal in the case of black brane system which is asymptotically Friedmann-Lemaître-Robertson-Walker universe in an infinite distance.As a result, we can show that the gravitationally interacting branes explain free energy, entropy, temperature and other physical quantities in these systems. This implies that the microstates of this kind of brane system can be also understood in the p-soup model.	10.1007/jhep05(2017)079	[ "https://arxiv.org/pdf/1702.01566v2.pdf" ]	119,413,219	1702.01566	d10862bd3ab03fc4a4836cca161f070dea3e8bd3	Microstates of black holes in expanding universe from interacting branes 31 Mar 2017 Shotaro Shiba [email protected] Theory Center High Energy Accelerator Research Organization (KEK) 1-1 Oho305-0801TsukubaIbarakiJapan Microstates of black holes in expanding universe from interacting branes 31 Mar 2017Preprint typeset in JHEP style -HYPER VERSION Thermodynamics of the near extremal black p-branes can be described by collective motions of gravitationally interacting branes. This proposal is called the p-soup model. In this paper, we check this proposal in the case of black brane system which is asymptotically Friedmann-Lemaître-Robertson-Walker universe in an infinite distance.As a result, we can show that the gravitationally interacting branes explain free energy, entropy, temperature and other physical quantities in these systems. This implies that the microstates of this kind of brane system can be also understood in the p-soup model. Introduction Microstates of black holes are still an outstanding problem in theoretical physics. This discussion was initiated by Strominger and Vafa [1]. In their picture, branes are static in noncompact spacetime and strings on the branes provide dynamical degrees of freedom. Such studies have been mainly developed in the intersecting black branes, especially in the D1-D5 system [2]. (See also a review [3].) However, it is still unclear whether they can be generalized to other various types of black holes. Recently we proposed another description of the black hole microstates [4,5,6]. In our picture, branes are moving at the speed proportional to Hawking temperature. They have kinetic energy and strongly gravitationally interacting with each other, then compose a bound state at low energy. We can regard this bound state as a black brane. More concretely, we impose the following settings for the system: • The characteristic velocity v of the branes should satisfy the condition v ∝ πT r, where r is the characteristic size of the system and T is Hawking temperature. • If the effective action for the branes is expanded in a series of gravitational coupling, all the terms should be of the same order. The first setting can be understood as the condition which Matsubara modes satisfy in systems at finite temperature. The second setting is a kind of virial theorem for systems with strong gravitational coupling. We call this proposal the p-soup model. In the previous papers, using these settings, we discussed the systems of parallel Dor M-branes [7] and intersecting D-or M-branes [8,9]. We analyzed these systems in our picture, then we could correctly estimate free energy, horizon size and other physical quantities. The results are consistent with those of the corresponding black branes. Here we note that while the branes are moving in our picture, these corresponding black branes are static solutions in supergravity. In some cases, momentum can be introduced in an isometry direction, but the time dependence is so limited. In this paper, we discuss more nontrivial time-dependent systems. A time-dependent solution in supergravity can be obtained as a simple generalization of a static black branes [10]. Therefore, we analyze the microstates of such solutions in the p-soup model, and check if our discussion can be applicable to the time-dependent brane systems. As a result, we can successfully show that gravitationally interacting branes explain correct physical quantities of the time-dependent black branes. Compared with the static black branes, unfortunately, some uncertainty appears. Including such a subtle point, we discuss these systems in detail. This paper is organized as follows. In §2, we review the time-dependent black brane solutions in Einstein-Maxwell-dilaton theory. In §3, we analyze such black brane systems based on the p-soup model and check if we can reproduce the supergravity results (summarized in appendix A). In §4, we conclude our discussion. Time-dependent black brane solution We consider D-dimensional gravitational theory coupled to dilaton φ and (n A + 1)-form field. The action is S D = 1 16πG D d D x √ −g R − 1 2 (∂φ) 2 − A 1 2(n A + 2)! e a A φ F 2 n A +2 (2.1) where we set a 2 A = 4 − 2(n A + 1)(D − n A − 3) D − 2 , (2.2) so that we have asymptotically flat spacetime solutions. The solutions are understood as intersecting brane systems. In the extremal limit, the metric can be written as ds 2 D = A H q A +1 D−2 A − A H −1 A dt 2 + D−d α=1 A H −δ (α) A A dy 2 α + d−1 i=1 dx 2 i (2.3) where H A is a harmonic function in (d − 1) dimensions x i . q A is the spatial dimension of brane A. δ (α) A equals 1 if the brane A is expanded in the direction y α , and otherwise δ (α) A = 0. The index A denotes species of branes. 1 The harmonic functions H A are usually time-independent: H A S = 1 + Q A S r d−3 . (2.4) However, we can generalize them by making some of them time-dependent 2 [10,11,12]: H A T = t t A T + Q A T r d−3 . (2.5) Here Q A S and Q A T are brane charges. t A T is constant and determines time dependence of each brane A T . Note that r 2 := i x 2 i , so then H A never depends on y α . This means that all the branes are winding or smeared in all the y α directions. Then we assume here that the y α directions are compactified on a torus T D−d and only the x i directions remain noncompact. Let us consider dimensional reduction of all the y α directions. The x i directions have spherical symmetry, so the metric in Einstein frame is ds 2 d = A H 1 d−2 A A H −1 A dt 2 + dr 2 + r 2 dΩ 2 d−2 (2.6) where A H A = n S A S =1 H A S n T A T =1 H A T = A Q A r (d−3)(n S +n T ) A S r d−3 Q A S + 1 A T t t A T r d−3 Q A T + 1 =: R r (d−3)(n S +n T ) . (2.7) Here n S and n T are the numbers of species of static and time-dependent branes, respectively. And we define R (d−3)(n S +n T ) = A S r d−3 Q A S + 1 A T t t A T r d−3 Q A T + 1 , r (d−3)(n S +n T ) = r (d−3)(n S +n T ) A Q A . (2.8) 1 If branes of the same species are expanded in different directions, we distinguish them here. 2 If there are more than one species of time-dependent branes (nT > 1), we need an additional potential for the dilaton ∼ e −αφ (where α is a non-negative constant) in the action (2.1) to obtain such solutions. The author would like to thank Nobuyoshi Ohta to point this out. In the following discussions, for simplicity, we set Q A S = Q S , Q A T = Q T , t A T = t 0 (2.9) for all A S and A T . Then we obtain the expressions R (d−3)(n S +n T ) = r d−3 Q S + 1 n S t t 0 r d−3 Q T + 1 n T ,r d−3 = r d−3 Q , (2.10) where we define Q := (Q n S S Q n T T ) 1 n S +n T . Region at infinite distance Let us now comment on the asymptotic behavior at an infinite distance r → ∞. We can rewrite the metric (2.6) as ds 2 d = −Ξ d−3 dt 2 + a 2 Ξ (dr 2 + r 2 dΩ 2 d−2 ) (2.11) where we defineΞ := 1 + Q S r d−3 − n S d−2 1 + t 0 t Q T r d−3 − n T d−2 (2.12) andt t 0 = t t 0 1− (d−3)n T 2(d−2) , a = t t 0 n T 2(d−2) . (2.13) Then in the limit of r → ∞, this metric becomes ds 2 d = −dt 2 + a 2 dr 2 + r 2 dΩ 2 d−2 . (2.14) This means that the time coordinatet is the proper time at infinity and the metric in infinity is asymptotically Friedmann-Lemaître-Robertson-Walker (FLRW) metric with the scale factor a ∝t n T 2(d−2)−(d−3)n T . Therefore, for 0 < n T ≤ 2(d−2) d−3 , we have expanding FLRW universe. For n T > 2, we have accelerating universe (if d > 3). Especially in the case of n T = 2(d−2) d−3 , the scale factor is divergent. This can be understood to describe the exponential expansion, i.e. de Sitter spacetime. Later we will concentrate on the cases of n T + n S = 2(d−2) d−3 , so there this de Sitter case is equivalently the n S = 0 case, where all the branes are time-dependent [13]. Near horizon region Now we look at the near horizon region r d−3 /Q S ≪ 1, r d−3 /Q T ≪ 1. If we also impose the condition tr d−3 /t 0 Q T ≪ 1, the metric (2.6) becomes 15) and the discussion becomes parallel to the time-independent cases n T = 0 [9]. Therefore, let us here concentrate on the case of ds 2 d = Q r d−3 n S +n T d−2 r d−3 Q n S +n T dt 2 + dr 2 + r 2 dΩ 2 d−2 ,(2.r d−3 Q S ≪ 1 , r d−3 Q T ≪ 1 , t t 0 r d−3 Q T ≃ 1 (2.16) to keep nontrivial time dependence. This means that when we define the variables as t := t t 0 ,r d−3 := r d−3 Q , (2.17) the near horizon limit can be defined by [11,12] t →t ǫ , r d−3 →r d−3 ǫ , ǫ → 0 . (2.18) In this limit, all the terms in the metric (2.6) remain finite, only if n T + n S = 2(d − 2) d − 3 . (2.19) Since d and n T + n S should be positive integers, all the combinations we need to consider are only (d, n T + n S ) = (1, 1), (4, 4), (5, 3). (2.20) In the following discussions, we will concentrate on the latter two cases. 3 Hereafter we fix n S as the relation (2.19) is satisfied and impose n T = 0. Then let us consider the metric in the near horizon limit. In this limit we obtain R (d−3)(n T +n S ) = (qr d−3 + 1) n S (qtr d−3 + 1) n T → (qtr d−3 + 1) n T (2.21) whereq := Q/Q S and q := Q/Q T , which are assumed to be of order one. Using this relation, we can change the radial coordinate. The metric in this limit becomes ds 2 nh Q 2 d−3 = − f (R) t 2 R 2(d−3) dt 2 − 4(d − 2) (d − 3) 2 n T 1 t R 2(d−2) n T +1 R 2(d−2) n T − 1 dtdR + 4(d − 2) 2 (d − 3) 2 n 2 T R 4(d−2) n T R 2(d−2) n T − 1 2 dR 2 + R 2 dΩ 2 d−2 (2.22) where we define f (R) := τ 2 R 2(d−2) n T − 1 2 − R 2(d−2) (d − 3) 2 , τ = t 0 qQ 1 d−3 . (2.23) Moreover, we can change the time coordinate so that the metric becomes a static form: ds 2 nh Q 2 d−3 = − f (R) R 2(d−3) dT 2 + 4(d − 2) 2 (d − 3) 2 n 2 T τ 2 R 4(d−2) n T f (R) dR 2 + R 2 dΩ 2 d−2 (2.24) where T = ± ln \|t\| + R 2(d − 2) (d − 3) 2 n T R 2d−5 1 − R − 2(d−2) n T f (R) dR . (2.25) This means that all the branes are static in this coordinate (T, R). Note that, however, this metric cannot be given in ordinary static black brane systems. Thus we have obtain new examples to study the microstates of black branes in the p-soup model. Analysis based on p-soup model In the p-soup model, microstates of black branes are given by infinitely many (elementary) branes which are moving at the speed proportional to Hawking temperature and are gravitationally interacting with each other. Therefore, in order to analyze black brane systems in this model, we need to write down effective action to describe the interactions among these branes. Let us see details of the interactions. First we choose one of the branes in the system as a probe and consider the probe brane action S probe,A = −µ A d q A +1 ξ − det γ µν + Ê A . (3.1) Here µ A is the brane tension.Ê A is the pullback of gauge potential to the brane worldvolume. γ µν is the worldvolume metric induced from the spacetime metric (2.3): γ µν = ∂ µ Z M ∂ ν Z N g M N e − ǫ A a A q A +1 φ (3.2) where ǫ A = 1, −1 for branes with electric/magnetic charges, respectively. Note that we set the background metric for the probe brane g M N to the original black brane solution (2.3) itself. This can be justified since black brane systems are composed of infinitely many branes. Even after we remove one of them (as a probe) from such a system, its metric must be nearly unchanged. In the D-dimensional metric (2.3), the harmonic functions H A S , H A T depend on only time and x i . It means that when we calculate the worldvolume metric, we can neglect time dependence of brane's behavior in the y α directions. As we saw in §2, all the branes are winding or smeared on the torus T D−d in these directions, so this assumption is justified. Therefore, we can take the static gauge for the coordinates ξ on the probe brane worldvolume. In addition, we assume that motion of the branes depends only on time t. That is, position of the probe brane in the target spacetime Z M = Z M (t). Using these settings and integrating over the torus T q A which the probe brane is winding around, the probe brane action becomes S probe,A = −m A dt   1 H A 1 − d r dt 2 A ′ H A ′ − 1 H A − 1   . (3.3) In the noncompact d-dimensional spacetime, the probe brane behaves as a BPS particle with the mass m A = µ A V A (no sum of A). V A is volume of the torus T q A . r is position of the probe brane in the d dimensions. Probe brane action in static coordinates In the p-soup model, as we mentioned in Introduction, velocities of the branes are important parameters. Naively, in the probe brane action (3.3) we can define the velocity v = d r/dt. However, the background here is time-dependent, so this velocity seems not suitable to describe behaviors of the branes. In order to avoid such problems, let us move to the time-independent frame, that is, the (T, R) coordinates (2.24). Using eqs. and dr dt = − R 2(d−2) n T − 1 1 d−3 (d − 3)q 1 d−3t d−2 d−3 1 − (d − 3) 2 f (R) R 2(d−2) g(R) dR dT 1 − g(R) dR dT . (3.5) Herer is, more precisely, a vector in the noncompact d dimensions. R denotes position of the probe brane in the R coordinate, and we define g(R) := 2(d − 2) (d − 3) 2 n T R 2d−5 1 − R − 2(d−2) n T f (R) . (3.6) Note that thet dependence in eq.(3.5) disappears when we consider the combination d r dt 2 A ′ H A ′ = 1 τ 2 R 2(d−2) (d − 3) 2 R 2(d−2) n T − 1 2 1 − (d − 3) 2 f (R) R 2(d−2) g(R) dR dT 1 − g(R) dR dT 2 . (3.7) Here H A ′ = H A ′ R , since in the action (3.3) we consider the integration on worldvolume of the probe brane. On the measure in the integration, we obtain the expression dt H A = dt H A t 0 (3.8) and dt H A S = Q Q Sr d−3 dt = R 2(d−2) n T − 1 Q T Q S dt t , dt H A T = Q Q Tr d−3 R 2(d−2) n T dt = 1 − R − 2(d−2) n T dt t (3.9) for a static brane A S and a time-dependent brane A T , respectively. Then using the relation (2.25), or equivalently, dt t = dT − g(R)dR = 1 − g(R) dR dT dT ,(3.10) we can successfully eliminated the time coordinatet. Now the probe brane action (3.3) can be written in the static coordinate (T, R), except the rest mass term −m A dt. This term doesn't affect brane's behavior, so we will neglect it in the following analysis. To summarize, the probe brane action for a static brane A S is S probe,A S = −m A S Q T Q S t 0 dT 1 − g(R) dR dT R 2(d−2) n T − 1 1 − h(R) 2 (d − 3) 2 τ 2 − 1 (3.11) and that for a time-dependent brane A T is S probe,A T = −m A T t 0 dT 1 − g(R) dR dT 1 − R − 2(d−2) n T 1 − h(R) 2 (d − 3) 2 τ 2 − 1 (3.12) where we define h(R) := R d−2 R 2(d−2) n T − 1 1 − (d − 3) 2 f (R) R 2(d−2) g(R) dR dT 1 − g(R) dR dT . (3.13) Let us here pay attention to the dependence on gravity coupling in d dimensions κ 2 d . The relation to Q S and Q T are given by 14) where N A is the number of q A -branes and Ω d−2 is the volume of a unit (d − 2)-sphere. Q S = 2m A S N A S (d − 3)Ω d−2 κ 2 d , Q T = 2m A T N A T (d − 3)Ω d−2 κ 2 d ,(3. Therefore, the expansion for small τ −1 = qQ 1 d−3 /t 0 means the expansion for small κ 2 d−3 d . Since the last factor in the actions (3.11) and (3.12) can be expanded as d . Finally we comment on the period of the T direction. Naively, it should be the inverse temperature β = 1/T . By taking into account the normalization, we can accurately obtain the period of the T direction 1 − h(R) 2 (d − 3) 2 τ 2 − 1 = − ∞ n=1 (2n − 3)!! 2 n n! h(R) 2n (d − 3) 2n τ −2n ,(3.1 T ren = (d − 3)n T 2(d − 2) R d−3− 2(d−2) n T τ 1 Q 1 d−3 1 T . (3.16) Here the first two factors come from the renormalization of Killing vector (A.7), and the next Q factor comes from the normalization in the static metric (2.24). Let us note that the temperature is low in our system so that moving branes compose a bound state and we can use the background metric (2.3) in the extremal limit. Effective action Now we discuss the details of interactions among branes and write down effective action to describe them. In the previous subsection, we studied the interaction between a probe brane and the background. Using this information, we can investigate how each brane interacts with other branes in this system. Let us look at the factor τ −2n in the expansion form (3.15). This can be written as τ −2n = qQ 1 d−3 t 0 2n = Q n S S Q n T −2 T t 2 0 n ∝ κ 4n d−3 d . (3.17) We have already fixed n S , but we use it here. From the dependence on Q S and Q T , we can find that this factor describes an interaction among n S n static branes and (n T − 2)n time-dependent branes. By taking into account a probe brane, we find both the probe brane actions (3.11) and (3.12) describe • interactions among n S n static branes and (n T − 2)n + 1 time-dependent branes where n = 1, 2, . . . , ∞. The number of interacting branes are (n S + n T − 2)n + 1 = 2n d−3 + 1, and since G d = κ 2 d /8π, we find 2n d−3 gravitons are exchanged among these branes. Note that we consider only the d = 4, 5 cases, as we saw in eq.(2.20), so the numbers of branes and gravitons are integers. When n T = 1, the number of interacting time-dependent branes becomes negative. This may mean that our picture cannot describe the n T = 1 case, and the p-soup model is valid only for the n T ≥ 2 cases. However, as we will see, the final results are correct also for the n T = 1 case. In order to describe the interactions among the branes, the position of each brane R in the probe brane actions, or the vector from the center of the black brane background, should be replaced by the relative position of arbitrary two branes R i − R j =: R ij .S eff = dT t 0 ∞ n=1 L n (3.19) where L n ∼ κ 2 d (d − 3) d−2 Ω d−2 2n d−3 {s 1 ,...,sn S n} {t 1 ,...,t (n T −2)n+1 } n S n a=1 m sa (n T −2)n+1 b=1 m t b ×   a =1 h( R s 1 sa ) b h( R s 1 t b ) + a h( R sat 1 ) b =1 h( R t 1 t b ) + · · ·   . (3.20) Hereafter '∼' means an equality up to numerical (especially rational) factors. h( R ij ) describes the interaction between two branes out of the interacting 2n d−3 + 1 branes, which is defined as h( R ij ) = 1 − g(R ij ) dR ij dT R 2(d−2) n T ij f(R ij ) d−3 2n h(R ij ) d−3 (3.21) where R ij := \| R ij \|. The '· · ·' term includes the interactions of all the other combinations of the branes. Note that in eq.(3.21) we define a function However, for the remaining analysis, we can get enough information on the interactions from the effective action (3.19), so we don't do it here. This would be an interesting future work. f(R) = c 0 + c 1 1 − R − 2(d−2) n T + c 2 1 − R − 2(d−2) Although such ambiguity is in this analysis, the physical quantities can be estimated, as we will see in the following subsections. It is because R is dimensionless and assumed to be of order one in the region (2.16) we concentrate on. Evaluation of horizon radius and temperature Let us now estimate the physical quantities of our brane systems using the p-soup model. As we mentioned in Introduction, first we need to set the characteristic scales of size and velocity in the systems: R ij ∼ R , d R ij dT ∼ dR dT (3.23) for all the branes i, j. This setting simplifies the following calculations. Next we impose the condition for these characteristic scales: dR dT ∼ πT ren R ,(3.24) which may mean that we look at Matsubara modes of brane's behaviors. Finally we impose the strong coupling condition, or virial theorem, such that L 1 ∼ L 2 ∼ · · · ∼ ∞ n=1 L n ,(3.25) since in the p-soup model the branes are strongly gravitationally interacting. The final condition (3.25) means that eq.(3.7) should be of order one, so we can rewrite it as h(R) 2 (d − 3) 2 τ 2 = R 2(d−2) τ 2 R 2(d−2) n T − 1 2 1 (d − 3) 2 − f (R)g(R) R 2(d−2) dR dT − f (R)g(R) 2 R 2(d−2) dR dT 2 + . . . 2 ∼ 1. (3.26) Here we expand the last factor of eq.(3.7) in a series of dR/dT . Note that the velocity dR/dT ≪ 1, because our brane system is assumed to be at low temperature T ren ≪ 1. Then the first term should satisfy 1 (d − 3) 2 τ 2 R 2(d−2) R 2(d−2) n T − 1 2 ∼ 1 ,(3.27) which evaluates the characteristic size R. As we saw in eq.(2.20), only the n T ≤ 4 cases are considered here, so we can solve this condition for all the cases: R ∼                        √ 1+4(d−3) 2 τ 2 ±1 2τ 1 d−2 for n T = 1 1 ± 1 (d−3)τ 1 d−2 for n T = 2 18 1 3 (d−3) 1 3 τ 1 3 ( √ 81−12(d−3) 2 τ 2 ±9) 1 3 12 1 3 (d−3) 2 3 τ 2 3 +( √ 81−12(d−3) 2 τ 2 ±9) 2 3 3 d−2 for n T = 3 (d−3)τ 2 (1 ± 1 − 4 (d−3)τ ) 2 d−2 , (d−3)τ 2 ( 1 + 4 (d−3)τ − 1) 2 d−2 for n T = 4 (3.28) where the double signs correspond in the n T = 3 case. This result reproduces the horizon radius of corresponding time-dependent black brane system, which can be easily checked: the horizon is at f (R) = 0, where the two sides of eq.(3.27) become equal. This means that we correctly reproduce a result from supergravity. The remaining terms of eq.(3.26) are proportional to f (R) R 2(d−2) g(R) dR dT n (3.29) for n = 1, 2, · · · . Now we know that we are looking at the branes which are slowly moving dR/dT ≪ 1 at the near horizon region f (R) ∼ 0, g(R) ≫ 1. Here it seems natural to impose the condition R 2(d−2) f (R) ∼ g(R) dR dT ∼ 1 . (3.30) This means that all the terms in eq.(3.26) are of order one. Then the second term should satisfy f (R)g(R) R 2(d−2) dR dT = 2(d − 2) (d − 3) 2 n T 1 1 − R − 2(d−2) n T 1 R dR dT ∼ 1 . (3.31) Using the setting (3.24), the temperature of our system can be evaluated as , where a 0,1,2,3 are arbitrary rational numbers. T ∼ n 2 T π 1 − R − 2(d−2) n T R d−3− 2(d−2) n T τ Q 1 d−3 ∼ n 2 T πRQ 1 d−3 1 − R − 2(d−2) In the cases of static black branes, as we showed in our previous papers [5,9], we have only uncertainty of overall rational factors. In the time-dependent black branes, on the other hand, we have another uncertainty of factors including R. However, this factor changes only the coefficients of R − 2(d−2) n T in evaluated quantities, and R is dimensionless and of order one. Therefore, we can still claim that the results of order estimation in the time-dependent black branes are consistent with the supergravity results. Finally note again that the temperature should be low in our brane system. More precisely, our system should be in the near extremal limit T ≪ 1/RQ 1 d−3 in supergravity. This condition means R ≃ 1, and it is consistent with the near horizon limit (2.16). To summarize, the p-soup model can correctly tell us about the horizon radius and temperature of the time-dependent black branes. In this picture, the branes are slowly moving at the near horizon region, which ensures that the system is at low temperature. Evaluation of free energy and entropy Let us continue to estimate the physical quantities of our brane system. The effective action (3.19) is evaluated as S eff ∼ t 0 T ren L 1 , (3.33) where we use the strong coupling condition (3.25). Then the partition function can be estimated as Z ∼ e −S eff , and the free energy is defined as F = −T log Z. Therefore, the free energy of our system can be evaluated as F ∼ R d−3− 2(d−2) n T t 0 τ Q 1 d−3 L 1 ∼ Ω d−2 κ 2 d QR d−3 f(R) = Ω d−2 κ 2 d QR d−3 c 0 + c 1 1 − R − 2(d−2) n T + c 2 1 − R − 2(d−2) n T 2 , (3.34) where we use the conditions (3.27) and (3.30). This is perfectly consistent with the result from supergravity (A.13). The coefficients c 0,1,2 cannot be determined in this analysis, but we find here that they should depend on only the parameters d and n T . This would be discussed in a future work. Finally the entropy of our system is evaluated as S = − ∂F ∂T ∼ πΩ d−2 κ 2 d Q d−2 d−3 R d−2 . (3.35) In the most right-hand side, the fractional expression where the numerator and denominator are polynomials in R of the same degree is set to one. This can be justified, since our analysis has uncertainty of the factors including R, as we discussed in the previous subsection. Such a factor is included in this uncertainty. Therefore, we can correctly reproduce Bekenstein-Hawking entropy (A.10). In this way, we can show that the p-soup model can explain various thermodynamic quantities of the time-dependent black branes. Conclusion and discussions In this paper, we discuss the p-soup proposal for a class of time-dependent black branes. Although they have many different properties from static black branes, we can analyze them in a very similar way. This may be partly because we can choose the time-independent frame (2.24), but their metrics in this frame are completely different from those of static black branes, so this is undoubtedly a new nontrivial application of the p-soup model. As a result, we find that the bound states of (elementary) branes in these systems exhibit the thermodynamic properties of the corresponding time-dependent black branes. This means that the p-soup analysis is applicable also for these systems and that we get another evidence that the p-soup model describes (at least a part of) the microstates of a large class of black holes. However, compared with the cases of static black branes, our analysis holds subtleties. For example, in the n T = 1 case, the p-soup picture of interacting branes seems not to be valid. In all the cases there is some ambiguity about the factors including R. The latter uncertainty is closely related to the undetermined coefficients in eq.(3.22), so we should analyze it more in detail and construct more plausible discussions in a future work. Finally let us comment on the class of time-dependent black branes. In the asymptotic region, the noncompact spacetime becomes FLRW universe. Therefore, we can expect that this system is applied to some discussions in cosmology. In particular, when all the branes are time-dependent (i.e. n S = 0), this universe has exponential expansion like inflation. When a part of the branes are time-dependent (i.e. n S , n T = 0), we have the universe with power law expansion. Such properties may help us to draw up a scenario of making our own universe from branes. Then, based on the p-soup model, it would be also interesting future works to discuss the systems of interacting branes creating various types of universe. In §2, we changed the coordinates and obtained the expression in the static form (2.24). In this coordinate (T, R), this Killing vector is rewritten as ξ µ = (∂/∂T ) µ . Note that this vector becomes null at the horizon f (R) = 0. The surface gravity associated with the Killing vector ξ µ is κ 2 ± = ∓ 1 2 (∇ µ ξ ν )(∇ µ ξ ν ) . (A.5) Then the surface gravities of horizons are evaluated as κ ± = ± (d − 3)n T 4(d − 2) f ′ (R ± ) τ R d−3+ 2(d−2) n T ± (A.6) where R = R ± are radii of the horizons. As we saw in eq.(3.28), we have two event horizons in n T = 1, 2, 3 cases, and three event horizons in n T = 4 case. We can choose a suitable sign in eq.(A.6) for each horizon. However, when we calculate the surface gravity in a time-dependent spacetime, we need to care about the normalization of Killing vector. In the case of a spherically symmetric spacetime, we should renormalize the Killing vector such that [11] ξ µ = ∂ ∂T µ → ξ µ nh = (d − 3)n T 2(d − 2) R d−3− 2(d−2) n T τ ∂ ∂T µ (A.7) where the renormalization factor is (g T T g RR ) − 1 2 Q 2 d−3 in the metric (2.24). Therefore, the surface gravities on the horizons should be evaluated using this Killing vector, and we Here we have eliminated τ by the condition f (R ± ) = 0. obtain κ ± nh = ± (d − 3) 2 n 2 T 8(d − 2) 2 f ′ (R ± ) ( 2 . 221) and (2.25), we can evaluate the norm of the velocity v as we use the indices i, j, . . . which denote each brane in the system. When we need to distinguish between static and time-dependent branes, we use the indices s a for each static brane and t b for each time-dependent brane. The effective action should be written as a sum of all the interactions in the probe brane actions (3.11), (3.12), then we can write it down as necessary because the second last factors of the probe brane actions (3.11) and (3.12) are slightly different from each other. The coefficients c 0,1,2 may be determined by the details of interactions among the static and time-dependent branes. In order to do it, we should give up the conditions (2.9) and see the interactions of each brane more in detail. consistent with the result from supergravity (A.9) up to an overall rational factor and coefficient of each term. In other words, we can reproduce the supergravity result up to the factor a 0 −a 1 R − 2(d−2) . (A.9) In the near horizon limit (2.18), it becomes(qtr d−3 + 1) d−3 d−2 n T q 2 τ 2 dt 2 + (qtr d−3 + 1)This metric is invariant under the Killing vector ξ µ defined byds 2 nh Q 2 d−3 = −r 2(d−3) n T d−2 r 2 (dr 2 +r 2 dΩ 2 d−2 ) . (A.3) ξ µ :=t ∂ ∂t µ −r ∂ ∂r µ . (A.4) In the d = 1 case, all the spatial directions are compactified. At this moment we don't have any idea to discuss this case. AcknowledgmentsThe author would like to thank Takeshi Morita for useful comments. This work is partially supported by Grant-in-Aid for Scientific Research (No. 16K17711) from Japan Society for the Promotion of Science (JSPS).A. Results from supergravityThe d-dimensional metric in Einstein frame (2.6) can be written asNext we discuss the black hole entropy. This can be calculated using the Bekenstein-Hawking entropy formula. Fortunately, the angular part of the metric (2.24) is so simple, then we can easily obtainFinally we discuss the free energy. It can be calculated as F = E − T S, where the energy E is ADM mass in ordinary supergravity calculations. However, in our case the metric is globally time-dependent, and it doesn't approach to a flat spacetime in asymptotic region. This means there is no globally conserved energy.Instead, let us here discuss quasilocal energy such as Misner-Sharp energy[14]. 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T Morita, S Shiba, arXiv:1305.0789Thermodynamics of black M-branes from SCFTs. 1307100hep-thT. Morita and S. Shiba, "Thermodynamics of black M-branes from SCFTs," JHEP 1307 (2013) 100 [arXiv:1305.0789 [hep-th]]. Microstates of D1-D5(-P) black holes as interacting D-branes. T Morita, S Shiba, arXiv:1410.8319Phys. Lett. B. 747164hep-thT. Morita and S. Shiba, "Microstates of D1-D5(-P) black holes as interacting D-branes," Phys. Lett. B 747 (2015) 164 [arXiv:1410.8319 [hep-th]]. Thermodynamics of Intersecting Black Branes from Interacting Elementary Branes. T Morita, S Shiba, arXiv:1507.03507JHEP. 150970hep-thT. Morita and S. Shiba, "Thermodynamics of Intersecting Black Branes from Interacting Elementary Branes," JHEP 1509 (2015) 070 [arXiv:1507.03507 [hep-th]]. Dynamics of intersecting brane systems -Classification and their applications. K Maeda, N Ohta, K Uzawa, arXiv:0903.5483JHEP. 090651hep-thK. Maeda, N. Ohta and K. 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[ "UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY", "UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY" ]	[ "Sébastien Boucksom ", "ANDTomoyuki Hisamoto ", "Mattias Jonsson " ]	[]	[]	Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ15]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in[Der15,BHJ15]. As a partial converse, we show that uniform Kstability implies coercivity of the Mabuchi functional when restricted to Bergman metrics. Date: December 1, 2016. arXiv:1603.01026v3 [math.DG] 30 Nov 2016 2 SÉBASTIEN BOUCKSOM, TOMOYUKI HISAMOTO, AND MATTIAS JONSSON Székelyhidi [Szé06]proposed to use a version of K-stability in which, for any test configuration (X , L) for (X, L), the Donaldson-Futaki invariant DF(X , L) is bounded below by a positive constant times a suitable norm of (X , L). (See also [Szé15] for a related notion.)Following this lead, we defined in the prequel [BHJ15] to this paper, (X, L) to be uniformly K-stable if there exists δ > 0 such thatfor any normal and ample test configuration (X , L). Here J NA (X , L) is a non-Archimedean analogue of Aubin's J-functional. It is equivalent to the L 1 -norm of (X , L) as well as the minimum norm considered by Dervan[Der15]. The norm is zero iff the normalization of (X , L) is trivial, so uniform K-stability implies K-stability.In[BHJ15]we advocated the point of view that a test configuration defines a non-Archimedean metric on L, that is, a metric on the Berkovich analytification of (X, L) with respect to the trivial norm on the ground field C. Further, we defined non-Archimedean analogues of many classical functionals in Kähler geometry. One example is the functional J NA above. Another is M NA , a non-Archimedean analogue of the Mabuchi K-energy functional M . It agrees with the Donaldson-Futaki invariant, up to an explicit error term, and uniform K-stability is equivalent tofor any ample test configuration (X , L). In [BHJ15] we proved that canonically polarized manifolds and polarized Calabi-Yau manifolds are always uniformly K-stable.A first goal of this paper is to exhibit precise relations between the non-Archimedean functionals and their classical counterparts. From now on we do not a priori assume that the reduced automorphism group of (X, L) is discrete. We prove Theorem A. Let (X , L) be an ample test configuration for a polarized complex manifold (X, L). Consider any smooth strictly positive S 1 -invariant metric Φ on L defined near the central fiber, and let (φ s ) s be the corresponding ray of smooth positive metrics on L. Denoting by M and J the Mabuchi K-energy functional and Aubin J-functional, respectively, we then haveThe corresponding equalities also hold for several other functionals, see Theorem 3.6. More generally, we prove that these asymptotic properties hold in the logarithmic setting, for subklt pairs (X, B) and with weaker positivity assumptions, see Theorem 4.2.At least when the total space X is smooth, the assertion in Theorem A regarding the Mabuchi functional is closely related to several statements appearing in the literature [PRS08, Corollary 2], [PT09, Corollary 1], [Li12, Remark 12, p.38], [Tia14, Lemma 2.1], following the seminal work [Tia97]. A special case appears already in[DT92,p.328]. However, to the best of our knowledge, neither the general and precise statement given here nor its proof is available in the literature.As in [PRS08], the proof of Theorem A uses Deligne pairings, but the analysis here is more delicate since the test configuration X is not smooth. Using resolution of singularities, we can make X smooth, but then we lose the strict positivity of Φ. It turns out that the situation can be analyzed by estimating integrals of the form Xτ e 2Ψ\| Xτ as τ → 0, where UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS 3 X → C is an snc test configuration for X, and Ψ is a smooth metric on the (logarithmic) relative canonical bundle of X near the central fiber, see Lemma 3.11. Donaldson [Don99] (see also[Mab87,Sem92]) has advocated the point of view that the space H of positive metrics on L is an infinite-dimensional symmetric space. One can view the space H NA of positive non-Archimedean metrics on L as (a subset of) the associated (conical) Tits building. Theorem A gives justification to this paradigm.The asymptotic formulas in Theorem A allow us to study coercivity properties of the Mabuchi functional. As an immediate consequence of Theorem A, we have Corollary B. If the Mabuchi functional is coercive in the sense that M ≥ δJ − C on H for some positive constants δ and C, then (X, L) is uniformly K-stable, that is,holds for any normal ample test configuration (X , L).Coercivity of the Mabuchi functional is known to hold if X is a Kähler-Einstein manifold without vector fields. This was first established in the Fano case by [PSSW08]; an elegant proof can be found in[DR15]. As a special case of a very recent result of Berman, Darvas and Lu [BDL16], coercivity of the Mabuchi functional also holds for general polarized varieties admitting a metric of constant scalar curvature and having discrete reduced automorphism group. Thus, if (X, L) admits a constant scalar curvature metric and Aut(X, L)/C * is discrete, then (X, L) is uniformly K-stable. The converse statement is not currently known in general, but see below for the Fano case.Next, we study coercivity of the Mabuchi functional when restricted to the space of Bergman metrics. For any m ≥ 1 such that mL is very ample, let H m be the space of Fubini-Study type metrics on L, induced by the embedding of X → P Nm via mL.Theorem C. Fix m such that (X, mL) is linearly normal, and δ > 0. Then the following conditions are equivalent:(i) there exists C > 0 such that M ≥ δJ − C on H m .(ii) DF(X λ , L λ ) ≥ δJ NA (X λ , L λ ) for all 1-parameter subgroups λ of GL(N m , C); (iii) M NA (X λ , L λ ) ≥ δJ NA (X λ , L λ ) for all 1-parameter subgroups λ of GL(N m , C). Here (X λ , L λ ) is the test configuration for (X, L) defined by λ.Note that a different condition equivalent to (i)-(iii) appears in [Pau13, Theorem 1.1]. The equivalence of (ii) and (iii) stems from the close relation between the Donaldson-Futaki invariant and the non-Archimedean Mabuchi functional. In view of Theorem A, the equivalence between (i) and (iii) can be viewed as a generalization of the Hilbert-Mumford criterion. The proof uses in a crucial way a deep result of Paul [Pau12], which states that the restrictions to H m of the Mabuchi functional and the J-functional have log norm singularities (see §5).Since every ample test configuration arises as a 1-parameter subgroup λ of GL(N m , C) for some m, Theorem C implies Corollary D. A polarized manifold (X, L) is uniformly K-stable iff there exist δ > 0 and a sequence C m > 0 such that M ≥ δJ − C m on H m for all sufficiently divisible m.	10.4171/jems/894	[ "https://arxiv.org/pdf/1603.01026v4.pdf" ]	54,182,600	1603.01026	355c2b70141f34ff4e3c5cce472a146e27a3aa2c	UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY Sébastien Boucksom ANDTomoyuki Hisamoto Mattias Jonsson UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ15]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in[Der15,BHJ15]. As a partial converse, we show that uniform Kstability implies coercivity of the Mabuchi functional when restricted to Bergman metrics. Date: December 1, 2016. arXiv:1603.01026v3 [math.DG] 30 Nov 2016 2 SÉBASTIEN BOUCKSOM, TOMOYUKI HISAMOTO, AND MATTIAS JONSSON Székelyhidi [Szé06]proposed to use a version of K-stability in which, for any test configuration (X , L) for (X, L), the Donaldson-Futaki invariant DF(X , L) is bounded below by a positive constant times a suitable norm of (X , L). (See also [Szé15] for a related notion.)Following this lead, we defined in the prequel [BHJ15] to this paper, (X, L) to be uniformly K-stable if there exists δ > 0 such thatfor any normal and ample test configuration (X , L). Here J NA (X , L) is a non-Archimedean analogue of Aubin's J-functional. It is equivalent to the L 1 -norm of (X , L) as well as the minimum norm considered by Dervan[Der15]. The norm is zero iff the normalization of (X , L) is trivial, so uniform K-stability implies K-stability.In[BHJ15]we advocated the point of view that a test configuration defines a non-Archimedean metric on L, that is, a metric on the Berkovich analytification of (X, L) with respect to the trivial norm on the ground field C. Further, we defined non-Archimedean analogues of many classical functionals in Kähler geometry. One example is the functional J NA above. Another is M NA , a non-Archimedean analogue of the Mabuchi K-energy functional M . It agrees with the Donaldson-Futaki invariant, up to an explicit error term, and uniform K-stability is equivalent tofor any ample test configuration (X , L). In [BHJ15] we proved that canonically polarized manifolds and polarized Calabi-Yau manifolds are always uniformly K-stable.A first goal of this paper is to exhibit precise relations between the non-Archimedean functionals and their classical counterparts. From now on we do not a priori assume that the reduced automorphism group of (X, L) is discrete. We prove Theorem A. Let (X , L) be an ample test configuration for a polarized complex manifold (X, L). Consider any smooth strictly positive S 1 -invariant metric Φ on L defined near the central fiber, and let (φ s ) s be the corresponding ray of smooth positive metrics on L. Denoting by M and J the Mabuchi K-energy functional and Aubin J-functional, respectively, we then haveThe corresponding equalities also hold for several other functionals, see Theorem 3.6. More generally, we prove that these asymptotic properties hold in the logarithmic setting, for subklt pairs (X, B) and with weaker positivity assumptions, see Theorem 4.2.At least when the total space X is smooth, the assertion in Theorem A regarding the Mabuchi functional is closely related to several statements appearing in the literature [PRS08, Corollary 2], [PT09, Corollary 1], [Li12, Remark 12, p.38], [Tia14, Lemma 2.1], following the seminal work [Tia97]. A special case appears already in[DT92,p.328]. However, to the best of our knowledge, neither the general and precise statement given here nor its proof is available in the literature.As in [PRS08], the proof of Theorem A uses Deligne pairings, but the analysis here is more delicate since the test configuration X is not smooth. Using resolution of singularities, we can make X smooth, but then we lose the strict positivity of Φ. It turns out that the situation can be analyzed by estimating integrals of the form Xτ e 2Ψ\| Xτ as τ → 0, where UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS 3 X → C is an snc test configuration for X, and Ψ is a smooth metric on the (logarithmic) relative canonical bundle of X near the central fiber, see Lemma 3.11. Donaldson [Don99] (see also[Mab87,Sem92]) has advocated the point of view that the space H of positive metrics on L is an infinite-dimensional symmetric space. One can view the space H NA of positive non-Archimedean metrics on L as (a subset of) the associated (conical) Tits building. Theorem A gives justification to this paradigm.The asymptotic formulas in Theorem A allow us to study coercivity properties of the Mabuchi functional. As an immediate consequence of Theorem A, we have Corollary B. If the Mabuchi functional is coercive in the sense that M ≥ δJ − C on H for some positive constants δ and C, then (X, L) is uniformly K-stable, that is,holds for any normal ample test configuration (X , L).Coercivity of the Mabuchi functional is known to hold if X is a Kähler-Einstein manifold without vector fields. This was first established in the Fano case by [PSSW08]; an elegant proof can be found in[DR15]. As a special case of a very recent result of Berman, Darvas and Lu [BDL16], coercivity of the Mabuchi functional also holds for general polarized varieties admitting a metric of constant scalar curvature and having discrete reduced automorphism group. Thus, if (X, L) admits a constant scalar curvature metric and Aut(X, L)/C * is discrete, then (X, L) is uniformly K-stable. The converse statement is not currently known in general, but see below for the Fano case.Next, we study coercivity of the Mabuchi functional when restricted to the space of Bergman metrics. For any m ≥ 1 such that mL is very ample, let H m be the space of Fubini-Study type metrics on L, induced by the embedding of X → P Nm via mL.Theorem C. Fix m such that (X, mL) is linearly normal, and δ > 0. Then the following conditions are equivalent:(i) there exists C > 0 such that M ≥ δJ − C on H m .(ii) DF(X λ , L λ ) ≥ δJ NA (X λ , L λ ) for all 1-parameter subgroups λ of GL(N m , C); (iii) M NA (X λ , L λ ) ≥ δJ NA (X λ , L λ ) for all 1-parameter subgroups λ of GL(N m , C). Here (X λ , L λ ) is the test configuration for (X, L) defined by λ.Note that a different condition equivalent to (i)-(iii) appears in [Pau13, Theorem 1.1]. The equivalence of (ii) and (iii) stems from the close relation between the Donaldson-Futaki invariant and the non-Archimedean Mabuchi functional. In view of Theorem A, the equivalence between (i) and (iii) can be viewed as a generalization of the Hilbert-Mumford criterion. The proof uses in a crucial way a deep result of Paul [Pau12], which states that the restrictions to H m of the Mabuchi functional and the J-functional have log norm singularities (see §5).Since every ample test configuration arises as a 1-parameter subgroup λ of GL(N m , C) for some m, Theorem C implies Corollary D. A polarized manifold (X, L) is uniformly K-stable iff there exist δ > 0 and a sequence C m > 0 such that M ≥ δJ − C m on H m for all sufficiently divisible m. Introduction Let (X, L) be a polarized complex manifold, i.e. smooth projective complex variety X endowed with an ample line bundle L. A central problem in Kähler geometry is to give necessary and sufficient conditions for the existence of canonical Kähler metrics in the corresponding Kähler class c 1 (L), for example, constant scalar curvature Kähler metrics (cscK for short). To fix ideas, suppose the reduced automorphism group Aut(X, L)/C * is discrete. In this case, the celebrated Yau-Tian-Donaldson conjecture asserts that c 1 (L) admits a cscK metric iff (X, L) is K-stable. That K-stability follows from the existence of a cscK metric was proved by Stoppa [Stop09], building upon work by Donaldson [Don05], but the reverse direction is considered wide open in general. This situation has led people to introduce stronger stability conditions that would hopefully imply the existence of a cscK metric. Building upon ideas of Donaldson [Don05], Following Paul and Tian [PT06,PT09], we say that (X, mL) is CM-stable when there exist C, δ > 0 such that M ≥ δJ − C on H m . Corollary E. If (X, L) is uniformly K-stable, then (X, mL) is CM-stable for any sufficiently divisible positive integer m. Hence the reduced automorphism group is finite. Here the last statement follows from a result by Paul [Pau13, Corollary 1.1]. Let us now comment on the relation of uniform K-stability to the existence of Kähler-Einstein metrics on Fano manifolds. In [CDS15], Chen, Donaldson and Sun proved that a Fano manifold X admits a Kähler-Einstein metric iff it is K-polystable; see also [Tia15]. Since then, several new proofs have appeared. Datar and Székelyhidi [DSz15] proved an equivariant version of the conjecture, using Aubin's original continuity method. Chen, Sun and Wang [CSW15] gave a proof using the Kähler-Ricci flow. In [BBJ15], Berman and the first and last authors of the current paper used a variational method to prove a slightly different statement: in the absence of vector fields, the existence of a Kähler-Einstein metric is equivalent to uniform K-stability. In fact, the direct implication uses Corollary B above. In §6 we outline a different proof of the fact that a uniformly K-stable Fano manifold admits a Kähler-Einstein metric. Our method, which largely follows ideas of Tian, relies on Székelyhidi's partial C 0 -estimates [Szé16] along the Aubin continuity path, together with Corollary D. As noted above, uniform K-stability implies that the reduced automorphism group of (X, L) is discrete. In the presence of vector fields, there should presumably be a natural notion of uniform K-polystability. We hope to address this in future work. There have been several important developments since a first draft of the current paper was circulated. First, Z. Sjöström Dyrefelt [SD16] and, independently, R. Dervan and J. Ross [DR16], proved a transcendental version of Theorem A. Second, as mentioned above, it was proved in [BBJ15] that in the case of a Fano manifold without holomorphic vector fields, uniform K-stability is equivalent to coercivity of the Mabuchi functional, and hence to the existence of a Kähler-Einstein metric. Finally, the results in this paper were used in [BDL16] to prove that an arbitrary polarized pair (X, L) admitting a cscK metric must be K-polystable. The organization of the paper is as follows. In the first section, we review several classical energy functionals in Kähler geometry and their interpretation as metrics on suitable Deligne pairings. Then, in §2, we recall some non-Archimedean notions from [BHJ15]. Specifically, a non-Archimedean metric is an equivalence class of test configurations, and the non-Archimedean analogues of the energy functionals in §1 are defined using intersection numbers. In §3 we prove Theorem A relating the classical and non-Archimedean functionals via subgeodesic rays. These results are generalized to the logarithmic setting in §4. Section 5 is devoted to the relation between uniform K-stability and CM-stability. In particular, we prove Theorem C and Corollaries D and E. Finally, in §6, we show how to use Székelyhidi's partial C 0 -estimates along the Aubin continuity path together with CM-stability to prove that a uniformly K-stable Fano manifold admits a Kähler-Einstein metric. for helpful conversations. He was partially supported by the ANR projects GRACK, MACK and POSITIVE. The second author was supported by JSPS KAKENHI Grant Number 25-6660 and 15H06262. The last author was partially supported by NSF grant DMS-1266207, the Knut and Alice Wallenberg foundation, and the United States-Israel Binational Science Foundation. Deligne pairings and energy functionals In this section we recall the definition and main properties of the Deligne pairing, as well as its relation to classical functionals in Kähler geometry. 1.1. Metrics on line bundles. We use additive notation for line bundles and metrics. If, for i = 1, 2, φ i is a metric on a line bundle L i on X and a i ∈ Z, then a 1 φ 1 + a 2 φ 2 is a metric on a 1 L 1 + a 2 L 2 . This allows us to define metrics on Q-line bundles. A metric on the trivial line bundle will be identified with a function on X. If σ is a (holomorphic) section of a line bundle L on a complex analytic space X, then log \|σ\| stands for the corresponding (possibly singular) metric on L. For any metric φ on L, log \|σ\| − φ is therefore a function, and \|σ\| φ := \|σ\|e −φ = exp(log \|σ\| − φ) is the length of σ in the metric φ. We normalize the operator d c so that dd c = i π ∂∂, and set (somewhat abusively) dd c φ := −dd c log \|σ\| φ for any local trivializing section σ of L. The globally defined (1, 1)-form (or current) dd c φ is the curvature of φ, normalized so that it represents the (integral) first Chern class of L. If X is a complex manifold of dimension n and η is a holomorphic n-form on X, then \|η\| 2 := i n 2 2 n η ∧η defines a natural (smooth, positive) volume form on X. More generally, there is a bijection between smooth metrics on the canonical bundle K X and (smooth, positive) volume forms on X, which associates to a smooth metric φ on K X the volume form e 2φ locally defined by e 2φ := \|η\| 2 /\|η\| 2 φ for any local section η of K X . If ω is a positive (1, 1)-form on X and n = dim X, then ω n is a volume form, so − 1 2 log ω n is a metric on −K X in our notation. The Ricci form of ω is defined as the curvature Ric ω := −dd c 1 2 log ω n of ω of this metric; it is thus a smooth (1, 1)-form in the cohomology class c 1 (X) of −K X . If φ is a smooth positive metric on a line bundle L on X, we denote by S φ ∈ C ∞ (X) the scalar curvature of the Kähler form dd c φ; it satisfies S φ (dd c φ) n = n Ric(dd c φ) ∧ (dd c φ) n−1 . (1.1) 1.2. Deligne pairings. While the construction below works in greater generality [Elk89,Zha96,MG00], we will restrict ourselves to the following setting. Let π : Y → T be a flat, projective morphism between smooth complex algebraic varieties, of relative dimension n ≥ 0. Given line bundles L 0 , . . . , L n on Y , consider the intersection product L 0 · . . . · L n · [Y ] ∈ CH dim Y −(n+1) (Y ) = CH dim T −1 (Y ). Its push-forward belongs to CH dim T −1 (T ) = Pic(T ) since T is smooth, and hence defines an isomorphism class of line bundle on T . The Deligne pairing of L 0 , . . . , L n selects in a canonical way a specific representative of this isomorphism class, denoted by L 0 , . . . , L n Y /T . The pairing is functorial, multilinear, and commutes with base change. It further satisfies the following key inductive property: if Z 0 is a non-singular divisor in Y , flat over T and defined by a section σ 0 ∈ H 0 (Y, L 0 ), then we have a canonical identification L 0 , . . . , L n Y /T = L 1 \| Z 0 , . . . , L n \| Z 0 Z 0 /T . (1.2) For n = 0, L 0 Y /T coincides with the norm of L 0 with respect to the finite flat morphism Y → T . These properties uniquely characterize the Deligne pairing. Indeed, writing each L i as a difference of very ample line bundles, multilinearity reduces the situation to the case where the L i are very ample. We may thus find non-singular divisors Z i ∈ \|L i \| with i∈I Z i non-singular and flat over T for each set I of indices, and we get L 0 , . . . , L n Y /T = L n \| Z 0 ∩···∩Z n−1 Z 0 ∩···∩Z n−1 /T . 1.3. Metrics on Deligne pairings. We use [Elk90,Zha96,Mor99] as references. Given a smooth metric φ j on each L j , the Deligne pairing L 0 , . . . , L n Y /T can be endowed with a continuous metric φ 0 , . . . , φ n Y /T , smooth over the smooth locus of π, the construction being functorial, multilinear, and commuting with base change. It is basically constructed by requiring that φ 0 , . . . , φ n Y /T = φ 1 \| Z 0 , . . . , φ n \| Z 0 Z 0 /T − Y /T log \|σ 0 \| φ 0 dd c φ 1 ∧ · · · ∧ dd c φ n (1.3) in the notation of (1.2), with Y /T denoting fiber integration, i.e. the push-forward by π as a current. By induction, the continuity of the metric φ 0 , . . . , φ n reduces to that of 3) is well-defined is the following symmetry property: if σ 1 ∈ H 0 (Y, L 1 ) is a section with divisor Z 1 such that both Z 1 and Z 0 ∩ Z 1 are non-singular and flat over T , then Y /T log \|σ 0 \| φ 0 dd c φ 1 ∧ · · · ∧ dd c φ n ,Y /T log \|σ 0 \| φ 0 dd c φ 1 ∧ α + Z 0 /T log \|σ 1 \| φ 1 α = Y /T log \|σ 1 \| φ 1 dd c φ 0 ∧ α + Z 1 /T log \|σ 0 \| φ 0 α with α = dd c φ 2 ∧ · · · ∧ dd c φ n . By the Lelong-Poincaré formula, the above equality reduces to π * (log \|σ 0 \| φ 0 dd c log \|σ 1 \| φ 1 ∧ α) = π * (log \|σ 1 \| φ 1 dd c log \|σ 0 \| φ 0 ∧ α) , which holds by Stokes' formula applied to a monotone regularization of the quasi-psh functions log \|σ i \| φ i . Metrics on Deligne pairings satisfy the following two crucial properties, which are direct consequences of (1.3). (i) The curvature current of φ 0 , . . . , φ n Y /T satisfies dd c φ 0 , . . . , φ n Y /T = Y /T dd c φ 0 ∧ · · · ∧ dd c φ n , (1.4) where again Y /T denotes fiber integration. (ii) Given another smooth metric φ 0 on L 0 , we have the change of metric formula φ 0 , φ 1 , . . . , φ n Y /T − φ 0 , φ 1 , . . . , φ n Y /T = Y /T (φ 0 − φ 0 )dd c φ 1 ∧ · · · ∧ dd c φ n . (1.5) 1.4. Energy functionals. Let (X, L) be a polarized manifold, i.e. a smooth projective complex variety X with an ample line bundle L. Set V := (L n ) andS := −nV −1 (K X · L n−1 ), where n = dim X. Denote by H the set of smooth positive metrics φ on L. For φ ∈ H, set MA(φ) := V −1 (dd c φ) n . Then MA(φ) is a probability measure equivalent to Lebesgue measure, and X S φ MA(φ) =S by (1.1). We recall the following functionals in Kähler geometry. Fix a reference metric φ ref ∈ H. Our notation largely follows [BBGZ13,BBEGZ11]. (i) The Monge-Ampère energy functional is given by E(φ) = 1 n + 1 n j=0 V −1 X (φ − φ ref )(dd c φ) j ∧ (dd c φ ref ) n−j . (1.6) (ii) The J-functional is a translation invariant version of E, defined as J(φ) := X (φ − φ ref ) MA(φ ref ) − E(φ). (1.7) The closely related I-functional is defined by I(φ) := X (φ − φ ref ) MA(φ ref ) − X (φ − φ ref ) MA(φ). (1.8) (iii) For any closed (1, 1)-form θ, the θ-twisted Monge-Ampère energy is given by E θ (φ) = 1 n n−1 j=0 V −1 X (φ − φ ref )(dd c φ) j ∧ (dd c φ ref ) n−1−j ∧ θ. (1.9) Taking θ := −n Ric(dd c φ ref ), we obtain the Ricci energy R := −E n Ric(dd c φ ref ) . (iv) The entropy of φ ∈ H is defined as H(φ) := 1 2 X log MA(φ) MA(φ ref ) MA(φ), (1.10) that is, (half) the relative entropy of the probability measure MA(φ) with respect to MA(φ ref ). We have H(φ) ≥ 0, with equality iff φ − φ ref is constant. (v) The Mabuchi functional (or K-energy) can now be defined via the Chen-Tian formula [Che00] (see also [BB14,Proposition 3.1]) as M (φ) = H(φ) + R(φ) +SE(φ). (1.11) These functionals vanish at φ ref and satisfy the variational formulas: δE(φ) = MA(φ) = V −1 (dd c φ) n δE θ (φ) = V −1 (dd c φ) n−1 ∧ θ δR(φ) = −nV −1 (dd c φ) n−1 ∧ Ric(dd c φ ref ) δH(φ) = nV −1 (dd c φ) n−1 ∧ (Ric(dd c φ ref ) − Ric(dd c φ)) δM (φ) = (S − S φ ) MA(φ) In particular, φ is a critical point of M iff dd c φ is a cscK metric. The functionals I, J and I − J are comparable in the sense that 1 n J ≤ I − J ≤ nJ (1.12) on H. For φ ∈ H we have J(φ) ≥ 0, with equality iff φ − φ ref is(φ + c) = H(φ) for c ∈ R. For E and R we instead have E(φ + c) = E(φ) + c and R(φ + c) = R(φ) −Sc, respectively. 1.5. Energy functionals as Deligne pairings. The functionals above can be expressed using Deligne pairings, an observation going back at least to [PS04]. Note that any metric φ ∈ H induces a smooth metric 1 2 log MA(φ) on K X . The following identities are now easy consequences of the change of metric formula (1.5). Lemma 1.2. For any φ ∈ H we have (n + 1)V E(φ) = φ n+1 X − φ n+1 ref X ; V J(φ) = φ, φ n ref X − φ n+1 ref X − 1 n + 1 φ n+1 X − φ n+1 ref X ; V I(φ) = φ − φ ref , φ n ref X − φ − φ ref , φ n X ; V R(φ) = 1 2 log MA(φ ref ), φ n X − 1 2 log MA(φ ref ), φ n ref X ; V H(φ) = 1 2 log MA(φ), φ n X − 1 2 log MA(φ ref ), φ n X ; V M (φ) = 1 2 log MA(φ), φ n X − 1 2 log MA(φ ref ), φ n ref X +S n + 1 φ n+1 X − φ n+1 ref X , where X denotes the Deligne pairing with respect to the constant map X → {pt}. Remark 1.3. The formulas above make it evident that instead of fixing a reference metric φ ref ∈ H, we could view E, H + R and M as metrics on suitable multiples of the complex lines L n+1 X , K X , L n X , and (n + 1) K X , L n X +S L n+1 X , respectively. 1.6. The Ding functional. Now suppose X is a Fano manifold, that is, L := −K X is ample. Any metric φ on L then induces a positive volume form e −2φ on X. The Ding functional [Din88] on H is defined by D(φ) = L(φ) − E(φ), where L(φ) = − 1 2 log X e −2φ . This functional has proven an extremely useful tool for the study of the existence of Kähler-Einstein metrics, which are realized as the critical points of D, see e.g. [Berm16,BBJ15]. Test configurations as non-Archimedean metrics In this section we recall some notions and results from [BHJ15]. Let X be a smooth projective complex variety and L a line bundle on X. 2.1. Test configurations. As in [BHJ15] we adopt the following flexible terminology for test configurations. Definition 2.1. A test configuration X for X consists of the following data: (i) a flat, projective morphism of schemes π : X → C; (ii) a C * -action on X lifting the canonical action on C; (iii) an isomorphism X 1 X. We denote by τ the coordinate on C, and by X τ the fiber over τ . These conditions imply that X is reduced and irreducible [BHJ15, Proposition 2.6]). If X , X are test configurations for X, then there is a unique C * -equivariant birational map X X compatible with the isomorphism in (iii). We say that X dominates X if this birational map is a morphism; when it is an isomorphism we somewhat abusively identify X and X . Any test configuration X is dominated by its normalization X . An snc test configuration for X is a smooth test configuration X whose central fiber X 0 has simple normal crossing support (but is not necessarily reduced). When X is a test configuration, we define the logarithmic canonical bundle as K log X := K X + X 0,red . Setting K log C := K C + [0] , we define the relative logarithmic canonical bundle as K log X /C := K log X − π * K log C = K X /C + X 0,red − X 0 ; this is well behaved under base change τ → τ d , see [BHJ15,§4.4]. Despite the terminology, K X , K X /C , K log X and K log X /C are only Weil divisor classes in general; they are line bundles when X is smooth. Definition 2.2. A test configuration (X , L) for (X, L) consists of a test configuration X for X, together with the following additional data: (iv) a C * -linearized Q-line bundle L on X ; (v) an isomorphism (X 1 , L 1 ) (X, L). A pull-back of a test configuration (X , L) is a test configuration (X , L ) where X dominates X and L is the pull-back of L. In particular, the normalization ( X ,L) is the pull-back of (X , L) with ν : X → X the normalization morphism. A test configuration (X , L) is trivial if X = X × C with C * acting trivially on X. This implies that (X , L + cX 0 ) = (X, L) × C for some constant c ∈ Q. A test configuration for (X, L) is almost trivial if its normalization is trivial. We say that (X , L) is ample (resp. semiample, resp. nef) when L is relatively ample (resp. relatively semiample, resp. nef). The pullback of a semiample (resp. nef) test configuration is semiample (resp. nef). If L is ample, then for every semiample test configuration (X , L) there exists a unique ample test configuration (X amp , L amp ) that is dominated by (X , L) and satisfies µ * O X = O Xamp , where µ : X → X amp is the canonical morphism; see [BHJ15,Proposition 2.17]. Note that, while X can often be chosen smooth, X amp will not be smooth, in general. It is, however, normal whenever X is. Fix m ≥ 1 such that mL is very ample, and consider the corresponding closed embedding X → P Nm−1 with N m := h 0 (X, mL). Then every 1-parameter subgroup (1-PS for short) λ : C * → GL(N m , C) induces an ample test configuration (X λ , L λ ) for (X, L). By definition, X λ is the Zariski closure in PV × C of the image of the closed embedding X × C * → PV × C * mapping (x, τ ) to (λ(τ )x, τ ). Note that (X λ , L λ ) is trivial iff λ is a multiple of the identity. We emphasize that X λ is not normal in general. In fact, every ample test configuration may be obtained as above. Using one-parameter subgroups, we can produce test configurations that are almost trivial but not trivial, as observed in [LX14,Remark 5]. See [BHJ15, Proposition 2.12] for an elementary proof of the following result. Proposition 2.3. For every m divisible enough, there exists a 1-PS λ : C * → GL(N m , C) such that the test configuration (X λ , L λ ) is nontrivial but almost trivial. 2.3. Valuations and log discrepancies. By a valuation on X we mean a real-valued valuation v on the function field C(X) (trivial on the ground field C). The trivial valuation v triv is defined by v triv (f ) = 0 for f ∈ C(X) * . A valuation v is divisorial if it is of the form v = c ord F , where c ∈ Q >0 and F is a prime divisor on a projective normal variety Y admitting a birational morphism onto X. We denote by X div the set of valuations on X that are either divisorial or trivial, and equip it with the weakest topology such that v → v(f ) is continuous for every f ∈ C(X) * . The log discrepancy A X (v) of a valuation in X div is defined as follows. First, A X (v triv ) = 0. For v = c ord F a divisorial valuation as above, we set A X = c(1 + ord F (K Y /X )), where K Y /X is the relative canonical (Weil) divisor. Now consider a normal test configuration X of X. Since C(X ) C(X)(τ ), any valuation w on X restricts to a valuation r(w) on X. Let E be an irreducible component of the central fiber X 0 = b E E. Then ord E is a C * -invariant divisorial valuation on C(X ) and satisfies ord E (t) = b E . If we set v E := r(b −1 E ord E ), then v E is a valuation in X div . Conversely, every valuation v ∈ X div has a unique C * -invariant preimage w under r normalized by w(τ ) = 1, and w is associated to an irreducible component of the central fiber of some test configuration for X, cf. [BHJ15, Theorem 4.6]. Note that ord E is a divisorial valuation on X × C. By [BHJ15, Proposition 4.11], the log discrepancies of ord E and v E are related as follows: A X×C (ord E ) = b E (1 + A X (v E )). 2.4. Compactifications. For some purposes it is convenient to compactify test configurations. The following notion provides a canonical way of doing so. Definition 2.4. The compactificationX of a test configuration X for X is defined by gluing together X and X × (P 1 \ {0}) along their respective open subsets X \ X 0 and X × (C \ {0}), using the canonical C * -equivariant isomorphism X \ X 0 X × (C \ {0}). The compactificationX comes with a C * -equivariant flat morphismX → P 1 , still denoted by π. By construction, π −1 (P 1 \ {0}) is C * -equivariantly isomorphic to X × (P 1 \ {0}) over P 1 \ {0}. Similarly, a test configuration (X , L) for (X, L) admits a compactification (X ,L), wherē L is a C * -linearized Q-line bundle onX . Note thatL is relatively (semi)ample iff L is. The relative canonical differential and relative canonical differential are now defined by KX /P 1 := KX − π * K P 1 K loḡ X /P 1 := K loḡ X − π * K log P 1 = KX /P 1 + X 0,red − X 0 . 2.5. Non-Archimedean metrics. Following [BHJ15, §6] (see also [BJ16b]) we introduce: Definition 2.5. Two test configurations (X 1 , L 1 ), (X 2 , L 2 ) for (X, L) are equivalent if there exists a test configuration (X 3 , L 3 ) that is a pull-back of both (X 1 , L 1 ) and (X 2 , L 2 ). An equivalence class is called a non-Archimedean metric on L, and is denoted by φ. We denote by φ triv the equivalence class of the trivial test configuration (X, L) × C. A non-Archimedean metric φ is called semipositive if some (or, equivalently, any) representative (X , L) of φ is nef. Note that this implies that L is nef. When L is ample, we say that a non-Archimedean metric φ on L is positive if some (or, equivalently, any) representative (X , L) of φ is semiample. We denote by H NA the set of all non-Archimedean positive metrics on L. By [BHJ15, Lemma 6.3], every φ ∈ H NA is represented by a unique normal, ample test configuration. The set of non-Archimedean metrics on a line bundle L admits two natural operations: (i) a translation action of Q, denoted by φ → φ + c, and induced by (X , L) → (X , L + cX 0 ); (ii) a scaling action of the semigroup N * of positive integers, denoted by φ → φ d and induced by the base change of (X , L) by τ → τ d . When L is ample (resp. nef) these operations preserve the set of positive (resp. semipositive) metrics. The trivial metric φ triv is fixed by the scaling action. As in §1.1 we use additive notation for non-Archimedean metrics. A non-Archimedean metric on O X induces a bounded (and continuous) function on X div . Remark 2.6. As explained in [BHJ15, §6.8], a non-Archimedean metric φ on L, as defined above, can be viewed as a metric on the Berkovich analytification [Berk90] of L with respect to the trivial absolute value on the ground field C. See also [BJ16b] for a more systematic analysis, itself building upon [BFJ16,BFJ15a]. 2.6. Intersection numbers and Monge-Ampère measures. Following [BHJ15, §6.6] we define the intersection number (φ 0 · . . . · φ n ) of non-Archimedean metrics φ 0 , . . . , φ n on line bundles L 0 , . . . , L n on X as follows. Pick representatives (X , L i ) of φ i , 0 ≤ i ≤ n, with the same test configuration X for X and set (φ 0 · . . . · φ n ) := (L 0 · . . . ·L n ), where (X ,L i ) is the compactification of (X , L i ). It follows from the projection formula that this does not depend of the choice of the L i . Note that (φ n+1 triv ) = 0. When L 0 = O X , so that L 0 = O X (D) for a Q-Cartier Q-divisor D = r E E supported on X 0 , we can compute the intersection number as (φ 0 · . . . · φ n ) = E r E (L 1 \| E · . . . · L n \| E ). To a non-Archimedean metric φ on a big and nef line bundle L on X we associate, as in [BHJ15, §6.7], a signed finite atomic Monge-Ampère measure on X div . Pick a representative (X , L i ) of φ, and set MA NA (φ) = V −1 E b E (L\| n E )δ v E , where E ranges over irreducible components of X 0 = E b E E, v E = r(b −1 E ord E ) ∈ X div , and V = (L n ). When the φ i are semipositive, the mixed Monge-Ampère measure is a probability measure. Definition 2.7. Let W be a set of non-Archimedean metrics on L that is closed under translation and scaling. A functional F : W → R is (i) homogeneous if F (φ d ) = dF (φ) for φ ∈ W and d ∈ N * ; (ii) translation invariant if F (φ + c) = F (φ) for φ ∈ W and c ∈ Q. When L is ample, a functional F on H NA may be viewed as a function F (X , L) on the set of all semiample test configurations (X , L) that is invariant under pull-back, i.e. F (X , L ) = F (X , L) whenever (X , L ) is a pull-back of a (X , L) (and, in particular, invariant under normalization). Homogeneity amounts to F (X d , L d ) = d F (X , L) for all d ∈ N * , and translation invariance to F (X , L) = F (X , L + cX 0 ) for all c ∈ Q. For each non-Archimedean metric φ on L, choose a normal representative (X , L) that dominates X × C via ρ : X → X × C. Then L = ρ * (L × C) + D for a uniquely determined Q-Cartier divisor D supported on X 0 . Write X 0 = E b E E and let (X ,L) be the compactification of (X , L). In this notation, we may describe our list of non-Archimedean functionals as follows. Assume L is big and nef. Let φ triv and ψ triv be the trivial metrics on L and K X , respectively. (i) The non-Archimedean Monge-Ampère energy of φ is E NA (φ) : = (φ n+1 ) (n + 1)V = L n+1 (n + 1)V . (ii) The non-Archimedean I-functional and J-functional are given by I NA (φ) : = V −1 (φ · φ n triv ) − V −1 ((φ − φ triv ) · φ n ) = V −1 (L · (ρ * (L × P 1 ) n ) − V −1 (D ·L n ). and J NA (φ) : = V −1 (φ · φ n triv ) − E NA (φ) = 1 V (L · (ρ * (L × P 1 ) n ) − 1 (n + 1)V (L n+1 ). (iii) The non-Archimedean Ricci energy is R NA (φ) : = V −1 (ψ triv · φ n ) = V −1 ρ * K log X×P 1 /P 1 ·L n . (iv) The non-Archimedean entropy is H NA (φ) : = X div A X (v) MA NA (φ) = V −1 K loḡ X /P 1 ·L n − V −1 ρ * K log X×P 1 /P 1 ·L n . (v) The non-Archimedean Mabuchi functional (or K-energy) is M NA (φ) : = H NA (φ) + R NA (φ) +SE NA (φ) = V −1 K loḡ X /P 1 ·L n +S (n + 1)V L n+1 . Note the resemblance to the formulas in §1.5. All of these functionals are homogeneous. They are also translation invariant, except for E NA and R NA , which satisfy E NA (φ + c) = E NA (φ) + c and R NA (φ + c) = R NA (φ) −Sc (2.1) for all φ ∈ H NA and c ∈ Q. The functionals I NA , J NA and I NA − J NA are comparable on semipositive metrics in the same way as (1.12). By [BHJ15, Lemma 7.7, Theorem 5.16], when φ is positive, the first term in the definition of J NA satisfies V −1 (φ · φ n triv ) = (φ − φ triv )(v triv ) = max X div (φ − φ triv ) = max E b −1 E ord E (D). Further, J NA (φ) ≥ 0, with equality iff φ = φ triv + c for some c ∈ Q, and J NA is comparable to both a natural L 1 -norm and the minimum norm in the sense of Dervan [Der15], see [BHJ15, Theorem 7.9, Remark 7.12]. For a normal ample test configuration (X , L) representing φ ∈ H NA we also denote the J-norm by J NA (X , L). 2.8. The Donaldson-Futaki invariant. As explained in [BHJ15], the non-Archimedean Mabuchi functional is closely related to the Donaldson-Futaki invariant. We have Proposition 2.8. Assume L is ample. Let φ ∈ H NA be the class of an ample test configuration (X , L) for (X, L), and denote by ( X ,L) its normalization, which is thus the unique normal, ample representative of φ. Then M NA (φ) = DF( X ,L) − V −1 ( X 0 − X 0,red ) ·L n (2.2) DF(X , L) = DF( X ,L) + 2V −1 E m E (E · L n ) , (2.3) where E ranges over the irreducible components of X 0 contained in the singular locus of X and m E ∈ N * is the length of ν * O X /O X at the generic point of E, with ν : X → X the normalization. In particular, DF(X , L) ≥ M NA (φ), and equality holds iff X is regular in codimension one and X 0 is generically reduced. Indeed, (2.2) and (2.3) follow from the discussion in [BHJ15, §7.3] and from [BHJ15, Proposition 3.15], respectively. Note that intersection theoretic formulas for the Donaldson-Futaki invariant appeared already in [Wan12] and [Oda13]. For a general non-Archimedean metric φ on L we can define DF(φ) = M NA (φ) + V −1 (X 0 − X 0,red ) ·L n = V −1 KX /P 1 ·L n +S (n + 1)V L n+1 for any normal representative (X , L) of φ. Clearly M NA (φ) ≤ DF(φ) when φ is semipositive. 2.9. The non-Archimedean Ding functional [BHJ15, §7.7]. Suppose X is weakly Fano, that is, L := −K X is big and nef. In this case, we define the non-Archimedean Ding functional on the space of non-Archimedean metrics on L by D NA (φ) = L NA (φ) − E NA (φ), where L NA is defined by L NA (φ) = inf v (A X (v) + (φ − φ triv )(v)) , the infimum taken over all valuations v on X that are divisorial or trivial. Recall from §2.5 that φ − φ triv is a non-Archimedean metric on O X and induces a bounded function on divisorial valuations. Note that L NA (φ+c) = L NA (φ)+c; hence D NA is translation invariant. We always have D NA ≤ J NA , see [BHJ15, Proposition 7.28]. When φ is semipositive, we have D NA (φ) ≤ M NA (φ), see [BHJ15, Proposition 7.32]. 2.10. Uniform K-stability. As in [BHJ15,§8] we make the following definition. Definition 2.9. A polarized complex manifold (X, L) is uniformly K-stable if there exists a constant δ > 0 such that the following equivalent conditions hold. ( The fact that J NA (φ) = 0 iff φ = φ triv +c implies that uniform K-stability is stronger than K-stability as introduced by [Tia97,Don02]. Our notion of uniform K-stability is equivalent to uniform K-stability defined either with respect to the L 1 -norm or the minimum norm in the sense of [Der15], see [BHJ15,Remark 8.3]. i) M NA (φ) ≥ δJ NA (φ) for every φ ∈ H NA (L);( In the Fano case, uniform K-stability is further equivalent to uniform Ding stability: Theorem 2.10. Assume L := −K X is ample and fix a number δ with 0 ≤ δ ≤ 1. Then the following conditions are equivalent: (i) M NA ≥ δJ NA on H NA ; (ii) D NA ≥ δJ NA on H NA . This is proved in [BBJ15] using the Minimal Model Program as in [LX14]. See [Fuj16] for a more general result, and also [Fuj15]. Non-Archimedean limits In this section we prove Theorem A and Corollary B. 3.1. Rays of metrics and non-Archimedean limits. For any line bundle L on X, there is a bijection between smooth rays (φ s ) s>0 of metrics on L and S 1 -invariant smooth metrics Φ on the pull-back of L to X × ∆ * , with ∆ * = ∆ * 1 ⊂ C the punctured unit disc. The restriction of Φ to X τ for τ ∈ ∆ * is given by pullback of φ log \|τ \| −1 under the map X τ → X given by the C * -action. Similarly, smooth rays (φ s ) s>s 0 correspond to S 1 -invariant smooth metrics on the pull-back of L to X × ∆ * r 0 , with r 0 = e −s 0 . A subgeodesic ray is a ray (φ s ) whose corresponding metric Φ is semipositive. Such rays can of course only exist when L is nef. Definition 3.1. We say that a smooth ray (φ s ) admits a non-Archimedean metric φ NA as non-Archimedean limit if there exists a test configuration (X , L) representing φ NA such that the metric Φ on L × ∆ * corresponding to (φ s ) s extends to a smooth metric on L over ∆. In other words, a non-Archimedean limit exists iff Φ has analytic singularities along X × {0}, i.e. splits into a smooth part and a divisorial part after pulling-back to a blow-up. Lemma 3.2. Given a ray (φ s ) s in H, the non-Archimedean limit φ NA ∈ H NA is unique, if it exists. Proof. Let ψ 1 and ψ 2 be non-Archimedean limits of (φ s ) s and let Φ be the smooth metric on L × ∆ * defined by the ray (φ s ). For i = 1, 2, pick a representative (X i , L i ) of ψ i such that Φ extends as a smooth metric on L i over ∆. After replacing (X i , L i ) by suitable pullbacks, we may assume X 1 = X 2 = : X and that X is normal. Then L 2 = L 1 + D for a Q-divisor D supported on X 0 . Now a smooth metric on L 1 induces a singular metric on L 1 + D that is smooth iff D = 0. Hence L 1 = L 2 , so that ψ 1 = ψ 2 . Remark 3.3. Following [Berk09, §2] (see also [Jon16,BJ16a]) one can construct a compact Hausdorff space X An fibering over the interval [0, 1] such that the fiber X An ρ over any point ρ ∈ (0, 1] is homeomorphic to the complex manifold X, and the fiber X An 0 over 0 is homeomorphic to the Berkovich analytification of X with respect to the trivial norm on C. Similarly, the line bundle L induces a line bundle L An over X An . If a ray (φ s ) s>0 admits a non-Archimedean limit φ NA , then it induces a continuous metric on L An whose restriction to L An ρ is given by φ log ρ −1 and whose restriction to X an 0 is given by φ NA . In this way, φ NA is indeed the limit of φ s as s → ∞. 3.2. Non-Archimedean limits of functionals. For the rest of §3, assume that L is ample. Definition 3.4. A functional F : H → R admits a functional F NA : H NA → R as a non-Archimedean limit if, for every smooth subgeodesic ray (φ s ) in H admitting a non-Archimedean limit φ NA ∈ H NA , we have lim s→+∞ F (φ s ) s = F NA (φ NA ). (3.1) Proposition 3.5. If F : H → R admits a non-Archimedean limit F NA : H NA → R, then F NA is homogeneous. Proof. Consider a semiample test configuration (X , L) representing a non-Archimedean metric φ NA ∈ H NA , and let (φ s ) s be a smooth subgeodesic ray admitting φ NA as a non-Archimedean limit. For d ≥ 1, let (X d , L d ) be the normalized base change induced by τ → τ d . The associated non-Archimedean metric φ NA d is then the non-Archimedean limit of the subgeodesic ray (φ ds ), so lim s→∞ s −1 F (φ ds ) = F NA (φ NA d ). On the other hand, we clearly have lim s→∞ (ds) −1 F (φ ds ) = lim s→∞ s −1 F (φ s ) = F NA (φ NA ). The result follows. Theorem 3.7. If L := −K X is ample, then the Ding functional D on H admits D NA on H NA as non-Archimedean limit. Remark 3.8. In §4 we will extend the two previous results to the logarithmic setting and with relaxed positivity assumptions. The main tool in the proof of Theorem 3.6 is the following result. Proof. The Deligne pairing F := L 0 , . . . , L n X /C is a line bundle on C, endowed with a C action and a canonical identification of its fiber at τ = 1 with the complex line L 0 , . . . , L n X . It extends to a line bundle L 0 , . . . ,L n X /P 1 on P 1 that is C -equivariantly trivial on P 1 {0}. Denoting by w ∈ Z the weight of the C * -action on the fiber at 0, we have w = deg L 0 , . . . ,L n X /P 1 = L 0 , . . . ,L n . Pick a nonzero vector v ∈ F 1 = L 0 , . . . , L n X . The C * -action produces a section τ → τ · v of F on C * , and σ := τ −w (τ · v) is a nowhere vanishing section of F on C, see [BHJ15, Corollary 1.4]. Since the metrics Φ i are smooth and S 1 -invariant, Ψ := Φ 0 , . . . , Φ n X /C is a continuous S 1 -invariant metric on F near 0 ∈ C. Hence the function log \|σ\| Ψ is bounded near 0 ∈ C. The S 1 -invariant metric Ψ defines a ray (ψ s ) of metrics on the line F 1 through \|v\| ψ s = \|τ · v\| Ψτ , for s = log \|τ \| −1 , where Ψ τ is the restriction of Ψ to F τ . Thus log \|v\| ψ s = log \|τ · v\| Ψτ = w log \|τ \| + log \|σ\| Ψτ = −sw + O(1). Proof of Theorem 3.6. Let (φ s ) s be a smooth subgeodesic ray in H admitting a non-Archimedean limit φ NA ∈ H NA . Pick a test configuration (X , L) representing φ NA such that X is smooth and X 0 has snc support. Thus L is relatively semiample and (φ s ) s corresponds to a smooth S 1 -invariant semipositive metric Φ on L over ∆. By Lemma 1.2, we have V (H(φ s ) + R(φ s )) = 1 2 log MA(φ s ), φ s , . . . , φ s X − ψ ref , φ ref , . . . , φ ref X , where ψ ref = 1 2 log MA(φ ref ), so we must show that 1 2 log MA(φ s ), φ s , . . . , φ s X − ψ ref , φ ref , . . . , φ ref X = s K loḡ X /P 1 ·L n + o(s). (3.2) The collection of metrics 1 2 log MA(Φ\| Xτ ) with τ = 0 defines a smooth metric Ψ on K log X /C over ∆ * , but the difficulty here (as opposed to the situation in [PRS08]) is that Ψ will not a priori extend to a smooth (or even locally bounded) metric on K log X /C over ∆. Indeed, since we have assumed that X is smooth, there is no reason why Φ is strictly positive. Instead, pick a smooth, S 1 -invariant reference metric Ψ ref on K log X /C over ∆, and denote by (ψ s ref ) s>0 the corresponding ray of smooth metrics on K X . By Lemma 3.9 we have ψ s ref , φ s , . . . , φ s X − ψ ref , φ ref , . . . , φ ref X = s K loḡ X /P 1 ·L n + O(1). Since 1 2 log MA(φ s ), φ s , . . . , φ s X − ψ s ref , φ s , . . . , φ s X = 1 2 X log MA(φ s ) e 2ψ s ref (dd c φ s ) n , Theorem 3.6 is therefore a consequence of the following result. Let us first prove an estimate of independent interest. See [BJ16a] for more precise results. Lemma 3.11. Let X be an snc test configuration for X and Ψ a smooth metric on K log with d denoting the dimension of the dual complex of X 0 , so that d + 1 is the largest number of local components of X 0 . Here A ∼ B means that A/B is bounded from above and below by positive constants. Proof. Since X 0 is an snc divisor, every point of X 0 admits local coordinates (z 0 , . . . , z n ) that are defined in a neighborhood of B := {\|z i \| ≤ 1} and such that z b 0 0 . . . z bp p = ετ with 0 ≤ p ≤ n and ε > 0. Here b i ∈ Z >0 is the multiplicity of X 0 along {z i = 0}. The integer d in the statement of the theorem is then the largest such integer p. By compactness of X 0 , it will be enough to show that B∩Xτ e 2Ψτ ∼ log \|τ \| −1 p . The holomorphic n-form η := 1 p + 1 p j=0 (−1) j b j dz 0 z 0 ∧ · · · ∧ dz j z j ∧ · · · ∧ dz p z p ∧ dz p+1 ∧ · · · ∧ dz n satisfies η ∧ dτ τ = dz 0 z 0 ∧ · · · ∧ dz p z p ∧ dz p+1 ∧ · · · ∧ dz n . Thus η defines a local frame of K log X /C on B, so the holomorphic n-form η τ := η\| Xτ satisfies C −1 \|η τ \| 2 ≤ e 2Ψτ ≤ C\|η τ \| 2 for a constant C > 0 independent of τ . Hence it suffices to prove B∩Xτ \|η τ \| 2 ∼ log \|τ \| −1 p . To this end, we parametrize B ∩X τ in (logarithmic) polar coordinates as follows. Consider the p-dimensional simplex σ = {w ∈ R p+1 ≥0 \| p j=0 b j w j = 1}, the p-dimensional (possibly disconnected) commutative compact Lie group T = {θ ∈ (R/Z) p+1 \| p j=0 b j θ j = 0}, and the polydisc D n−p ⊂ C n−p . We may cover C * by two simply connected open sets, on each of which we fix a branch of the complex logarithm. We then define a diffeomorphism χ τ from σ × T × D n−p to B ∩ X τ by setting z j = e w j log(ετ )+2πiθ j for 0 ≤ j ≤ p. A simple computation shows that χ * τ (\|η τ \| 2 ) = const log \|ετ \| −1 p dV, where dV denotes the natural volume form on σ × T × D n−p . It follows that, for \|τ \| 1, B∩Xτ \|η τ \| 2 ∼ σ×T ×D n−p χ * τ (\|η τ \| 2 ) ∼ log \|τ \| −1 p , which completes the proof. Proof of Lemma 3.10. On the one hand, we have on X τ is uniformly bounded from above. Indeed, if τ = e −s , we then see that V −1 X log MA(φ s ) e 2ψ s ref (dd c φ s ) n = X log MA(φ s ) e 2ψ s ref / X e 2ψ s ref MA(φ s ) − log X e 2ψ s ref ≥ − logX log MA(φ s ) e 2ψ s ref (dd c φ s ) n = Xτ (log V −1 + log g τ )(dd c Φ\| Xτ ) n is uniformly bounded from above, since (dd c Φ\| Xτ ) n has fixed mass V for all τ . To bound g τ from above, we use local coordinates (z j ) n 0 as in the proof of Lemma 3.11. With the notation in that proof, it suffices to prove that the function (Ω\| Xτ ) n /e 2Ψτ on X τ is uniformly bounded from above, where Ω := i 2 n j=0 dz j ∧ dz j . Indeed, we have dd c Φ ≤ CΩ for some constant C > 0. It then further suffices to prove the bound i n dz 0 ∧ dz 0 ∧ · · · ∧ dz j ∧ dz j ∧ · · · ∧ dz n ∧ dz n Xτ ≤ Ce 2Ψτ (3.4) for 0 ≤ j ≤ p and a uniform constant C > 0. To prove (3.4) we use the logarithmic polar coordinates in the proof of Lemma 3.10. Namely, if χ τ : σ × T × D n−p → B ∩ X τ is the diffeomorphism in that proof, we have χ * τ (e 2Ψτ ) ∼ (log \|τ \| −1 ) p dV. χ * τ (i n dz 0 ∧ dz 0 ∧ · · · ∧ dz j ∧ dz j ∧ · · · ∧ dz n ∧ dz n ) ∼ (log \|τ \| −1 ) p 0≤l≤p,l =j \|z l \| 2 dV. Thus (3.4) holds, which completes the proof. The logarithmic setting In this section we extend, for completeness, Theorem 3.6-and hence Theorem A and Corollary B-to the logarithmic setting. We will also relax the positivity assumptions used. Our conventions and notation largely follow [BBEGZ11]. 4.1. Preliminaries. If X is a normal projective variety of dimension n, and φ 1 , . . . , φ n are smooth metrics on Q-line bundles L 1 , . . . , L n on X, then we define dd c φ 1 ∧ · · · ∧ dd c φ n as the pushforward of the measure dd c φ 1 \| Xreg ∧ · · · ∧ dd c φ n \| Xreg from X reg to X. This is a signed Radon measure of total mass (L 1 · . . . · L n ), positive if the φ i are semipositive. A boundary on X is a Weil Q-divisor B on X such that the Weil Q-divisor class K (X,B) := K X + B is Q-Cartier. Note that B is not necessarily effective. We call (X, B) a pair. The log discrepancy of a divisorial valuation v = c ord F with respect to (X, B) is defined as in §2.3, using A (X,B) (v) = c(1 + ord F (K Y /(X,B) )). The pair (X, B) is subklt if A (X,B) (v) > 0 for all (nontrivial) divisorial valuations v. (It is klt when B is further effective.) A pair (X, B) is log smooth if X is smooth and B has simple normal crossing support. A log resolution of (X, B) is a projective birational morphism f : X → X, with X smooth, such that Exc(f ) + f −1 * (B) has simple normal crossing support. In this case, there is a unique snc divisor B on X such that f * B = B and K (X ,B ) = f * K (X,B) . In particular the pair (X , B ) is log smooth. The pair (X, B) is subklt iff (X , B ) is subklt, and the latter is equivalent to B having coefficients < 1. A smooth metric ψ on K (X,B) canonically defines a smooth positive measure µ ψ on X reg \B as follows. Let φ B be the canonical singular metric on O Xreg (B), with curvature current given by [B]. Then ψ − φ B is a smooth metric on K Xreg\B , and hence induces a smooth positive measure µ ψ := e 2(ψ−φ B ) on X reg \ B. The fact that (X, B) is subklt means precisely that the total mass of µ ψ is finite. Thus we can view µ ψ as a finite positive measure on X that is smooth on X reg \ B and gives no mass to B or X sing . The analogue of the Ricci energy R is defined on smooth metrics φ on L by R B (φ) := n−1 j=0 1 V Xreg (φ − φ ref )dd c ψ ref ∧ (dd c φ) j ∧ (dd c φ ref ) n−1−j . It satisfies R B (φ+c) = R B (φ)−S B c and is pullback invariant in the following sense. Suppose q : X → X is a birational morphism, with X projective normal, and define B by q * B = B and q * K (X, B) = K (X ,B ) . Set φ ref = q * φ ref and ψ ref := q * ψ ref . Then R B (φ) = R B (φ ), where φ = q * φ.H B (φ) := 1 2 Xreg log MA(φ) µ ref MA(φ) = 1 2 Xreg log MA(φ) e 2(ψ ref −φ B ) MA(φ).M B (φ) = H B (φ) − (E(φ) − Xreg (φ − φ ref ) MA(φ)). We have D B ≤ M B on smooth semipositive metrics. 4.3. Non-Archimedean functionals. The extensions of the non-Archimedean functionals in §2.7 to the logarithmic setting were studied in [BHJ15,§7]. Let us briefly review them. Consider a normal complex projective variety X and a big and nef Q-line bundle L on X. Let φ be a non-Archimedean metric on L, represented by a normal test configuration (X , L) for (X, L), that we assume dominates (X × C, L × C) via ρ : X → X × C. The formulas in §2.7 for E NA (φ), I NA (φ) and J NA (φ) are still valid. Given a boundary B on X we set R NA B (φ) : = V −1 (ψ triv · φ n ) = V −1 ρ * K log (X×P 1 ,B×P 1 )/P 1 ·L n . Now assume (X, B) is subklt and let B (resp.B) be the (component wise) Zariski closure of B × C * in X (resp.X ). Then H NA B (φ) : = X div A (X,B) (v) MA NA (φ) = V −1 K log (X ,B)/P 1 ·L n − V −1 ρ * K log (X×P 1 ,B×P 1 )/P 1 ·L n . and M NA B (φ) : = H NA B (φ) + R NA B (φ) +S B E NA (φ) = 1 V K log (X ,B)/P 1 ·L n +S B (n + 1)V L n+1 . While the definitions of H NA B (φ) and M NA B (φ) make sense for arbitrary non-Archimedean metrics φ, we will usually assume that φ is semipositive. All the functionals above have the same invariance properties as their Archimedean cousins. They are also homogeneous in the sense of Definition 2.7. Finally, when (X, B) is weakly log Fano, so that (X, B) is subklt and L := −K (X,B) is big and nef, the non-Archimedean Ding functional is defined by D NA B (φ) = L NA B (φ) − E NA (φ), where L NA B (φ) = inf v A (X,B) (v) + (φ − φ triv )(v) , the infimum taken over all valuations v on X that are divisorial or trivial. The Ding functional D NA B is translation invariant and pullback invariant. The formula for the Mabuchi functional simplifies in the log Fano case to M NA B (φ) = H NA B (φ) − (E NA (φ) − X div (φ − φ ref ) MA NA (φ)). We have D NA B ≤ min{M NA B , J NA } on semipositive metrics, see Propositions 7.28 and 7.32 in [BHJ15]. 4.4. Asymptotics. The following result generalizes Theorem 3.6 and shows that if F is one of the functionals E, I, J, H B , R B or M B on H, then F admits a non-Archimedean limit on H NA given by F NA . For future reference, we state the result in detail. Theorem 4.2. Let X be a normal projective variety, L a big and nef Q-line bundle on X, and (X , L) a test configuration for (X, L) inducing a non-Archimedean metric φ NA on L. Further, let Φ be a smooth, S 1 -invariant metric on L near X 0 , inducing a smooth ray (φ s ) s>s 0 of metrics on L. Fix a smooth reference metric φ ref on L. Then lim s→+∞ F (φ s ) s = F NA (φ NA ), (4.1) where F is any of the functionals E, I, J. Further, if B is a boundary on X and ψ ref is a smooth reference metric on K (X,B) , then (4.1) also holds for F = R B . Finally, if (X, B) is subklt and Φ is semipositive, then (4.1) holds for F = H B and F = M B . In addition, Berman proved that in the log Fano case, the Ding functional D B admits D NA B as non-Archimedean limit. Indeed, the following result follows from Proposition 3.8 and §4.3 in [Berm16]. Remark 4.4. Theorems 4.2 and 4.3 remain true even when Φ is not S 1 -invariant, in the following sense. For τ ∈ ∆ * , let φ τ be the metric on L defined as the pullback of Φ\| Xτ under the C * -action. Then we have lim τ →0 (log \|τ \| −1 ) −1 F (φ τ ) = F NA (φ NA ). 4.5. Proof of Theorem 4.2. By pullback invariance, we may assume that X is smooth. After further pullback, we may also assume that X is smooth and dominates X × C. In this case, the asymptotic formulas for E, I and J follow immediately from Lemma 3.9. When considering the remaining functionals, we may similarly, by pullback invariance, assume that the pair (X, B) is log smooth. The asymptotic formula for R B now follows from Lemma 3.9 since we can express R B (φ) in terms of Deligne pairings: R B (φ) = ψ ref , φ n X − ψ ref , φ n ref X , whereas the non-Archimedean counterpart is given by the intersection number R NA B (φ) = V −1 ρ * K log (X×P 1 ,B×P 1 )/P 1 ·L n X . Finally we consider the functionals H B and M B . Thus assume (X, B) is log smooth and subklt. We may further assume that the divisor X 0 + B has simple normal crossing support, where B is the (component-wise) Zariski closure of the pullback of B × C * in X . As V H B (φ) = 1 2 X log MA(φ) e 2(ψ ref −ψ B ) (dd c φ) n + i∈I c i X log \|σ i \| ψ i (dd c φ) n = 1 2 log MA(φ), φ n X − ψ ref , φ n X + ψ B , φ n X + i∈I c i ( φ n B i − ψ i , φ n X ) = 1 2 log MA(φ), φ n X − ψ ref , φ n X + i∈I c i φ n B i , for any smooth semipositive metric φ on L. This implies V (H B (φ) + R B (φ)) = 1 2 log MA(φ), φ n X − ψ ref , φ n ref X + i∈I c i φ n B i = V (H(φ) + R(φ)) + n i∈I c i (L n−1 · B i )E(φ\| B i ) + O(1). On the non-Archimedean side, we have V (H NA B (φ NA ) + R NA B (φ NA )) = K log (X ,B)/P 1 ·L n X = K loḡ X /P 1 ·L n X + B ·L n X = V (H NA (φ NA ) + R NA (φ NA )) + i∈I c i L \| n B i B i = V (H NA (φ NA ) + R NA (φ NA )) + n i∈I c i (L n−1 · B i )E NA (φ NA i ), where φ NA i is the non-Archimedean metric on L\| B i represented by L\| B i . It now follows from Theorem 3.6 that 1 lim s→∞ 1 s (H(φ s ) + R(φ s )) = H NA (φ NA ) + R(φ NA ), Applying Theorem 3.6 on B i and B i , we also get lim s→∞ 1 s E(φ s i ) = E NA (φ NA i ). Thus lim s→∞ 1 s (H B (φ s ) + R B (φ s )) = H NA B (φ NA ) + R B (φ NA ), which completes the proof of Theorem 4.2. 4.6. Coercivity and uniform K-stability. Let us finally extend Corollary B to the logarithmic setting. Consider a pair (X, B) and a big and nef line bundle L on X. The Donaldson-Futaki invariant of a normal test configuration (X , L) for (X, L) is given by DF B (X , L) : = 1 V (K (X .B)/P 1 ·L n ) +S B (L n+1 ) (n + 1)V = M NA B (φ) + 1 V ((X 0 − X 0,red ) · L n ) , 1 While Theorem 3.6 is stated in the case when L and L are ample and Φ is positive, the proof extends to the weaker positivity assumptions used here. where φ is the non-Archimedean metric on L represented by φ. Now assume L is ample. We then define (X, B); L) to be uniformly K-stable if the following two equivalent conditions hold: (i) there exists δ > 0 such that M NA B (φ) ≥ δJ NA (φ) for every φ ∈ H NA (L); (ii) there exists δ > 0 such that DF B (X , L) ≥ δJ NA (X , L) for any normal ample test configuration (X , L). The equivalence between the two conditions is proved in [BHJ15,Proposition 8.2]. Uniform K-stability and CM-stability From now on, X is smooth. In this section we explore the relationship between uniform K-stability and (asymptotic) CM-stability. In particular we prove Theorem C, Corollary D and Corollary E. 5.1. Functions with log norm singularities. In this section, G denotes a reductive complex algebraic group. Definition 5.1. We say that a function f : G → R has log norm singularities if there exist finitely many rational numbers a i , finite dimensional complex vector spaces V i endowed with a G-action and non-zero vectors v i ∈ V i such that f (g) = i a i log g · v i + O(1) for some choice of norms on the V i 's. Remark 5.2. By the equivalence of norms on a finite dimensional vector space, the description of f is independent of the choice of norms on the V i . In particular, given a maximal compact subgroup K of G, the norms may be assumed to be K-invariant, so that f descends to a function on the Riemannian symmetric space G/K. Remark 5.3. Taking appropriate tensor products, is is easy to see that every function f on G with log norm singularities may be written as f (g) = a (log g · v − log g · w ) + O(1), (5.1) where a ∈ Q >0 and v, w are vectors in a normed vector space V endowed with a G-action. The following generalization of the Kempf-Ness/Hilbert-Mumford criterion is closely related to results of [Pau13], which they simplify to some extent. Our elementary argument is inspired by the discussion on pp.241-243 of [Tho06]. Theorem 5.4. Let f be a function on G with log norm singularities. (i) For each 1-PS λ : C * → G, there exists f NA (λ) ∈ Q such that (f • λ)(τ ) = f NA (λ) log \|τ \| −1 + O(1) for \|τ \| ≤ 1. (ii) f is bounded below on G iff f NA (λ) ≥ 0 for all 1-PS λ. The chosen notation stems from the fact that f NA induces a function on the (conical) Tits building of G, i.e. the non-Archimedean analogue of G/K (compare [MFF,§2.2]). Before entering the proof, let us recall some basic facts about representations of algebraic tori. Let T (C * ) r be an algebraic torus, and introduce as usual the dual lattices Proof of Theorem 5.4. By Remark 5.3 we may assume f is of the form f (g) := log g · v − log g · w , where v, w are nonzero vectors in a finite dimensional normed vector space V equipped with a G-action. (i) Let first λ : C * → G be a 1-parameter subgroup, and denote by I v ⊂ Z the set of weights of v with respect to λ. We then have (ii) The direct implication follows immediately from (i). For the reverse implication we use the Cartan (or polar) decomposition G = KT K, where T ⊂ G is any maximal algebraic torus and K ⊂ G be a maximal compact subgroup. We then get an isomorphism T /K ∩ T N R , hence a group homomorphism Log \| · \| : T → N R , which in compatible bases for T C * r and N R R r is given by (t 1 , . . . , t r ) → (log \|t 1 \|, . . . , log \|t r \|). λ(τ ) · v = m∈Iv τ m v m , Note that log \|m(t)\| = m, Log \|t\| for all m ∈ M and t ∈ T , and Log \|λ(τ )\| = (log \|τ \|)λ in N R for each 1-PS λ : C * → T (i.e. λ ∈ N ). In this notation, we claim that f (k tk) = h k·v (Log \|t\|) − h k·w (Log \|t\|) + O(1), (5.2) for all k, k ∈ K and t ∈ T . To see that (5.2) holds, we may assume the norm on V is K-invariant. We then have for all k, k ∈ K and t ∈ T log (k tk) · v = log t · (k · v) = log m∈M k·v m(t)(k · v) m = log max m∈M k·v m(t)(k · v) m + O(1) = max m∈M k·v ( m, Log \|t\| + log (k · v) m ) ) + O(1). By the compactness of K, we further may find C = C(v) > 0 such that −C ≤ log (k · v) m ≤ C for all k ∈ K and all m ∈ M k·v . By the definition of the support function h k·v , we thus have max m∈M k·v ( m, Log \|t\| + log (k · v) m ) ) = h k·v (Log \|t\|) + O(1). We have thus proved that log (k tk) · v = h k·v (Log \|t\|) + O(1). A similar estimate of course holds with w in place of v, and (5.2) follows. As a consequence of 5.2, we get f NA (k −1 λk) = h k·v (λ) − h k·w (λ) (5.3) for all λ ∈ N . If we assume that f NA ≥ 0 on all 1-PS of G, then h k·v ≥ h k·w on N , hence on N Q by homogeneity, and hence on N R by density. From (5.2) and the Cartan decomposition G = KT K it follows, as desired, that f is bounded below on G. The proof is now complete. 5.2. Proof of Theorem C and Corollaries D and E. Replacing L with mL, we may assume for notational simplicity that m = 1. Set N := h 0 (L) and G := SL(N, C), so that each σ ∈ G defines a Fubini-Study type metric φ σ on L. Note that M − δJ is bounded below on H 1 GL(N, C)/ U(N ) iff M (φ σ )−δJ(φ σ ) bounded below for σ ∈ G, by translation invariance of M and J. The key ingredient is the following result of S. Paul [Pau12]. Theorem 5.5. The functionals E, J and M all have log norm singularities on G. Granted this result we can deduce Theorem C. The equivalence of (ii) and (iii) follows from the same argument as Proposition 8.2 in [BHJ15], so it suffices to show that (i) and (iii) are equivalent. By Theorem 5.5, the function f (σ) := M (φ σ ) − δJ(φ σ ) on G has log norm singularities. By Theorem 5.4, it is thus bounded below iff lim s→+∞ (f • λ)(e −s ) s ≥ 0 for each 1-parameter subgroup λ : C * → G. We obtain the desired result since by Theorem B, this limit is equal to M NA (φ λ ) − δJ NA (φ λ ), where φ λ ∈ H NA is the non-Archimedean metric on L defined by λ. Corollary D follows since every ample test configuration of (X, L) is induced by a 1-PS, see §2.2. The first assertion of Corollary E follows immediately, and the fact that the reduced automorphism group of (X, L) is finite is a consequence of [Pau13, Corollary 1.1]. Proof of Theorem 5.5. Recall from [Pau12] that to the linearly normal embedding X → PH 0 (X, L) * P N −1 are associated the X-resultant R, i.e. the Chow coordinate of X, and the X-hyperdiscriminant ∆, which cuts out the dual variety of X × P n−1 → P N −1 × P n−1 → P N n−1 , the second arrow being the Segre embedding. In our notation, we then have deg R = V (n + 1) and deg ∆ = V n(n + 1) −S [Pau12, Proposition 5.7], and [Pau12, Theorem A] becomes M (φ σ ) = V −1 log σ · ∆ − V −1 deg ∆ deg R log σ · R + O(1), (5.4) which proves the assertion for M (φ σ ). We next consider J(φ σ ) = X (φ σ − φ ref ) MA(φ ref ) − E(φ σX (φ σ − φ ref ) MA(φ ref ) = log σ + O(1). The assertion for J(φ σ ) follows. 5. 3. Discussion of [Tia14]. The statement of [Tia14, Lemma 3.1] sounds overoptimistic from the GIT point of view, as it would mean that CM-stability can be tested by only considering 1-parameter subgroups of a fixed maximal torus T . At least, the proof is incorrect, the problem being the estimate (3.1), which claims that φ τ k − φ τ is uniformly bounded with respect to τ ∈ T and k ∈ K. As the next example shows, this is not even true for a fixed k ∈ K. Example 5.6. Assume (s 1 , s 2 ) is a basis of H 0 (X, L), let k ∈ U (2) be the unitary transformation exchanging s 1 and s 2 , τ = (t, t −1 ), and pick a point x with s 1 (x) = 0. Then φ τ k (x) − φ τ (x) = 4 log \|τ \| is unbounded. In any case, the methods here do not seem to be able to deduce CM-stability from Kstability, because of the following fact (cf. [Li12,p.39]). Proposition 5.7. For each polarized manifold (X, L) and each m large and divisible enough, there exists a non-trivial 1-PS λ in GL(N m , C) such that J and M remain bounded on the corresponding Fubini-Study ray φ s := φ λ(e −s ) . Proof. As originally observed in [LX14] (cf. Proposition 2.3), (X, L) admits a non-trivial ample test configuration (X , L) that is almost trivial, i.e. with trivial normalization. As recalled in §2.2, for each m large and divisible enough, (X , L) can be realized as the test configuration induced by a 1-PS λ : C * → GL(N m , C), which is non-trivial since (X , L) is. Since the normalization of (X , L) is trivial, the associated non-Archimedean metric is of the form φ triv + c for some c ∈ Q, and hence M NA (φ λ ) = J NA (φ λ ) = 0. Since M and J have log norm singularities on GL(N m , C) by Theorem 5.5, M and J are indeed bounded on φ s by Theorem 5.4. Remarks on the Yau-Tian-Donaldson conjecture As explained in the introduction, we will here give a simple argument, following ideas of Tian, for the existence of a Kähler-Einstein metric on a Fano manifold X, assuming (X, −K X ) is uniformly K-stable and the partial C 0 -estimates due to Székelyhidi. 6.1. Partial C 0 -estimates and the continuity method. For the moment, consider an arbitrary polarized manifold (X, L). For each m such that mL is very ample, we have a 'Bergman kernel approximation' map P m : H → H m , defined by setting P m (φ) to be the Fubini-Study metric induced by the L 2 -scalar product on H 0 (X, mL) defined by mφ. Definition 6.1. A subset A ⊂ H satisfies partial C 0 -estimates at level m if there exists C > 0 such that \|P m (φ) − φ\| ≤ C for all φ ∈ A. Now assume X is Fano, and set L := −K X . Given a Kähler form α ∈ c 1 (X), consider Aubin's continuity method Ric(ω t ) = tω t + (1 − t)α. (6.1) It is well-known that there exists a unique maximal solution (ω t ) t∈[0,T ) , where 0 < T ≤ 1. The following important result, due to Székelyhidi [Szé16], confirms a conjecture of Tian. Theorem 6.2. The set A := {ω t \| t ∈ [0, T )} satisfies partial C 0 -estimates at level m, for arbitrarily large positive integers m. Given this result, we shall prove Theorem 6.3. Any uniformly K-stable Fano manifold admits a Kähler-Einstein metric. By working (much) harder, Datar and Székelyhidi [DSz15] have in fact been able to deduce from Theorem 6.2 a much better result dealing with K-polystability and allowing a compact group action. 6.2. CM-stability and partial C 0 -estimates. We first present in some detail well-known ideas due to Tian [Tia12,§4.3]. In this section, (X, L) is an arbitrary polarized manifold. Proposition 6.4. Assume that (X, mL) is CM-stable, and that A ⊂ H satisfies partial C 0 -estimates at level m. Then there exist δ, C > 0 such that M ≥ δJ − C on A. The proof, which is similar to the arguments in [Szé16,§5]. is based on two lemmas. Lemma 6.5. For any two metrics φ, ψ ∈ H, we have Proof. Recall that E(φ) − E(ψ) = 1 n + 1 n j=0 V −1 X (φ − ψ)(dd c φ) j ∧ (dd c ψ) n−j . As a consequence, \|E(φ) − E(ψ)\| ≤ sup \|φ − ψ\|, and (i) follows immediately. For (ii), we basically argue as in the proof of [Tia14, Lemma 3.1]. By the Chen-Tian formula 1.11, we have M (φ) − M (ψ) = H ψ (φ) +S (E(φ) − E(ψ)) + E Ric(dd c ψ) (ψ) − E Ric(dd c ψ) (φ). Here the entropy term H ψ (φ) is non-negative, and we have E Ric(dd c ψ) (φ) − E Ric(dd c ψ) (ψ) = n−1 j=0 V −1 X (φ − ψ)(dd c φ) j ∧ (dd c ψ) n−j−1 ∧ Ric(dd c ψ). Assume Ric(dd c ψ) ≤ Cdd c ψ for some constant C > 0. We may then write (dd c φ) j ∧ (dd c ψ) n−j−1 ∧ Ric(dd c ψ) = C(dd c φ) j ∧ (dd c ψ) n−j − (dd c φ) j ∧ (dd c ψ) n−j−1 ∧ (C dd c ψ − Ric(dd c ψ)), a difference of two positive measures of mass CV and CV + (L n−1 · K X ), respectively, and the desired estimate follows. The case where Ric(dd c ψ) ≥ −C dd c ψ is treated similarly (and will anyway not be used in what follows). We next recall a well-known upper bound for the Ricci curvature of restrictions of Fubini-Study metrics. Lemma 6.6. We have Ric(dd c φ) ≤ N m dd c φ for all φ ∈ H m . Proof. Choose a basis of H 0 (X, mL), and let ω be the corresponding Fubini-Study metric on P := PH 0 (X, mL) * . Its curvature tensor Θ(T P , ω) ∈ C ∞ (P, Λ 1,1 T * P ⊗ End(T P )) is Griffiths positive and satisfies Tr T P Θ(T P , ω) = Ric(ω) = N m ω. For each complex submanifold Y ⊂ P, the curvature of its tangent bundle T Y with respect to ω\| Y satisfies Θ(T Y , ω\| Y ) ≤ Θ(T P , ω)\| T Y as (1, 1)-forms on Y with values in the endomorphisms of T Y , as a consequence of a well-known curvature monotonicity property going back to Griffiths. We thus have Ric(ω\| Y ) = Tr T Y Θ(T Y , ω\| Y ) ≤ Tr T Y Θ(T P , ω)\| T Y . Using now Θ(T P , ω) ≥ 0, we have on the other hand Tr T Y Θ(T P , ω)\| T Y ≤ Tr T P Θ(T P , ω)\|Y = N m ω\| Y , and hence Ric(ω\| Y ) ≤ N m ω\| Y . Applying this to the images of X ⊂ P under projective transformations yields the desired result. Proof of Proposition 6.4. Since (X, mL) is CM-stable, there exist δ, C > 0 such that M (P m (φ)) ≥ δJ(P m (φ)) − C (6.2) for all φ ∈ H. By assumption on A, we also have \|P m (φ) − φ\| ≤ C for all φ ∈ A, and by Lemma 6.6, the Ricci curvature of dd c P m (φ) is uniformly bounded above. Hence Lemma 6.5 shows, as desired, that there exists C > 0 with M (φ) ≥ δJ(φ) − C for all φ ∈ A. 6.3. Proof of Theorem 6.3. Assume now that X is a Fano manifold and set L := −K X . Consider the continuity method (6.1). Pick metrics ψ and φ t on −K X such that α = dd c ψ and ω t = dd c φ t , respectively. After adding a constant to φ t , (6.1) may be written (dd c φ t ) n = e −2(tφt+(1−t)ψ) . (6.3) We recall the proof of the following well-known monotonicity property. Lemma 6.7. The function t → M (φ t ) is non-increasing. 1 π ∆ t − t φ t ∆ tφ t MA(φ t ) = 1 − t 2π X 1 π ∆ t − t ∂φ t ,∂φ t ωt MA(φ t ). Since Ric(ω t ) ≥ tω t , the∂-Laplacian ∆ t satisfies 1 π ∆ t ≥ t on (0, 1)-forms, and the last integral is thus nonnegative. Indeed, this follows from the Bochner-Kodaira-Nakano identity applied to C ∞ (X, Λ 0,1 T * X ) C ∞ (X, Λ n,1 T * X ⊗ K * X ) with the fiber metric ψ t = − 1 2 log ω n t on K * X = −K X , with curvature dd c ψ t = Ric(ω t ). constant. These properties are thus also shared by I and I − J. The functionals H, I, J, M are translation invariant in the sense that H Remark 1. 4 . 4In the definition of R, we could replace − Ric dd c φ ref by dd c ψ ref for any smooth metric ψ ref on K X . Similarly, in the definition of H, we could replace the reference measure MA(φ ref ) by e 2ψ ref . Doing so, and keeping the Chen-Tian formula, would only change the Mabuchi functional M by an additive constant. 2. 7 . 7Functionals on non-Archimedean metrics. Following [BHJ15, §7] we define non-Archimedean analogues of the functionals considered in §1.4. Fix a line bundle L. ii) DF(φ) ≥ δJ NA (φ) for every φ ∈ H NA (L); (iii) DF(X , L) ≥ δJ NA (X , L) for any normal ample test configuration (X , L).Here the equivalence between (ii) and (iii) is definitional, and (i) =⇒ (ii) follows immediately from DF ≤ M NA . The implication (ii) =⇒ (i) follows from the homogeneity of M NA together with the fact that DF(φ d ) = M NA (φ d ) for d sufficiently divisible. See [BHJ15, Proposition 8.2] for details. 3. 3 . 3Asymptotics of the functionals. The following result immediately implies Theorem A and Corollary B. Theorem 3. 6 . 6The functionals E, H, I, J, M and R on H admit non-Archimedean limits on H NA given, respectively, by E NA , H NA , I NA , J NA , M NA and R NA .In addition, we have the following result due to Berman[Berm16, Proposition 3.8]. See also [BBJ15, Theorem 3.1] for a more general result. Lemma 3. 9 . 9For i = 0, . . . , n, let L i be a line bundle on X with a smooth reference metric φ i,ref . Let also (X , L i ) be a smooth test configuration for (X, L i ), Φ i an S 1 -invariant smooth metric on L i near X 0 , and denote by (φ s i ) the corresponding ray of smooth metrics on L i . Then φ s 0 , . . . , φ s n X − φ 0,ref , . . . , φ n,ref X = s L 0 · . . . ·L n + O(1) as s → ∞. Here (X ,L i ) is the compactification of (X , L i ) for 0 ≤ i ≤ n and ·, . . . , · X denotes the Deligne pairing with respect to the constant morphism X → {pt}. By functoriality, the metric ψ s on F 1 is nothing but the Deligne pairing φ s 0 , . . . , φ s n . If weset ψ ref = φ 0,ref , . . . , φ n,ref X , it therefore follows that φ s 0 , . . . , φ s n X − φ 0,ref , . . . , φ n,ref X = log \|v\| ψ ref − log \|v\| ψ s = sw + O(1), which completes the proof. (n + 1 ) 1V (E(φ s ) − E(φ ref )) = φ s , . . . , φ s X − φ ref , . . . , φ ref X . NA (φ NA ),which proves the result for the Monge-Ampère energy E. The case of the functionals I, J and R is similarly a direct consequence of Lemma 1.2 and Lemma 3.9. In view of the Chen-Tian formulas for M and M NA , it remains to consider the case of the entropy functional H. In fact, it turns out to be easier to treat the functional H + R.By Lemma 1.2 we have Lemma 3 . 10 . 310We have X log MA(φ s ) e 2ψ s ref (dd c φ s ) n = O(log s) as s → ∞. near X 0 . 0Denote by e 2Ψτ the induced volume form on X τ for τ = 0. Then Xτ e 2Ψτ ∼ log \|τ \| −1 d as τ → 0, (3.3) first term on the second line is the relative entropy of the probability measure MA(φ s ) with respect to the probability measure e 2ψ s ref / X e 2ψ s ref . By Lemma 3.11 we have X e 2ψ s ref = O(s d ), where 0 ≤ d ≤ n. This gives the lower bound in Lemma 3.10. To get the upper bound, it suffices to prove that the function g τ := (dd c Φ\| Xτ ) n e 2Ψτ 4. 2 . 2Archimedean functionals. Let X be a normal complex projective variety of dimension n. Fix a big and nef Q-line bundle L on X and set V := (L n ) > 0. For a smooth metric φ on L, set MA(φ) = V −1 (dd c φ) n .Fix a smooth positive reference metric φ ref on L The energy functionals E, I and J are defined on smooth metrics on L exactly as in (1.6), (1.8) and (1.7), respectively; they are normalized by E(φ ref ) = I(φ ref ) = J(φ ref ) = 0. The functionals I and J are translation invariant, whereas E(φ + c) = E(φ) + c. All three functionals are pullback invariant in the following sense. Let q : X → X be a birational morphism, with X normal and projective, and set L := q * L. For any smooth metric φ on L, we have E(φ ) = E(φ), I(φ ) = I(φ) and J(φ ) = J(φ), where φ = q * φ and where the functionals are computed with respect to the reference metric φ ref := q * φ ref . Now consider a boundary B on X. SetS B := −nV −1 (K (X,B) · L n−1 ) and fix a smooth reference metric ψ ref on K (X,B) . When X is smooth and B = 0, we could pick ψ ref = 1 2 log MA(φ ref ), but in general, there seems to be no canonical way to get ψ ref from φ ref . Now assume (X, B) is subklt and let µ ref = µ ψ ref be the finite positive measure defined in §4.1. It is smooth and positive on X ref \ B, and may be assumed to have mass 1, after adding a constant to ψ ref . For a smooth semipositive metric φ on L, set We may have H B (φ ref ) = 0. However, H B is bounded from below and translation invariant. It is also pullback invariant in the sense above, with reference measure µ ref = µ ψ ref on X . Lemma 4 . 1 . 41If φ is a smooth semipositive metric on L, then H B (φ) < +∞.Proof. By pullback invariance we may assume that (X, B) is log smooth. In this case MA(φ) and µ ref are smooth measures on X that are strictly positive on X reg . Consider any point ξ ∈ B and pick local coordinates (z 1 , . . . , z n ) at ξ such that the irreducible components of B are given by{z i = 0}, 0 ≤ i ≤ p. Fix a volume form dV near ξ. Then µ ref = g p i=0 \|z i \| 2a i dV , and MA(φ) = hdV , with a i > −1, g > 0 and h ≥ 0 smooth. If f = h log( h g p i=0 \|z i \| −2a i ),then f is locally integrable with respect to dV . This completes the proof. As in §1.4 we define the Mabuchi functional on semipositive smooth metrics by M B := H B + R B +S B E. Then M B is translation invariant and pullback invariant in the sense above. At least formally, the critical points of M B satisfy n(Ric(dd c φ) − [B]) ∧ (dd c φ) n−1 =S B (dd c φ) n and should be conical cscK metrics, see [Li14]. Finally consider the (weak) log Fano case, in which L := −K (X,B) is big and nef. The Ding functional is then defined on smooth metrics as D B = L B − E, with L B (φ) φ+φ B ) . If we use ψ ref = −φ ref , then the formula for the Mabuchi functional simplifies to Theorem 4 . 3 . 43Let (X, B) be a subklt pair with L := −K (X,B) big and nef, (X , L) a test configuration for (X, L) inducing a non-Archimedean metric φ NA on L, and Φ a semipositive smooth, S 1 -invariant metric on L near X 0 , inducing a smooth ray (φ s ) s>s 0 of semipositive metrics on L. Then lim s→+∞1 s D B (φ s ) = D NA B (φ NA ). In fact, it is enough to assume Φ is semipositive and locally bounded in Theorem 4.3. in §3.3 it suffices to prove the asymptotic formula for the functional H B + R B . To this end, we express H B in terms of Deligne pairings. Write B = i c i B i , where B i , i ∈ I, are the irreducible components of B and c i ∈ Q. Fix a smooth metric ψ i on O X (B i ) for i ∈ I. Then ψ B := i c i ψ i is a smooth metric on O X (B), and it follows from (1.3) that Corollary 4 . 5 . 45Let (X, B) be a subklt pair and L an ample line bundle on X. Suppose that the Mabuchi functional is coercive in the sense that there exist positive constants δ and C such that M B (φ) ≥ δJ(φ) − C for every positive smooth metric φ on L. Then ((X, B); L) is uniformly K-stable; more precisely DF B (X , L) ≥ M B (φ) ≥ δJ NA (φ) for every positive non-Archimedean metric on L, where (X , L) is the unique normal ample representative of φ. M := Hom(T, C * ) Z r and N := Hom(C * , T ) Z r .Note that N is the group of 1-PS of T . For each finite-dimensional vector space V on which T acts and each m ∈ M , let V m ⊂ V be the subspace on which each t ∈ T acts by multiplication by m(t). The action of T on V being diagonalizable, we have a direct sum decomposition V = m∈M V m , and the set of weights of V is defined as the (finite) setM V ⊂ M of characters m ∈ M for which V m = 0.Given a non-zero vector v ∈ V , the set M v ⊂ M V of weights of v is defined as those m ∈ M for which the projection v m ∈ V m of v is non-zero. The weight polytope of v is defined as the convex hull P v ⊂ M R of M v in M R , whose support function h v : N R → R is the convex, positively homogeneous function defined byh v (λ) = max m∈Mv m, λ ,where the bracket denotes the dual pairing between M R and N R . (τ ) · v = max m∈Iv (m log \|τ \| + log v m ) + O(1) = − min m∈Iv m log \|τ \| −1 + O(1)for \|τ \| ≤ 1, and (i) follows with f NA (λ) = min I w − min I v . ) M (φ) ≥ M (ψ) − C sup \|φ − ψ\| for some C > 0 only depending on a one-sided bound (either upper or lower) for the Ricci curvature of the Kähler metric dd c ψ. 2.2. One-parameter subgroups. Suppose L is ample. Ample test configurations are then essentially equivalent to one-parameter degenerations of X. See [BHJ15, §2.3] for details on what follows. ) . )On the one hand, by [Pau04, Theorem 1] (or [Zha96, Theorem 1.6, Theorem 3.6]) we have On the other hand, choosing any norm on the space of complex N × N -matrices (in which G of course embeds), it is observed in the proof of [Tia14, Lemma 3.2] thatE(φ σ ) = 1 deg R log σ · R + O(1). (5.5) Acknowledgment. The authors would like to thank Robert Berman for very useful discussions. The first author is also grateful to Marco Maculan, Vincent Guedj and Ahmed ZeriahiWe may now complete the proof of Theorem 6.3. By Corollary E, (X, −mK X ) is CMstable for all m divisible enough. Theorem 6.2 and Proposition 6.4 therefore yield δ, C > 0 such that M (φ t ) ≥ δJ(φ t ) − C along Aubin's continuity method. Since M (φ t ) is bounded above by Lemma 6.7, it follows that J(φ t ) remains bounded. 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[ "Dual virtual element method in presence of an inclusion", "Dual virtual element method in presence of an inclusion" ]	[ "Alessio Fumagalli " ]	[]	[]	We consider a Darcy problem for saturated porous media written in dual formulation in presence of a fully immersed inclusion. The lowest order virtual element method is employ to derive the discrete approximation. In the present work we study the effect of cells with cuts on the numerical solution, able to geometrically handle in a more natural way the inclusion tips. The numerical results show the validity of the proposed approach.	10.1016/j.aml.2018.06.004	[ "https://arxiv.org/pdf/1709.03519v1.pdf" ]	52,082,056	1709.03519	00385347e2b619b87ea62815f22ef81e44cca25e	Dual virtual element method in presence of an inclusion Alessio Fumagalli Dual virtual element method in presence of an inclusion VEMDarcy flowfractured porous mediainclusion We consider a Darcy problem for saturated porous media written in dual formulation in presence of a fully immersed inclusion. The lowest order virtual element method is employ to derive the discrete approximation. In the present work we study the effect of cells with cuts on the numerical solution, able to geometrically handle in a more natural way the inclusion tips. The numerical results show the validity of the proposed approach. Introduction Single-phase flow in fractured porous media is a challenging problem involving different aspects, i.e. the derivation of proper mathematical models to describe fracture surrounding porous media flow and the subsequent discretization with ad-hoc numerical schemes. One of the most common approach is to consider fractures as co-dimensional objects and derive proper reduced models and coupling condition to describe the flow in the new setting. A hybrid dimensional description of the problem is thus introduced, see [1][2][3][4][5]. In presence of multiple fractures, forming a complex system of network, grid creation may become challenging and the number of cells or their shape may not be satisfactory for complex application, e.g., the benchmark study proposed in [6]. In the present work we simplify the problem considering a single fracture and substituting with an inclusion, i.e. the flow problem in the fracture is not considered but its effect of its normal flow. The inclusion is an internal condition for the problem. We consider two extreme cases: perfectly permeable fracture (infinite normal permeability) and impermeable fracture (zero normal permeability). The pressure imposed on both sides of the inclusion determines these two cases. The virtual element method (VEM), introduced in [7][8][9][10][11][12][13][14][15], is able to discretize the problem on grids with rather general cell shape. The theory developed in the aforementioned works considers star shaped cells. In the present study, the lowest order VEM is considered in presence of cells with an internal cut for the approximation at the tips of the inclusion. The paper is organised as follow. In Section 2 the mathematical model and its weak formulation are presented. Section 3 introduces the discrete formulation of the problem. Numerical examples are reported in Section 4 for both the extreme cases. The work finishes with conclusions Section 5. Mathematical model Let us set Ω ⊂ R 2 a regular domain representing a saturated porous media. We consider the Darcy model for single phase flow in a saturated porous media written in dual formulation, namely u + K∇p = 0 in Ω ∧ ∇ · u = f in Ω ∧ p = 0 on ∂Ω.(1a) The unknowns are: u the Darcy velocity and p the fluid pressure. In (1a) K represents the permeability matrix, symmetric and positive defined, and f a scalar source or sink term. To keep the presentation simple we consider only homogeneous boundary condition for the pressure at the outer boundary of Ω, denoted by ∂Ω. Coupled to (1a) we are interested to model an immersed inclusion γ, see Figure 1 as an example. With an abuse of notation, ∂Ω does not include γ which represents an internal boundary for Ω. It is possible to define a unique normal n associated to γ and Ω γ + γ − Figure 1: domain two different sides of γ with respect to the direction of n. We indicate them as γ + and γ − , with n + = n and n − = −n the associated normals. It is important to note that geometrically γ, γ + , and γ − are indeed the same object but introducing the two sides help us to impose different internal conditions. We have p = p + on γ + ∧ p = p − on γ − . (1b) We define the Hilbert spaces Q = L 2 (Ω) and V = H div (Ω). By standard arguments the weak formulation of (1) reads a(u, v) + b(v, p) = J(v) ∀v ∈ V ∧ b(u, q) = F(q) ∀q ∈ Q.(2) In 2 we have indicated by (·, ·) Ω : Q × Q → R the scalar product in Q. The bilinear forms and functionals are defined as a(·, ·) : V × V → R s.t. a(w, v) := (K −1 w, v) Ω , b(·, ·) : V × Q → R s.t. b(v, q) := −(∇ · v, q) Ω , J(·) : V → R s.t. J(v) := − v + · n + , p + γ + − v − · n − , p − γ − , F(·) : Q → R s.t. F(q) := −( f, q) Ω where the duality pairings are defined as ·, · γ ± : H − 1 2 (γ ± ) × H 1 2 (γ ± ) → R. We are assuming p ± ∈ H 1 2 (γ ± ) and f ∈ L 2 (Ω). Following [15] problem (2) is well posed. Discrete formulation The creation of a computational grid with multiple intersecting inclusions, or more generally fractures, is a challenging aspect possibly resulting in a high number and/or poorly shaped cells. To overcome this aspect we consider a VEM formulation with a clustering technique from a finer triangular grid. We consider now the discrete formulation of problem (2) employing the VEM. For simplicity we limit our analysis to the lowest order case. For a more general case we refer to [7][8][9][10][11][12][13][14]. Let be T (Ω) a generic tessellation of Ω made of non-overlapping polytopes. We allow non-star shaped cells by considering cells with an internal cut. We introduce the following approximation spaces for p and u. For a cell E ∈ T (Ω) with edges E(E), let us define Q h (E) := {q ∈ Q : q ∈ P 0 (E)} ∧ V h (E) := {v ∈ V : v · n\| e ∈ P 0 (e) ∀e ∈ E(E), ∇ · v ∈ P 0 (E), ∇ × v = 0} . (3) The norms and the global spaces, Q h and V h , are defined accordingly. With the definition (3) it is possible to compute immediately the discrete approximation of b(·, ·), J(·), and F(·). Since the shape of the functions in V h (E), for each cell E ∈ T (E), is not prescribed we cannot compute directly the bilinear form a(·, ·). In this case we need to introduce a sub-space of V h and a projection operator, the space is V h (E) := {v ∈ V h (E) : v = ∇v, for v ∈ P 1 (E)} It is worth noting that the local approximation of P 1 (E) is done by means of a suitable monomial expansion. The projection operator is defined locally as Π 0 : V(E) → V h (E) such that a(T 0 u, v) = 0 for all v ∈ V h (E), with T 0 := I −Π 0 the projection on the orthogonal space of V h . Considering the property of the projection operator, we make the following approximation a(u, v) ≈ a h (u, v) := a(Π 0 u, Π 0 v) + s(T 0 u, T 0 v), where the fist part is a consistency term and the second a stabilization term. The latter is approximated by the bilinear form s(·, ·) such that an equivalence property holds true, i.e. ∃ι * , ι * ∈ R independent from the discretization size h satisfying ι * a(Π 0 v, Π 0 v) ≤ s(T 0 v, T 0 v) ≤ ι * a(Π 0 v, Π 0 v) , see the aforementioned work for more details. With these choices the bilinear form a h (·, ·) is computable for each cell of the grid. Numerical example In this section we present a numerical study to investigate the error decay for the model presented previously. Two examples are shown with the same and different boundary conditions on the internal boundary γ. We consider the domain depicted in Figure 1 with Ω = [0, 1] 2 and γ = {(x, y) : 0.25 ≤ x ≤ 0.75 ∧ y = 0.5}. The reference solution, named p ref , is computed considering a grid of 77546 triangles very refined at the ending points of γ. The L 2 relative error presented below is defined as err(p) = \|\|p − p ref \|\| L 2 (Ω) \| max p ref − min p ref \| .(4) In both cases we generate a family of triangular grids with a different level of refinement, in particular, close to the ending points of γ. Later, an agglomeration technique is employed based on the cell measure: neighbour cells with small measure are glue together forming new cells. To stress the method presented previously we enforce the creation, at the tips of γ, of cells with internal cuts. Figure 2 shows a zoom of the considered grids at the left ending point of γ. It is worth noting that the computation of the mesh size h is itself complex. The implementation is done with PorePy: a simulation tool for fractured and deformable porous media written in Python. See github.com/pmgbergen/porepy for further information. All the examples of this section are available in the package. In the following pictures a "Blue to Red Rainbow" colour map is used. Continuous pressure condition We consider p + = p − = 1 as condition for γ + and γ − . The pressure solution of the reference and of the coarser grid is reported in Figure 3 on the left. We notice that the solution computed on the coarse grid with cells containing a cut resembles the reference solution, even at the tips of the inclusion. The error, computed for each cell of the reference grid, is represented in Figure 4. The error is distributed in the domain without any specific peak at the tips. In this case we can conclude that the cells with a cut do not deteriorate the quality of the solution. Finally, the error decay is given in Table 1 on the left confirming a first order of convergence. Discontinuous pressure condition We consider p + = 1 for γ + and p − = −1 for γ − as boundary condition for the inclusion. We expect a jump in the solution and a poor regularity at the tips of γ. The pressure solution of the reference and of a coarse grid is reported in Figure 3 on the right. The solution computed with the coarse grid is in accordance with the reference solution, even at the tips of the inclusion. The error, computed for each cell of the reference grid, is depicted in Figure 5. We point out that the error is focused at the tips of the inclusion, since a complex solution is now approximated with a single degree of freedom on each coarse cell containing the tip. Nevertheless, the errors listed in Table 1 confirm also in this case a first order of convergence, thus the quality of the computed solution. Conclusion In this work we studied the applicability of a dual virtual element method in presence of cells with a cut for a problem with an inclusion. The proposed scheme behaved accurately obtaining correct results and error decay. For a different conditions at the inclusion a pick of error was concentrated at the inclusion tips, because a complex solution was approximated with a constant. However, the solution was not deteriorate and the global error was acceptable. While in the case of equal condition at the inclusion, a more uniform error was observed inside the domain. This is a promising approach to enlight the computational cost in presence of multiple inclusions, or even fractures represented as co-dimensional domains, which is currently under investigation. Figure 2 : 2Detail of the computational grids on the left tip of γ, depicted in red. and coarse solution for the continuous case.(b) Reference and coarse solution for the discontinuous case. Figure 3 : 3On the left: pressure (range (0, 1)) for the continuous case. On the right: pressure (range(−1, 1)) for the discontinuous case. Figure 4 : 4In the centre: the error on Ω represented on the reference grid. On the right and left: a zoom of the error for the right and left tip, respectively. Range in (0, 0.25). Figure 5 : 5From the left: reference solution, coarse solution, and solution (range (0, 0.57)) for the continuous case. Table 1 : 1On the left: (4) for example in Subsection 4.1. On the right: (4) for the example in Subsection 4.2. AcknowledgmentWe acknowledge financial support for the ANIGMA project from the Research Council of Norway (project no. 244129/E20) through the ENERGIX program. The author wish to thank: Runar Berge, Inga Berre, Wietse Boon, Eirik Keilegavlen, and Ivar Stefansson for many fruitful discussions.Bibliography An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators. M Karimi-Fard, L J Durlofsky, K Aziz, SPE Journal. 92M. Karimi-Fard, L. J. Durlofsky, K. Aziz, An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators, SPE Journal 9 (2) (2004) 227-236. Modeling Fractures and Barriers as Interfaces for Flow in Porous Media. V Martin, J Jaffré, J E Roberts, 10.1137/S1064827503429363SIAM J. Sci. 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[ "Dependence of the separative power of an optimised Iguassu gas centrifuge on the velocity of rotor", "Dependence of the separative power of an optimised Iguassu gas centrifuge on the velocity of rotor" ]	[ "S V Bogovalov \nBogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia\n", "V D Borman \nBogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia\n", "V D Borisevich \nBogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia\n", "I V Tronin \nBogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia\n", "V N Tronin \nBogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia\n" ]	[ "Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia", "Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia", "Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia", "Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia", "Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin )\nNational research nuclear university (MEPHI)\nKashirskoje shosse, 31115409MoscowRussia" ]	[]	Purpose. The objective of the work is to determine dependence of the separative power of the optimised Iguassu gas centrifuge on the velocity of the rotor. Methodolgy. The dependence is determined by means of computer simulation of the gas flow in the gas centrifuge and numerical solution of the diffusion equation for the light component of the binary mixture of uranium isotopes. 2-D axisymmetric model with the sources/sinks of the mass, angular momentum and energy reproducing the affect of the scoops was explored for the computer simulation. Parameters of the model correspond to the parameters of the so called Iguassu centrifuge. The separative power has been optimised in relation to the pressure of the gas, temperature of the gas, the temperature drop along the rotor, power of the source of angular momentum and energy, feed flow and geometry of the lower baffle. In the result the optimised separative power depends only on the velocity, length and diameter of the rotor. Findings. The dependence on the velocity is described by the power law function with the power law index 2.6 which demonstrate stronger dependence on the velocity than it follows from experimental data. However, the separative power obtained with limitation on the pressure depends on the velocity on the power ≈ 2 which well agree with the experiments. Originality. For the first time the optimised separative power of the gas centrifuges have been calculated via numerical simulation of the gas flow and Email address: [email protected] (S.V. diffusion of the binary mixture of the isotopes.	10.1108/hff-03-2016-0133	[ "https://arxiv.org/pdf/1712.02633v1.pdf" ]	119,328,688	1712.02633	e295914745df55701794fbf79ca3e25377f8518f	Dependence of the separative power of an optimised Iguassu gas centrifuge on the velocity of rotor 7 Dec 2017 S V Bogovalov Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin ) National research nuclear university (MEPHI) Kashirskoje shosse, 31115409MoscowRussia V D Borman Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin ) National research nuclear university (MEPHI) Kashirskoje shosse, 31115409MoscowRussia V D Borisevich Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin ) National research nuclear university (MEPHI) Kashirskoje shosse, 31115409MoscowRussia I V Tronin Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin ) National research nuclear university (MEPHI) Kashirskoje shosse, 31115409MoscowRussia V N Tronin Bogovalov, V.D. Borman, V.D.Borisevich, I.V.Tronin, V.N.Tronin ) National research nuclear university (MEPHI) Kashirskoje shosse, 31115409MoscowRussia Dependence of the separative power of an optimised Iguassu gas centrifuge on the velocity of rotor 7 Dec 2017Preprint submitted to International Journal Numerical methods for Heat & Fluid flowApril 24, 2018gas centrifugeisotope separationdiffusion in strong centrifugal fieldseparative power PACS: 2860+s4732Ef5110+y5120+d Purpose. The objective of the work is to determine dependence of the separative power of the optimised Iguassu gas centrifuge on the velocity of the rotor. Methodolgy. The dependence is determined by means of computer simulation of the gas flow in the gas centrifuge and numerical solution of the diffusion equation for the light component of the binary mixture of uranium isotopes. 2-D axisymmetric model with the sources/sinks of the mass, angular momentum and energy reproducing the affect of the scoops was explored for the computer simulation. Parameters of the model correspond to the parameters of the so called Iguassu centrifuge. The separative power has been optimised in relation to the pressure of the gas, temperature of the gas, the temperature drop along the rotor, power of the source of angular momentum and energy, feed flow and geometry of the lower baffle. In the result the optimised separative power depends only on the velocity, length and diameter of the rotor. Findings. The dependence on the velocity is described by the power law function with the power law index 2.6 which demonstrate stronger dependence on the velocity than it follows from experimental data. However, the separative power obtained with limitation on the pressure depends on the velocity on the power ≈ 2 which well agree with the experiments. Originality. For the first time the optimised separative power of the gas centrifuges have been calculated via numerical simulation of the gas flow and Email address: [email protected] (S.V. diffusion of the binary mixture of the isotopes. Introduction Knowledge of dependence of the separative power on the parameters of the gas centrifuge (hereafter GC) is necessary for design of efficient GC for industrial production of enriched uranium. Up to now this knowledge has basically an empirical character. In spite of 60-th year history of exploration of the gas centrifuges for enriched uranium production, there is no fundamental understanding of the dependence of the separative power of the centrifuges on the parameters. The upper limit on the separative power of GC has been estimated firstly by Dirac [1]. The separative power δU max of any GC is limited by the value δU max = πρDL 2 ∆M V 2 2RT 2 ,(1) where ρD is the density of uranium hexafluoride (U F 6 ) times the coefficient of self-diffusion of uranium isotopes 238 U and 235 U . ∆M is the mass difference of molecules with different uranium isotopes , R is the gas-law constant, T is gas temperature, L is the length of the rotor of GC, V is the linear velocity of the rotor rotation. We are interested here in the separative power of GC optimised on all parameters which can be controlled by a designer. The separative power can be presented as a function of a lot of parameters δU (V, L, T, a, α 1 , α 2 , ...), where a is the rotor diameter and the series of parameters α i includes, for example, pressure at the wall of the rotor, feed flow F , feed cut θ, variation of temperature along the rotor δT and many others. Optimization of the gas centrifuge means a search for the maximum of this function at the variation of all the parameters α i . Such a search is performed for every series of V, L, T, a. Therefore, the separative power of the optimised GC depends only on the limited set of the parameters V, L, T, a. Such a formulation of the problem carries additional difficulties in the solution of the problem because it is necessary not only to calculate the separative power of the GC, but additionally to optimise (to find maximal value) in relation to all possible parameters at fixed V , L, a and T . At the beginning of 1960's an Onsager group from US developed a theory called the pancake approximation [2,3]. This approach gave the following equation for the optimised separative power δU = (0.038V − 11.5)L, kg · SW U/yr. In contrast to eq. (1) where the separative power increases as V 4 , in eq.( 2) the separative power grows linearly with V . This difference is crucially important for the gas centrifuge designers because equations (1) and (2) give essentially different predictions for the separative power at the growth of the rotor velocity. Experimental data, collected in Russia [4] gave the following empirical equation δU = 12L V 700 m/s 2 2a 12 cm 0.4 , kg · SW U/yr,(3) where L is measured in meters. Recently this result has been well confirmed by more extended experimental data [5]. After that, a new equation defining the separative power of GC has been proposed in [6] δU = V 2 L 33000 e E , kg · SW U/yr, where V is measured in m/s, L is the rotor length in meters, e E is a dimensionless experimental efficiency which take into account specific features of every centrifuge and varies in the limit 0.4 -1.2. This equation correctly reproduces the empirical law (3). However, since this equation has been obtained in the pancake model which does not take into account important features of the real flow in the GC the question about dependence of the optimised separative power of the GC on the parameters remains an open problem up to now. Solution of this problem is important from practical point of view indeed. Simple estimates show that the maximal separative power defined by equation (1) is 4-5 times higher than the optimal separative power (3) determined experimentally at V = 700 m/s and 2a = 12 cm. This remarkable difference is due to the different dependence of the separative power on V . Our final objective is to answer on a few fundamental questions. What are the physical reasons for V 2 dependence in (3)? What factors limit the growth of δU with V ? Is it possible to dispose these factors and to increase the separative power of the gas centrifuges a few times at the same velocity and length of the rotor? In other words, is it possible to design a gas centrifuge a few times more efficient compared to the existing ones? Recently, one more step in understanding of the dependence of the separative power on the parameters has been done in [7]. A simplified concurrent GC has been considered in this work. The optimised separative power of such a GC is defined by the equation δU = 12.7 V 700 m/s 2 300 K T L, kg · SWU/year(5) which well agrees with the empirical equation including the numerical coefficient. At first glance this result solves the problem. However, eq. (5) has been obtained for the simplified concurrent centrifuge while the centrifuges used in industry are countercurrent. An axial countercurrent circulation of gas is excited in these centrifuges specially to increase their efficiency. Therefore, it is not evident that the separative power of the countercurrent centrifuges can be described by the same equation as the concurrent centrifuges. The problem of the separative power of the concurrent centrifuges has been solved analytically. In the case of countercurrent centrifuges it is necessary to perform huge amount of computational work on numerical simulation and optimization of the GC. In this work for the first time we present results of calculations of the optimised separative power of the countercurrent GC as a function of velocity. The paper is organised as follows. In the second section, we present the scheme of the countercurrent centrifuge, basic equations and assumptions. In sec. 3 the solution is described in details. In sec. 4 the optimised separative power is calculated and finally, we discuss the solution in last sec.5. 2-D model of the Iguassu gas centrifuge The model Iguassu centrifuge [8] is widely explored for the numerical simulations. The modelling of the flow and separation is performed in the rotating frame system. The diameter of the rotor of the GC is d = 0.12 m. The length of the rotor L = 1 m. The modelling is performed in axisymmetric approximation. The affect of the waste scoop has been modelled as a source of mass, momentum and energy distributed over a local toroidal region. The computational domain consists of the working chamber and waste chamber located between two coaxial cylinders as it is shown in fig. 1. The outer cylinder corresponds to the rotor wall. The inner cylinder is the artificial wall introduced into the model to avoid the simulation in the rarefied gas where the hydrodynamical approximation is not valid [10]. The product chamber located below the working chamber was not included into the computational domain. Its influence was modelled by the pressure imposed at the rotor wall near the lower baffle of the working chamber. The product flow has been specified as a fraction θ of the feed flow. A linear temperature profile has been specified at the wall of the rotor. The minimal temperature T 0 has been specified above the temperature of sublimation of the working gas. At every step of the optimization procedure of the GC the temperature of the gas has been calculated at the pressure exceeding on 15% the current pressure at the wall. This temperature has been taken as T 0 . The sublimation of the gas is avoided due to this procedure. Internal boundary of the computational domain corresponds to the Knudsen zone were the gas pressure is of the order of 1 Pa. The feed flow has been specified at the surface of this boundary. Numerical solution Basic equations and numerical methods The solution of the problem of the optimised separation power on the parameters consists on two parts. First we have to solve full system of hydrodynamic equations defining the flow of the working gas with molar mass M and rotating with the angular velocity ω. The system of equations defining the dynamics of the gas in the rotating frame system is as follows [11]: where r i are the components of the cylindrical radius, c p and c v are the specific heat capacity for constant pressure and volume, P -pressure, ρdensity, T -temperature and v i are the components of the velocity, µ dynamic viscosity and χ is the thermal conductivity. ∂ρ ∂t + ∂ρv k ∂x k = 0,(6)ρ ∂v i ∂t +ρ v k ∂v i ∂x k − ω 2 r i − 2ε iml ω m v l = − ∂P ∂x i + ∂ ∂x k µ ∂v i ∂x k + 1 3 ∂v k ∂x i . (7) ∂ρ (c v T + v 2 /2 − ω 2 r 2 /2) ∂t = ∂ ∂x k (ρv k c p T + v 2 /2 − ω 2 r 2 /2 + +χ ∂T ∂x k + v i ( ∂v i ∂x k + ∂v k ∂x i − 1 3 δ ik ∂v l ∂x l ))(8) These equations are supplemented by the equation for concentration C of the U F 6 with light isotope of uranium 235 U . This equation has a form ∂ρC ∂t − ∂ ∂x i ρv i C − ρD ∂C ∂x i + ∆M M C(1 − C) ∂ ln p ∂x i = 0.(9) A specialised numerical codes has been developed in National research nuclear university (MEPHI) for solution of the full system of equations. The code is based on Godunov numerical scheme of the second order [12]. The numerical solution has been obtained on the computer cluster of the university. Verification of the numerical code The codes for numerical simulation of the hydrodynamical flows and diffusion of the light isotope in the gas centrifuge need special procedures of verification. The verification of our numerical code have been performed using the methodology developed in [13,14]. Optimization of the separative power of the gas centrifuge. The second step to be performed consists in optimizing the GC over the whole range of parameters. The separative power δU is calculated according to the well known equation proposed by Peierls [1] δU = F p V (C p ) + F w V (C w ) − F V (C f ),(10) where F p is the product flow of the gas enriched by uranium 235 U , C p is the concentration of the light component in the product flow, F w is the waste flow, C w is the concentration of the light component in the waste flow and F , C f are the feed flow and concentrations in the feed flow. The target of the optimization procedure was to reach maximum of δU at variation of the parameters of the gas centrifuge with some limitations on these parameters. The variable parameters were the pressure of the gas at the wall of the rotor p w , temperature of the lower cap end of the rotor T 0 , temperature variation between the upper and lower cap ends of the rotor ∆T , feed flow F , the breaking capacity of the waste scoop W and the radius r b of the product baffle. In addition we varied the radius R low of the hole of the lower baffle for a total 7 parameters. Finally, the optimised separative power depends only on the length L, velocity V , radius of the rotor a and the feed cut θ. U F 6 gas physical limitations must be considered when optimizing the different parameters. According to the equation of state, the gas goes through a sublimation point and forms a solid phase at the specified pressure and temperature. The sublimation is defined by the pressure and temperature. At a specified pressure there is a minimal temperature T sub (p). Above this temperature U F 6 exists in the gas phase. Thus we imposed the limitation that p < 0.85 · P sub (T ), where P sub (T ) is the pressure of sublimation at temperature T . The BOBYQA [15] direct search method within the NLopt package [16] has been used for the optimization of the centrifuge. Optimization provides us the values of pressure at the wall of the rotor p, feed flow F , temperature drop along the wall of the rotor ∆T and power of the braking of the gas by the scoop W at which the maximal value of the separative power of the GC is achieved. All the calculations were performed in parallel using computer cluster. Single evaluation of the GC separative power uses 8 processor cores and takes from 10 to 50 minutes depending on the parameters of the GC. It is nessesary to perform from 10 to 100 evaluations in order to the BOBYQA optimization method converges. Therefore, one optimizations takes from several hours to several days to converge. Optimization calculations with different rotor velocities were done in parallel. Results Dependence of the optimised separative power on the rotor velocity Dependence of the optimised separative power on the velocity of the Iguassu centrifuge is shown in fig. 2. The calculated dependence (crosses) is well described by the power law function with the index α = 2.62 ± 0.04. This dependence does not agree with the experimental measurements obtained for Russian centrifuges which gives the power law index α = 2. Analysis shows that the difference between our calculations and the experiment arises because of limitations on pressure. We did not impose any limitations on pressure at the walls of the rotor. In real experiments the pressure is always limited from above. Therefore, we have performed calculations of the dependence of the optimised separative power on velocity for 3 different fixed pressures in the rotor. These dependences are described by the power law as well. However, the power law index is different. At low pressure the power law index α = 2.20 ± 0.06 for p = 50 mmHg, α = 2.04 ± 0.04 for p = 200 mmHg and α = 1.89 ± 0.06 for p = 360 mmHg. This means that if the pressure is kept constant at some P limit the index appears very close to the value obtained in the experiments with Russian GCs. In general the dependence of the optimal separation power on velocity can be presented as follows δU (V ) = V 2.6 if p opt (V ) < p limit , V 2 if p opt (V ) > p limit ,(11) where p opt (V ) is the optimal pressure. 5.2. Dependence of the optimal pressure p opt on the velocity of the rotor One of the most important parameters of the optimal centrifuge is the pressure at the wall of the rotor. To our knowledge, the pressure in the optimal regime of exploration of the GC has been defined only in the work by [3]. According to this work, the optimal pressure depends on the velocity of the rotor as V 5 . The numerical simulation gives us unique opportunity to obtain the dependence of the optimal pressure on the rotor velocity. The results of calculations of this dependence are presented in fig. 3. This dependence can be approximated by a power law as well. The power law index equals to 3.54 ± 0.08. This index remarkably differs from 5 defined in the work [3]. Conclusion In the work we have performed numerical simulation of the hydrodynamics and diffusion of isotopes in 2-D model of the Iguassu gas centrifuge. Special procedure of optimization of the gas centrifuge has been performed with the objective to reach maximal separative power at a specified velocity, Figure 3: Dependence of the optimal pressure in the gas centrifuge on the velocity of the rotor length and diameter of the rotor. The calculations show that the separative power depends on the rotor velocity as V 2.6 . This dependence differs from the dependence δU ∼ V 2 determined experimentally for the Russian centrifuges and differs from the dependence determined for the concurrent centrifuges [7]. This difference is explained by the fact that no limitations on the optimal pressure in the GC was imposed as it is done conventionally in the real centrifuges. If the pressure in the centrifuge is kept constant, the dependence of the separative power on the rotor velocity becomes close to the experimentally determined. Acknowledgements The work has been performed under the support of the Ministry of education and science of Russia, grant no. 3.726.2014/K. The computer simulations have been partially performed on the computer cluster of National research nuclear university(MEPhI), Basov farm. This work was also performed within the framework of the Center "Physics of nonequilibrium atomic systems and composites" supported by MEPhI Academic Excellence Project (contracts No. 02.a03.21.0005, 27.08.2013). Figure 1 : 1Scheme of the computational domain Figure 2 : 2Dependence of the optimised separative power on the velocity of rotor at the conventional procedure of optimization (blue line). Dependence of the optimised separative power at the fixed pressure are shown in green (p = 50 mmHg), red (p = 200 mmHg) and magenta (p = 360 mmHg) lines. The theory of isotope separation as applied to the large-scale production of U 235. K Cohen, G.M.MurphyMcGraw-Hill; New YorkK.Cohen. The theory of isotope separation as applied to the large-scale production of U 235 . Ed. by G.M.Murphy, (McGraw-Hill: New York. 1951) Onsagers pancake approximation for the uid dynamics of a gas centrifuge. H G Wood, J B Morton, J. Fluid Mech. V. 101. P. 1.Wood, H.G.; Morton, J.B. Onsagers pancake approximation for the uid dynamics of a gas centrifuge. J. Fluid Mech., V. 101. P. 1. (1980) Optimization studies for gas centrifuges. F Doneddu, P Roblin, H G Wood, Sep. Sci. Technol. 35812071221Doneddu, F.; Roblin, P.; Wood, H.G. Optimization studies for gas cen- trifuges. Sep. Sci. Technol., 35 (8): 12071221.(2000) Gas centrifuges. A P Senchenkov, S A Senchenkov, V D Borisevich, ISOTOPES. Properties. Production. Application. Ed., Baranov, V. Yu., Fizmatlit: Moscow1168208in RussianSenchenkov, A.P.; Senchenkov, S.A.; Borisevich, V.D. Gas centrifuges. In: ISOTOPES. Properties. Production. Application. Ed., Baranov, V. Yu., Fizmatlit: Moscow, Vol. 1: 168208 (in Russian).(2005) Comparison of the circulation efficiency in gas centrifuges with different geometric and speed characteristics for uranium enrichment. V D Borisevich, O N Godisov, D V Yatsenko, Atomic energy. 116Borisevich V.D., Godisov O.N., Yatsenko D.V. Comparison of the cir- culation efficiency in gas centrifuges with different geometric and speed characteristics for uranium enrichment. Atomic energy, 116, 5, 363 -371 (2015) Gas centrifuge theory and development: A review of U. 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Takeda, Source-Sink Flow in a Gas Cen- trifuge, Journal of Fluid Mechanics 69(1), 197208(1975) Effect of vacuum core boundary conditions on separation in the gas centrifuge. E Von Halle, H G Wood, R A Lowry, Nuclear Technology. 626325334E. Von Halle, H.G. Wood, R.A. Lowry, Effect of vacuum core boundary conditions on separation in the gas centrifuge, Nuclear Technology 62 (6), 325334(1983) . L D Landau, E M Lifshitz, Fluid Mechanics, Fiz-matlit. Landau L. D., Lifshitz E. M., Fluid Mechanics (Butterworth- Heinemann, Oxford, 1987; Fiz-matlit, Moscow,2001) Numerical solution of multidimensional problems of gas dynamics, in russian. S K Godunov, NaukaS.K.Godunov, Numerical solution of multidimensional problems of gas dynamics, in russian, 1976, Nauka. Verification of numerical codes for modeling of the flow and isotope separation in gas centrifuges. S V Bogovalov, V D Borisevich, V D Borman, Computers and Fluids. 86S.V. Bogovalov, V.D. Borisevich, V.D. Borman, et al. , Verification of numerical codes for modeling of the flow and isotope separation in gas centrifuges, Computers and Fluids 86, 177-184 (2013) Verification of software codes for simulation of unsteady flows in a gas centrifuge. V I Abramov, S V Bogovalov, V D Borisevich, Computational Mathematics and Mathematical Physics. 536V.I. Abramov, S.V. Bogovalov, V.D. Borisevich, et al., Verification of software codes for simulation of unsteady flows in a gas centrifuge, Com- putational Mathematics and Mathematical Physics 53(6), 789-797(2013) The BOBYQA algorithm for bound constrained optimization without derivatives. M J D Powell, NA2009/06Department of Applied Mathematics and Theoretical Physics. technical reportM. J. D. Powell, The BOBYQA algorithm for bound constrained opti- mization without derivatives, Department of Applied Mathematics and Theoretical Physics, Cambridge England, technical report NA2009/06 (2009). The NLopt nonlinear-optimization package. G Steven, Johnson, Steven G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt.	[]
[ "Quantum Tunneling and Trace Anomaly", "Quantum Tunneling and Trace Anomaly" ]	[ "Rabin Banerjee \nS. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India\n", "Bibhas Ranjan \nS. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India\n", "Majhi \nS. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India\n" ]	[ "S. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India", "S. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India", "S. N\nBose National Centre for Basic Sciences\nJD Block\nSector III, Salt LakeKolkata-700098India" ]	[]	We compute the corrections, using the tunneling formalisim based on a quantum WKB approach, to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. The results are related to the trace anomaly and are shown to be equivalent to findings inferred from Hawking's original calculation based on path integrals using zeta function regularization. Finally, exploiting the corrected temperature and periodicity arguments we also find the modification to the original Schwarzschild metric which captures the effect of quantum corrections. *	10.1016/j.physletb.2009.03.019	[ "https://arxiv.org/pdf/0808.3688v3.pdf" ]	16,026,777	0808.3688	b9bfdf7e247f7e2d8accdfc216a21726a4762750	Quantum Tunneling and Trace Anomaly 13 Mar 2009 March 13, 2009 Rabin Banerjee S. N Bose National Centre for Basic Sciences JD Block Sector III, Salt LakeKolkata-700098India Bibhas Ranjan S. N Bose National Centre for Basic Sciences JD Block Sector III, Salt LakeKolkata-700098India Majhi S. N Bose National Centre for Basic Sciences JD Block Sector III, Salt LakeKolkata-700098India Quantum Tunneling and Trace Anomaly 13 Mar 2009 March 13, 2009 We compute the corrections, using the tunneling formalisim based on a quantum WKB approach, to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. The results are related to the trace anomaly and are shown to be equivalent to findings inferred from Hawking's original calculation based on path integrals using zeta function regularization. Finally, exploiting the corrected temperature and periodicity arguments we also find the modification to the original Schwarzschild metric which captures the effect of quantum corrections. * Introduction Hawking [1] gave the idea that black holes are not perfectly black, rather they radiate energy continuously. Since his original calculation, several derivations of Hawking effect were subsequently presented in the literature [2,3,4,6,7,8,9,10]. A particularly intuitive method is the visualisation of the source of radiation as tunneling of s-waves [5,11,12,13,14]. Although the results in the tunneling formulation agree with Hawking's original calculation [1], their connection is obscure. Also, most of the computations are only confined to the semiclassical approximation 1 . It is not obvious how to go beyond this approximation and whether the results still continue to agree. Both issues are addressed here. In this paper we explicitly compute the corrections to the semiclassical expressions for the thermodynamic entities of the Schwarzschild black hole in the tunneling approach. An exact equivalence between our approach and Hawking's [15] original calculation based on path integrals using zeta function regularization will be established. We briefly summarise our methodology. To begin with, we will first give a new approach to get the quantum Hamilton-Jacobi equation from the definition of the quantum canonical momentum. Then following the method suggested in [16], the corrected form of the one particle action is computed for the Schwarzschild black hole. Exploiting the "detailed balance" condition [11,14], the modified form of the Hawking temperature is obtained and from there, using the Gibbs form of the first law of thermodynamics, the famous logarithmic and inverse powers of area corrections to the Bekenstein-Hawking area law, shown earlier in [17,18,19], are reproduced. We find, using a constant scale transformation to the metric, that the coefficient of the logarithmic correction is related to the trace anomaly. Precisely the same result is inferred from Hawking's computations involving the one loop correction to the gravitational action [15] due to fluctuations in presence of scalar fields in the spacetime. Finally, exploiting the corrected temperature and periodicity arguments [20], we derive the modifications to the original Schwarzschild black hole metric. Confining to the O(h) contribution of this modified metric we show our results are similar to those given in the existing literature [21,22] obtained by including the one loop back reaction effect. In section 2 a quantum version of the Hamilton-Jacobi formalism is developed from which the corrected forms of the thermodynamic entities are obtained. In the leading (O(h)) approximation the corrections involve logarithmic terms. Higher order corrections contain inverse powers of the black hole mass or area. All these corrections contain undetermined constants as their normalization. In section 3, we identify the normalization of the leading (logarithmic) correction with the trace anomaly. An explicit value of this normalization is thereby obtained. We also demonstrate the equivalence of our findings with Hawking's original analysis [15]. Section 4 reveals, using periodicity arguments, a correction to the original Schwarzschild metric as a consequence of the modified Hawking temperature. Our concluding remarks are given in section 5. 2 Corrected forms of Hawking temperature and entropy from quantum Hamilton-Jacobi equation The tunneling method involves calculating the imaginary part of the action for the (classically forbidden) process of s-wave emission across the horizon which in turn is related to the Boltzmann factor for emission at the Hawking temperature [5,11,13]. We consider a massless scalar particle in a general class of static, spherically symmetric spacetime of the form ds 2 = −f (r)dt 2 + dr 2 g(r) + r 2 dΩ 2(1) satisfying the massless Klein-Gordon equation which, in the operator form, is written as g µνp µpν φ = 0 (2) where φ is some massless scalar field. Now considering the eigenvalue equationp µ φ = −ih∇ µ φ = p µ φ with the classical canonical momentum p µ = −∂ µ S, (2) is expanded as g tt ∂ 2 t S + g rr ∂ 2 r S − g tt Γ σ tt (∂ σ S) − g rr Γ σ rr (∂ σ S) − ī h g tt (∂ t S) 2 − ī h g rr (∂ r S) 2 = 0(3) where '∇ µ ' is the covariant derivative and S(r, t) is the one particle action for the scalar particle. But for the metric (1) we have the following non-vanishing inverse metric coefficients and connection terms g tt = − 1 f ; g rr = g; Γ r tt = f ′ g 2 ; Γ r rr = − g ′ 2g .(4) Substituting these values in (3) we obtain i f (r)g(r) ∂S ∂t 2 − i f (r)g(r) ∂S ∂r 2 −h f (r)g(r) ∂ 2 S ∂t 2 +h f (r)g(r) ∂ 2 S ∂r 2 +h 2 ∂f (r) ∂r g(r) f (r) + ∂g(r) ∂r f (r) g(r) ∂S ∂r = 0(5) which is the desired quantum Hamilton-Jacobi equation. Settingh = 0 yields the usual classical Hamilton-Jacobi equation. The above relation can also be obtained by a direct substitution of the ansatz, φ(r, t) = exp − ī h S(r, t)(6) in (2) [16]. In this sense the quantum Hamilton-Jacobi equation (5) implies the quantum Klein-Gordon equation (2). Now expanding S(r, t) in powers ofh, we find, S(r, t) = S 0 (r, t) + ih i S i (r, t).(7) where i = 1, 2, 3, ....... In this expansion the terms from O(h) onwards are treated as quantum corrections over the semiclassical value S 0 . Substituting (7) in (5) and equating terms involving identical powers inh, we obtain the same set of equations for all S's [16], ∂S a ∂t = ± f (r)g(r) ∂S a ∂r ; a = 0, 1, 2, 3, .....(8) To obtain a solution for S(r, t) we first solve for S 0 (r, t). Now since the metric (1) is stationary, it has timelike Killing vectors and therefore we consider a solution for S 0 (r, t) of the form, S 0 (r, t) = ωt +S 0 (r)(9) where ω is interpreted as the conserved quantity associated with the timelike Killing vector. Substituting this in the first equation of the set (8) (i.e. for a = 0) a solution forS 0 is obtained. Inserting this back in (9) yields, S 0 (r, t) = ωt ± ω dr f (r)g(r) . The +(−) sign is for ingoing (outgoing) particle and the limits of the integration are chosen such that the particle goes through the event horizon r = r H . Likewise, the other pieces S i (r, t) appearing in (8) are also functions of (t ± r * ) where r * = dr √ f g . To see we take, following (9), the following ansatz, S i (r, t) = ω i t +S i (r)(11) Inserting this in (8) yields,S i (r) = ±ω i r * = ±ω i dr √ f g(12) Combining all the terms we obtain, S(r, t) = 1 + ih i γ i (t ± r * )ω(13) where γ i = ω i ω . From the expression within the first parentheses of the above equation it is found that γ i must have dimension ofh −i . Again in units of G = c = k B = 1 the Planck constanth is of the order of square of the Planck Mass M P and so from dimensional analysis γ i must have the dimension of M −2i where M is the mass of the black hole. Specifically, for Schwarzschild type black holes having mass as the only macroscopic parameter, these considerations show that (13) has the form, S(r, t) = 1 + i β ih i M 2i (t ± r * )ω(14) where β i 's are dimensionless constant parameters. Now keeping in mind that the ingoing probability P (in) = \|φ (in) \| 2 has to be unity in the classical limit (i.e.h → 0), instead of zero or infinity [16], we obtain, using (6) and (14), Im t = −Im dr f (r)g(r) . Therefore the outgoing probability is P out = \|φ (out) \| 2 = exp − 4 h ω 1 + i β ih i M 2i Im dr f (r)g(r) . Then use of the principle of "detailed balance" [11,14] P out = exp − ωβ (corr.) P in = exp − ωβ (corr.) yields the corrected inverse Hawking temperature [16] β (corr. ) = β H 1 + i β ih i M 2i ,(17) where T H = β −1 H =h 8πM is the standard semiclassical Hawking temperature. The non leading terms are the corrections to temperature due to quantum effect. From the first law of thermodynamics dS bh = β (corr.) dM it is easy to find the corrected form of the Bekenstein-Hawking entropy which in this case is given by [16] S bh = 4πM 2 h + 8πβ 1 ln M − 4πhβ 2 M 2 + higher order terms inh.(18) Expressing the above equation in terms of the usual semiclassical area A = 16πM 2 yields, S bh = A 4h + 4πβ 1 ln A − 64π 2h β 2 A + higher order terms inh(19) which is the corrected form of the Bekenstein-Hawking area law. The first term in (18) or (19) is the usual semiclassical result while the second term is the logarithmic correction [13,16,17,18,19] which in this case comes fromh order correction to the action S(r, t) and so on. In the next section we will discuss a method of fixing the coefficient β 1 . Connection with trace anomaly Naively one would expect T µ µ , the trace of the energy momentum tensor, to vanish for a zero rest mass field. However this is not the case since it is not possible to simultaneously preserve conformal and diffeomorphism symmetries at the quantum level. As the latter symmetry is usually retained there is, in general, a violation of the conformal invariance which is manifested by a non vanishing trace of the energy-momentum tensor. We now show that the coefficient β 1 appearing in (18) is related to this trace anomaly. We begin by studying the behaviour of the action, upto orderh, under an infinitesimal constant scale transformation, parametrised by k, of the metric coefficients, g µν = kg µν ≃ (1 + δk)g µν .(20) Under this the metric coefficients of (1) change asf = kf,ḡ = k −1 g. Also, in order to preserve the scale invariance of the Klein-Gordon equation (2) ∂ µ ( √ −gg µν ∂ ν )φ = 0, φ should transform asφ = k −1 φ. On the other hand, φ has the dimension of mass and since in our case the only mass parameter is the black hole mass M , the infinitesimal change of it is given by, M = k −1 M ≃ (1 − δk)M.(21) A similar result was also obtained by Hawking [15] from other arguments. Now the form of the imaginary part of S 0 (r, t) for the outgoing particle, derived from (10) using (15) is given by, ImS (out) 0 = −2ωIm dr f (r)g(r) (22) where ω gets identified with the energy (i.e. mass M ) of a stable black hole [13]. Therefore ω also transforms like (21) under (20). Considering only theh order term in (14) and using (21) we obtain, under the scale transformation, ImS (out) 1 = β 1 M 2 ImS (out) 0 = − 2β 1 M 2ω Im dr fḡ ≃ − 2β 1 M 2 (1 − δk) ωIm dr √ f g ≃ β 1 M 2 (1 + δk)ImS (out) 0 .(23) Therefore, δImS (out) 1 = ImS 1 (out) − ImS (out) 1 ≃ δk β 1 M 2 ImS (out) 0(24) leading to, δImS (out) 1 δk = β 1 M 2 ImS (out) 0 .(25) Now use of the definition of the energy-momentum tensor and (25) yields, Im d 4 x √ −gT µ µ = 2δImS (out) 1 δk = 2β 1 M 2 ImS (out) 0 .(26) Thus, in the presence of a trace anomaly, the action is not invariant under the scale transformation. This relation connects the coefficient β 1 with the trace anomaly. Since for the Schwarzschild black hole f (r) = g(r) = 1 − 2M r , from (22) we obtain ImS (out) 0 = −4πωM . Substituting this in (26) β 1 can be expressed as β 1 = − M 8πω Im d 4 x √ −gT µ µ .(27) For a stable black hole, as mentioned below (22), ω = M and the above equation simplifies to, β 1 = − 1 8π Im d 4 x √ −gT µ µ .(28) Using this in (18) the leading correction to the semiclassical contribution is obtained, S bh = 4πM 2 h − Im d 4 x √ −gT µ µ ln M.(29) We now show that the above result is exactly equivalent to Hawking's [15] original calculation by path integral approach based on zeta function regularization where he has modified the path integral by including the effect of fluctuations due to the presence of scalar field in the black hole spacetime. The path integral has been calculated under the saddle point approximation leading to the following expression for the logarithm of the partition function, ln Z = − 4πM 2 h + ζ(0) ln M (30) where the zero of the zeta function is given by [15] ζ(0) = −Im d 4 x √ −gT µ µ .(31) The first term in (30) is the usual semiclassical contribution. The second term is a one loop effect coming from the fluctuation of the scalar field. From this expression the corrected entropy can be inferred, S bh = ln Z + β H M = 4πM 2 h − Im d 4 x √ −gT µ µ ln M(32) which reproduces (29). As before, β H is the semiclassical result for the inverse Hawking temperature β H = 8πM h . To obtain an explicit value for β 1 (28) it is necessary to calculate the trace anomaly. For a scalar background this is given by [23], T µ µ = 1 2880π 2 [R µναλ R µναλ − R µν R µν + ∇ µ ∇ µ R].(33) Inserting this in (28) yields, β 1 = − 1 8π 1 2880π 2 Im ∞ r=2m π θ=0 2π φ=0 −8iπM t=0 48M 2 r 6 r 2 sinθdrdθdφdt = 1 360π .(34) As a check we recall that the general form for β 1 is given by [17] β 1 = − 1 360π − N 0 − 7 4 N 1 2 + 13N 1 + 233 4 N 3 2 − 212N 2(35) where 'N s ' denotes the number of fields with spin 's'. The result (34) is reproduced from (35) by setting N 0 = 1 and N 1 2 = N 1 = N 3 2 = N 2 = 0. Corrected Schwarzschild metric In this section, exploiting the corrected temperature and Hawking's periodicity arguments [20], the modification to the Schwarzschild metric will be calculated. The corrected inverse Hawking temperature is given by (17). Now if one considers this β (corr.) as the new periodicity in the euclidean time coordinate τ , then following Hawking's arguments [20] the euclidean form of the metric will be given by, ds 2 (corr.) = x 2 dτ 4M 1 + i β ih i M 2i 2 + r 2 r 2 h 2 dx 2 + r 2 dΩ 2 .(36) This metric is regular at x = 0, r = r h and τ is regarded as an angular variable with period β (corr.) . Here r h is the corrected event horizon for the black hole whose value will be derived later. Taking f (corr.) (r) as the corrected metric coefficient for dτ 2 we define a transformation of the form x = 4M 1 + i β ih i M 2i f 1 2 (corr.) (r) under which the above metric simplifies to ds 2 (corr.) = f (corr.) (r)dτ 2 + dr 2 g (corr.) (r) + r 2 dΩ 2(37) where the form of the g (corr.) (r) is g (corr.) (r) = r 2 h r 2 2 1 (2M ) 2 1 + i β ih i M 2i −2 f (corr.) f ′−2 (corr.) .(38) Now to get the exact form of these metric coefficients we will use the asymptotic limit: as r → ∞, f (corr.) (r) → 1 and g (corr.) (r) → 1. With these boundary conditions f ′ (corr.) (r) has the following general form, f ′ (corr.) (r) = r 2 h r 2 1 2M 1 + i β ih i M 2i −1 1 + ∞ n=1 C n r −n(39) where C n s are some constants. After integration f (corr.) (r) = − r 2 h 2M 1 + i β ih i M 2i −1 1 r + ∞ n=1 C n r −(n+1) n + 1 + constant.(40) The asymptotic limit determines the integration constant as 1. The constants C n will be determined by imposing the condition that when there is no quantum correction (h → 0) then this metric coefficient should reduces to its original form. This condition shows that each C n must be equal to zero. So the corrected metric coefficients are f (corr.) (r) = g (corr.) (r) = 1 − r 2 h 2M r 1 + i β ih i M 2i −1 .(41) For a static black hole the event horizon is given by g tt (r h ) = g rr (r h ) = 0. Therefore in this case the event horizon is given by r h = 2M 1 + i β ih i M 2i .(42) Finally, inserting (42) in (41) and changing τ → it lead to the corrected form of the metric (37) as, ds 2 (corr.) = − 1 − 2M r 1 + i β ih i M 2i dt 2 + dr 2 1 − 2M r 1 + i β ih i M 2i + r 2 dΩ 2 .(43) This modified metric includes all quantum corrections. Expectedly forh → 0 it reduces to the standard Schwarzschild metric. A discussion on the comparison of the above results with the earlier works [21,22] is feasible. If we confine ourself to O(h) only, with β 1 defined by (28), then equations(42) and (43)reduce to r h = 2M 1 + β 1h M 2(44) and ds 2 (corr.) = − 1 − 2M r 1 + β 1h M 2 dt 2 + dr 2 1 − 2M r 1 + β 1h M 2 + r 2 dΩ 2 .(45) These have a close resemblance with the results obtained before [21,22] by solving the semiclassical Einstein equations containing the one loop renormalized energy-momentum tensor. This tensor acts as a source of curvature (back reaction effect) modifying the metric and the horizon radius by terms proportional to the trace anomaly, analogous to (44, 45). Summary and discussions Let us summarise our findings. We gave a new approach to get the quantum Hamilton-Jacobi equation from the definition of the quantum canonical momentum. Using this equation and adopting the tunneling method [16], the corrected form of the inverse Hawking temperature was obtained. Exploiting the first law of thermodynamics, the logarithmic and inverse powers of area corrections to the Bekenstein-Hawking area law were reproduced. Arguments based on a constant scale transformation to the metric revealed that the coefficient of the logarithmic correction was proportional to the trace anomaly. The explicit value of this coefficient was computed in the case of a scalar background. An exact equivalence of our O(h) result with Hawking's [15] original result obtained by computing the one loop effective action employing zeta function regularization was discussed. Finally, the modification to the original Schwarzschild metric was derived on the basis of the corrected temperature and periodicity arguments [20]. Upto O(h) correction, this modification was similar to the existing results [21,22] obtained by including one loop back reaction effect. We observe, therefore, that the equivalence between Hawking's original calculations [15] and the tunneling formalism in the Hamilton-Jacobi approach is valid even beyond the semiclassical approximation. In the original scheme the one loop effects were computed from a zeta function regularization of the path integral obtained from fluctuations of a scalar field. Here, on the other hand, these effects were obtained from a modification, beyond the usual WKB approximation, in the one particle action corresponding to a massless scalar particle. Let us next discuss the precise connection between Hawking's analysis [15] and the tunneling formalism. We feel that the tunneling mechanism adapted by us is closest in spirit to the original computations [15], thereby explaining the equivalence of the results. The point is that formal considerations [24] imply, upto O(h), an equivalence of the path integral with the WKB ansatz φ = exp[− ī h S] adopted here. What we established was the precise equivalence of the O(h) one particle action with the path integral obtained by the zeta function regularization approach. As far as the evaluation of the one particle action was concerned, no specific regularization was necessary. For path integrals, however, meaningful expressions can only be abstracted by using an appropriate regularization. In this sense the zeta function regularization of path integrals in curved background was singled out. This view is compatible with [15] where it was shown that other regularizations-like dimensional regularization-led to ambiguous results. We would like to mention that non thermal corrections to the Hawking effect were earlier discussed in [25] following a different approach. Usually the tunneling probability takes the form Γ ∼ e −β H ω from which the Hawking temperature is identified as β −1 H . 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[ "Use and misuse of ChPT in the heavy-light systems * Use and misuse of ChPT for the heavy-light systems", "Use and misuse of ChPT in the heavy-light systems * Use and misuse of ChPT for the heavy-light systems" ]	[ "Damir Bećirević [email protected] ", "Svjetlana Fajfer [email protected] ", "Jernej Kamenik [email protected] ", "Damir Bećirević ", "\nJ. Stefan Institute\nDepartment of Physics\nLaboratoire de Physique Théorique (Bât. 210\nUniversité Paris-Sud\nCentre d', Jamova 39P.O.Box 3000F-91405Orsay, Orsay-CedexFrance\n", "\nUniversity of Ljubljana\nJadranska 191000LjubljanaSlovenia\n", "\nINFN\nLaboratori Nazionali di Frascati I00044Frascati, RegensburgItaly, Germany\n" ]	[ "J. Stefan Institute\nDepartment of Physics\nLaboratoire de Physique Théorique (Bât. 210\nUniversité Paris-Sud\nCentre d', Jamova 39P.O.Box 3000F-91405Orsay, Orsay-CedexFrance", "University of Ljubljana\nJadranska 191000LjubljanaSlovenia", "INFN\nLaboratori Nazionali di Frascati I00044Frascati, RegensburgItaly, Germany" ]	[ "The XXV International Symposium on Lattice Field Theory" ]	We discuss the range of validity of chiral perturbation theory when applied to the systems of heavy-light mesons. Having in mind the recent experimental evidence according to which the heavy-light scalar and axial states are closer to the ground states than anticipated, we revisited the prediction for the chiral behavior of the B 0 q − B 0 q mixing amplitude and examined the impact of nearness of the (1/2) + states. We conclude that the standard ChPT expressions with N F = 3 light flavours are not useful in guiding the extrapolation of hadronic quantities computed on the lattice. Instead those derived in HMChPT with N F = 2, i.e., including only the pion loops, are still adequate as long as they are applied to the pions lighter than 350 MeV, or the quarks lighter than a third of the physical strange quark mass.	10.22323/1.042.0063	[ "https://arxiv.org/pdf/0710.3496v1.pdf" ]	16,030,432	0710.3496	5c267871bc8a4bc77c4442817fe95d778e96bc11	Use and misuse of ChPT in the heavy-light systems * Use and misuse of ChPT for the heavy-light systems 18 Oct 2007 July 30-4 August 2007 Damir Bećirević [email protected] Svjetlana Fajfer [email protected] Jernej Kamenik [email protected] Damir Bećirević J. Stefan Institute Department of Physics Laboratoire de Physique Théorique (Bât. 210 Université Paris-Sud Centre d', Jamova 39P.O.Box 3000F-91405Orsay, Orsay-CedexFrance University of Ljubljana Jadranska 191000LjubljanaSlovenia INFN Laboratori Nazionali di Frascati I00044Frascati, RegensburgItaly, Germany Use and misuse of ChPT in the heavy-light systems * Use and misuse of ChPT for the heavy-light systems The XXV International Symposium on Lattice Field Theory 18 Oct 2007 July 30-4 August 2007* Supported in part by the EU Contract No. MRTN-CT-2006-035482, "Flavianet". † Speaker. We discuss the range of validity of chiral perturbation theory when applied to the systems of heavy-light mesons. Having in mind the recent experimental evidence according to which the heavy-light scalar and axial states are closer to the ground states than anticipated, we revisited the prediction for the chiral behavior of the B 0 q − B 0 q mixing amplitude and examined the impact of nearness of the (1/2) + states. We conclude that the standard ChPT expressions with N F = 3 light flavours are not useful in guiding the extrapolation of hadronic quantities computed on the lattice. Instead those derived in HMChPT with N F = 2, i.e., including only the pion loops, are still adequate as long as they are applied to the pions lighter than 350 MeV, or the quarks lighter than a third of the physical strange quark mass. We discuss the range of validity of chiral perturbation theory when applied to the systems of heavy-light mesons. Having in mind the recent experimental evidence according to which the heavy-light scalar and axial states are closer to the ground states than anticipated, we revisited the prediction for the chiral behavior of the B 0 q − B 0 q mixing amplitude and examined the impact of nearness of the (1/2) + states. We conclude that the standard ChPT expressions with N F = 3 light flavours are not useful in guiding the extrapolation of hadronic quantities computed on the lattice. Instead those derived in HMChPT with N F = 2, i.e., including only the pion loops, are still adequate as long as they are applied to the pions lighter than 350 MeV, or the quarks lighter than a third of the physical strange quark mass. The XXV International Symposium on Lattice Field Theory July 30-4 August 2007 Regensburg, Germany Introduction In this note we summarise the findings of our research presented in ref. [1]. The fact that the lattice QCD community is heavily dependent on the formulae derived in chiral perturbation theory (ChPT) to extrapolate the directly accessible results for almost any phenomenologically relevant quantity to the physical -nearly chiral-limit, requires a clear assessment of the validity of ChPT. Only then one can be able to claim "a high precision physical result" deduced from the lattice QCD simulations combined with ChPT. Before touching on the peculiarities related to the heavy-light mesons, we will make a short trip to the sector of light mesons because the first warnings of how far we should (should not) push ChPT in terms of precision requirements already show there. Light mesons The viability of chiral expansion in the theory with N F = 3 light flavours (u, d, and s) has been questionable since the very beginning of the theory. The main reason is that the strange quark mass is half-a-way between the chiral limit and Λ QCD , which might significantly lower the ChPT order parameters [ lim m u,d,s →0 ( f π , qq )], with respect to their N F = 2 counterparts, and thus spoil the perturbative nature of the chiral expansion (a number of situations in which the next to leading order term in the chiral expansion is larger than the leading one). Of course, the issue can be settled after confronting the experimental data to the ChPT formulae, but the quality of the actual Kπscattering data is still not good enough to resolve this issue (a special worry is related to the low energy constants L 4 and L 6 ). As of now, it is safe to say that the formulae derived in ChPT with N F = 3 are not sufficiently reliable to seek O(1 %) precision when extrapolating the lattice data to the physical limit. The dashed (thick) ellipses are obtained by using the old (new) experimental K e4 -data. The two sets of ellipses refer to two different analyses [2,3] while the small one, in the centre of the plot, is the 2-loop ChPT prediction (Courtesy of S.Descotes-Genon). The situation with N F = 2 is different simply because that theory has been tested. Back in 2001, such a test looked like a triumph: the experimentally measured S-wave scattering lengths of ππ-system, emerging from K e4 decay, were fully consistent with the NNLO ChPT formula. It was deduced that the the Gell-Mann-Oakes-Renner formula (GMOR), m 2 π = 2B 0 m u,d + . . ., is saturated by the leading term to more than 94% [2]. However, that situation changed since the new and improved K e4 data, collected in NA48, appeared. If one repeats exactly the same analysis as that of ref. [2] with those new data then "only" about 85% of the GMOR formula is saturated by the first term, which can be read off from the plot shown in fig. 1. In spite of the speculations that the electromagnetic and isospin breaking corrections might patch up the new conclusion we believe it is fair to summarise that ChPT with pions only (N F = 2) passed the experimental tests but the level of accuracy is still controversial. Instead, the actual situation in the N F = 3 case remains unclear. Heavy-light mesons ChPT has been extensively applied to describe the dynamics of light constituents in the heavylight mesons. In particular, heavy-meson ChPT (HMChPT) has been constructed in the static heavy quark limit [4], with only one extra parameter, g, the coupling of a pseudo-Goldstone boson (PGB) to a doublet of lowest lying heavy-light mesons [ j P ℓ = (1/2) − ], namely the pseudoscalar (J P = 0 − ) and vector (1 − ) mesons. HMChPT with N F = 2 and N F = 3 light flavours inherit the problems discussed above. An extra complication appeared after the experimenters reported on the observation of the orbitally excited states [ j P ℓ = (1/2) + ], both scalar (0 + ) and axial (1 + ) ones. The observations are made in the case of mesons with the charmed heavy quark [5], to which the confirmation came from the unquenched lattice study in the static heavy quark limit [6]. In summary the splitting between the excited and ground states is only ∆ S s ≡ m D * 0s − m D s = m D 1s − m D * s = 350 MeV, ∆ S q ≈ 430(30) MeV , (1.1) for the strange and the non-strange case respectively. Leaving aside the reasons why ∆ S s = ∆ S q , it is clear that ∆ S s,q < Λ χ , m η , m K , and thus the basic assumption that no resonances appear between zero and Λ χ is simply not correct in HMChPT. 1 In view of importance of the HMChPT guidance to extrapolating the lattice results to reach the phenomenologically relevant physical quantities, such as f B , or B 0 − B 0 mixing parameters, we revised the derivation of these expressions in HMChPT and studied the impact of ∆ S on the chiral behavior of the B 0 d − B 0 d mixing amplitude. The special attention is given to B 0 d − B 0 d , rather than B 0 s − B 0 s , because of its better potential for the new physics search [8]. Computation of f B √ m B →f q In HMChPT, in the static heavy quark limit (m Q → ∞), the pseudoscalar decay constant is of dimension 3/2, and schematically we write lim m B →∞ f B q √ m B q →f q . Its light quark dependence is expected to be described by the chiral loops shown in fig. 2, and the result readŝ f d =f 0 1 − 1 + 3g 2 4(4π f ) 2 3m 2 π log m 2 π µ 2 + 2m 2 K log m 2 K µ 2 + 1 3 m 2 η log m 2 η µ 2 + c.t. ,(2.1) wheref 0 is the heavy-light meson decay constant in the chiral limit, g is the coupling of the heavylight mesons doublet to a PGB (also in the chiral limit) are the new parameters are the two constants coming from the weak current and from the lagrangian. "c.t." stands for the local counterterms the µ-dependence of which cancels against the one present in the chiral logarithms. A tacit assumption in deriving the above formula is that there is a clear separation by which the chiral logarithms describe the long distance dynamics whereas the local counterterms encode the information on short distance physics. The lattice results obtained at larger m q > m d are expected to provide a viable method to fix those counterterms, i.e., to compute the associated low energy constants. An equivalent statement is that HMChPT can be used to guide the extrapolation off q computed for m q > m d to the physical limit. 0− 0− 1− 0− Specifying the separation scale means distinguishing the particles which can propagate in the chiral loops from those which cannot. In eq. (2.1) that scale is evidently assumed to be Λ χ m η . 2 But the fact that ∆ S s,q m K,η , and thus also ∆ S s,q < Λ χ , means that one must include the propagation of the scalars in the loops as well. The lagrangian and the weak current that include the effect of (1/2) + states are given in ref. [1]. The only two new diagrams that contribute tof d are those shown in fig. 3 and there are only two new couplings that appear in the new -extended to include scalarexpression, f q =f 0 1 + ∑ i t i qa t i † aq 2(4π f ) 2 3g 2 lim x→0 d dx [xJ 1 (m 2 i , x)] − I 1 (m 2 i ) − h 2 J 1 (m 2 i , ∆ S ) + J 2 (m 2 i , ∆ S ) +∆ S d d∆ S J 1 (m 2 i , ∆ S ) + J 2 (m 2 i , ∆ S ) − 2hf + 0 f 0 I 1 (m 2 i ) + I 2 (m 2 i , ∆ S ) + c.t. ,(2.2) are h, the coupling of the PGB to one (1/2) − and one (1/2) + heavy-light meson, andf + 0 , the weak decay constant of the orbitally excited heavy-light meson in the chiral limit. If one takes the limit m i < ∆ S , which only applies to pions, there is an amusing automatic separation between the ∆ S -dependent terms from the independent ones. The result is that the HMChPT with the pion loops "survive" whereas the K-and η-loops are drowned in a whole lot of new terms of which it is significant the presence of those proportional to ∆ 2 S log(∆ 2 S /µ 2 ), thus competitive in size with the K-and η-contributions. This definitely restricts the applicability of the HMChPT formulae to the light quark masses corresponding to pions lighter than ∆ S . That can be converted to a condition that the light quark masses to which the HMChPT formulae apply should be lighter than a third of the physical strange quark mass, i.e., m q < m phys. s /3, and only to such data the HMChPT with N F = 2 flavours can be used, or f q =f 0 1 − 1 + 3g 2 2(4π f ) 2 3 2 m 2 π log m 2 π µ 2 + c f (µ)m 2 π , (2.3) with c f (µ) being the combination of low energy constants that multiply m 2 π and which, together with g andf 0 , should be fixed by fitting the lattice data to this expression. Otherwise, i.e., if we do not restrict to m q < m phys. s /3, the number of parameters in the expression rises from 3 to 13. Bag Parameters The Standard Model (SM) bag parameter in the static heavy quark limit is defined via B 0 q \| O(ν)\|B 0 q = (8/3)f q (ν) 2 B q (ν), where the ∆F = 2 operator O = (b v γ L µ q)(b v γ L µ q) ≡ γ L µ ⊗ γ µ L . Assuming its UVphysics (ν-scale dependence) is being taken care of, say in HQET perturbation theory, one can study its bosonised form in HMChPT, O = ∑ X β 1X Tr (ξ H Q ) q γ µ (1 − γ 5 )X × Tr (ξ HQ) q γ µ (1 − γ 5 )X + ct. , where X ∈ {1, γ 5 , γ ν , γ ν γ 5 , σ νρ }, and the field H(v) describes the (1/2) − -doublet, consisting of the pseudoscalar and vector heavy-light mesons. Once the scalar/axial fields are introduced in HM-ChPT the expressions become much more complicated and essentially useless for a meaningful numerical study. In ref. [1] we showed that for m q < m phys. s /3, we again can separate the contribution of the pion loop corrections and lump everything else into the finite counterterms. Before giving the explicit expressions, let us show on a specific example how that separation occurs. 3 Consider a typical (dimensionally regularised) integral that appears in these calculations and expand around the decoupling limit of the positive parity states: −2(4π) 2 v µ v ν × iµ ε d 4−ε p (2π) 4−ε p µ p ν (p 2 − m 2 π )[∆ 2 S − (vp) 2 ] = − 2(4π 2 ) ∆ 2 S v µ v ν iµ ε d 4−ε p (2π) 4−ε p µ p ν p 2 − m 2 π + O(1/∆ 2 S ) −→ − m 4 π 2∆ 2 S log m 2 π µ 2 + . . . (3.1) where the dots stand for terms of higher order in m 2 π /∆ 2 S . This expansion separates the N F = 2 chiral loops from the rest, diagram by diagram, as demonstrated in ref. [1]. At the end we arrive at the useful formulas B qf 2 q = B 0f 2 0 1 − 3g 2 + 2 (4π f ) 2 m 2 π log m 2 π µ 2 + c O 1 (µ)m 2 π , B q = B 0 1 − 1 − 3g 2 2(4π f ) 2 m 2 π log m 2 π µ 2 + c B (µ)m 2 π . Supersymmetric B 0 − B 0 mixing amplitude In SUSY not only W -boson propagates in the loop, and thus not only the left-left operator survives at low energies. In the static heavy quark limit there are in fact 4 operators, the matrix elements of which can contribute to the B 0 −B 0 mixing amplitude. In HMChPT that number further reduces to 3 because the two operators differ only by the gluon exchange which cannot alter the chiral logarithms [9]. The remaining operators are O 2,4 = 1 1 L ⊗ 1 1 L,R , where 1 1 L/R =b v (1 ∓ γ 5 )q. The bosonised forms of these operators were first correctly figured out in ref. [10], which we then confirmed by deriving them in a somewhat different way [1]. The resulting expressions for the bag parameters, defined as B 0 q \| O 2,4 \|B 0 q = (b 2,4 /3)f 2 q B 2,4 , with b 2 = −5, b 4 = 7, are of course different from the SM bag-parameter. In the theory with N F = 2 light quarks (u, d) we have B 2,4qf 2 q = B Tree 2,4f 2 0 1 − 3g 2 (3 −Y ) + 3 ± 1 2(4π f ) 2 m 2 π log m 2 π µ 2 + c O 2,4 (µ)m 2 π , ⇒ B 2,4q = B Tree 2,4 1 + 3g 2 Y ∓ 1 2(4π f ) 2 m 2 π log m 2 π µ 2 + c B 2,4 (µ)m 2 π ,(4.1) which structurally differs from eq. (3.2) in that the coefficient multiplying the logarithmic contribution here involves a yet another low-energy constant, Y , and which is also to be extracted from the fit with the lattice data. In our paper [1] we went through the same steps as above: we presented the expressions obtained in the theory with N F = 3, then included the effects of the scalars in the loops and showed that a decoupling of the pion piece with respect to the kaon, eta and the contribution of excited heavy-light mesons occurs in this case too. Therefore the useful formulas are those written in eq. (4.1). Conclusion ChPT is nowadays accepted as an effective theory of QCD at very low energies. However, it is a theory solely based on the spontaneous chiral symmetry breaking pattern, SU (N F ) L ⊗SU (N F ) R → SU (N F ) V , and tells us nothing about confinement. Its elementary objects are PGB's, and not quarks and gluons like in QCD. Therefore an appropriate matching of ChPT to QCD amounts to solving the confinement problem in QCD, as well as that of the spontaneous chiral symmetry breaking at the more fundamental level. In other words, the matching between ChPT and QCD -as of now-is unclear. This is to be contrasted to the case of heavy quark effective theory (HQET) where such a matching at any given order in the 1/m Q -expansion can be made and is in fact systematically improvable, order-by-order in α S (m Q ). Both HQET (expansion in 1/m n Q ) and ChPT (expansion in p 2n π ) share the same worry, i.e., how good is their convergence in realistic situations in which one retains only one or two subleading terms in the expansion. 4 In ChPT with N F = 3 that issue is still a subject of controversies, while in the N F = 2 case the experimental tests have been made and the results are quite encouraging although it is still unclear how to interpret this test in terms of accuracy. From the point of view of lattice QCD practitioners, the chiral behavior predicted by an effective theory for the quantities that are of high phenomenological interest, such as f B , B 0 − B 0 mixing amplitude, or the B → π form factors, is a very important guideline when extrapolating the results collected at unphysical light quark masses down to the physical d/u-quark mass. The hope is that a result of such an extrapolation has smaller systematic errors. Spurred by the recent experimental evidence indicating that the heavy-light excited states belonging to the (1/2) + doublet are much lighter than expected, we revisited the predictions based on the use of HMChPT with N F = 3 light flavours and in the static heavy quark limit. We showed that in practical applications only the formulas involving the pion loops, i.e. the theory with N F = 2, should be used. Otherwise the number of low energy constants to be fixed from the lattice data becomes prohibitively large and the contributions due to the presence of the near heavy-light excitations are comparable in size to the ones that are due to kaon-and/or η-loops. We showed on explicit examples how this decoupling occurs and provided the expressions for f B , the standard and SUSY B 0 − B 0 mixing amplitude, as well as for the couplings g, h [11], the Isgur-Wise function [12], and the scalar meson decay constant [1]. et al. (+ prelim. NA48) Updated Descotes et al. (Glob. fit + prel. NA48) Updated Descotes et al. (Ext. fit + prel. NA48) Colangelo et al. (E865 data) Descotes et al. (Global fit) Descotes et al. (Extended fit) Figure 1 : 1S-wave scattering lengths a I 0 (isospin I = 0, 2). Figure 2 : 2The chiral loop corrections tof d : the box denotes the weak current vertex, double line is the heavylight meson and the dashed line is the PGB propagator. Figure 3 : 3Inclusion of the scalars in the chiral corrections tof d . Notice that m 0 + − m 0 − = ∆ S < Λ χ . there is one new counterterm coefficient, c B (µ), and B 0 = lim m q →0 B q , both of which should be fixed from lattice data collected with the light quark m q < m phys. s /3. The fact that ∆ S s < ∆ S q is still craving for an explanation. It cannot be reconciled with results from the lattice simulations, nor with the HMChPT considerations[7]. The propagation of 1 − heavy-light state does not give any contribution because we take it to be degenerate in mass with the pseudoscalar meson, which holds true in the static heavy quark limit. This practice is pretty standard in soft collinear effective theory. It is perhaps worth mentioning that unlike in ChPT, where expansion is in p 2n π , in HMChPT it goes like p n π . Chiral behavior of the B 0 s,d − B 0 s,d mixing amplitude in the standard model and beyond. D Becirevic, S Fajfer, J Kamenik, hep-ph/0612224JHEP. 07063D. 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[ "STUDIES ON THE CHAZY EQUATIONS", "STUDIES ON THE CHAZY EQUATIONS" ]	[ "Yusuke Sasano " ]	[]	[]	In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by tanh(t). For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational Bäcklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the firstorder ordinary differential equations by birational transformations. For this system, we give two birational Bäcklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system.	null	[ "https://arxiv.org/pdf/0708.3537v16.pdf" ]	16,823,915	0708.3537	3d8f17008fefff1c05985c8cf5f4be641af72a7c	STUDIES ON THE CHAZY EQUATIONS 1 Dec 2009 Yusuke Sasano STUDIES ON THE CHAZY EQUATIONS 1 Dec 2009 In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by tanh(t). For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational Bäcklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the firstorder ordinary differential equations by birational transformations. For this system, we give two birational Bäcklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system. Introduction In 1910, Chazy studied Painlevé type equation with third order (see [6,8]) explicitly given by (1) d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 . Here u denotes unknown complex variable. It is known that this equation fails some Painlevé test [1,6,8]. Nevertheless, Chazy gave this special attention [1,2,7]. This equation has a solution [1] (2) u(t) = 4 d dt logθ 1 ′ (0, t), and special solutions (3)        u 1 (t) = a (t − t 0 ) 2 − 6 t − t 0 (a, t 0 ∈ C), u 2 (t) = − 1 t − t 0 (t 0 ∈ C). u := X v := Y u 1 (t) u 2 (t) v 1 (t) v 2 (t) v 3 (t) P 3 Classical equation (1) New equation (4) In this paper, by making a change of variables X = u 6 , Y = − du dt u , Z = − d 2 u dt 2 du dt + u 3 , the system (1) is transformed as follows: dX dt = −XY, dY dt = (2X + Y − Z)Y, dZ dt = Z 2 + 8XZ − 20XY − 20X 2 . By elimination of X, Z and setting v := Y , we obtain (4) d 3 v dt 3 = −3 v 2 − dv dt dv dt + 3 2 v 4 − v 3 − 2 d 2 v dt 2 5v 3 + 2 d 2 v dt 2 2 v 2 + 2 dv dt . We find its meromorphic solutions given by (5)                v 1 (t) = −1 t − t 0 + a 1 + a 2 (t − t 0 ) + a 1 (a 2 1 + a 2 ) 2 (t − t 0 ) 2 + · · · , v 2 (t) = −2 t − t 0 + a 1 (t − t 0 ) 2 + 2 21 a 2 1 (t − t 0 ) 5 + · · · , v 3 (t) = 1 t − t 0 , where a 1 , a 2 are free parameters and t 0 is an initial position. The solutions v 1 (t), v 2 (t) are new (see Figure 1). The solutions u 2 (t), v 3 (t) are common in each system. We also show that this system admits new special solutions (see Section 5): (6) (X, Y, Z) = (c 1 , 0, −6c 1 tanh(6(c 1 t − c 1 c 2 )) − 4c 1 ) (c 1 , c 2 ∈ C). We remark that the classical Darboux-Halphen system (see [16]) (7)                dx dt = yz − x(y + z), dy dt = xz − y(x + z), dz dt = xy − z(x + y) is equivalent to the equation (1) if one sets (8) u = −2(x + y + z), du dt = 2(xz + yz + xy), d 2 u dt 2 = −12xyz. This system is invariant under the transformation: π : (x, y, z) → (y, z, x) (π 3 = 1). It is well-known that this system has the following rational solutions:                  x 1 (t) = 1 (t − t 0 ) , y 1 (t) = 1 (t − t 0 ) , z 1 (t) = 1 (t − t 0 ) (t 0 ∈ C), and                  x 2 (t) = 1 (t − t 0 ) , y 2 (t) = 1 (t − t 0 ) , z 2 (t) = a (t − t 0 ) 2 + b (t − t 0 ) (a, b, t 0 ∈ C). Reviews of Chazy's work on III and results of further research can be found in [1,2,7]. Chazy-III, or an equivalent system of first-order equations, appears in several physics contexts, for example, self-dual Yang-Mills equations [3]. Clarkson and Olver [7] obtained III among the group-invariant reductions of the partial differential equation, w xxx = w y w xx − w x w yy , which has applications in boundary-layer theory. These authors and C. M. Cosgrove also gave a theory of higher-order equations having properties similar to III. We also study the Chazy IX equation: (9) d 3 u dt 3 = 54u 4 + 72u 2 du dt + 12 du dt 2 + δ (δ ∈ C), where u denotes unknown complex variable and δ is its constant parameter. In 2000, C. M. Cosgrove constructed the general solution of the Chazy IX equation in terms of hyperelliptic functions (see [8]). For this equation, we give two birational Bäcklund transformations: g 0 (u; δ) → ( √ 5 − 3){108u 4 + 18(5 + √ 5)u 2 u ′ + 6(3 + √ 5)(u ′ ) 2 + 3( √ 5 − 1)uu ′′ + 2δ 6( √ 5 − 1){3( √ 5 − 1)uu ′ + u ′′ } ; 7 + 3 √ 5 2 δ , g 1 (u; δ) → (− √ 5 − 3){108u 4 + 18(5 − √ 5)u 2 u ′ + 6(3 − √ 5)(u ′ ) 2 + 3(− √ 5 − 1)uu ′′ + 2δ 6(− √ 5 − 1){3(− √ 5 − 1)uu ′ + u ′′ } ; 7 − 3 √ 5 2 δ ,(10) where u ′ = du dt , u ′′ = d 2 u dt 2 . These Bäcklund transformations are new. We show that the birational transformation (11)            x = u, y = du dt + 3 2 ( √ 5 − 1)u 2 , z = d 2 u dt 2 + 3( √ 5 − 1) u du dt takes the Chazy IX equation to the system of the first-order ordinary differential equations: (12)                dx dt = − 3 2 ( √ 5 − 1)x 2 + y, dy dt = z, dz dt = 3( √ 5 + 3)y 2 + 3( √ 5 − 1)xz + δ. We make this system in the polynomial class from the viewpoint of accessible singularity and local index. For this system, we give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables (t, s) involving the Chazy IX equation: (13)                                  dx = − 3 2 ( √ 5 − 1)x 2 + y dt + −12( √ 5 + 2)x 2 y + ( √ 5 + 1) 3xz − 2y 2 − 2 3 δ ds, dy =zdt + {−6(2 √ 5 − 5)x 2 z + 12 √ 5xy 2 − ( √ 5 + 1)yz + (3 √ 5 − 5)δx}ds, dz ={3( √ 5 + 3)y 2 + 3( √ 5 − 1)xz + δ}dt + {48xyz − 24y 3 − ( √ 5 + 1)z 2 + 2( √ 5 − 3)δy}ds. We remark that when s = 0, we can obtain the Chazy IX system. We will show that this system satisfies the compatibility condition, and admits a travelling wave solution. We also study the Chazy X equation (see [8]): X.a : d 3 u dt 3 =6u 2 du dt + 3 11 (9 + 7 √ 3) du dt + u 2 2 − 1 22 (4 − 3 √ 3)α du dt + 1 44 (3 − 5 √ 3)αu 2 − 1 352 (9 + 7 √ 3)α 2 , X.b : d 3 u dt 3 =6u 2 du dt + 3 11 (9 − 7 √ 3) du dt + u 2 2 − 1 22 (4 + 3 √ 3)α du dt + 1 44 (3 + 5 √ 3)αu 2 − 1 352 (9 − 7 √ 3)α 2 ,(14) where u denotes unknown complex variable and α is its constant parameter. The full version of Class X with α = 0 was found by C. M. Cosgrove (see [8]), the parameter α being overlooked in Chazy's original work. Chazy looked for recessive terms in his Class X but missed them because of some minor calculation error. Bureau (see [20]), having derived the 13 reduced Chazy classes from first principles, did not attempt to attach recessive terms to this class. Exton (see [21]), who incorrectly concluded that Chazy Classes IX and X were unstable equations, also apparantly did not look for resessive terms. The splitting of Chazy Class X deserves a comment. Because its coefficients contain the irrationality √ 3 in an essential way, changing the sign of √ 3 yields a distinct equation. The two versions, which we denote Chazy-X.a and Chazy-X.b, are related by a Bäcklund transformation, which was also found by C. M. Cosgrove (see [8]). In 2000, C. M. Cosgrove constructed the general solution of the Chazy X equation in terms of hyperelliptic functions (see [8]). For these equations, we give two birational Bäcklund transformations from Chazy-X.a to Chazy-X.b: g 0 (u; α) → ( 3 + √ 3 1056{(3 + √ 3)uu ′ − u ′′ } {288(6 + √ 3)u 2 u ′ − 176(3 + √ 3)uu ′′ + 96(9 + 7 √ 3)u 4 + 64(−3 + 5 √ 3)(u ′ ) 2 − 8(−3 + 5 √ 3)αu 2 + 16(−4 + 3 √ 3)αu ′ − (9 + 7 √ 3)α 2 }; α), g 1 (u; α) → (4 + √ 3){8(−4 + √ 3)u 3 − 8(−4 + √ 3)uu ′ − 8(−4 + √ 3)u ′′ − (−7 + 5 √ 3)αu} 13{8u 2 + 8u ′ + ( √ 3 − 1)α} ; (−2 + √ 3)α ,(15) where u ′ = du dt , u ′′ = d 2 u dt 2 . The transformation g 1 is new. Thanks to these Bäcklund transformations, in this paper we discuss the case of Chazy-X.a equation. We show that the birational transformation (16)              x = u, y = du dt − 3 + √ 3 2 u 2 , z = d 2 u dt 2 − (3 + √ 3)u du dt takes the Chazy X.a equation to the system of the first-order ordinary differential equations: (17)                dx dt = 3 + √ 3 2 x 2 + y, dy dt = z, dz dt = 2 11 (−3 + 5 √ 3)y 2 − (3 + √ 3)xz + 1 22 (−4 + 3 √ 3)αy − 1 352 (9 + 7 √ 3)α 2 . We make this system in the polynomial class from the viewpoint of accessible singularity and local index. For this system, we give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover the Chazy X.a equation. The Chazy polynomial class In [6], Chazy attempted the complete classification of all third-order differential equations of the form: (18) d 3 u dt 3 = F t, u, du dt , d 2 u dt 2 , where F is a polynomial in u, du dt and d 2 u dt 2 and locally analytic in t, having the Painlevé property. Chazy proved that Painlevé-type differential equations of the third-order in the polynomial class must take the form: d 3 u dt 3 =Qu d 2 u dt 2 + R du dt 2 + Su 2 du dt + T u 4 + A(t) d 2 u dt 2 + B(t)u du dt + C(t) du dt + D(t)u 3 + E(t)u 2 + F (t)u + G(t),(19) where, after a suitable normalization, Q, R, S and T are certain rational or algebraic numbers, and the remaining coefficients are locally analytic functions of the complex variable t to be determined. The canonical reduced equations defining each of the Chazy classes is as follows: I : d 3 u dt 3 = −6 du dt 2 ; II : d 3 u dt 3 = −2u d 2 u dt 2 − 2 du dt 2 ; III : d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 ; IV : d 3 u dt 3 = −3u d 2 u dt 2 − 3 du dt 2 − 3u 2 du dt ; V : d 3 u dt 3 = −2u d 2 u dt 2 − 4 du dt 2 − 2u 2 du dt ; V I : d 3 u dt 3 = −u d 2 u dt 2 − 5 du dt 2 − u 2 du dt ; V II : d 3 u dt 3 = −u d 2 u dt 2 − 2 du dt 2 + 2u 2 du dt ; V III : d 3 u dt 3 = 6u 2 du dt ; IX : d 3 u dt 3 = 12 du dt 2 + 72u 2 du dt + 54u 4 ; X.a : d 3 u dt 3 = 6u 2 du dt + 3 11 (9 + 7 √ 3) du dt + u 2 2 ; X.b : d 3 u dt 3 = 6u 2 du dt + 3 11 (9 − 7 √ 3) du dt + u 2 2 ;(20) XI : d 3 u dt 3 = −2u d 2 u dt 2 − 2 du dt 2 + 24 N 2 − 1 du dt + u 2 2 ; XII : d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 − 4 N 2 − 36 6 du dt − u 2 2 ; XIII : d 3 u dt 3 = 12u du dt . In Chazy-XI, N is a positive integer not equal to 1 or a multiple of 6. In Chazy-XII, N is a positive integer not equal to 1 or 6. It is well-known that the KdV equation belongs to Chazy Class XIII, the modified KdV equation belongs to Class VIII, and the potential KdV equation (as well as the soliton equation u xxt = −6u x u t + α) belongs to Class I. Accessible singularity and local index Let us review the notion of accessible singularity. Let B be a connected open domain in C and π : W −→ B a smooth proper holomorphic map. We assume that H ⊂ W is a normal crossing divisor which is flat over B. Let us consider a rational vector fieldṽ on W satisfying the conditionṽ ∈ H 0 (W, Θ W (− log H)(H)). Fixing t 0 ∈ B and P ∈ W t 0 , we can take a local coordinate system (x 1 , . . . , x n ) of W t 0 centered at P such that H smooth can be defined by the local equation x 1 = 0. Sincẽ v ∈ H 0 (W, Θ W (− log H)(H)), we can write down the vector fieldṽ near P = (0, . . . , 0, t 0 ) as follows: ṽ = ∂ ∂t + g 1 ∂ ∂x 1 + g 2 x 1 ∂ ∂x 2 + · · · + g n x 1 ∂ ∂x n . This vector field defines the following system of differential equations (21) dx 1 dt = g 1 (x 1 , . . . , x n , t), dx 2 dt = g 2 (x 1 , . . . , x n , t) x 1 , · · · , dx n dt = g n (x 1 , . . . , x n , t) x 1 . Here g i (x 1 , . . . , x n , t), i = 1, 2, . . . , n, are holomorphic functions defined near P = (0, . . . , 0, t 0 ). Definition 3.1. With the above notation, assume that the rational vector fieldṽ on W satisfies the condition (A)ṽ ∈ H 0 (W, Θ W (− log H)(H)). We say thatṽ has an accessible singularity at P = (0, . . . , 0, t 0 ) if x 1 = 0 and g i (0, . . . , 0, t 0 ) = 0 for every i, 2 ≤ i ≤ n. If P ∈ H smooth is not an accessible singularity, all solutions of the ordinary differential equation passing through P are vertical solutions, that is, the solutions are contained in the fiber W t 0 over t = t 0 . If P ∈ H smooth is an accessible singularity, there may be a solution of (21) which passes through P and goes into the interior W − H of W. Here we review the notion of local index. Let v be an algebraic vector field with an accessible singular point − → p = (0, . . . , 0) and (x 1 , . . . , x n ) be a coordinate system in a neighborhood centered at − → p . Assume that the system associated with v near − → p can be written as d dt         x 1 x 2 . . . x n−1 x n         = 1 x 1                        a 11 0 0 . . . 0 a 21 a 22 0 . . . 0 . . . . . . . . . 0 0 a (n−1)1 a (n−1)2 . . . a (n−1)(n−1) 0 a n1 a n2 . . . a n(n−1) a nn                 x 1 x 2 . . . x n−1 x n         +         x 1 h 1 (x 1 , . . . , x n , t) h 2 (x 1 , . . . , x n , t) . . . h n−1 (x 1 , . . . , x n , t) h n (x 1 , . . . , x n , t)                        , (h i ∈ C(t)[x 1 , . . . , x n ], a ij ∈ C(t))(22) where h 1 is a polynomial which vanishes at − → p and h i , i = 2, 3, . . . , n are polynomials of order at least 2 in x 1 , x 2 , . . . , x n , We call ordered set of the eigenvalues (a 11 , a 22 , · · · , a nn ) local index at − → p . We are interested in the case with local index (23) (1, a 22 /a 11 , . . . , a nn /a 11 ) ∈ Z n . These properties suggest the possibilities that a 1 is the residue of the formal Laurent series: (24) y 1 (t) = a 11 (t − t 0 ) + b 1 + b 2 (t − t 0 ) + · · · + b n (t − t 0 ) n−1 + · · · (b i ∈ C), and the ratio (1, a 22 /a 11 , . . . , a nn /a 11 ) is resonance data of the formal Laurent series of each y i (t) (i = 2, . . . , n), where (y 1 , . . . , y n ) is original coordinate system satisfying (x 1 , . . . , x n ) = (f 1 (y 1 , . . . , y n ), . . . , f n (y 1 , . . . , y n )), f i (y 1 , . . . , y n ) ∈ C(t)(y 1 , . . . , y n ). If each component of (1, a 22 /a 11 , . . . , a nn /a 11 ) has the same sign, we may resolve the accessible singularity by blowing-up finitely many times. However, when different signs appear, we may need to both blow up and blow down. The α-test, (25) t = t 0 + αT, x i = αX i , α → 0, yields the following reduced system: d dT         X 1 X 2 . . . X n−1 X n         = 1 X 1         a 11 (t 0 ) 0 0 . . . 0 a 21 (t 0 ) a 22 (t 0 ) 0 . . . 0 . . . . . . . . . 0 0 a (n−1)1 (t 0 ) a (n−1)2 (t 0 ) . . . a (n−1)(n−1) (t 0 ) 0 a n1 (t 0 ) a n2 (t 0 ) . . . a n(n−1) (t 0 ) a nn (t 0 )                 X 1 X 2 . . . X n−1 X n         ,(26) where a ij (t 0 ) ∈ C. Fixing t = t 0 , this system is the system of the first order ordinary differential equation with constant coefficient. Let us solve this system. At first, we solve the first equation: (27) X 1 (T ) = a 11 (t 0 )T + C 1 (C 1 ∈ C). Substituting this into the second equation in (26), we can obtain the first order linear ordinary differential equation: (28) dX 2 dT = a 22 (t 0 )X 2 a 11 (t 0 )T + C 1 + a 21 (t 0 ). By variation of constant, in the case of a 11 (t 0 ) = a 22 (t 0 ) we can solve explicitly: (29) X 2 (T ) = C 2 (a 11 (t 0 )T + C 1 ) a 22 (t 0 ) a 11 (t 0 ) + a 21 (t 0 )(a 11 (t 0 )T + C 1 ) a 11 (t 0 ) − a 22 (t 0 ) (C 2 ∈ C). This solution is a single-valued solution if and only if a 22 (t 0 ) a 11 (t 0 ) ∈ Z. In the case of a 11 (t 0 ) = a 22 (t 0 ) we can solve explicitly: (30) X 2 (T ) = C 2 (a 11 (t 0 )T + C 1 ) + a 21 (t 0 )(a 11 (t 0 )T + C 1 )Log(a 11 (t 0 )T + C 1 ) a 11 (t 0 ) (C 2 ∈ C). This solution is a single-valued solution if and only if a 21 (t 0 ) = 0. Of course, a 22 (t 0 ) a 11 (t 0 ) = 1 ∈ Z. In the same way, we can obtain the solutions for each variables (X 3 , . . . , X n ). The conditions a jj (t) a 11 (t) ∈ Z, (j = 2, 3, . . . , n) are necessary condition in order to have the Painlevé property. (31) d 3 u dt 3 = u d 2 u dt 2 − 2 du dt 2 + 6u 2 du dt . Here u denotes unknown complex variable. We will show that this equation is the integrable non-Painlevé equation.                x = u, y = − du dt u + u, z = − d 2 u dt 2 du dt + 2u takes the equation (31) to the system (33)                dx dt = x 2 − xy, dy dt = y 2 − xy + xz − yz, dz dt = z 2 − 3xz − 4xy. Here x, y, z denote unknown complex variables. We will show that this system violates the condition (23). Let us take the coordinate system (p, q, r) centered at the point (p, q, r) = (0, 0, 0): p = 1 x , q = y x , r = z x . The system (33) is rewritten as follows: d dt    p q r    = 1 p         −1 0 0 0 −2 1 0 −4 −4       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =    1 0 0 0 2 √ −3 3 3+ √ −3 6 0 − 2 √ −3 3 3− √ −3 6       p q r    to arrive at d dt    X Y Z    = 1 X         −1 0 0 0 −3 + √ −3 0 0 0 −3 − √ −3       X Y Z    + · · ·      . In this case, the ratio ( −3+ √ −3 −1 , −3− √ −3 −1 ) = (3 − √ −3, 3 + √ −3) is not in Z 2 . Example 3.4. For an application of the condition (23), let us consider (34) d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 + a 6 du dt − u 2 2 (a ∈ C). Here u denotes unknown complex variable. Proposition 3.5. The birational transformation (35)                    x = u 2 du dt , y = du dt u , z = d 2 u dt 2 du dt takes the equation (34) to the system (36)                dx dt = 2xy − xz, dy dt = −y 2 + yz, dz dt = −3xy 2 + 36axy 2 − 12ax 2 y 2 + ax 3 y 2 + 2xyz − z 2 . Here x, y, z denote unknown complex variables. We will consider when this system satisfies the condition (23). Let us take the coordinate system (p, q, r) centered at the point (p, q, r) = (0, 0, 0): p = x − 3a − √ 9a 2 − a a , q = y z − 1 2 , r = 1 z . The system (36) is rewritten as follows: d dt    p q r    = 1 p         0 − 2(−3a+ √ 9a 2 −a) a 0 1−9a−3 √ 9a 2 −a 4 − a+2 √ 9a 2 −a 2a 0 0 0 − 1 2       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =      − √ a(9a−1) 4−42a+14 √ a(9a−1) a a−2 √ a(9a−1) 0 √ a(9a−1) 4−42a+14 √ a(9a−1) 2−18a 2−18a+ √ a(9a−1) 0 0 0 1         p q r    to arrive at d dt    X Y Z    = 1 Z         − 1 2 0 0 0 − √ a(9a−1) a 0 0 0 − 1 2       X Y Z    + · · ·      . In this case, the ratio − 1 2 − 1 2 , − √ a(9a−1) a − 1 2 is in Z 2 if and only if (37) 4a(9a − 1) = N 2 a 2 (N ∈ Z). This equation can be solved as follows: (38) a = − 4 N 2 − 36 (N ∈ N). This case coincides with Chazy-XII equation. In the next section, in order to consider the phase spaces for each system, let us take the compactification [z 0 : z 1 : z 2 : z 3 ] ∈ P 3 of (x, y, z) ∈ C 3 with the natural embedding (x, y, z) = (z 1 /z 0 , z 2 /z 0 , z 3 /z 0 ). Moreover, we denote the boundary divisor in P 3 by H. Extend the regular vector field on C 3 to a rational vector fieldṽ on P 3 . It is easy to see that P 3 is covered by four copies of C 3 : U 0 = C 3 ∋ (x, y, z), U j = C 3 ∋ (X j , Y j , Z i ) (j = 1, 2, 3), via the following rational transformations X 1 = 1/x, Y 1 = y/x, Z 1 = z/x, X 2 = x/y, Y 2 = 1/y, Z 2 = z/y, X 3 = x/z, Y 3 = y/z, Z 3 = 1/z.(39) Chazy-III equation Chazy-III equation is given by d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 . Here u denotes unknown complex variable.                  x = u 6 , y = − du dt u + u 6 , z = − d 2 u dt 2 du dt + u 3 takes the equation (1) to the system (41)                dx dt = x 2 − xy, dy dt = y 2 − xy + xz − yz, dz dt = z 2 + 8xz − 20xy. Here x, y, z denote unknown complex variables. Let us do the Painlevé test. To find the leading order behaviour of a singularity at t = t 0 one sets                x ∝ a 0 (t − t 0 ) , y ∝ b 0 (t − t 0 ) , z ∝ c 0 (t − t 0 ) , from which it is easily deduced that (1) (a 0 , b 0 , c 0 ) = (−1, 0, 0),(2)(a 0 , b 0 , c 0 ) = (0, −1, 0),(3)(a 0 , b 0 , c 0 ) = (0, 0, −1),(4)(a 0 , b 0 , c 0 ) = (0, −2, −1). Case (a 0 , b 0 , c 0 ) = (−1, 0, 0) In this case, we find          x(t) = −1 (t − t 0 ) (t 0 ∈ C),y(t) = 0, z(t) = 0. Case (a 0 , b 0 , c 0 ) = (0, −1, 0) In this case, we find            x(t) = a 2 (t − t 0 ) − a 2 c 1 2 (t − t 0 ) 2 + · · · , y(t) = −1 (t − t 0 ) + c 1 2 + (28a 2 + c 2 1 ) 4 (t − t 0 ) + (8a 2 + c 2 1 )c 1 8 (t − t 0 ) 2 + · · · , z(t) = c 1 + (20a 2 + c 2 1 )(t − t 0 ) + (14a 2 + c 2 1 )c 1 (t − t 0 ) 2 + · · · , where (a 2 , c 1 ) are free parameters. Case (a 0 , b 0 , c 0 ) = (0, 0, −1) In this case, we find                x(t) = a 1 − a 1 b 2 2 (t − t 0 ) 2 + · · · , y(t) = a 1 + b 2 (t − t 0 ) + 11a 1 b 2 2 (t − t 0 ) 2 + · · · , z(t) = −1 (t − t 0 ) − 4a 1 − 12a 2 1 (t − t 0 ) − 4a 1 b 2 (t − t 0 ) 2 + · · · , where (a 1 , b 2 ) are free parameters. Case (a 0 , b 0 , c 0 ) = (0, −2, −1) In this case, we find                  x(t) = a 3 (t − t 0 ) 2 − 4a 2 3 5 (t − t 0 ) 5 + · · · ,y(t) = −2 (t − t 0 ) + 17a 3 5 (t − t 0 ) 2 − 44a 2 3 175 (t − t 0 ) 5 + · · · , z(t) = −1 (t − t 0 ) + 8a 3 (t − t 0 ) 2 + 172a 2 3 35 (t − t 0 ) 5 + · · · , where a 3 is a free parameter. Proposition 4.2. The system (41) becomes again a system in the polynomial class in each coordinate system: U j = C 3 ∋ {(x j , y j , z j )}, j = 0, 1, . . . , 4 via the following birational transformations: 0) x 0 = x, y 0 = y, z 0 = z, 1) x 1 = − x − y 2x , y 1 = x, z 1 = (x − y)(x + 3y − 2z) 4x , 2) x 2 = xy, y 2 = 1 y , z 2 = z, 3) x 3 = x, y 3 = (y − x)z, z 3 = 1 z , 4) x 4 = −(x − y)xz, y 4 = − 1 (x − y)z , z 4 = 1 z .(42) Corollary 4.3. The system (41) is equivalent to the following systems: (43)                dx 1 dt = x 2 1 y 1 − z 1 , dy 1 dt = −2x 1 y 2 1 , dz 1 dt = 2y 1 (6x 3 1 y 1 + 5x 1 z 1 + 6z 1 ), and(44)               dx 2 dt = (x 2 y 2 2 − 1)x 2 z 2 , dy 2 dt = −(x 2 y 2 2 − 1)(y 2 z 2 − 1), dz 2 dt = z 2 2 + 8x 2 y 2 z 2 − 20x 2 , and(45)               dx 3 dt = −x 3 y 3 z 3 , dy 3 dt = y 3 (−20x 3 y 3 z 2 3 − 20x 2 3 z 3 + y 3 z 3 + 10x 3 ), dz 3 dt = −1 − 8x 3 z 3 + 20x 2 3 z 2 3 + 20x 3 y 3 z 3 3 , and(46)               dx 4 dt = −10x 2 4 (2x 4 y 2 4 z 4 + 2z 2 4 − y 4 ), dy 4 dt = 20x 2 4 y 3 4 z 4 + 20x 4 y 4 z 2 4 − 10x 4 y 2 4 − z 4 , dz 4 dt = −1 − 8x 4 y 4 z 4 + 20x 2 4 y 2 4 z 2 4 + 20x 4 z 3 4 . The following Lemma shows that the rational vector fieldṽ associated with the system (41) has six accessible singular points on the boundary divisor H ⊂ P 3 . Lemma 4.4. The rational vector fieldṽ has six accessible singular points: (47)                            P 1 = {(X 1 , Y 1 , Z 1 )\|X 1 = Y 1 = Z 1 = 0}, P 2 = {(X 2 , Y 2 , Z 2 )\|X 2 = Y 2 = Z 2 = 0}, P 3 = {(X 3 , Y 3 , Z 3 )\|X 3 = Y 3 = Z 3 = 0}, P 4 = {(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = 2}, P 5 = {(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = −10}, P 6 = {(X 2 , Y 2 , Z 2 )\|X 2 = Y 2 = 0, Z 2 = 1 2 }, where P 4 is multiple point of order 2. This lemma can be proven by a direct calculation. Next let us calculate its local index at each point. Singular point Type of local index P 1 (−1, 3, 2) P 2 (2, 1, 1) P 3 (1, 2, 1) P 5 (0, 12, −12) P 6 (3, 1, −2) Example 4.5. Let us take the coordinate system (p, q, r) centered at the point P 1 : p = X 1 = 1 x , q = Y 1 = y x , r = Z 1 = z x . The system (41) is rewritten as follows: d dt    p q r    = 1 p         −1 0 0 0 −2 1 0 −20 7       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =    1 0 0 0 −4 1 0 5 −1       p q r    to arrive at d dt    X Y Z    = 1 X         −1 0 0 0 3 0 0 0 2       X Y Z    + · · ·      . In this case, the local index is (−1, 3, 2). This suggests the possibilities that −1 is the residue of the formal Laurent series: (48) x(t) = −1 (t − t 0 ) + a 1 + a 2 (t − t 0 ) + · · · + a n (t − t 0 ) n−1 + · · · (a i ∈ C), and the ratio ( 3 −1 , 2 −1 ) = (−3, −2) is resonance data of the formal Laurent series of (y(t), z(t)), respectively. We see that the formal Laurent series which passes through P 1 have no free parameters. There is only one solution which passes through P 1 explicitly given by (49) x(t) = − 1 (t − t 0 ) , y(t) = 0, z(t) = 0 (t 0 ∈ C). This is a rational solution. Example 4.6. Let us take the coordinate system (p, q, r) centered at the point P 5 : p = X 1 = 1 x , q = Y 1 − 1 = y x − 1, r = Z 1 + 10 = z x + 10. The system (41) is rewritten as follows: d dt    p q r    = 1 p         0 0 0 0 12 0 0 −30 −12       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =    1 0 0 0 − 1 4 0 0 5 4 1       p q r    to arrive at d dt    X Y Z    = 1 X         0 0 0 0 12 0 0 0 −12       X Y Z    + · · ·      . In this case, the local index is (0, 12, −12). We see that the residue of the formal Laurent series: (50) x(t) = a 0 (t − t 0 ) + a 1 + a 2 (t − t 0 ) + · · · + a n (t − t 0 ) n−1 + · · · (a i ∈ C) is equal to a 0 = 0. By a direct calculation, we see that there are no solutions which pass through P 5 . Example 4.7. Let us take the coordinate system (p, q, r) centered at the point P 3 : p = X 3 = x z , q = Y 3 = y z , r = Z 3 = 1 z . The system (41) is rewritten as follows: d dt    p q r    = 1 r         −1 0 0 1 −2 0 0 0 −1       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =    1 0 0 1 1 0 0 0 1       p q r    to arrive at d dt    X Y Z    = 1 Z         −1 0 0 0 −2 0 0 0 −1       X Y Z    + · · ·      . In this case, the local index is (−1, −2, −1). This suggests the possibilities that −1 is the residue of the formal Laurent series: (51) z(t) = −1 (t − t 0 ) + c 1 + c 2 (t − t 0 ) + · · · + c n (t − t 0 ) n−1 + · · · (c i ∈ C), and the ratio ( −1 −1 , −2 −1 ) = (1, 2) is resonance data of the formal Laurent series of (x(t), y(t)), respectively. There exist meromorphic solutions with three free parameters which passes through P 3 . Example 4.8. Let us take the coordinate system (p, q, r) centered at the point P 6 : p = X 2 = x y , q = Y 2 = 1 y , r = Z 2 − 1 2 = z y − 1 2 . The system (41) is rewritten as follows: d dt    p q r    = 1 q         − 3 2 0 0 0 − 1 2 0 − 63 4 0 1       p q r    + · · ·      satisfying (22). To the above system, we make the linear transformation    X Y Z    =    63 10 0 0 0 1 0 − 63 10 0 1       p q r    to arrive at d dt    X Y Z    = 1 Y         − 3 2 0 0 0 − 1 2 0 0 0 1       X Y Z    + · · ·      . In this case, the local index is (− 3 2 , − 1 2 , 1). This suggests the possibilities that − 1 2 is the residue of the formal Laurent series: (52) y(t) = − 1 2 (t − t 0 ) + b 1 + b 2 (t − t 0 ) + · · · + b n (t − t 0 ) n−1 + · · · (b i ∈ C), and the ratio ( − 3 2 − 1 2 , 1 − 1 2 ) = (3, −2) is resonance data of the formal Laurent series of (x(t), z(t)), respectively. There exist meromorphic solutions with two free parameters which passes through P 6 . Particular solutions of the system (41) We see that the system (41) admits a particular solution x = 0. Moreover (y, z) satisfy (53)      dy dt = y 2 − yz, dz dt = z 2 . The equation dz dt = z 2 can be solved as follows: (54) z(t) = − 1 t + c 1 (c 1 ∈ C). By substituting this solution to the equation dy dt = y 2 − yz, we obtain (55) dy dt = y 2 + y t + c 1 . This equation can be solved by (56) y(t) = − 2(t + c 1 ) t 2 + 2c 1 t − 2c 2 (c 1 , c 2 ∈ C). We also see that by making a change of variables (41) is transformed as follows: X := x, Y := y − x, Z := z the system(57)                dX dt = −XY, dY dt = (2X + Y − Z)Y, dZ dt = Z 2 + 8XZ − 20XY − 20X 2 . By elimination of X, Z and setting v := Y , we obtain d 3 v dt 3 = −3 v 2 − dv dt dv dt + 3 2 v 4 − v 3 − 2 d 2 v dt 2 5v 3 + 2 d 2 v dt 2 2 v 2 + 2 dv dt . This system admits a particular solution Y = 0. Moreover (X, Z) satisfy (58)      dX dt = 0, dZ dt = Z 2 + 8XZ − 20X 2 . By substituting X = c 1 (c 1 ∈ C) to the equation dZ dt = Z 2 + 8XZ − 20X 2 , we obtain (59) dZ dt = Z 2 + 8c 1 Z − 20c 2 1 . This system can be solved by (60) Z(t) = −6c 1 tanh(6(c 1 t − c 1 c 2 )) − 4c 1 (c 2 ∈ C). The Chazy-IX equation Chazy-IX equation is given by (61) d 3 u dt 3 = 54u 4 + 72u 2 du dt + 12 du dt 2 + δ (δ ∈ C). In this paper, at first we transform the equation (61) to a system of differential equations by birational transformations. For this system, we give two new Bäcklund transformations. We also give the holomorphy condition of this system. Thanks to this condition, we obtain a new partial differential system in two variables (t, s) involving this system, This system satisfies the compatibility condition, and admits a travelling wave solution. Theorem 6.1. The birational transformation ϕ 0 (62)            x = u, y = du dt + 3 2 ( √ 5 − 1)u 2 , z = d 2 u dt 2 + 3( √ 5 − 1)u du dt takes the equation (61) to the system (63)                dx dt = − 3 2 ( √ 5 − 1)x 2 + y, dy dt = z, dz dt = 3( √ 5 + 3)y 2 + 3( √ 5 − 1)xz + δ. Before we will prove Theorem 6.1, we review the case of the second Painlevé equation. The case of the second Painlevé system In this section, we review the case of the second Painlevé system: (64) d 2 u dt 2 = 2u 3 + tu + α (α ∈ C) . Let us make its polynomial Hamiltonian from the viewpoint of accessible singularity and local index. Step 0: We make a change of variables. (65) x = u, y = du dt . Step 1: We make a change of variables. (66) x 1 = 1 x , y 1 = y x 2 . In this coordinate system, we see that this system has two accessible singular points: (67) (x 1 , y 1 ) = {(0, 1), (0, −1)} . Around the point (x 1 , y 1 ) = (0, 1), we can rewrite the system as follows. Step 2: We make a change of variables. (68) x 2 = x 1 , y 2 = y 1 − 1. In this coordinate system, we can rewrite the system satisfying the condition (22): d dt x 2 y 2 = 1 x 2 −1 0 0 −4 x 2 y 2 + · · · , and we can obtain the local index (−1, −4) at the point {(x 2 , y 2 ) = (0, 0)}. The ratio of the local index at the point {(x 2 , y 2 ) = (0, 0)} is a positive integer. We aim to obtain the local index (−1, −2) by successive blowing-up procedures. Step 3: We blow up at the point {(x 2 , y 2 ) = (0, 0)}. (69) x 3 = x 2 , y 3 = y 2 x 2 . Step 4: We blow up at the point {(x 3 , y 3 ) = (0, 0)}. (70) x 4 = x 3 , y 4 = y 3 x 3 . In this coordinate system, we see that this system has the following accessible singular point: (71) (x 4 , y 4 ) = (0, t/2). Step 5: We make a change of variables. (72) x 5 = x 4 , y 5 = y 4 − t/2. In this coordinate system, we can rewrite the system as follows: d dt x 5 y 5 = 1 x 5 −1 0 α − 1/2 −2 x 5 y 5 + · · · , and we can obtain the local index (−1, −2). Here, the relation between (x 5 , y 5 ) and (x, y) is given by      x 5 = 1 x , y 5 = y − x 2 − t 2 . Finally, we can choose canonical variables (q, p). Step 9: We make a change of variables. (73) q = 1 x 5 , p = y 5 , and we can obtain the system      dq dt = q 2 + p + t 2 , dp dt = −2qp + α − 1 2 with the polynomial Hamiltonian H II : (74) H II = q 2 p + 1 2 p 2 + t 2 p − α − 1 2 q. We remark that we can discuss the case of the accessible singular point (x 1 , y 1 ) = (0, −1) in the same way as in the case of (x 1 , y 1 ) = (0, 1). 8. Proof of Theorem 6.1 By the same way of the second Painlevé system, we can prove Theorem 6.1. Proof. At first, we rewrite the equation (61) to the system of the first-order ordinary differential equations. Step 0: We make a change of variables. (75) x = u, y = du dt , z = d 2 u dt 2 . Step 1: We make a change of variables. (76) x 1 = 1 x , y 1 = y x 2 , z 1 = z x 3 . In this coordinate system, we see that this system has three accessible singular points: (77) (x 1 , y 1 , z 1 ) = (0, −1, 2), 0, 3 2 (1 − √ 5), 9(3 − √ 5) , 0, 3 2 (1 + √ 5), 9(3 + √ 5) . Around the point (x 1 , y 1 , z 1 ) = 0, 3 2 (1 − √ 5), 9(3 − √ 5) , we can rewrite the system as follows. Step 2: We make a change of variables. (78) x 2 = x 1 , y 2 = y 1 − 3 2 (1 − √ 5), z 2 = z 1 − 9(3 − √ 5). In this coordinate system, we can rewrite the system satisfying the condition (22): d dt    x 2 y 2 z 2    = 1 x 2         3 2 ( √ 5 − 1) 0 0 0 6( √ 5 − 1) 1 0 −9( √ 5 − 3) 9 2 ( √ 5 − 1)       x 2 y 2 z 2    + · · ·      . and we can obtain the local index 3 2 ( √ 5 − 1), 3( √ 5 − 1), 15 2 ( √ 5 − 1) at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)} are all positive integers (79) 3( √ 5 − 1) 3 2 ( √ 5 − 1) , 15 2 ( √ 5 − 1) 3 2 ( √ 5 − 1) = (2, 5). This is the reason why we choose this accessible singular point. We aim to obtain the local index (1, 0, 2) by successive blowing-up procedures. Step 3: We blow up at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. (80) x 3 = x 2 , y 3 = y 2 x 2 , z 3 = z 2 x 2 . Step 4: We blow up at the point {(x 3 , y 3 , z 3 ) = (0, 0, 0)}. (81) x 4 = x 3 , y 4 = y 3 x 3 , z 4 = z 3 x 3 . In this coordinate system, we see that this system has the following accessible singular locus: (82) (x 4 , y 4 , z 4 ) = (0, y 4 , −3( √ 5 − 1)y 4 ). Step 5: We blow up along the curve {(x 4 , y 4 , z 4 ) = (0, y 4 , −3( √ 5 − 1)y 4 )}. (83) x 5 = x 4 , y 5 = y 4 , z 5 = z 4 + 3( √ 5 − 1)y 4 x 4 . Step 6: We make a change of variables.. (84) x 6 = 1 x 5 , y 6 = y 5 , z 6 = z 5 , and we can obtain the system (63). Thus, we have completed the proof of Theorem 6.1. We note on the remaining accessible singular points. Around the point (x 1 , y 1 , z 1 ) = 0, 3 2 (1 + √ 5), 9(3 + √ 5) , we can rewrite the system as follows. Step 2: We make a change of variables. (85) x 2 = x 1 , y 2 = y 1 − 3 2 (1 + √ 5), z 2 = z 1 − 9(3 + √ 5). In this coordinate system, we can rewrite the system satisfying the condition (22): d dt    x 2 y 2 z 2    = 1 x 2         − 3 2 ( √ 5 + 1) 0 0 0 −6( √ 5 + 1) 1 0 9( √ 5 + 3) − 9 2 ( √ 5 + 1)       x 2 y 2 z 2    + · · ·      . and we can obtain the local index − (86) −3( √ 5 + 1) − 3 2 ( √ 5 + 1) , − 15 2 ( √ 5 + 1) − 3 2 ( √ 5 + 1) = (2, 5). We remark that we can discuss this case in the same way as in the case of (x 1 , y 1 , z 1 ) = 0, 3 2 (1 − √ 5), 9(3 − √ 5) . Around the point (x 1 , y 1 , z 1 ) = (0, −1, 2), we can rewrite the system as follows. Step 2: We make a change of variables. (87) x 2 = x 1 , y 2 = y 1 + 1, z 2 = z 1 − 2. In this coordinate system, we can rewrite the system satisfying the condition (22): In this case, the local index involves a negative integer. So, we need to blow down. d dt    x 2 y 2 z 2    = 1 x 2         Birational symmetry of Chazy-IX equation In this section, let us consider the birational Bäcklund transformations of the Chazy IX equation. Two birational Bäcklund transformations are new.            X = x + 2{3( √ 5 + 3)y 2 + δ} 3( √ 5 − 1)z , Y = −y, Z = −z (u, u ′ , u ′′ ) (x, y, z) (p, q, r) = (p, p ′ , p ′′ ) (X, Y, Z) g 0 ϕ 0 ϕ 1 s 0 Figure 2. takes the system (63) to the system (90)                dX dt = − 3 2 ( √ 5 − 1)X 2 − (9 + 4 √ 5)Y, dY dt = Z, dZ dt = 3( √ 5 + 3)Y 2 + 3( √ 5 − 1)XZ + δ. Proposition 9.2. The birational transformation ϕ 1 (91)              p = 1 2 ( √ 5 − 3)X, q = 3( √ 5 − 2)X 2 + 1 2 (7 + 3 √ 5)Y, r = 9(3 √ 5 − 7)X 3 − 6( √ 5 + 2)XY + 1 2 (3 √ 5 + 7)Z takes the system (90) to the system (92)                dp dt = q, dq dt = r, dr dt = 54p 4 + 72p 2 q + 12q 2 + 7 + 3 √ 5 2 δ. Then, we can obtain the Chazy IX equation: (93) d 3 p dt 3 = 54p 4 + 72p 2 dp dt + 12 dp dt 2 + 7 + 3 √ 5 2 δ (δ ∈ C). The compositions of the transformations (89) and (91) are a Bäcklund transformation of the Chazy IX equation (see Figure 2). g 0 (u; δ) → ( √ 5 − 3){108u 4 + 18(5 + √ 5)u 2 u ′ + 6(3 + √ 5)(u ′ ) 2 + 3( √ 5 − 1)uu ′′ + 2δ 6( √ 5 − 1){3( √ 5 − 1)uu ′ + u ′′ } ; 7 + 3 √ 5 2 δ ,(94)where u ′ = du dt , u ′′ = d 2 u dt 2 . This Bäcklund transformation is new. (u, u ′ , u ′′ ) (x, y, z) (X, Y, Z) g 1 ϕ 0 π (p, q, r) (u, v, w) ϕ 2 s 1 = (p, p ′ , p ′′ )       X = x, Y = y − 3 √ 5x 2 , Z = z − 6 √ 5xy − 9( √ 5 − 5)x 3 takes the system (63) to the system (96)                dX dt = − 3 2 (− √ 5 − 1)X 2 + Y, dY dt = Z, dZ dt = 3(− √ 5 + 3)Y 2 + 3(− √ 5 − 1)XZ + δ. This transformation changes the sign of √ 5 in the system (63) (cf. [8]). Proposition 9.5. The birational transformation s 1 (97)            u = X − 2{3(− √ 5 + 3)Y 2 + δ} 3( √ 5 + 1)Z , v = −Y, w = −Z takes the system (96) to the system (98)                du dt = − 3 2 (− √ 5 − 1)u 2 + (−9 + 4 √ 5)v, dv dt = w, dw dt = 3(− √ 5 + 3)v 2 + 3(− √ 5 − 1)uw + δ. Proposition 9.6. The birational transformation ϕ 2 (99)              p = 1 2 (− √ 5 − 3)u, q = 3(− √ 5 − 2)u 2 + 1 2 (7 − 3 √ 5)v, r = 9(−3 √ 5 − 7)u 3 − 6(− √ 5 + 2)uv + 1 2 (−3 √ 5 + 7)w takes the system (98) to the system (100)                dp dt = q, dq dt = r, dr dt = 54p 4 + 72p 2 q + 12q 2 + 7 − 3 √ 5 2 δ. Then, we can obtain the Chazy IX equation: (101) d 3 p dt 3 = 54p 4 + 72p 2 dp dt + 12 dp dt 2 + 7 − 3 √ 5 2 δ (δ ∈ C). The compositions of the transformations (95), (97) and (99) are a Bäcklund transformation of the Chazy IX equation (see Figure 3). Theorem 9.7. The Chazy IX equation (61) is invariant under the biratioral transformation: g 1 (u; δ) → (− √ 5 − 3){108u 4 + 18(5 − √ 5)u 2 u ′ + 6(3 − √ 5)(u ′ ) 2 + 3(− √ 5 − 1)uu ′′ + 2δ 6(− √ 5 − 1){3(− √ 5 − 1)uu ′ + u ′′ } ; 7 − 3 √ 5 2 δ ,(102) where u ′ = du dt , u ′′ = d 2 u dt 2 . This Bäcklund transformation is new. Holomorphy conditions of Chazy-IX equation In this section, we give the holomorphy condition of the system (63). Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. Theorem 10.1. Let us consider the following differential system in the polynomial class:        dx = f 1 (x, y, z)dt + g 1 (x, y, z)ds, dy = f 2 (x, y, z)dt + g 2 (x, y, z)ds, dz = f 3 (x, y, z)dt + g 3 (x, y, z)ds. We assume that (A1) deg(f i ) = 3 and deg(g i ) = 3 with respect to x, y, z. (A2) The right-hand side of this system becomes again a polynomial in each coordinate system (x i , y i , z i ) (i = 1, 2). 1) x 1 = 1 x , y 1 = y, z 1 = − zx + 2{3( √ 5 + 3)y 2 + δ} 3( √ 5 − 1) x, 2) x 2 = 1 x , y 2 = y − 3 √ 5x 2 , z 2 = −((z − 6 √ 5xy − 9( √ 5 − 5)x 3 )x − 2{−( √ 5 − 3)(135x 4 + 3y 2 ) + (90 − 54 √ 5)x 2 y + δ} 3( √ 5 + 1) )x.(103) Then such a system coincides with (104)                                  dx = − 3 2 ( √ 5 − 1)x 2 + y dt + −12( √ 5 + 2)x 2 y + ( √ 5 + 1) 3xz − 2y 2 − 2 3 δ ds, dy =zdt + {−6(2 √ 5 − 5)x 2 z + 12 √ 5xy 2 − ( √ 5 + 1)yz + (3 √ 5 − 5)δx}ds, dz ={3( √ 5 + 3)y 2 + 3( √ 5 − 1)xz + δ}dt + {48xyz − 24y 3 − ( √ 5 + 1)z 2 + 2( √ 5 − 3)δy}ds. These transition functions satisfy the condition: dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz (i = 1, 2). We remark that when s = 0, we can obtain the system (63). T anh 3 2 ( √ 5 − 2)c 2 t + 3( √ 5 + 1)c 3 2 s + c 1 , − 1 4 ( √ 5 − 3)c 2 2 , 0; − 3 8 ( √ 5 − 3)c 4 2 ) (c i ∈ C).(106) The Chazy X equation The Chazy-X.a equation is given by d 3 u dt 3 =6u 2 du dt + 3 11 (9 + 7 √ 3) du dt + u 2 2 − 1 22 (4 − 3 √ 3)α du dt + 1 44 (3 − 5 √ 3)αu 2 − 1 352 (9 + 7 √ 3)α 2 .(107) In this section, at first we transform the equation (107) to a system of differential equations by birational transformations. For this system, we give two Bäcklund transformations from Chazy-X.a to Chazy-X.b. One of them is new. We also give the holomorphy condition of this system. Thanks to this condition, we can recover the Chazy-X.a system. Theorem 11.1. The birational transformation ϕ 0 (108)              x = u, y = du dt − 3 + √ 3 2 u 2 , z = d 2 u dt 2 − (3 + √ 3)u du dt takes the Chazy X.a equation to the system of the first-order ordinary differential equations: (109)                dx dt = 3 + √ 3 2 x 2 + y, dy dt = z, dz dt = 2 11 (−3 + 5 √ 3)y 2 − (3 + √ 3)xz + 1 22 (−4 + 3 √ 3)αy − 1 352 (9 + 7 √ 3)α 2 . By the same way of the second Painlevé system, we can prove Theorem 11.1. Proof. At first, we rewrite the equation (107) to the system of the first-order ordinary differential equations. Step 0: We make a change of variables. (110) x = u, y = du dt , z = d 2 u dt 2 . Step 1: We make a change of variables. (111) x 1 = 1 x , y 1 = y x 2 , z 1 = z x 3 . In this coordinate system, we see that this system has three accessible singular points: (x 1 , y 1 , z 1 ) = (0, −1, 2), 0, 3 + √ 3 2 , 3(2 + √ 3) , 0, −1 − 2 √ 3 11 , 2 121 (13 + 4 √ 3) .(112) Around the point (x 1 , y 1 , z 1 ) = 0, 3+ √ 3 2 , 3(2 + √ 3) , we can rewrite the system as follows. Step 2: We make a change of variables. (113) x 2 = x 1 , y 2 = y 1 − 3 + √ 3 2 , z 2 = z 1 − 3(2 + √ 3). In this coordinate system, we can rewrite the system satisfying the condition (22): d dt    x 2 y 2 z 2    = 1 x 2         − 3+ √ 3 2 0 0 0 −2(3 + √ 3) 1 0 3(2 + √ 3) − 3(3+ √ 3) 2       x 2 y 2 z 2    + · · ·      . and we can obtain the local index − 3+ (114) −(3 + √ 3) − 3+ √ 3 2 , − 5 2 (3 + √ 3) − 3+ √ 3 2 = (2, 5). This is the reason why we choose this accessible singular point. We aim to obtain the local index (1, 0, 2) by successive blowing-up procedures. Step 3: We blow up at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. (115) x 3 = x 2 , y 3 = y 2 x 2 , z 3 = z 2 x 2 . Step 4: We blow up at the point {(x 3 , y 3 , z 3 ) = (0, 0, 0)}. (116) x 4 = x 3 , y 4 = y 3 x 3 , z 4 = z 3 x 3 . In this coordinate system, we see that this system has the following accessible singular locus: (117) (x 4 , y 4 , z 4 ) = (0, y 4 , (3 + √ 3)y 4 ). Step 5: We blow up along the curve {(x 4 , y 4 , z 4 ) = (0, y 4 , (3 + √ 3)y 4 )}. (118) x 5 = x 4 , y 5 = y 4 , z 5 = z 4 − (3 + √ 3)y 4 x 4 . Step 6: We make a change of variables.. (119) x 6 = 1 x 5 , y 6 = y 5 , z 6 = z 5 , and we can obtain the system (109). Thus, we have completed the proof of Theorem 11.1. We note on the remaining accessible singular points. Around the point (x 1 , y 1 , z 1 ) = (0, −1, 2), we can rewrite the system as follows. Step 2: We make a change of variables. (120) x 2 = x 1 , y 2 = y 1 + 1, z 2 = z 1 − 2. In this coordinate system, we can rewrite the system satisfying the condition (22): We remark that we can discuss this case in the same way as in the case of (x 1 , y 1 , d dt    x 2 y 2 z 2    = 1 x 2         1 0 0 0 4 1 0 0 3       x 2 y 2 z 2    + · · ·     z 1 ) = 0, 3+ √ 3 2 , 3(2 + √ 3) . Around the point (x 1 , y 1 , z 1 ) = 0, −1−2 √ 3 11 , 2 121 (13 + 4 √ 3) , we can rewrite the system as follows. Step 2: We make a change of variables. (122) x 2 = x 1 , y 2 = y 1 + 1 + 2 √ 3 11 , z 2 = z 1 − 2 121 (13 + 4 √ 3). In this coordinate system, we can rewrite the system satisfying the condition (22): d dt    x 2 y 2 z 2    = 1 x 2         1+2 √ 3 11 0 0 0 4(1+2 √ 3) 11 1 0 72(13+4 √ 3) 121 3(1+2 √ 3) 11       x 2 y 2 z 2    + · · ·      . and we can obtain the local index 1+2 In this case, the local index involves a negative integer. So, we need to blow down. Birational symmetry of Chazy X equation In this section, let us consider the birational Bäcklund transformations from Chazy-X.a to Chazy-X.b. One of them is new.            X = x − 64(−3 + 5 √ 3)y 2 + 16(−4 + 3 √ 3)αy − (9 + 7 √ 3)α 2 176(3 + √ 3)z , Y = −y, Z = −z takes the system (109) to the system (125)                dX dt = 3 + √ 3 2 X 2 + 1 11 (−43 + 24 √ 3)Y + 21 − 13 √ 3 66 α, dY dt = Z, dZ dt = 2 11 (−3 + 5 √ 3)Y 2 − (3 + √ 3)XZ − 1 22 (−4 + 3 √ 3)αY − 1 352 (9 + 7 √ 3)α 2 . Proposition 12.2. The birational transformation ϕ 1 (126)                p = (2 + √ 3)X, q = 2 + √ 3 11(3 + √ 3) {33(2 + √ 3)X 2 + (−57 + 29 √ 3)Y − (−4 + 3 √ 3)α}, r = 6(2 + √ 3) 11(3 + √ 3) 3 {33(33 + 19 √ 3)X 3 − 6(13 + 4 √ 3)XY + (−27 + √ 3)Z − (9 + 7 √ 3)αX} takes the system (125) to the system (127)                        dp dt = q, dq dt = r, dr dt = 6p 2 q + 3 11 (9 − 7 √ 3) q + p 2 2 − 1 22 (4 + 3 √ 3)αq + 1 44 (3 + 5 √ 3)αp 2 − 1 352 (9 − 7 √ 3)α 2 . Then, we can obtain the Chazy X.b equation: d 3 u dt 3 =6u 2 du dt + 3 11 (9 − 7 √ 3) du dt + u 2 2 − 1 22 (4 + 3 √ 3)α du dt + 1 44 (3 + 5 √ 3)αu 2 − 1 352 (9 − 7 √ 3)α 2 ,(128) The compositions of the transformations (124) and (126) are a Bäcklund transformation from Chazy-X.a to Chazy-X.b (see Figure 1). g 0 (u; α) → ( 3 + √ 3 1056{(3 + √ 3)uu ′ − u ′′ } {288(6 + √ 3)u 2 u ′ − 176(3 + √ 3)uu ′′ + 96(9 + 7 √ 3)u 4 + 64(−3 + 5 √ 3)(u ′ ) 2 − 8(−3 + 5 √ 3)αu 2 + 16(−4 + 3 √ 3)αu ′ − (9 + 7 √ 3)α 2 }; α),(129) where u ′ = du dt , u ′′ = d 2 u dt 2 . Proposition 12.4. The birational transformation π (130)            X = x, Y = y + 1 2 (5 + √ 3)x 2 + 1 8 ( √ 3 − 1)α, Z = z − (5 + √ 3)x 3 − 1 4 (2 √ 3 − 1)αx takes the system (109) to the system (131)                dX dt = −X 2 + Y − 1 8 ( √ 3 − 1)α, dY dt = (5 + √ 3)XY + Z, dZ dt = −(15 + 7 √ 3)X 2 Y + 2 11 (−3 + 5 √ 3)Y 2 − (3 + √ 3)XZ − 3 4 αY. Proposition 12.5. The birational transformation s 1 (132)          u = X − ( √ 3 − 4)Z 13Y , v = Y, w = −Z takes the system (131) to the system (133)                du dt = −(4 + √ 3)u 2 + 8(89 + 46 √ 3)v − 11(11 + 7 √ 3)α 1144 , dv dt = (5 + √ 3)uv − 1 13 (−4 + √ 3)w, dw dt = (15 + 7 √ 3)u 2 v + (3 + √ 3)uw + 1 44 v{8(3 − 5 √ 3)v + 33α}. Proposition 12.6. The birational transformation ϕ 2 (134)            p = (4 + √ 3)u, q = −(19 + 8 √ 3)u 2 + 1 11 (38 + 21 √ 3)v − 1 8 (5 + 3 √ 3)α, r = 2(100 + 51 √ 3)u 3 + 1 143 (89 + 46 √ 3)w + u 44 {−4(177 + 101 √ 3)v + 11(29 + 17 √ 3)α} takes the system (133) to the system (135)                        dp dt = q, dq dt = r, dr dt = 6p 2 q + 3 11 (9 − 7 √ 3) q + p 2 2 − 1 22 (4 + 3 √ 3)βq + 1 44 (3 + 5 √ 3)βp 2 − 1 352 (9 − 7 √ 3)β 2 , where β = (−2 + √ 3)α. Then, we can obtain the Chazy X.b equation: d 3 u dt 3 =6u 2 du dt + 3 11 (9 − 7 √ 3) du dt + u 2 2 − 1 22 (4 + 3 √ 3)β du dt + 1 44 (3 + 5 √ 3)βu 2 − 1 352 (9 − 7 √ 3)β 2 , β = (−2 + √ 3)α. The compositions of the transformations (130), (132) and (134) are a Bäcklund transformation from Chazy-X.a to Chazy-X.b (see Figure 2). Theorem 12.7. The Chazy X.a equation (107) can be transformed into the Chazy X.b equation by the birational transformation: g 1 (u; α) → (4 + √ 3){8(−4 + √ 3)u 3 − 8(−4 + √ 3)uu ′ − 8(−4 + √ 3)u ′′ − (−7 + 5 √ 3)αu} 13{8u 2 + 8u ′ + ( √ 3 − 1)α} ; (−2 + √ 3)α ,(136)where u ′ = du dt , u ′′ = d 2 u dt 2 . This Bäcklund transformation from Chazy-X.a to Chazy-X.b is new. Holomorphy conditions of Chazy X equation In this section, we give the holomorphy condition of the system (109). Thanks to this holomorphy condition, we can recover the system (109). Theorem 13.1. Let us consider the following ordinary differential system in the polynomial class:                dx dt = f 1 (x, y, z), dy dt = f 2 (x, y, z), dz dt = f 3 (x, y, z). We assume that (A1) deg(f i ) = 3 with respect to x, y, z. (A2) The right-hand side of this system becomes again a polynomial in each coordinate system (x i , y i , z i ) (i = 1, 2). 1) x 1 = 1 x , y 1 = y, z 1 = − zx − 4 11 (−4 + 3 √ 3)y 2 + 1 66 (−21 + 13 √ 3)αy − 1 + 2 √ 3 176 α 2 x, 2) x 2 = 1 x , y 2 = − y + 5 + √ 3 2 x 2 + √ 3 − 1 8 α x + −17 + √ 3 13 x 3 − −4 + √ 3 13 z − −10 + 9 √ 3 52 αx x, z 2 = z − (5 + √ 3)x 3 − 1 4 (2 √ 3 − 1)αx.(137) Then such a system coincides with (138)                    dx dt = − a(t) 3 + √ 3 3 + √ 3 2 x 2 + y , dy dt = − a(t) 3 + √ 3 z, dz dt = − a(t) 3 + √ 3 2 11 (−3 + 5 √ 3)y 2 − (3 + √ 3)xz + 1 22 (−4 + 3 √ 3)αy − 1 352 (9 + 7 √ 3)α 2 , where a(t) ∈ C(t). Setting a(t) = −(3 + √ 3), we obtain the system (109). These transition functions satisfy the condition: dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz (i = 1, 2). Particular solutions of Chazy X equation In this section, we study a solution of the system (109) which is written by the use of known functions. (x, y, z; α) = − 2 (3 + √ 3)t + 2c , 0, 0; 0 ,(139) and admits two special solutions: (x, y, z; α) =    − (−3 + √ 3) 3 + √ 3 √ αT an 3 1 4 4 3 + √ 3 √ α(t + 8c) 4 × 3 3 4 , √ 3 8 α, 0; α    , (x, y, z; α) =   − (3 + √ 3) 3 2 √ αT anh 1 4 5 + 3 √ 3 √ α(t + 24c) 12 3 + 2 √ 3 , − 3 + 2 √ 3 24 α, 0; α   ,(140) where c ∈ C is an integral constant. Appendix A In this section, we study the Chazy I equation (see [8]): I : d 3 u dt 3 =6 − du dt 2 + A(t) du dt + u 2 + B(t)u + C(t) ,(141) where u denotes unknown complex variable, and the coefficient functions A(t), B(t) and C(t) satisfy the relations: (142)                d 2 A dt 2 = 6A 2 , d 2 B dt 2 = 6AB, d 2 C dt 2 = B 2 + 2AC. At first, making a change of variables (143) x = u, y = du dt , z = d 2 u dt 2 , we obtain the system of the first-order ordinary differential equations: (144)                dx dt = y, dy dt = z, dz dt = 6 −y 2 + A(t) y + x 2 + B(t)x + C(t) . Let us make its phase space by gluing two copies of C 3 × C via the birational transformations. The following Lemma shows that this rational vector fieldṽ associated with the system (144) has seven accessible singular points on the boundary divisor H ⊂ P 3 . Lemma 15.1. The rational vector fieldṽ associated with the system (144) has three accessible singular points: (145)        P 1 = {(X 1 , Y 1 , Z 1 )\|X 1 = Z 1 = 0, Y 1 = A(t)}, P 2 = {(X 1 , Y 1 , Z 1 )\|X 1 = Z 1 = 0, Y 1 = − A(t)}, P 3 = {(X 3 , Y 3 , Z 3 )\|X 3 = Y 3 = Z 3 = 0}, where the point P 3 has multiplicity of order 5 and (X i , Y i , Z i ) are given by (39). This lemma can be proven by a direct calculation. Next let us calculate its local index at each point. Singular point Type of local index P 1 (0, 0, 0) P 2 (0, 0, 0) Example 15.2. Let us take the coordinate system (p, q, r) centered at the point P 1 : p = X 1 = 1 x , q = Y 1 − A(t) = y x − A(t), r = Z 1 = z x . The system (144) is rewritten as follows: d dt    p q r    = 1 p            0 0 0 −A(t) − A ′ (t) 2 √ A(t) 0 0 6(A(t) A(t) + B(t)) −12 A(t) 0        p q r    + · · ·        satisfying (22). In this case, the local index is (0, 0, 0). We see that the residue of the formal Laurent series: (146) x(t) = a 0 (t − t 0 ) + a 1 + a 2 (t − t 0 ) + · · · + a n (t − t 0 ) n−1 + · · · (a i ∈ C) is equal to a 0 = 0. By a direct calculation, we see that there are no solutions which pass through P 1 . In the case of P 2 , we also discuss the same way. In order to do analysis for the accessible singular point P 3 , we need to replace a suitable coordinate system because this point has multiplicity of order 5. At first, let us do the Painlevé test. To find the leading order behaviour of a singularity at t = t 0 one sets                x ∝ a (t − t 0 ) m , y ∝ b (t − t 0 ) n , z ∝ c (t − t 0 ) p , from which it is easily deduced that m = 1, n = 2, p = 3. Each order of pole (m, n, p) suggests a suitable coordinate system to do analysis for the accessible singular point P 3 , which is explicitly given by (X, Y, Z) = 1 x , y x 2 , z x 3 . In this coordinate, the singular points are given as follows: P (1) 3 = {(X, Y, Z) = (0, −1, 2)} . Next let us calculate its local index at the point P 3 . Singular point Type of local index P (1, 1, 6) Now, we try to resolve the accessible singular point P 3 . Step 0: We take the coordinate system centered at P (1) 3 : p = X, q = Y + 1, r = Z − 2. In this coordinate, the system (144) is rewritten as follows:       p q r    to arrive at d dt    X Y Z    = 1 X         1 0 0 0 1 0 0 0 6       X Y Z    + · · ·      . By considering the ratio of the local index (1, 1, 6), we obtain the resonances 1 1 , 6 1 = (1, 6). This property suggests that we will blow up one time to the direction q and six times to the direction r. Step 1: We blow up at the point P 3 : p (1) = p, q (1) = q p , r (1) = r p . Step 2: We blow up along the curve {(p (1) , q (1) , r (1) ) = (p (1) , q (1) , −3q (1) )}: p (2) = p (1) , q (2) = q (1) , r (2) = r (1) + 3q (1) p (1) . Step 3: We blow up along the curve (p (2) , q (2) , r (2) ) = (p (2) , q (2) , 3 4 (q (2) ) 2 ) : p (3) = p (2) , q (3) = q (2) , r (3) = r (2) − 3 4 (q (2) ) 2 p (2) . Step 4: We blow up along the curve{( p (3) , q (3) , r (3) ) = (p (3) , q (3) , 1 8 ((q (3) ) 3 − 16A(t)q (3) − 16B(t))}: p (4) = p (3) , q (4) = q (3) , r (4) = r (3) − 1 8 ((q (3) ) 3 − 16A(t)q (3) − 16B(t)) p (3) . Step 5: We blow up along the curve {(p (4) , q (4) , r (4) ) = (p (4) , q (4) , 1 64 (3(q (4) ) 4 − 80A(t)(q (4) ) 2 − 32(2A ′ (t) + 3B(t))q (4) − 64B ′ (t) − 192C(t))}: p (5) = p (4) , q (5) = q (4) , r (5) = r (4) − 1 64 (3(q (4) ) 4 − 80A(t)(q (4) ) 2 − 32(2A ′ (t) + 3B(t))q (4) − 64B ′ (t) − 192C(t)) p (4) . Step 6: We blow up along the curve {(p (5) , q (5) , r (5) ) = (p (5) , q (5) , r (5) − g(q (5) )): p (6) = p (5) , q (6) = q (5) , r (6) = r (5) − g(q (5) ) p (5) , where the symbol g(q (5) ) is given by g(q (5) ) = 1 128 (3(q (5) ) 5 − 128A(t)(q (4) ) 3 − 48(3B(t) + 4)(q (4) ) 2 + 64( 8A(t) 2 − 3C(t)A ′ (t) − 4B ′ (t) − 2A ′′ (t))q (5) − 384C ′ (t) − 128B ′′ (t) + 512A(t)B(t)). Now, we have blowed up one time to the direction q and six times to the direction r. In this coordinate, the system (144) is rewritten as follows: (147) (6) , q (6) , r (6) ), dq (6) dt = g 2 (p (6) , q (6) , r (6) ),                  dp (6) dt = g 1 (pdr (6) dt = h 1 (t)(q (6) ) 2 + h 2 (t)q (6) + h 3 (t) p (6) + g 3 (p (6) , q (6) , r (6) ), where g i (p (6) , q (6) , r (6) ) ∈ C(t)[p (6) , q (6) , r (6) ] (i = 1, 2, 3) and h 1 (t)(q (6) ) 2 +h 2 (t)q (6) +h 3 (t) ∈ C(t)[q (6) ]. Each right-hand side of the system (147) is a polynomial if and only if (148)                d 2 A dt 2 = 6A 2 , d 2 B dt 2 = 2 3 (9A(t)B(t) + 12A ′ (t)A(t) − A ′′′ (t)) , d 2 C dt 2 = 1 3 (3B(t) 2 + 6A(t)C(t) + 6A ′ (t)B(t) + 6A(t)B ′ (t) − B ′′′ (t)). By solving these equations, we can obtain the relations (142). Thus, we have completed the proof of the following theorem. Theorem 15.3. The phase space X for the system (144) is obtained by gluing two copies of C 3 × C: U j × C = C 3 × C ∋ {(x j , y j , z j , t)}, j = 0, 1 via the following birational transformations: 0) x 0 = x, y 0 = y, z 0 = z, 1) x 1 = 1 x , y 1 = y + x 2 x , z 1 = −((( z x 3 − 2 x + 3 y + x 2 x x − 3 4 y + x 2 x 2 x − 1 8 y + x 2 x 3 − 16A(t) y + x 2 x − 16B(t) )x − 1 64 3 y + x 2 x 4 − 80A(t) y + x 2 x 2 − 32(2A ′ (t) + 3B(t)) y + x 2 x − 64B ′ (t) − 192C(t) )x − 1 128 {3 y + x 2 x 5 − 128A(t) y + x 2 x 3 − 48(3B(t) + 4) y + x 2 x 2 + 64(8A(t) 2 − 3C(t)A ′ (t) − 4B ′ (t) − 2A ′′ (t)) y + x 2 x − 384C ′ (t) − 128B ′′ (t) + 512A(t)B(t)})x.(149) We remark that these transition functions satisfy the condition: dx 1 ∧ dy 1 ∧ dz 1 = dx ∧ dy ∧ dz. Appendix B The Chazy-VIII equation is given by d 3 u dt 3 =6u 2 du dt + (−2α 2 t 2 + βt + γ) du dt + α + 2αu 2 + (−4α 2 t + β)u,(150) where u denotes unknown complex variable and α, β and γ are its constant parameters. In [8], we distinguish two canonical subcases. If α = 0, we may set β = 0 by a translation in t. This gives equation VIII.a, whose solution is (151) u = w(t) − αt, where w(t) satisfies the Painlevé IV equation (152) w ′′ = 1 2w (w ′ ) 2 + 3 2 w 3 − 4αtw 2 + 1 2 (4α 2 t 2 + γ)w + K w ′ := d dt , K being a constant of integration. If α = 0, we call the equation Chazy-VIII.b. A first integral of VIII.b is the Painlevé II equation: (153) u ′′ = 2u 3 + (βt + γ)u + K ′ := d dt , K being a constant of integration. Theorem 16.1. The birational transformation ϕ 0 (154)            x = u, y = du dt − u 2 − 1 2 {(−2α 2 t + β)t + 2α + γ}, z = d 2 u dt 2 − 2u 3 − {(−2α 2 t + β)t + 2α + γ}u takes the Chazy VIII equation to the system of the first-order ordinary differential equations: (155)                dx dt = x 2 + y + β 2 t − α 2 t 2 + α + γ 2 , dy dt = −2xy + z + 2α 2 t − β 2 , dz dt = −2αy − 2α 2 . By the same way of the second Painlevé system, we can prove Theorem 16.1. Proof. At first, we rewrite the equation (150) to the system of the first-order ordinary differential equations. Step 0: We make a change of variables. (156) x = u, y = du dt , z = d 2 u dt 2 . Step 1: We make a change of variables. (157) x 1 = 1 x , y 1 = y x 2 , z 1 = z x 3 . In this coordinate system, we see that this system has two accessible singular points: (158) (x 1 , y 1 , z 1 ) = {(0, 1, 2), (0, −1, 2)} . Around the point (x 1 , y 1 , z 1 ) = (0, 1, 2), we can rewrite the system as follows. Step 2: We make a change of variables. (159) x 2 = x 1 , y 2 = y 1 − 1, z 2 = z 1 − 2. In this coordinate system, we can rewrite the system satisfying the condition (22): This is the reason why we choose this accessible singular point. We aim to obtain the local index (−1, −2, 0) by successive blowing-up procedures. d dt    x 2 y 2 z 2    = 1 x 2         −1 0 0 0 −4 1 0 0 −3       x 2 y 2 z 2    + · · ·      . Step 3: We blow up at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. (161) x 3 = x 2 , y 3 = y 2 x 2 , z 3 = z 2 x 2 . Step 4: We blow up at the point {(x 3 , y 3 , z 3 ) = (0, 0, 0)}. (162) x 4 = x 3 , y 4 = y 3 x 3 , z 4 = z 3 x 3 . In this coordinate system, we see that this system has the following accessible singular point: (163) (x 4 , y 4 , z 4 ) = 0, 1 2 (2α − 2α 2 t 2 + βt + γ), 2α − 2α 2 t 2 + βt + γ . Step 5: We blow up along the curve {(x 4 , y 4 , z 4 ) = (0, 1 2 (2α − 2α 2 t 2 + βt + γ), 2α − 2α 2 t 2 + βt + γ)}. (164) x 5 = x 4 , y 5 = y 4 − 1 2 (2α − 2α 2 t 2 + βt + γ), z 5 = z 4 − (2α − 2α 2 t 2 + βt + γ) x 4 . Step 6: We make a change of variables. (165) x 6 = 1 x 5 , y 6 = y 5 , z 6 = z 5 , and we can obtain the system (155). Thus, we have completed the proof of Theorem 16.1. We note on the remaining accessible singular points. Around the point (x 1 , y 1 , z 1 ) = (0, −1, 2), we can rewrite the system as follows. Step 2: We make a change of variables. (166) x 2 = x 1 , y 2 = y 1 + 1, z 2 = z 1 − 2. d dt    x 2 y 2 z 2    = 1 x 2         1 0 0 0 4 1 0 0 3       x 2 y 2 z 2    + · · ·      . and we can obtain the local index (1,4,3) We remark that we can discuss this case in the same way as in the case of (x 1 , y 1 , z 1 ) = (0, 1, 2). Now, let us consider the birational Bäcklund transformations of the Chazy-VIII system. These Bäcklund transformations are new. Proposition 16.2. The birational transformation s 0 (168)            X = x − 2z + 4α 2 t − β 2y , Y = y, Z = −z takes the system (155) to the system (169)                dX dt = X 2 + Y − α 2 t 2 + β 2 t + 3α + γ 2 , dY dt = Z − 2XY − 2α 2 t + β 2 , dZ dt = 2αY + 2α 2 . Proposition 16.3. The birational transformation s 1 (170)              X = − x + z + 4αx − 2α 2 t + β 2 y + 2x 2 − 2α 2 t 2 + βt + γ , Y = −(y + 2x 2 − 2α 2 t 2 + βt + γ), Z = −(−z − 4αx) takes the system (155) to the system Figure 4. The transformation π can be obtained by the compositions of the transformations (168), (173) and (172). (171)                dX dt = X 2 + Y − α 2 t 2 + β 2 t − 3α + γ 2 , dY dt = Z − 2XY − 2α 2 t + β 2 , dZ dt = −2αY + 2α 2 . (x, y, z) (X, Y, Z) (x, y, z) (X, Y, Z) π s 0 s 1 −1 ϕ Here, its inverse transformation s 1 −1 is given by is a Bäcklund transformation from the system (169) to the system (171). (172)                        x = − X + 2Z − 4α 2 t + β 2Y , y = − Y − 2X 2 + 2α 2 t 2 − βt − γ + 4X(Z − 2α 2 t) Y + 4βXY − 4Z 2 + 4Z(4α 2 t − β) − 16α 4 t 2 + 8α 2 βt − β 2 2Y 2 , z =Z + 4αX − 2α(2Z − 4α 2 t + β) Y . Theorem 16.5. The system (155) is invariant under the following birational transformations: Next, we give the holomorphy condition of the system (155). Thanks to this holomorphy condition, we can recover the system (155). s 0 : (x, y, z; 0, 0, γ) → x − z y , y, −z; 0, 0, γ ,s 1 : (x, y, z; 0, 0, γ) → − x + z y + 2x 2 + γ , −(y + 2x 2 + γ), z; 0, 0, γ , π : (x, y, z; α, β, γ) → (−x, −y − 2x 2 + 2α 2 t 2 − βt − γ, −z − 4αx; −α, β, γ).(174) Theorem 16.6. Let us consider the following ordinary differential system in the polynomial class:                dx dt = f 1 (x, y, z), dy dt = f 2 (x, y, z), dz dt = f 3 (x, y, z). We assume that (A1) deg(f i ) = 3 with respect to x, y, z. (A2) The right-hand side of this system becomes again a polynomial in each coordinate system (x i , y i , z i ) (i = 1, 2). 1) x 1 = 1 x , y 1 = − yx − 1 2 (2z + 4α 2 t − β) x, z 1 = z, 2) x 2 = 1 x , y 2 = − (y + 2x 2 − 2α 2 t 2 + βt + 4α + γ)x + z − 2α 2 t + β 2 x, z 2 = z + 4αx.(175) Then such a system coincides with the system (155). These transition functions satisfy the condition: dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz (i = 1, 2). Finally, we study a solution of the system (155) which is written by the use of known functions. Proposition 16.7. The system (155) admits a rational solution: (x, y, z; α, β, γ) = 0, − β 2 t − γ 2 , 0; 0, β, γ ,(176) and admits a special solution: (x, y, z; α, β, γ) = x, 0, 1 2 (β − 4α 2 t); α, β, γ ,(177) where the equation in x satisfies as follows: (178) dx dt = x 2 + β 2 t − α 2 t 2 + α + γ 2 . Setting x := − d dt logX = − X ′ X , this equation can be transformed into the second-order linear differential equation: (179) d 2 X dt 2 = 1 2 (2α 2 t 2 − βt − 2α − γ)X. This equation can be solved by the ParabolicCylinder functions. We remark that the solution (177) also can be obtained as the fixed point of the Bäcklund transformation π (see Proposition 16.5). Appendix C In this section, we present a 6-parameter family of ordinary differential systems in dimension three explicitly given by (180)                dx dt = x 2 − xy − xz + (−α 3 + α 4 − α 5 + α 6 )x + α 3 y + α 5 z + α 3 α 5 − α 4 α 5 − α 3 α 6 , dy dt = y 2 − xy − yz + α 1 x + (−α 1 + α 2 + α 5 − α 6 )y + α 6 z − α 1 α 5 + α 1 α 6 − α 2 α 6 , dz dt = z 2 − xz − yz + α 2 x + α 4 y + (α 1 − α 2 + α 3 − α 4 )z − α 2 α 3 − α 1 α 4 + α 2 α 4 . Here x, y, z denote unknown complex variables and α i (i = 1, 2, . . . , 6) are complex parameters. Proposition 17.1. This system is invariant under the following transformations: s 0 (x, y, z; α 1 , . . . , α 6 ) →(y, x, z; α 3 , α 4 , α 1 , α 2 , α 6 , α 5 ), s 1 (x, y, z; α 1 , . . . , α 6 ) →(z, y, x; α 6 , α 5 , α 4 , α 3 , α 2 , α 1 ), s 2 (x, y, z; α 1 , . . . , α 6 ) →(x, z, y; α 2 , α 1 , α 5 , α 6 , α 3 , α 4 ), π(x, y, z; α 1 , . . . , α 6 ) →(y, z, x; α 4 , α 3 , α 6 , α 5 , α 1 , α 2 ). (181) Theorem 17.2. After a series of explicit blowing-ups at eight points including four infinitely near points on the boundary divisor H ∼ = P 2 in P 3 , we obtain the smooth projective 3-foldX and a morphism ϕ :X → P 3 . Its canonical divisor KX is given by KX = −4E 0 − 2 4 i=1 E i ,(182) where the symbol E 0 denotes the proper transform of boundary divisor H of P 3 by ϕ and E i denote the exceptional divisors, which are isomorphic to F 1 . Moreover,X − (−KX ) red satisfies (183)X − (−KX ) red = X . Theorem 17.3. The phase space X for the system (180) is obtained by gluing five copies of C 3 : U j ∼ = C 3 ∋ {(x j , y j , z j )}, j = 0, 1, . . . , 4 via the following birational transformations: 0) x 0 = x, y 0 = y, z 0 = z, 1) x 1 = 1 x , y 1 = −(y − α 1 )x, z 1 = (z − α 2 )x, 2) x 2 = (x − α 3 )y, y 2 = 1 y , z 2 = −(z − α 4 )y, 3) x 3 = −(x − α 5 )z, y 3 = (y − α 6 )z, z 3 = 1 z , 4) x 4 = 1 x , y 4 = −(y − x + α 2 − α 4 + α 5 − α 6 )x, z 4 = (z − x + α 1 + α 3 − α 4 − α 6 )x.(184) These transition functions satisfy the condition: dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz (i = 1, 2, 3, 4). Proposition 17.4. This system has (185) I := xz − yz − α 2 x + α 4 y − (α 5 − α 6 )z as its first integral. By using (185), elimination of x from the system (180) gives the second-order ordinary differential system in the variables (y, z); (186)                      dy dt =y 2 − yz + (−α 1 + α 2 + α 5 − α 6 )y + α 6 z − α 1 α 5 + α 1 α 6 − α 2 α 6 − (y − α 1 )(I + yz − α 4 y + (α 5 − α 6 )z) z − α 2 , dz dt =z 2 − 2yz + 2α 4 y + (α 1 − α 2 + α 3 − α 4 − α 5 + α 6 )z − α 1 α 4 + α 2 α 4 − α 2 α 3 − I. Proposition 17.5. The canonical transformation (187)    X = y − α 1 z − α 2 , Y = z − α 2 takes the system (186) to the Hamiltonian system (188)                  dX dt = ∂H ∂Y =2X 2 Y + (α 2 − α 4 )X 2 − 2XY + (α 1 − α 2 − α 3 + α 4 + α 5 − α 6 )X + α 6 − α 1 , dY dt = − ∂H ∂X = − 2XY 2 + Y 2 − 2(α 2 − α 4 )XY − (α 1 − α 2 − α 3 + α 4 + α 5 − α 6 )Y − I − α 1 α 2 + α 1 α 4 − α 2 α 5 + α 2 α 6 with the polynomial Hamiltonian H :=X 2 Y 2 + (α 2 − α 4 )X 2 Y − XY 2 + (α 1 − α 2 − α 3 + α 4 + α 5 − α 6 )XY − (−I − α 1 α 2 + α 1 α 4 − α 2 α 5 + α 2 α 6 )X + (α 6 − α 1 )Y. This system is an autonomous version of the fifth Painlevé system. Appendix D In this section, we present a 3-parameter family of ordinary differential systems in dimension three explicitly given by (189)                dx dt = x 2 − xy − (α 1 − 2α 3 )x + (α 1 − α 3 )y − (α 1 − α 3 )α 3 , dy dt = y 2 − xy + xz − yz + (α 1 − α 2 )x − (α 1 − α 2 + α 3 )y + α 3 z + (α 1 − α 2 )α 3 , dz dt = z 2 − 3xz + 3α 2 x + (3α 1 − 2α 2 − 3α 3 )z − α 2 (3α 1 − α 2 − 3α 3 ). Here x, y, z denote unknown complex variables and α i (i = 1, 2, 3) are complex parameters. Proposition 18.1. This system has I =2x 3 (z − α 2 ) + x 2 {y 2 − 2y(z + α 1 − α 2 ) − 2(2α 1 − 3α 3 )z − 6α 2 α 3 + α 2 1 + 4α 1 α 2 } − 2x(α 1 − α 3 ){y 2 − 2y(z + α 1 − α 2 ) − (α 1 − 3α 3 )z + α 2 1 + α 1 α 2 − 3α 2 α 3 } + (α 1 − α 3 ) 2 {y 2 − 2y(z + α 1 − α 2 ) + 2α 3 z} (190) as its first integral. The following Lemma shows that the rational vector fieldṽ associated with the system (189) has seven accessible singular points on the boundary divisor H ⊂ P 3 . Lemma 18.2. The rational vector fieldṽ associated with the system (189) has seven accessible singular points: (191)                                      P 1 = {(X 1 , Y 1 , Z 1 )\|X 1 = Y 1 = Z 1 = 0}, P 2 = {(X 2 , Y 2 , Z 2 )\|X 2 = Y 2 = Z 2 = 0}, P 3 = {(X 3 , Y 3 , Z 3 )\|X 3 = Y 3 = Z 3 = 0}, P 4 = (X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 4 3 , Z 1 = 8 3 , P 5 = (X 2 , Y 2 , Z 2 )\|X 2 = Y 2 = 0, Z 2 = 1 2 , P 6 = {(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = 3}, P 7 = {(X 1 , Y 1 , Z 1 )\|Y 1 = 1, X 1 = Z 1 = 0}, where (X i , Y i , Z i ) are given by (39). This lemma can be proven by a direct calculation. Next let us calculate its local index at each point. Singular point Type of local index P 1 (1, 2, 4) P 2 (2, 1, 1) P 3 (1, 2, 1) P 4 (1, 6, 4) P 5 (3, 1, −2) P 6 (0, 2, −3) P 7 (0, −1, 2) By resolving the accessible singular points, we can obtain the phase space X for the system (189). Theorem 18.3. The phase space X for the system (189) is obtained by gluing six copies of C 3 : U j = C 3 ∋ {(x j , y j , z j )}, j = 0, 1, . . . , 5 via the following birational transformations: 0) x 0 = x, y 0 = y, z 0 = z, 1) x 1 = 1 x , y 1 = (y − α 1 )x, z 1 = x 3 (z − α 2 ), 2) x 2 = (x − (α 1 − α 3 ))y, y 2 = 1 y , z 2 = z, 3) x 3 = x, y 3 = (y − x − α 3 )z, z 3 = 1 z , 4) x 4 = 1 x , y 4 = − 2 3 x 5 (16x − 6y − 3z − 10α 1 + 3α 2 + 16α 3 ), z 4 = x 3 (8x − 4y − z − 4α 1 + α 2 + 8α 3 ), 5) x 5 = −(x − y + α 3 )(x − α 1 + α 3 )z, y 5 = − 1 (x − y + α 3 )z , z 5 = 1 z .(192) Theorem 18.4. Let us consider a system of the first-order ordinary differential equations in the polynomial class: dx dt = f 1 (x, y, z), dy dt = f 2 (x, y, z), dz dt = f 3 (x, y, z). We assume that (A1) deg(f i ) = 2 with respect to x, y, z. (A2) The right-hand side of this system becomes again a polynomial in each coordinate system (x i , y i , z i ) (i = 1, 2, 3, 4). Then such a system coincides with the system (189). Appendix E The Chazy-XI equation with N = 3 is explicitly given by (193) d 3 u dt 3 = 3u 4 + 6u 2 du dt + du dt 2 − 2u d 2 u dt 2 . Here u denotes unknown complex variable. Proposition 19.1. The equation (193) is equivalent to the system of the first-order ordinary differential equations: (194)                dx dt = x 2 − 2xy − 2yz, dy dt = y 2 − 2xy, dz dt = xz. Here x, y, z denote unknown complex variables. Proposition 19.2. This system admits rational solutions: (195)                x(t) = − 1 t + c 1 , y(t) = 3(t 2 + 2c 1 t + c 2 1 ) t 3 + 3c 1 t 2 + 3c 2 1 t + 3c 2 , z(t) = 1 t + c 1 (c 1 , c 2 ∈ C), and(196)           x(t) = − c 2 c 2 t − c 1 , y(t) = 0, z(t) = − 1 c 2 t − c 1 (c 1 , c 2 ∈ C). Proposition 19.3. This system has (197) I = (x + z)y 2 z 3 as its first integral. By using this, elimination of x from the system (194) gives the second-order ordinary differential system for (y, z); namely,        dy dt = y 2 − 2(I − y 2 z 4 ) yz 3 , dz dt = I − y 2 z 4 y 2 z 2 .(198) By making a change of the variables (199)        X = − 1 yz 2 , Y = 1 z , we obtain an autonomous version of Painlevé IV system: (200)      dX dt = Y 2 , dY dt = −IX 2 + 1. Now, we give a generalization of the system (200) in addition to constant complex parameters: (201)      ds dt = c 2 − as + α 1 , dc dt = −s 2 + ac + α 2 . Here s, c denote unknown complex variables, and α i , a are constant complex parameters. Theorem 19.4. The system (201) has extended affine Weyl group symmetry of type A (1) 2 , whose generators w i , π are given by w 0 ( * ) → s − α 1 + α 2 s + c + a , c + α 1 + α 2 s + c + a ; −α 2 , −α 1 , w 1 ( * ) → s − (−1) via the following birational and symplectic transformations: 0) x 0 = s, y 0 = c, 1) x 1 = 1 s , y 1 = −((c + s + a)s − α 1 − α 2 )s, 2) x 2 = 1 s , y 2 = −{(c − (−1) 1 3 s + (−1) 2 3 a)s − (−1) 1 3 ((−1) 1 3 α 1 − α 2 )}s, 3) x 3 = 1 s , y 3 = −{(c + (−1) 2 3 s − (−1) 1 3 a)s − (−1) 2 3 ((−1) 2 3 α 1 + α 2 )}s. The following Lemma shows that this rational vector fieldṽ associated with the system (194) has four accessible singular loci on the boundary divisor H ⊂ P 3 . Lemma 19.6. The rational vector fieldṽ associated with the system (194) has four accessible singular loci: (202)              C 1 ∪ C 3 = {(X 1 , Y 1 , Z 1 )\|X 1 = Y 1 = 0} ∪ {(X 3 , Y 3 , Z 3 )\|Y 3 = Z 3 = 0} ∼ = P 1 , P 2 = {(X 2 , Y 2 , Z 2 )\|X 2 = Y 2 = Z 2 = 0}, P 4 = {(X 1 , Y 1 , Z 1 )\|X 1 = Z 1 = 0, Y 1 = 1}, P 5 = {(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 3, Z 1 = −1}, where (X i , Y i , Z i ) are given by (39). This lemma can be proven by a direct calculation. Next let us calculate its local index at each point. Singular point Type of local index P 1 (−1, −3, 0) P 2 (−3, −1, −1) P 4 (1, 3, 2) P 5 (−1, 3, −6) P 6 (0, −3c, −c) Here, the notations P 1 , P 3 and P 6 are given by (203)        P 1 = {(X 1 , Y 1 , Z 1 )\|X 1 = Y 1 = Z 1 = 0} ∈ C 1 , P 3 = {(X 3 , Y 3 , Z 3 )\|X 3 = Y 3 = Z 3 = 0} ∈ C 3 , P 6 = {(X 3 , Y 3 , Z 3 )\|Y 3 = Z 3 = 0, X 3 = c} ∈ C 3 . Theorem 19.7. The phase space X for the system (194) is obtained by gluing six copies of C 3 : U j = C 3 ∋ {(x j , y j , z j )}, j = 0, 1, . . . , 5 via the following birational transformations: 0) x 0 = x, y 0 = y, z 0 = z, 1) x 1 = 1 x , y 1 = x 2 y, z 1 = z x , 2) x 2 = (x + z)y 2 , y 2 = 1 y , z 2 = z, 3) x 3 = x z , y 3 = yz 2 , z 3 = 1 z , 4) x 4 = 1 x , y 4 = (y − x + 2z)x 2 , z 4 = xz, 5) x 5 = 1 x , y 5 = 1 x 2 y , z 5 = (x + z)x 3 y 2 .(204) Appendix F In 1881, Halphen studied an integrable third-order ordinary differential system [17,18]: (205)                dx dt = x 2 + γ(x − y) 2 + β(z − x) 2 + α(y − z) 2 , dy dt = y 2 + γ(x − y) 2 + β(z − x) 2 + α(y − z) 2 , dz dt = z 2 + γ(x − y) 2 + β(z − x) 2 + α(y − z) 2 , where x, y, z denote unknown complex variables and α, β, γ are complex parameters. It is known that this system can be solved by hypergeometric functions (see [17,18]). Proposition 20.1. This system is invariant under the following transformations: s 0 (x, y, z; α, β, γ) →(y, x, z; β, α, γ), s 1 (x, y, z; α, β, γ) →(z, y, x; γ, β, α), s 2 (x, y, z; α, β, γ) →(x, z, y; α, γ, β), π(x, y, z; α, β, γ) →(y, z, x; β, γ, α). In [16], it is shown that when the three parameters α, β, γ are equal or when two of the parameters are 1/3 this system reduced to the generalized Chazy equation which is a classically known third-order scalar polynomial ordinary differential equation: XII : d 3 u dt 3 = 2u d 2 u dt 2 − 3 du dt 2 − 4 N 2 − 36 6 du dt − u 2 2 ,(207) where N is a positive integer not equal to 1 or 6. The general solution of the system (90) is densely branched for generic α, β, γ and so does not pass the Painlevé property. In this section, we study the system (205) from the viewpoint of its accessible singularities and local index. The following Lemma shows that this rational vector fieldṽ associated with the system (205) has seven accessible singular points on the boundary divisor H ⊂ P 3 . Lemma 20.2. The rational vector fieldṽ associated with the system (205) has seven accessible singular points: (208)                                                                        P 1 ={(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = 1}, P 2 = (X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = 2α + 2β + 1 − √ 4α + 4β + 1 2(α + β) , P 3 = (X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 1, Z 1 = 2α + 2β + 1 + √ 4α + 4β + 1 2(α + β) , P 4 = (X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 2α + 2γ + 1 − √ 4α + 4γ + 1 2(α + γ) , Z 1 = 1 , P 5 = (X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 2α + 2γ + 1 + √ 4α + 4γ + 1 2(α + γ) , Z 1 = 1 , P 6 ={(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 2β + 2γ + 1 − √ 4β + 4γ + 1 2(β + γ) , Z 1 = 2β + 2γ + 1 − √ 4β + 4γ + 1 2(β + γ) }, P 7 ={(X 1 , Y 1 , Z 1 )\|X 1 = 0, Y 1 = 2β + 2γ + 1 + √ 4β + 4γ + 1 2(β + γ) , Z 1 = 2β + 2γ + 1 + √ 4β + 4γ + 1 2(β + γ) }, where (X i , Y i , Z i ) are given by (39). This lemma can be proven by a direct calculation. Next let us calculate its local index at each point. Singular point Local index (a (i) 1 , a (i) 2 , a (i) 3 ) P 1 (−1, 1, 1) P 2 (− 4α+4β+1− √ 4α+4β+1 2(α+β) , 2 1+ √ 4α+4β+1 , − 4α+4β+1− √ 4α+4β+1 2(α+β) ) P 3 (− 4α+4β+1+ √ 4α+4β+1 2(α+β) , − 2 −1+ √ 4α+4β+1 , − 4α+4β+1+ √ 4α+4β+1 2(α+β) ) P 4 (− 4α+4γ+1− √ 4α+4γ+1 2(α+γ) , − 4α+4γ+1− √ 4α+4γ+1 2(α+γ) , 2 1+ √ 4α+4γ+1 ) P 5 (− 4α+4γ+1+ √ 4α+4γ+1 2(α+γ) , − 4α+4γ+1+ √ 4α+4γ+1 2(α+γ) , − 2 −1+ √ 4α+4γ+1 ) P 6 (− 4β+4γ+1− √ 4β+4γ+1 2(β+γ) , − 4β+4γ+1− √ 4β+4γ+1 2(β+γ) , − 2 1+ √ 4β+4γ+1 ) P 7 (− 4β+4γ+1+ √ 4β+4γ+1 2(β+γ) , 1+ √ 4β+4γ+1 2(β+γ) , − 4β+4γ+1+ √ 4β+4γ+1 2(β+γ) ) It is easy to see that the system (205) admits a rational solution: (209) x(t) = − 1 t − t 0 , y(t) = − 1 t − t 0 , z(t) = − 1 t − t 0 (t 0 ∈ C), which passes through P 1 . Let us take the coordinate system (p, q, r) centered at the point P 6 : p = 1 x , q = y x − 2β + 2γ + 1 − √ 4β + 4γ + 1 2(β + γ) , r = z x − 2β + 2γ + 1 − √ 4β + 4γ + 1 2(β + γ) . Making a linear transformation to arrive at    dX dt dY dt dZ dt    = 1 X         − 4β+4γ+1− √ 4β+4γ+1 2(β+γ) 0 0 0 − 4β+4γ+1− √ 4β+4γ+1 2(β+γ) K 1 0 0 − 2 1+ √ 4β+4γ+1       X Y Z    + · · ·      , where K 1 is given by K 1 := − 2β + 2γ + 1 − √ 4β + 4γ + 1 2(β + γ) − √ 2(β + γ) 1 + 4(β + γ) + 2(β + γ) 2 + (2β + 2γ + 1) √ 4β + 4γ + 1 . Let us take the coordinate system (p, q, r) centered at the point P 7 : p = 1 x , q = y x − 2β + 2γ + 1 + √ 4β + 4γ + 1 2(β + γ) , r = z x − 2β + 2γ + 1 + √ 4β + 4γ + 1 2(β + γ) . Making a linear transformation to arrive at    dX dt dY dt dZ dt    = 1 X         − 4β+4γ+1+ √ 4β+4γ+1 2(β+γ) 0 0 0 1+ √ 4β+4γ+1 2(β+γ) 0 0 K 2 − 4β+4γ+1+ √ 4β+4γ+1 2(β+γ)       X Y Z    + · · ·      , where K 2 is given by K 2 := − 2β + 2γ + 1 + √ 4β + 4γ + 1 2(β + γ) − √ 2(β + γ) 1 + 4(β + γ) + 2(β + γ) 2 − (2β + 2γ + 1) √ 4β + 4γ + 1 . if and only if the parameters α, β and γ satisfy the conditions: (215)                  1 √ 4α + 4β + 1 = l, 1 √ 4α + 4γ + 1 = m, 1 √ 4β + 4γ + 1 = n, where (l, m, n) ∈ Z 3 . This equation can be solved by (216)                  α = 1 8 1 l 2 + 1 m 2 − 1 n 2 − 1 , β = 1 8 1 l 2 − 1 m 2 + 1 n 2 − 1 , γ = 1 8 − 1 l 2 + 1 m 2 + 1 n 2 − 1 , where (l, m, n) ∈ N 3 . Under the condition (216), we see that K 1 = 0, K 2 = 0. Next, we give a generalization in dimension four of the second Halphen equation given by (217)                        dx dt = x 2 + α(x − y) 2 + β(x − z) 2 + χ(x − w) 2 + δ(y − z) 2 + ε(y − w) 2 + γ(z − w) 2 , dy dt = y 2 + α(x − y) 2 + β(x − z) 2 + χ(x − w) 2 + δ(y − z) 2 + ε(y − w) 2 + γ(z − w) 2 , dz dt = z 2 + α(x − y) 2 + β(x − z) 2 + χ(x − w) 2 + δ(y − z) 2 + ε(y − w) 2 + γ(z − w) 2 , dw dt = w 2 + α(x − y) 2 + β(x − z) 2 + χ(x − w) 2 + δ(y − z) 2 + ε(y − w) 2 + γ(z − w) 2 , where x, y, z, w denote unknown complex variables and α, β, χ, δ, ε, γ are complex parameters. where m i ∈ Z (i = 1, 2, . . . , 7). This equation can be solved by (222)                                              Figure 1 . 1Each figure denotes 3-dimensional projective space (X, Y, Z) ∈ C 3 ⊂ P 3 . The solutions u 1 , u 2 and v 3 are rational solutions which pass through an accessible singular point (see Section 3) in the boundary divisor H, respectively. The solutions v 1 , v 2 are new meromorphic solutions which pass through an accessible singular point in the boundary divisor H, respectively. The equation (1) can be written by the variable u and new equation (4) can be written by the variable v := − du dt u . For the equation (1) we can not find the solutions with initial conditions in the part surrounding the dotted line. The solutions u 2 (t), v 3 (t) are common in each equation. Example 3. 2 . 2For an example of the condition (23), let us consider Proposition 3. 3 . 3The birational transformation (32) at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)} are all positive integers can obtain the local index (1, −3, 10) at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) Proposition 9. 1 . 1The birational transformation s 0 (89) Theorem 9. 3 . 3The Chazy IX equation (61) is invariant under the biratioral transformation: Figure 3 . 3Proposition 9.4. The birational transformation π (95) at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)} are all positive integers . and we can obtain the local index(1,4,3) at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) = at the point {(x 2 , y 2 , z 2 ) = (0, 0, 0)}. The continued ratio of the local index at the point {(x 2 , y 2 , z 2 ) Proposition 16. 4 . 4The transformation (173) ϕ : (X, Y, Z; α, β, γ) → (X, Y, Z; −α, β, γ) These transformations are new. It should be clear from the form of s 0 and s 1 that the transformations become auto-Bäcklund transformations for the system (155) only if α = β = 0. The transformation π can be obtained by the compositions of the transformations (168), (173) and (172). ( 213 ) 213Proposition 20.3. Each local index at each accessible singular point P i , (i = 1, 2, . . . , 7) satisfies the condition: Theorem 12.3. The Chazy X.a equation (107) can be transformed into the Chazy X.b equation by the birational transformation: Proposition 20.4. This system is invariant under the following transformations: s 1 ( * ) →(y, x, z, w; α, δ, ε, β, χ, γ), s 2 ( * ) →(z, y, x, w; δ, β, γ, α, ε, χ), s 3 ( * ) →(w, y, z, x; ε, γ, χ, δ, α, β), s 4 ( * ) →(x, z, y, w; β, α, χ, δ, γ, ε), s 5 ( * ) →(x, w, z, y; χ, β, α, γ, ε, δ), s 6 ( * ) →(x, y, w, z; α, χ, β, ε, δ, γ), π( * ) →(y, z, w, x; δ, ε, α, γ, β, χ),where the symbol ( * ) denotes (x, y, z, w; α, β, χ, δ, ε, γ), and s 2 i = 1, π 4 = 1.Lemma 20.5. The system (217) has fifteen accessible singular points on the boundary divisor H ⊂ P 4 .It is easy to see that the system (217) admits a rational solution:For the system (217), each local index at each accessible singular point P i , (i = 1, 2, . . . , 15) satisfies the condition:if and only if the parameters α, β, χ, δ, ε and γ satisfy the conditions: M J Ablowitz, P A Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. C. U. P., Cambridge149M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, L. M. S. Lect. Notes Math. 149, C. U. P., Cambridge, (1991). Integration of some nonlinear systems of ordinary differential equations, Annali di Matematica. F Bureau, 94F. Bureau, Integration of some nonlinear systems of ordinary differential equations, Annali di Matem- atica. 94 (1972), 345-359. Reductions of self-dual Yang-Mills fields and classical systems. S Chakravarty, M J Ablowitz, P A Clarkson, Phys. Rev. Lett. 65S. 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[ "Differential Private Discrete Noise Adding Mechanism: Conditions, Properties and Optimization", "Differential Private Discrete Noise Adding Mechanism: Conditions, Properties and Optimization" ]	[ "Shuying Qin ", "Jianping He [email protected].‡: ", "Chongrong Fang [email protected].‡: ", "James Lam [email protected]. ", "\nThe Dept. of Automation\nMinistry of Education of China\nShanghai Engineering Research Center of Intelligent Control and Management\nThe Dept. of Mechanical Engineering\nthe Key Laboratory of System Control and Information Processing\nShanghai Jiao Tong University\nShanghaiChina\n", "\nThe University of Hong Kong\nPokfulam RoadHong Kong\n" ]	[ "The Dept. of Automation\nMinistry of Education of China\nShanghai Engineering Research Center of Intelligent Control and Management\nThe Dept. of Mechanical Engineering\nthe Key Laboratory of System Control and Information Processing\nShanghai Jiao Tong University\nShanghaiChina", "The University of Hong Kong\nPokfulam RoadHong Kong" ]	[]	Differential privacy is a standard framework to quantify the privacy loss in the data anonymization process. To preserve differential privacy, a random noise adding mechanism is widely adopted, where the trade-off between data privacy level and data utility is of great concern. The privacy and utility properties for the continuous noise adding mechanism have been well studied. However, the related works are insufficient for the discrete random mechanism on discretely distributed data, e.g., traffic data, health records. This paper focuses on the discrete random noise adding mechanisms. We study the basic differential privacy conditions and properties for the general discrete random mechanisms, as well as the trade-off between data privacy and data utility. Specifically, we derive a sufficient and necessary condition for discrete -differential privacy and a sufficient condition for discrete ( , δ)-differential privacy, with the numerical estimation of differential privacy parameters. These conditions can be applied to analyze the differential privacy properties for the discrete noise adding mechanisms with various kinds of noises.Then, with the differential privacy guarantees, we propose an optimal discrete -differential private noise adding mechanism under the utility-maximization framework, where the utility is characterized by the similarity of the statistical properties between the mechanism's input and output. For this setup, we find that the class of the discrete noise probability distributions in the optimal mechanism is Staircase-shaped.	10.48550/arxiv.2203.10323	[ "https://arxiv.org/pdf/2203.10323v1.pdf" ]	247,595,227	2203.10323	48daf5ffaae600dc8fc317c7654cabe4ae2108e5	Differential Private Discrete Noise Adding Mechanism: Conditions, Properties and Optimization 19 Mar 2022 Shuying Qin Jianping He [email protected].‡: Chongrong Fang [email protected].‡: James Lam [email protected]. The Dept. of Automation Ministry of Education of China Shanghai Engineering Research Center of Intelligent Control and Management The Dept. of Mechanical Engineering the Key Laboratory of System Control and Information Processing Shanghai Jiao Tong University ShanghaiChina The University of Hong Kong Pokfulam RoadHong Kong Differential Private Discrete Noise Adding Mechanism: Conditions, Properties and Optimization 19 Mar 20221 Preliminary results have been accepted by 2022 American Control Conference [1]. 2Index Terms Differential privacyDiscrete random mechanismNoise adding processWasserstein distance Differential privacy is a standard framework to quantify the privacy loss in the data anonymization process. To preserve differential privacy, a random noise adding mechanism is widely adopted, where the trade-off between data privacy level and data utility is of great concern. The privacy and utility properties for the continuous noise adding mechanism have been well studied. However, the related works are insufficient for the discrete random mechanism on discretely distributed data, e.g., traffic data, health records. This paper focuses on the discrete random noise adding mechanisms. We study the basic differential privacy conditions and properties for the general discrete random mechanisms, as well as the trade-off between data privacy and data utility. Specifically, we derive a sufficient and necessary condition for discrete -differential privacy and a sufficient condition for discrete ( , δ)-differential privacy, with the numerical estimation of differential privacy parameters. These conditions can be applied to analyze the differential privacy properties for the discrete noise adding mechanisms with various kinds of noises.Then, with the differential privacy guarantees, we propose an optimal discrete -differential private noise adding mechanism under the utility-maximization framework, where the utility is characterized by the similarity of the statistical properties between the mechanism's input and output. For this setup, we find that the class of the discrete noise probability distributions in the optimal mechanism is Staircase-shaped. I. INTRODUCTION A. Background Data anonymization, namely, preventing information from being re-identified [2], is an important approach to protect data privacy in data publishing. Random noise adding mechanism is a classic method to achieve data anonymization. Usually, few random mechanisms can fully protect privacy, i.e., the privacy loss is non-negligible. To quantify the privacy loss, many privacy frameworks are proposed, include information-theoretic privacy [3], differential privacy [4], and privacy based on secure multiparty computation [5], etc. The frameworks differ mostly in the privacy guarantee strength. Due to the strong privacy guarantee brought by differential privacy, this brand-new privacy framework has received wide attention. It is introduced by Dwork et al. [6], where the idea is inspired by the probabilistic encryption. The innovation lies in that it is a property towards the anonymization process (i.e., a random noise adding mechanism) rather than the datasets. Thanks to such a useful property, the differential privacy framework is widely employed in many areas, such as distributed optimization [7], [8], control and network systems [9]- [12], filtering [13], [14] and others [15]- [18], etc. Meanwhile, it is able to obtain privacy guarantees and analyze how much information is leaked, e.g., when processing the telemetry data [19], census data [20], and medical data [21], etc. Note that the differential privacy preservation achieved by the random noise adding mechanisms is at the cost of the data utility. Many scholars are dedicated to studying the trade-off between the privacy level and utility for the random noise adding mechanisms. B. Motivations There is a large amount of discrete data in practice, e.g., census records, traffic data, etc. The data privacy is to be preserved by the random noise adding mechanisms. To ensure the interpretability of the protected numerical discrete data, the random added noises in the mechanisms should be discretely distributed, i.e., discrete noise adding mechanisms. When the random noises satisfy the continuous Lipschitz and continuous differentiability, researchers have carried out a series of differential privacy condition studies on the continuous data. However, the conditions of the Lipschitz continuity and the differentiability are not guaranteed under the discrete scenarios. It is unclear what would be the problem if the continuous differential privacy conditions were directly applied to the discrete noise adding mechanisms. Besides, it remains unknown whether the properties for the well-known continuous noise adding mechanisms (e.g., the Laplacian and the Gaussian mechanisms) can be maintained for the discrete ones, and whether more differential privacy properties are available. These issues call for discrete privacy-critical studies to ensure the deployment of discrete differential private mechanisms. Moreover, to improve the utility for the published data, the trade-off between the differential privacy level and utility for the discrete random noise adding mechanisms should be considered. Most of the existing studies [22]- [24] model the utility function as a general function depending on the noise added to the query output. This utility measure is reasonable but indirect, since the utility is maximized in terms of minimizing the noises (e.g., the magnitude, the variance). To make the utility metric more intuitive, one challenge is whether there exists a utility function on the level of distortion after noise addition to the data. A new insight is given by the similarity degree of the statistical properties between the original data and the noise added data (i.e., the mechanism's input and output). Given that the inputs and outputs are random variables, the similarity degree can be captured by the probability distributions, a more comprehensive characterisation than the statistical information such as variance and expectation, etc. Specifically, the degree of similarity is usually quantified by the distance function between the probability distributions [25]. The commonly used distance functions of interest include Kullback-Leibler (KL) divergence [26], Jensen-Shannon (JS) divergence [27], and Wasserstein distance [28]. In this paper, we adopt the Wasserstein distance as the utility metric. In contrast to the KL divergence, it satisfies the basic properties of distance (non-negativity, identity of indiscernible, symmetry and triangle inequality). Compared with the JS divergence, it takes into account the geometric properties between two probability distributions. With the innovative utility model, it would be desirable to provide an implementable discrete random noise adding mechanism for the utility optimization problem. However, the explicit expressions for the general form of the Wasserstein distance are rare, except the one-dimensional Gaussian cases. Besides, even adopting the one-dimensional case directly as an optimization objective, it is still a non-convex optimization problem. Therefore, it is necessary to find an equivalent form of the primal problem to transform the unconventional optimization problem into a solvable one. C. Contributions Motivated by the above observations, in this paper, we study the differential privacy conditions, properties and utility optimization for the discrete random noise adding mechanisms. Beginning with the definition of the discrete data, we first clarify the discrete random noise adding mechanism. As for the differential privacy analysis, we find that the conditions for the discrete differential private mechanism are further simplified compared with the conditions for the continuous mechanism [29]. The differential privacy parameters estimation results remain a certain similarity. Also, compared with the continuous random noise adding mechanism, the differential privacy properties hold well in most discrete scenarios, e.g., the discrete Gaussian, Laplacian and Exponential mechanisms. Especially, our results for the discrete Gaussian noise adding mechanism are consistent with the literature [30] to some extent. More concretely, we obtain the same differential privacy properties and similar differential privacy parameters estimation. Moreover, as for the trade-off between the privacy level and utility, we select the Wasserstein distance as a new utility measure. The innovation lies in that the utility model measures the distance between the input and output of the proposed mechanism, by taking the geometric properties of these two discrete distributions into account. Then, we propose an equivalent form of the primal problem that transforms the non-convex optimization problem into a linear programming problem. Finally, we obtain the optimal discrete -differential private mechanism with the Simplex Method [31]. In Table I, we compare various works on the differential privacy and the utility properties for the random noise adding mechanisms. The differences between this paper and its conference version [1] include i) the analysis of the differential privacy properties for the discrete Exponential noise adding mechanism, ii) the optimization of the discrete noise adding mechanisms, i.e., maximizing the data utility under the differential privacy guarantees, iii) the sufficient simulations on the optimal discrete noise adding mechanism. The main contributions are summarized as follows. • (Conditions.) We investigate general differential privacy conditions for the discrete noise adding mechanisms, i.e., a sufficient and necessary condition for -differential privacy, and a sufficient one for ( , δ)-differential privacy. Moreover, we obtain a numerical method to estimate the two privacy parameters and δ. • (Properties.) We analyze the differential privacy properties and provide the privacy guarantees for the representative discrete noise adding mechanisms with the obtained theories. In detail, we investigate the mechanisms under the discrete Gaussian, Laplacian, Staircaseshaped, Uniform, Exponential distributed noises, respectively. • (Optimization.) We study the utility-maximization optimization for the -differential private mechanisms. Defining the utility as the Wasserstein distance between the mechanism input and output probability distributions, we derive an optimal discrete Staircase-shaped noise adding mechanism. Further, we conduct extensive simulations to verify its optimality. D. Organization The remainder of this paper is organized as follows. The related works are investigated in Section II. Section III states necessary preliminaries. In Section IV, we give theoretical differential privacy conditions and parameters estimation, perform further analysis on the differential privacy properties, and propose a discrete differential private mechanism with the maximum utility. Section V provides evaluations for the mechanism optimality. Finally, conclusions are given in Section VI. II. RELATED WORK Since Dwork [6] first introduced the differential privacy definition in 2006, it has become the flagship data privacy definition. Shortly after it was proposed, numerous attack models and different scenarios are adapted to the variants and extensions of the differential privacy [32], [33]. More recently, Desfontaines et al. [34] gave a systematic taxonomy of the existing differential privacy definitions (approximately 225 kinds). They compared the definitions from seven dimensions, and showed how the new differential privacy definitions are formed with the combination of different dimensions. This work allowed new practitioners to have a general idea of the differential privacy research area. The majority differential privacy researches focus on the continuous random noise adding mechanisms, in a bid to achieve anonymity protection for the continuously distributed data. Regarding the differential privacy analysis for the general continuous random noise adding mechanisms, He et al. [29] proposed a sufficient and necessary differential privacy condition, with the privacy parameters estimation. The basic theories can be applied to analyze various random noises. Then, they performed in-depth analysis on the differential privacy properties, and applied obtained theories on consensus algorithms. Apart from the related analysis for the general continuous random mechanisms, differential privacy is widely discussed under a specific continuous random noise adding mechanism [35]. For instance, the continuous Gaussian noise adding mechanism preserves ( , δ)-differential privacy for the query functions with infinite dimensions and real values [36]. Besides, the random mechanism with the continuous Laplacian distributed [6] noise guarantees -differential privacy. So far, the differential privacy properties for the continuous random noise adding mechanism have been widely studied, but it is unknown how the results are suitable for the discrete one (adding discrete random noise on discretely distributed data). Recently, researchers have paid attention to the discrete random differential private mechanisms. For instance, the Exponential mechanism is a well-known discrete noise adding mechanism that guarantees -differential privacy [35], [37]. It aims to protect non-numerical discrete data. In this mechanism, a scoring function is introduced for the query output, and then the final probability of the output is determined by the score. The analysis of the mechanism is mature, but it does not necessarily apply to the mechanism that protects numerical discrete data. Furthermore, Canonne et al. [30] studied the differential privacy properties for the discrete Gaussian noise adding mechanism. They obtained that the discrete Gaussian mechanism guarantees essentially the same level of privacy and accuracy as the continuous one. Apart from the related properties, Koskela et al. [38] proposed a Fourier transform based numerical method to compute the differential privacy parameters for discrete-valued mechanisms. Specifically, they evaluated the privacy loss for the discrete ( , δ)-differential private mechanisms, and provided the lower and upper ( , δ)-differential privacy bounds for the subsampled discrete Gaussian mechanism. Despite the excellent properties, we cannot apply the specific conclusions to general discrete noise adding mechanisms, which makes the analysis of differential privacy in distinct scenarios more difficult. In addition to the extensive research on the differential privacy properties, some works further consider the fundamental trade-off between the privacy level and utility for the random noise adding mechanisms, which are the two vital properties for the mechanisms. Gupte et al. [23] found that the optimal differential private mechanism is achieved by adding Geometric distributed noise, on the basis of a fixed query sensitivity. Based on the decision theory, they took the information loss caused by the random noise uncertainty as the utility measure. More concretely, the objective function was to minimize the worst case of the noise variance or the expected magnitude. In this line of research, [24] generalized the fixed sensitivity to an arbitrary value and derived the optimal noise with the Staircase-shaped distribution. The utility metric they adopted was the same as the one in [23]. This utility model is rational and risk-averse, but it is hard to determine how the added noise affects the original data with this widely-used model directly. Based on the related work mentioned above, in this paper, we investigate the discrete differential private noise adding mechanism. III. PRELIMINARIES In this section, we mainly introduce the discrete random noise adding mechanisms, the differential privacy definition and our utility metric for the random mechanisms. A. Preliminaries of Discrete Random Mechanisms First, we specify the discrete quantitative data discussed in this paper, by introducing a set of discrete numbers with interval ∆, which is given by Z ∆ = {k \|k = k 0 ∆, k 0 ∈ Z} , ∆ ∈ R + . Denote Z + ∆ as a set of positive numbers in Z ∆ , and Z n ∆ as a set of n-dimensional column vectors L = [x 1 , x 2 , . . . , x n ] T , where x i ∈ Z ∆ , i ∈ V = {1, 2, . . . , n}. With the basic concept of the discrete data Z ∆ , we then introduce a discrete random noise adding mechanism, which is utilized to achieve privacy protection. Specifically, this mechanism is a randomized function that takes the original data as input and returns an output after adding 8 random noises. Let Ω, Θ, S ⊆ Z n ∆ represent the n-dimensional input, noise, output space, respectively. Note that S Ω ⊕ Θ, where ⊕ refers to the sum of elements with the same dimension. Then, the general discrete random noise adding mechanism A : Ω → S is given by A (x) = x + h (ϑ) , ∀x ∈ Ω, ϑ ∈ R n , h(ϑ) ∈ Θ,(1) where ϑ = [ϑ 1 , . . . , ϑ n ] T and the output A (x) is a n-dimensional random variable. Note that if the added noises are not discretely distributed at initial, then the interpretability and validity of the original discrete data will be destroyed. To avoid this case, we define the function h : R n → Z n ∆ as a discretization function, which maps the continuous added noise ϑ ∈ R n to the discrete one. In terms of probability distributions, we propose a general discretization method for the variable ϑ i in every dimension, which is shown as p ϑ i (k) = k+∆ k f (ϑ i ) dϑ i , i ∈ V,(2) where k ∈ Z ∆ and f (ϑ i ) refers to the probability density function of the original continuous random noise. The term p ϑ i (k) is the probability of the discretized random variable ϑ i when ϑ i = k. In summary, the discrete random noise adding mechanism A represents the process of adding discrete random noise to the discretely distributed data. To make the expression more concise, we abbreviate it as the discrete random mechanism A. Further, we distinguish the mechanisms based on the added discrete noise distributions. For instance, we call the approach to perturb the data by adding discrete Gaussian distributed random noise as the Gaussian mechanism. Similarly, we define the Laplacian mechanism, the Staircase mechanism, and the Exponential mechanism, etc. B. Background on Differential Privacy In this subsection, we introduce the differential privacy (DP) properties for the discrete random mechanism A. In other words, if the mechanism realizes the privacy protection of the numerical discrete data measured by differential privacy, we call it a differential private mechanism. First, we adopt the adjacency definition to illustrate the protected data. Consider two ndimensional data that differ only in one dimension. Our goal is to preserve the privacy of this single record. That is, we are concerned with the value of the record rather than its presence in the data. Based on this observation, we give the definition of m-adjacency for two discrete vectors by referring to the studies in [29], [39]- [41]. Definition 1 (m-adjacency). Given m ∈ Z + ∆ , the pair of vectors x, y ∈ Z n ∆ is m-adjacent, if for a given i 0 ∈ V , we have ∀i ∈ V, \|x i − y i \| ≤    m, i = i 0 ; 0, i = i 0 .(3) From (3), we obtain that the pair of m-adjacent vectors x and y has the same size and differs only in one record with the same dimension, where the difference is no more than m. Next, we present the definition of ( , δ)-differential privacy for a discrete random mechanism A. Definition 2 (( , δ)-differential privacy). A discrete random mechanism A is ( , δ)-DP if for any pairs of m-adjacent vectors x and y, and for all O ⊆ S, we have Pr {A (x) ∈ O} ≤ e Pr {A (y) ∈ O} + δ.(4) Intuitively speaking, a DP mechanism will not reveal more than a bounded amount of information about the data in the probabilistic perspective. Note that and δ are two key DP parameters. The positive number measures the privacy maintained by the discrete random mechanism. More concretely, the term e quantifies the privacy loss across the mechanism outputs [42]. With the parameter → 0, the mechanism causes less privacy loss, i.e., achieves better degree of privacy protection. Moreover, for cases where the upper bound does not hold (privacy loss larger than e ), the parameter δ functions to compensate for outputs by allowing a small probability of error. Specifically, if the strong DP property holds (δ = 0), we denote -DP to replace ( , 0)-DP for a simplified expression. By referring to [34], more detailed DP assumptions and explanations are given in Table II. In the following sections, we analyze the DP properties for any given discrete random mechanism (i.e., -DP, or ( , δ)-DP), and give numerical estimation methods for the two DP parameters. C. Wasserstein Distance In the privacy-preserving process implemented by the discrete random mechanism A, in addition to the degree of privacy protection, we also focus on another crucial mechanism property, utility, which is characterized by the similarity of the statistical properties of the mechanism's input and output. In this subsection, we present a general definition of the utility measure, Wasserstein distance. and differ only in one record. The difference is no more than m. Privacy Level expressed in x, y Associate the data in each dimension with the same acceptable level of risk. Randomness: Pr{A(·) ∈ O} Only comes from the mechanism itself, (i.e., the added random noise θ). The input follows a certain probability distribution. Computational Power Assume infinite for attackers. Definition 3 (p-Wasserstein distance). The p-Wasserstein distance between two probability mea- sures u and v on R d is W p (u, v) = inf X∼u,Y ∼v (E X − Y p ) 1 p , p ≥ 1, where X and Y are two d-dimensional random vectors with marginals u and v. The infimum is taken over all joint distributions of the random variables X and Y , provided that the p-th moments exists. Intuitively, the distance W p (u, v) is the minimal effort required to reconstruct u's mass distribution into the v's. The effort is quantified by moving every unit of mass from x to y with the cost x − y p . In this paper, we focus on the special case of 1-Wasserstein distance on R d=1 . By referring to [28], the explicit formulae of the Wasserstein distance with p = 1, d = 1 is shown as: W p=1 (X, Y ) = R \|F X (t) − F Y (t)\| dt,(5) where F X (·), F Y (·) are the cumulative distribution functions (CDF) of the continuous random variables X and Y , respectively. Further, we extend (5) to the discrete situations as the basis of our utility model, with more detailed information illustrated in Section IV-C. x, y ∈ Z n ∆ A pair of m-adjacent vectors V A set of dimensions, = {1, 2, · · · , n} Ω ⊆ Z n ∆ The set of inputs of random mechanisms S ⊆ Z n ∆ The set of possible outputs of random mechanisms O ⊆ S The subset of possible outputs, Oi is a set of i-th column element in O, i ∈ V A : Ω → S A discrete random mechanism (probabilistic) A(·) The output of the mechanism A Θ ⊆ Z n ∆ The set of noises added to the mechanism inputs θ ∈ Θ The noise added to the mechanism input, where θi is the i-th element of the noise, i ∈ V p x/θ/x+θ The input / noise / output probability distribution p x/θ/x+θ (·) The Probability Mass Function (PMF) p θ i (k) The probability value of θi at point k, which is a simplified expression of P r(θi = k) P x/θ/x+θ (·) The Cumulative Distribution Function (CDF) IV. MAIN RESULTS In this section, we first propose the DP conditions for the discrete random mechanism A, followed by the estimation methods for the DP parameters and δ. Next, we analyze the DP properties for five representative mechanisms. Then, we consider the trade-off between the privacy level and utility, deriving a -DP mechanism with the maximum utility. In this paper, we consider the added noise is discrete by default, i.e., either it has been discretized by the method shown in (2) or it is originally discretely distributed. The simplified discrete random mechanism A is rewritten as: A (x) = x + θ,(6) where x ∈ Ω ⊆ Z n ∆ , θ ∈ Θ ⊆ Z n ∆ . Here, we use θ to substitute h (ϑ), a function of continuous random variables 1 mentioned in the general discrete random mechanism (1). A. DP Conditions and Parameters Estimation In this subsection, a sufficient and necessary condition for the -DP mechanism and a sufficient condition for the ( , δ)-DP mechanism are given by Theorem 1 and Theorem 2, respectively, with the numerical DP parameters estimation. First of all, we consider the -DP conditions for the discrete random mechanism A. Theorem 1. The discrete random mechanism A satisfies -DP if and only if (iff) there exists a positive constant c b such that sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ ,∀i∈V p θ i (k − m 0 ) p θ i (k) = c b ,(7) where k ∈ Z ∆ . Moreover, we have that c b is an increasing function of m. The privacy parameter is estimated by = log(c b ).(8) Proof. Please see the proof in the Appendix A. We make some explanation about the relationship between the privacy cost and the adjacency m. Theorem 1 shows that the privacy loss e = c b decreases with smaller adjacency m for the pair of two input vectors. It is consistent with our intuition that the original data with more similarity (smaller m) lead to lower privacy loss (e ), i.e., guaranteeing better privacy (smaller ). Furthermore, the existence of the least upper bound c b in (7) implies that the denominator p θ i (k) cannot be zero. For this setup, we obtain a necessary condition for -DP, i.e., ∀i ∈ V, p θ i (k) > 0, k ∈ Z ∆ .(9) Remark 1. We further explore the similarities and differences between the discrete DP conditions in Theorem 1 and the continuous results in [29]. First, the criteria for the discrete random -DP mechanisms (7) has the same essence as the continuous ones. It means that any adjacent 1 Notice that the random noise ϑ is continuously distributed and we denote h (ϑ) as a discretization process. The discretization result is a discretely distributed random variable θ ∈ Θ, which is the basis for the implementation of the discrete random mechanism A. To make the mechanism expression more concise, we replace the term h (ϑ) with θ. Both of them essentially represent the random variables with discrete distributions. We can use the simplified mechanism (6) to analyze the DP properties. probability ratio for the noise probability distribution should have an upper bound c b . With the DP parameter estimation (8), it implies that the privacy loss e will not be infinite in the process of protecting any distinct data. Meanwhile, the discrete conditions are the simplification of the continuous ones. For the continuous random noise distributions, due to the uncountability of the real number set, the potential infinite local maximum and minimum should be considered in any given interval. But for the discrete probability distributions, we only need pay attention to whether the probability value at single point is zero (as described in (9)). The difference shows that the DP parameter is highly related to how we discretize a continuous probability distribution. It is further explained in Section IV-B. In summary, Theorem 1 allows us to verify whether a given discrete random mechanism A is -DP or not, only relying on the properties of the added discrete noise probability distributions. This idea is distinguished from the existing work [43], which validates the DP properties for the mechanisms by the original DP definition. Next, we consider a more relax notion, ( , δ)-DP, for cases where the -DP conditions cannot be strictly met. In detail, we propose a sufficient condition to verify the ( , δ)-DP properties for the discrete random mechanism A, along with the estimation of the value of DP parameters and δ. Theorem 2. Let Θ ⊆ Z n ∆ be the set of discrete random variable θ. Suppose that Θ 0 and Θ 1 are two subsets of Θ, which satisfies Θ = Θ 0 Θ 1 and Θ 0 Θ 1 = ∅. Assume θ∈Θ 0 p θ i (k) ≤ δ,(10) and the condition (7) holds when θ ∈ Θ 1 , i.e., sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ ,θ∈Θ 1 p θ i (k − m 0 ) p θ i (k) = c b ,(11) where ∀i ∈ V, k ∈ Z ∆ . Then the discrete random mechanism A is ( , δ)-DP, and the privacy parameter is given by = log (c b ) .(12) Proof. Please see the proof in the Appendix B. To further verify the rationality of Theorem 2, we consider the extreme limitations of the two DP parameters and δ, according to (10) and (11), respectively: • Θ 1 → Θ and Θ 0 → ∅. It evolves into the -DP since δ = lim Θ 0 →∅ θ∈Θ 0 p θ i (k) = 0, i.e., Theorem 1 is satisfied. 1. Then, one implies that any mechanism A satisfies (0, 1)-DP. • Θ 0 → Θ and Θ 1 → ∅. Then we have δ = lim Θ 0 →Θ θ∈Θ 0 p θ i (k) = 1 and c b = lim Θ 1 →∅ sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ ,θ∈Θ 1 p θ i (k − m 0 ) p θ i (k) = 1, Note that only discussing the general limitations of the DP parameters in the second case is not sufficient, since it can be applied to arbitrary discrete random mechanisms, making the probability error δ meaningless here. Thus, it is worth to estimate the tight bound of and δ for every discrete random mechanism, which will be further discussed in Section IV-B. Now, we have obtained the conditions for both discrete -DP and ( , δ)-DP mechanisms. The corresponding DP parameters estimation approaches are summarized in Table IV. B. DP Properties and Privacy Guarantees In this subsection, we apply the obtained conditions to discuss the DP properties for two kinds of discrete random mechanisms. The first kind of mechanism is achieved by adding discrete noises that are discretized from the continuous ones. Here, four representative mechanisms are selected, i.e., the Gaussian, the Laplacian, the Staircase and the Uniform mechanisms. The second one is obtained through adding noises that are originally discretely distributed. The most commonly adopted mechanism is the Exponential mechanism. For each mechanism, we derive the DP properties ( -DP or ( , δ)-DP), followed by the esimated DP parameters based on Theorem 1 and Theorem 2. Recalling the mechanism definition in Section III-A, we denote every mechanism as the abbreviation of a random mechanism under the corresponding discrete probability distribution. In other words, the discrete data are anonymously protected by adding the specific kind of discrete random noise. Since most noises are given by continuous probability density functions (PDF), we first discretize them to obtain the discrete probability mass functions (PMF) based on the proposed discretization methods shown in (2). Note that the following analysis of DP properties is suitable for any discrete random mechanism regardless of the discretization methods. 1) The Gaussian mechanism: This mechanism is realized by adding the discrete Gaussian distributed noise, where the PMF shown in Fig. 1 is given by: p θ i (k) = k+∆ k 1 √ 2πσ 2 e − (θ i −µ) 2 2σ 2 dθ i , k ∈ Z ∆ ,(13) where the parameters µ and σ are the mean and the standard deviation of the original continuous distribution, respectively. and δ = max i∈V 1 √ 2πσ 2 Φ i k+∆ k e − (θ i −µ) 2 2σ 2 dθ i ,(15)where Φ i = (−∞, −M ] ∪ [M, ∞) ⊆ Z ∆ . Proof. Please see the proof in the Appendix C. Remark 2. We find that applying the DP conditions for the general discrete random mechanisms to the analysis of the specific Gaussian mechanism yields some similar DP conclusions (Theorem 3) with those in [30]. First, [30] also proved that the discrete Gaussian mechanism can only provide ( , δ)-DP guarantees despite different discretization methods, where the nonzero DP parameter δ = Pr θ i > σ 2 ∆ − ∆ 2 − e · Pr θ i > σ 2 ∆ + ∆ 2 determines that it cannot guarantee pure -DP. Then, we consider the similarity of the strict upper bounds on the permissible privacy probability error, δ. For instance, when we take the adjacency m = 1 (the same as the sensitivity in [30]), we estimate the parameter from (43) by = max m 0 ∈{0,1} 1 2σ 2 (m 0 + 1) (2M + 1 − m 0 ) = 2M σ 2 . Substituting it with the estimation approach of δ proposed in [30], i.e., δ ≤ 1 √ 2πσ 2 e − σ 2 2 2σ 2 , we have δ ≤ 1 √ 2πσ 2 e − 2M 2 σ 2 .(16) Besides, the estimation value based on our results (44) is δ ≤ 2 √ 2πσ 2 · +∞ k=M k+1 k e − k 2 2σ 2 .(17) The parameter estimations shown in (16) and (17) are slightly different, due to the distinct discretization methods. Note that the discrete Gaussian distribution in [30] comes from a natural analogue of the continuous Gaussian, i.e., p θ i (k) = e −(k−µ) 2 /2σ 2 k ∈Z e −(k −µ) 2 /2σ 2 ,(18) which is a more accurate but complicated discretization method. Especially, we find that the DP guarantees for the Gaussian mechanisms derived from both discretization methods are almost the same under a large standard deviation σ. Moreover, it is worth mentioning that our DP parameter estimation method (17) is more general, as it relies less on the specific noise probability distribution, thanks to the easier discretization approach in (2) than (18). 2) The Laplacian mechanism: Next, we analyze the DP properties for the Laplacian mechanism. The Laplacian distribution is discretized under (2) from the PDF (f (z) = 1 2λ e − \|z−µ\| λ ). With simplification, we have p θ i (k) =        1 − e −∆/λ 2 e µ−k λ , k ≥ µ; e ∆/λ − 1 2 e k−µ λ , k ≤ µ − ∆, where k ∈ Z ∆ , µ, λ are the same position and scale parameters as the continuous distribution, respectively. From Fig. 2, it is easy to obtain that the discrete Laplacian mechanism guarantees the -DP So we conclude that the Laplacian mechanism A is a discrete -DP mechanism, where the DP parameter is estimated by = log e m+∆ λ = m + ∆ λ ,(19) which is highly related to the scale parameter λ. Then, one implies that the mechanisms' DP properties have strong correlations with the parameters and properties of the specific discrete probability distributions. 3) The Staircase mechanism: Since the study in [24] pointed out that the continuous -DP Staircase mechanism performs best in maintaining the data utility, we are interested in this mechanism and hope to verity its DP properties with our general DP conditions. First, we derive discrete Staircase-shaped distribution by discretizing the continuous one in [29]. The probability distribution in Fig. 3 is obtained by p θ i (k) =      1 − ρ 2a ρ j , ja∆ ≤ k < (j + 1)a∆; 1 − ρ 2a ρ j , − (j + 1)a∆ ≤ k < −ja∆,(20) where k ∈ Z ∆ , a ∈ Z + , ρ ∈ {x \|0 < x < 1, x ∈ R} , j ∈ N. Here a and ρ represent the width and height of the Staircase-shaped distribution, respectively. It is easy to check (20) as a valid PMF, since +∞ k=−∞ p (k) = 2 +∞ j=0 a · 1 − ρ 2a ρ j = 1. Based on Theorem 1, we find that the Staircase mechanism is -DP, the same result as [24]. More concretely, for any given adjacency m, where m ∈ {m 0 \|ca∆ < m 0 ≤ (c + 1) a∆, a, c, ∈ Z + }, there exists a corresponding upper bound c b satisfying: sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ p θ i (k − m 0 ) p θ i (k) = 1 ρ c = c b . Then, the corresponding DP parameter is shown as: = log(c b ) = log ρ − m a∆ ,(21) where the term m a∆ represents the smallest integer that is not less than m a∆ . Furthermore, we have that the Staircase mechanism can preserve any given DP levels, with the design of the stair width a and the stair height ρ. 4) The Uniform mechanism: The DP properties for the discrete Uniform distributed noise adding mechanism can be easily obtained. The PMF of the discrete Uniform noise follows: p θ i (k) = ∆ b − a + ∆ , a ≤ k 0 ≤ b, k 0 ∈ Z ∆ . Obviously, the Uniform mechanism violates the -DP precondition in (9), since certain probability values are zero (intuitively shown in Fig. 2), which will lead to infinity privacy loss. Thus, we have that the Uniform mechanism is ( , δ)-DP. With the sufficient ( , δ)-DP condition in Theorem 2, the DP parameters are given by: = log(c b ) = log(1) = 0, δ = m∆ b − a + ∆ .(22) 5) The Exponential mechanism: Different from the above four discrete random mechanisms, the Exponentially distributed noise added in this mechanism is inherently discrete, i.e., the preprocessing of discretization is not needed. The PMF of the discrete Exponential noise in is given by: p θ i (k) = ηe −ηk , k ∈ Z + ∆ ,(23) where η is the rate parameter of the Exponential distribution. According to Theorem 1, we have that the Exponential mechanism is -DP. Because for any m 0 ∈ [−m, m] , m 0 ∈ Z ∆ and k > m 0 , we obtain that p θ i +m 0 (k) p θ i (k) = ηe −η(k−m 0 ) ηe −ηk = e ηm 0 < e ηm , i.e., the probability ratio is bounded. Meanwhile, the estimation of the DP parameter is = log e ηm = ηm. Thus, we conclude that the privacy cost is proportional to the adjacency m, which is consistent with the result in [35]. Remark 3. Despite the similar DP properties with [35], the Exponential mechanism we discuss here is slightly different from the existing Exponential mechanisms. The main reason is that the discrete data we are protecting can be represented numerically. The common Exponential mechanism protects non-numerical discrete data. To achieve differential privacy, the mechanism returns the originally determined result x with a certain probability value, which is highly related to a scoring function u. The function u gives every result x a score, where a higher score means a higher output probability. Formally, the Exponential mechanism A u, with the quality score u(x) and the privacy parameter is given by: A u, (x, m) ∼ e u(x) 2m . However, the Exponential mechanism A discussed in this paper aims to protect discretely distributed numerical data. Denote the Exponential distribution in (23) as Exp(η). Then, the Exponential mechanism A is shown as: A (x) = x + θ, θ ∼ Exp(η), which is in line with the definition of a general discrete random mechanism A in (6). Hence, we can apply the DP conditions in Section IV-A to analyze the Exponential mechanism. In summary, the DP properties for the several typical mechanisms ( (14), (15), (19), (21), (22), (24)) are listed in Table V. Based on the detailed analysis as well as the comparisons with the existing work, we verify the validity of our conclusions. Gaussian ( , δ)-DP 1 2σ 2 (m+∆) (2M +∆−m− 2µ) 2 +∞ k=M p θ i (k) Laplacian -DP m+∆ λ 0 Staircase -DP log ρ − m a∆ 0 Uniform ( , δ)-DP 0 m∆ b−a+∆ Exponential -DP ηm 0 • Note that the DP properties are related with the discrete noise distributions, the discretization interval ∆, as well as the adjacency m. C. The Optimal DP Mechanism Based on the two proposed DP conditions for discrete random mechanisms in Section IV-A, we further study the trade-off between the privacy level and the utility. In this subsection, a utility-maximization ( Wasserstein distance-minimization) non-convex optimization problem is formulated, subject to the DP constraints. Then, we give an equivalent linear programming form of the problem to make it solvable. At last, we derive an optimal -DP Staircase mechanism. Since the utility metric is based on the statistical properties of the mechanism's input and output, we begin with the explanation for the inputs and outputs. Here, we model the input data as a random variable being generated by a specific probability distribution. Recalling the simplified discrete random mechanism in (6), we denote x, θ, x+θ as the random variables of the mechanism input, added noise and output, respectively. From the probabilistic perspective, we believe that the mechanism output is consistent for the same input (shown in (39)). Then, for the original n-dimensional data x, we only focus on the i 0 -th dimension data ( (3), which need to be protected. In this problem, for more concise expressions, we omit the subscript i 0 , i.e., all the three random variables are one-dimensional discrete random variables, with the corresponding PMFs as p x (·), p θ (·), p x+θ (·) : Z ∆ → R, respectively. Moreover, from the probability theory, the mechanism output is regarded as the summation of two random variables. The corresponding PMF is computed as: x i 0 , θ i 0 , x i 0 + θ i 0 , i 0 ∈ V ) mentioned inp x+θ (k) = i p x (i) p θ (k − i), k, i ∈ Z ∆ .(25) With the above explanation, we construct the utility optimization problem as following. • Utility model: The utility model in this paper is a minimization framework. We aim to maximize the mechanism utility (or minimize the utility loss) by maintaining the similarity between input and output probability distributions as much as possible, i.e., reducing their Wasserstein distance. Extending the Wasserstein distance (5) based on the continuous distributions, we obtain the discrete Wasserstein distance, a distance function about the discrete random mechanism input and output. It is formulated as: W 0 (x, x + θ) = k∈Z ∆ \|P x (k) − P x+θ (k)\|, where P (·) is the CDF for discrete random variables. Thus, the objective is to minimize the Wasserstein distance between the input and output, i.e., min p θ (k) W 0 (x, x + θ).(26) The optimization variable is the noise PMF, p θ (·). • Constraints: The primal constraint comes from the -DP guarantees of a discrete random mechanism. Once given the privacy cost , based on the sufficient and necessary condition in Theorem 1, we have c b = sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ p θ (k − m 0 ) p θ (k) = e .(27) Besides, the existence of the upper bound c b implies that the probability value at any point should be nonzero. Combined with the non-negativity of the probability value, we obtain the second constraint: p θ (k) > 0, ∀k ∈ Z ∆ .(28) The last constraint is obvious that it should satisfy the basic properties of probability, i.e., the total sum of the probability values should be one: k∈Z ∆ p θ (k) = 1.(29) • Optimization: Combining the objective function (26) and three constraints (27)- (29), we formulate the following primal optimization problem: P 0 : min p θ (k) W 0 (x, x + θ) (30a) s.t. ≤ log(c b ); (30b) k∈Z ∆ p θ (k) = 1; (30c) p θ (k) > 0, ∀k ∈ Z ∆ . (30d) Further, to make this non-convex optimization problem P 0 solvable, we propose an equivalent problem form. The main idea is to convert the original problem P 0 into a conventional convex optimization problem P 1 . First, we introduce two column vectors as the probability distributions, i.e., the input probability distribution 31) and the noise probability distribution p x = . . . , p x (k−∆), p x (k), p x (k+∆), . . . T(p θ = . . . , p θ (k−∆), p θ (k), p θ (k+∆), . . . T ,(32) where k ∈ Z ∆ and p x (k), p θ (k) denote the probability value at the point k of the input and the noise, respectively. Both distributions satisfy the basic probability property, i.e., \|p x \|=1 and \|p θ \|=1. With these two notions, we give the equivalent problem P 1 , where the equivalence is proved in Theorem 4. P 1 : min p θ W 1 (x, x + θ) s.t. Ap θ ≤ b, \|p θ \| = 1, p θ 0, where W 1 (x, x + θ) = k p T x M k p θ . The term p x is an arbitrarily given input distribution and p θ is the noise distribution that we are interested in. The matrix M k and A, b are shown in (49) and (52), respectively. Theorem 4. The problem P 0 is equivalent to P 1 , which means that they have the same optimal solutions. Proof. Please see the proof in the Appendix D. One difficulty of solving the problem P 1 is that the optimization variable p θ is still involved in the absolute value. Next, we try to make the optimization variable p θ (k) independent of the calculation with the absolute values, which is at the cost of certain results accuracy. Based on the absolute value inequality, it is easy to have W 1 (x, x + θ) = k p T x M k p θ (33) ≤ k p T x M k p θ = W 2 (x, x + θ). Then, we obtain an approximate optimization problem with the standard linear programming form: P 2 : min p θ W 2 (x, x + θ) s.t. Ap θ ≤ b, \|p θ \| = 1, p θ 0, where the objective function W 2 is given in (33) and the constraints are the same as the ones in the problem P 1 . Finally, we solve the primal optimization problem with P 2 . This standard linear programming problem is realized by the Simplex Method [31]. From P 2 , one implies that the optimal solution is determined by the input distribution p x involved in the objective function, as well as the privacy cost and the adjacency m in the constraints. Through extensive simulations, we get the optimal discrete random mechanism is realized by the class of Staircase-shaped probability distributions, shown in Fig. 6. The parameters of the optimal distribution (i.e., the height and the width of the stairs) are unfixed, due to the three factors mentioned above. The in-depth analysis on how parameters affects the mechanism is provided in Section V. V. SIMULATION In this section, we validate the utility guaranteed by the optimal -DP Staircase mechanism. A. Simulation Scenario First, we give a brief description of the simulation scenario, especially the mechanism inputs and noises. In this paper, the original discrete data is modeled as a random variable, so we generate the mechanism input with the designated discrete distribution. Based on the assumptions underlying the mechanism randomness in Table II, we keep the input data constant in each simulation, to guarantee that the mechanism randomness comes only from the added noise. As for the mechanism noise, the probability distribution is determined by the specific discrete random mechanism. In this subsection, we consider three representative -DP mechanisms, i.e., the Laplacian mechanism, the Staircase mechanism with a lower stair width (a = 5), and the one with a higher stair width (a = 20). Once we set the DP parameter and the adjacency parameter m, the parameters of each discrete noise distribution can be uniquely determined according to Table IV. Besides, for simplification, we set the minimum discrete interval ∆ = 1 in the following simulations. B. The Effect of Three Factors on the Mechanism Utility With the above explanation, next, we discuss how the parameters mentioned in Section IV-C (the input distribution p x , the privacy cost , and the adjacency m) affect the utility of the optimal -DP mechanism, respectively. • The effect of the input distribution p x Given that the Gaussian and the Poisson distribution are two commonly used distributions [44], [45], we select these two as the input instances. The discrete Gaussian PMF is a discretization result, with the mean parameter µ, and the variance σ 2 : p x (k) = k+1 k 1 √ 2πσ e − (x−µ) 2 2σ 2 dx, k ∈ Z ∆=1 . and the discrete Poisson PMF with parameter γ is given by: distributions are all Staircase-shaped, with the height and width of the stairs influenced by the input distributions, as we have expected. p x (k) = γ k k! e −γ , k ∈ Z ∆=1 . • The effect of the privacy cost In this part, we assume a moderate adjacency m = 15. In the trade-off problem, higher privacy cost (a bigger ) implies less utility loss, as reflected in smaller Wasserstein distance. This trend can be verified in Fig. 8, with the Gaussian and the Poisson distributed inputs, respectively. Both two figures compare the utility performance of four -DP mechanisms. Notice that the result with the Laplacian noise (green cross line) and the Staircase-shaped noise with smaller stair width (blue triangle line) are comparable, especially in the high privacy regime (smaller ). This is because the two distributions are similar under the parameter settings subject to the same DP constraints. Further, as increases, the optimality of our mechanism is better represented, with the Wasserstein distance approaching zero, i.e., the statistical properties of the mechanism's input and output can be retained. • The effect of the adjacency m With the other DP parameter set as = 2, the effect of the adjacency on the Wasserstein distance is shown in Fig. 9. We also select four mechanisms for comparison. It is observed that the Wasserstein distance and the adjacency are positively correlated. If the elements in two datasets differ significantly, then even with the optimal -DP mechanism, the utility guarantee is limited. Since the larger differences require the noises of greater amplitudes, the mechanism utility is significantly sacrificed. Remark 4. In the above simulation, we compare the mechanism utility with the optimal Staircaseshaped distribution with unfixed parameters (red circle line), and two standard Staircase-shaped distributions with explicit parameters (blue triangle line and purple square line). Although they all satisfy the -DP constraints, the standard fixed Staircase mechanism performs slightly worse in guaranteeing the utility. Since the utility measure we define is related to the mechanism input, we couple the optimal noise probability distribution with the input. The effectiveness is confirmed by the simulation. Due to the arbitrariness of the inputs, we are unable to give a closed-form expression of the optimal noise distribution independent of the input. Instead, we obtain the optimal mechanism with unfixed parameters, by solving an equivalent problem through linear programming. C. Verification of the Mechanism Optimality To further validate the mechanism optimality, we compare the statistical properties of the mechanism utility with three other mechanisms, under the same -DP guarantees. To eliminate the uncertainty of the discrete random noise, we conduct 100 simulation runs for each simulation, and do frequency statistics on the mechanism utility (characterized by the Wasserstein distance). Notice that the smaller Wasserstein distance implies the higher mechanism utility. and adjacency parameters settings. Overall, our proposed mechanism has a higher probability of the small Wasserstein distance, i.e., the higher mechanism utility. Further, to compare the mechanism utility more clearly, we summarize the results (the average, maximum, minimum Wasserstein distance) in Fig. 10. In some cases, the utility guaranteed by the optimal mechanism is similar to the existing mechanism. For example, with higher privacy protection ( = 0.5), its utility is similar to the Staircase mechanism; in the low privacy regime, the performance is close to the Laplacian mechanism. Note that the utility for the optimal mechanism is not always the highest, partially due to the uncertainty of the discrete random noise. In conclusion, the optimal -DP mechanism ensures the maximum mechanism utility in the vast majority of cases. VI. CONCLUSION For the discrete random noise adding mechanisms, we considered the DP conditions, properties and the trade-off between the mechanism utility and privacy level. For the general DP mechanisms, a sufficient and necessary condition for -DP and a sufficient condition for ( , δ)-DP were derived, followed by the DP parameters estimation. Afterwards, based on the conditions, we analyzed the DP properties for several typical mechanisms. Furthermore, we took the Wasserstein distance between mechanism inputs and outputs as the utility metric, and built the trade-off issue as a utility-maximization optimization problem. The proposed optimal mechanism is Staircaseshaped, with the parameters depending on the mechanism inputs and the differential privacy requirements. Extensive simulations were performed to verify its optimality. Future directions include the DP conclusions extension for more individuals, and exploring the correlation between the differential privacy with the homomorphic encryption. p θ i (k − m 0 ) p θ i (k) = ∞, k ∈ Z ∆ , i.e., for any given large constant M , there exists k 0 ∈ Z ∆ such that p θ i (k 0 − m 0 ) p θ i (k 0 ) ≥ M, where m 0 ∈ Z ∆ \ {0}. Construct a pair of m 0 -adjacent state vectors x i , y i ∈ Z ∆ satisfying ∀i ∈ V, \|x i − y i \| ≤    m 0 , i = i 0 ; 0, i = i 0 . Based on the discrete property of Z ∆ and the sign of m 0 , we divide m 0 ∈ [−m, m] into three parts: m 0 ∈ {−∆, ∆}, {j\|2∆ ≤ j ≤ m, j ∈ Z ∆ } and {j\| − m ≤ j ≤ −2∆, j ∈ Z ∆ }. Denote O ⊆ S, where O i is a set of the i-th column element in O. Note that DP is guaranteed if (4) holds for any given O. In the following three parts, we construct the output range O i 0 respectively to derive the contradiction for the necessity proof. • m 0 ∈ {−∆, ∆} Define O i 0 = {k\|k = y i 0 + k 0 }. From (6), we have Pr {A (x i 0 ) ∈ O i 0 } Pr {A (y i 0 ) ∈ O i 0 } = p x i 0 +θ i 0 (k) p y i 0 +θ i 0 (k) = p x i 0 +θ i 0 (y i 0 + k 0 ) p y i 0 +θ i 0 (y i 0 + k 0 ) = p θ i (k 0 − m 0 ) p θ i (k 0 ) = M.(34)• m 0 ∈ {j\|2∆ ≤ j ≤ m, j ∈ Z ∆ } Define O i 0 = {k\|y i 0 + k 0 ≤ k ≤ y i 0 + k 0 + m 0 − ∆}. Since p θ i 0 is bounded, there exists a constant C ≥ 1, s.t., k 0 +m 0 −∆ k=k 0 p θ i 0 (k) = p θ i 0 (k 0 ) + k 0 +m 0 −∆ k=k 0 +∆ p θ i 0 (k) ≤ C · p θ i 0 (k 0 ) . Then, one follows that Pr {A (x i 0 ) ∈ O i 0 } Pr {A (y i 0 ) ∈ O i 0 } = y i 0 +k 0 +m 0 −∆ k=y i 0 +k 0 p x i 0 +θ i 0 (k) y i 0 +k 0 +m 0 −∆ k=y i 0 +k 0 p y i 0 +θ i 0 (k) = k 0 −∆ k=k 0 −m 0 p θ i 0 (k) k 0 +m 0 −∆ k=k 0 p θ i 0 (k) = p θ i 0 (k 0 − m 0 ) + k 0 −∆ k=k 0 −m 0 +∆ p θ i 0 (k) k 0 +m 0 −∆ k=k 0 p θ i 0 (k) ≥ p θ i 0 (k 0 − m 0 ) k 0 +m 0 −∆ k=k 0 p θ i 0 (k) ≥ p θ i 0 (k 0 − m 0 ) C · p θ i 0 (k 0 ) = M C .(35)• m 0 ∈ {j\| − m ≤ j ≤ −2∆, j ∈ Z ∆ } Define O i 0 = {k\|y i 0 + k 0 + m 0 + ∆ ≤ k ≤ y i 0 + k 0 }. Similarly, we have Pr {A (x i 0 ) ∈ O i 0 } Pr {A (y i 0 ) ∈ O i 0 } = y i 0 +k 0 k=y i 0 +k 0 +m 0 +∆ p x i 0 +θ i 0 (k) y i 0 +k 0 k=y i 0 +k 0 +m 0 +∆ p y i 0 +θ i 0 (k) = k 0 −m 0 k=k 0 +∆ p θ i 0 (k) k 0 k=k 0 +m 0 +∆ p θ i 0 (k) = p θ i 0 (k 0 − m 0 ) + k 0 −m 0 −∆ k=k 0 +∆ p θ i 0 (k) k 0 k=k 0 +m 0 +∆ p θ i 0 (k) ≥ p θ i 0 (k 0 − m 0 ) C · p θ i 0 (k 0 ) = M C .(36) Note that M were to take any value and (34), (35), (36) would violate the ( , δ)-DP definition (δ = 0) in (4). Thus, through the contradictions, we prove that (7) is a necessary condition for the -DP mechanism A. ⇒: Next, we prove the sufficiency. Based on (6), we have Pr{A(x) ∈ O} = Pr {x + θ ∈ O} = Pr {x i 0 + θ i 0 ∈ O i 0 } n i=1,i =i 0 Pr {x i + θ i ∈ O i }(37) and Pr{A(y) ∈ O} = Pr {y + θ ∈ O} = Pr {y i 0 + θ i 0 ∈ O i 0 } n i=1,i =i 0 Pr {y i + θ i ∈ O i } .(38) Due to x i = y i , i = i 0 , we have n i=1,i =i 0 Pr {x i + θ i ∈ O i } = n i=1,i =i 0 Pr {y i + θ i ∈ O i } .(39) Besides, with the condition in (7), it follows that Pr {x i 0 + θ i 0 ∈ O i 0 } = O i 0 p x i 0 +θ i 0 (k) = O i 0 p y i 0 +m 0 +θ i 0 (k) ≤ O i 0 c b · p y i 0 +θ i 0 (k) =c b Pr {y i 0 + θ i 0 ∈ O i 0 } .(40) Combining (37)- (40), it yields that Pr{A(x) ∈ O} ≤ c b Pr{A(y) ∈ O} = e log(c b ) Pr{A(y) ∈ O},(41) which satisfies the definition of -DP. Furthermore, comparing (4) and (41), we can easily obtain the estimation of the DP parameter , i.e., = log(c b ). From (7), we note that the upper bound c b relies on the adjacency m 0 . When m 1 ≤ m 2 holds, we have sup ∀m 0 ∈[−m 1 ,m 1 ] p θ i +m 0 (k) p θ i (k) ≤ sup ∀m 0 ∈[−m 2 ,m 2 ] p θ i +m 0 (k) p θ i (k) , where m 0 , m 1 , m 2 , k ∈ Z ∆ . Hence, we refer that c b is an increasing function of m. APPENDIX B PROOF OF THEOREM 2 Proof. Construct a pair of m-adjacent state vectors x and y with x i 0 = y i 0 + m and x i = y i (when i = i 0 ), where x i , y i ∈ Z ∆ . Then, we obtain the following result: Pr{A(x) ∈ O} = n i=1 Pr {A(x i ) ∈ O i } = Pr {A(x i 0 ) ∈ O i 0 } n i=1,i =i 0 Pr {A(x i ) ∈ O i } = Pr {A(x i 0 ) ∈ O i 0 \|θ ∈ Θ 1 } + Pr {A(x i 0 ) ∈ O i 0 \|θ ∈ Θ 0 } × n i=1,i =i 0 Pr {A(x i ) ∈ O i } = O i 0 ,θ∈Θ 1 p x i 0 +θ i 0 (k) + O i 0 ,θ∈Θ 0 p x i 0 +θ i 0 (k) × n i=1,i =i 0 Pr {A(x i ) ∈ O i } = O i 0 ,θ∈Θ 1 p y i 0 +m+θ i 0 (k) + O i 0 ,θ∈Θ 0 p x i 0 +θ i 0 (k) × n i=1,i =i 0 Pr {A(y i ) ∈ O i } ≤c b O i 0 ,θ∈Θ 1 p y i 0 +θ i 0 (k) n i=1,i =i 0 Pr {A(y i ) ∈ O i } + Θ 0 p θ i 0 (k) n i=1,i =i 0 Pr {A(y i ) ∈ O i } =c b Pr {A(y i 0 ) ∈ O i 0 } n i=1,i =i 0 Pr {A(y i ) ∈ O i } + Θ 0 p θ i 0 (k) n i=1,i =i 0 Pr {A(y i ) ∈ O i } ≤c b Pr{A(y) ∈ O} + δ.(42) Combining (4) and (42), we conclude that c b = e ⇒ = log(c b ). Thus, we have completed the proof. APPENDIX C PROOF OF THEOREM 3 Proof. First, we prove that the Gaussian mechanism is not -DP. Due to the symmetry of the term m 0 , we take m 0 ∈ {j\|0 ≤ j ≤ m, j ∈ Z ∆ } in this proof. The following proof can be applied to the situation m 0 < 0 similarly. Based on the relationship between the probability point k and the PMF parameter µ, we divide the point k into three parts. Case 1: k ≥ µ + m 0 . Here the PMF is decreasing. Based on the PMF in (13), we obtain that p θ i (k − m 0 ) p θ i (k) = k+∆ k 0 e − (θ i −m 0 −µ) 2 2σ 2 dθ i k+∆ k e − (θ i −µ) 2 2σ 2 dθ i ≤ ∆·e − 1 2σ 2 (k−m 0 −µ) 2 ∆·e − 1 2σ 2 (k+∆−µ) 2 = e 1 2σ 2 (m 0 +∆)(2k+∆−m 0 −2µ) = Q. Case 2: k ≤ µ − ∆. Here the PMF is increasing. Similarly, we have p θ i (k − m 0 ) p θ i (k) ≤ ∆ · e − 1 2σ 2 (k+∆−m 0 −µ) 2 ∆ · e − 1 2σ 2 (k−µ) 2 =e 1 2σ 2 (m 0 −∆)(2k+∆−m 0 −2µ) = R. Case 3: µ ≤ k ≤ µ + m 0 − ∆. It shows that p θ i (k − m 0 ) p θ i (k) ≤ ∆ · e − 1 2σ 2 (k+∆−m 0 −µ) 2 ∆ · e − 1 2σ 2 (k+∆−µ) 2 =e 1 2σ 2 m 0 (2k+2∆−m 0 −2µ) = S. It shows that with the upper bounds of k in Case 1,2, the upper bounds R and S exist. However, we have Q → ∞ as k increases. According to Theorem 1, there does not exist a bounded parameter c b to guarantee the finite privacy loss. Thus, we have that the Gaussian mechanism is not -DP. Next, we apply Theorem 2 to prove that the Gaussian mechanism guarantees ( , δ)-DP. Given Meanwhile, the probability of error δ is bounded by δ ≤ θ i ∈Φ i p θ i (k) = 1 √ 2πσ 2 θ i ∈Φ i k+∆ k e − (θ i −µ) 2 2σ 2 dθ i ,(44)where Φ i = (−∞, −M ] ∪ [M, ∞) ⊆ Z ∆ is the i-th dimensional noise range of O i . Finally, one infers that the Gaussian mechanism is ( , δ)-DP, where the DP parameters and δ are estimated by (43) and (44), respectively. APPENDIX D PROOF OF THEOREM 4 Proof. The key idea of the equivalence proof is to make the optimization variables p θ explicitly involved in the problem, which is a basis to convert the primal problem into a conventional convex optimization problem. To make the derivation more clear, we represent the mechanism input probability distribution as a finite one: p x = p x (k 0 +∆),p x (k 0 +2∆),. . . ,p x (k 0 +n∆) T ,(45) where k 0 ∈ Z ∆ and n ∈ R. Note that k 0 is an arbitrary value and n → ∞. We can have the finite input (45) replace the infinite one (31). Our goal is to prove that W 0 = W 1 , where W 0 = k \|P x (k) − P x+θ (k)\|, W 1 = k p T x M k p θ . First, we discuss the equivalence of the objective variables. With the introduction of the noise distribution p θ in (32), the original objective variable p θ (k) aiming at every probability value is contained in this column vector p θ , which is expressed equivalently but more concisely. Next, we prove the equivalence of two objective functions, W 0 and W 1 . We aim to convert the CDF into PMF, which contains the optimization variables more explicitly. • The input cumulative probability at point k: P x (k). P x (k) = p x (−∞) + · · · + p x (k − ∆) + p x (k). Then, for k ∈ k k 0 + ∆ ≤ k ≤ k 0 + n∆, k ∈ Z ∆ , P x (k) = p x (k 0 + ∆) + p x (k 0 + 2∆) + · · · + p x (k) = k i=k 0 +∆ p x (i)·(· · · + p θ (k−∆) + p θ (k) + · · ·) =1 = p T x           − 1 − . . . − 1 − − 0 − . . .           M 1 k p θ = p T x   1 r 0 n−r   M 1 k p θ , where the row number r of the matrix M 1 k with all elements 1 is related to the relationship between k and k 0 , i.e., r = 1 + [k − (k 0 + ∆)]/∆. The column number of the matrix M 1 k depends on the noise distribution p θ . Denote the matrix M 1 k as the result derived from the input P x at point k, and the matrix M 2 as the one from the output P x+θ , which will be obtained later. Specifically, we have M 1 k = 0 for k < k 0 + ∆ and M 1 k = 1 for k > k 0 + n∆, where 0 and 1 are two matrixes full of elements 0 and 1, respectively. • The output cumulative probability at point k: P x+θ (k). P x+θ (k) = p x+θ (−∞)+· · · + p x+θ (k − ∆) + p x+θ (k). With the finite representation of the input distribution p x in (45), the output probability at point k in (25) is reformulated as following, where the upper and the lower bounds of the summation is further clarified: p x+θ (k) = k 0 +n∆ i=k 0 +∆ p x (i) p θ (k − i), k, i ∈ Z ∆ .(48) Then, we substitute every element in (47) with (48), and make simplification with the goal of p x and p θ . Consequently, we have P x+θ (k) = k−k 0 −(n+1)∆ i=−∞ k 0 +n∆ j=k 0 +∆ p x (j) p θ (i) + k−k 0 −∆ i=k−k 0 −n∆ k−i j=k 0 +∆ p x (j) p θ (i) =p T x           [k 1 ] [k 2 ] · · · 1 1 · · · 1 1 0 · · · · · · 1 1 · · · 1 0 0 · · · . . . . . . . . . . . . · · · . . . . . . . . . · · · 1 1 0 · · · 0 0 · · ·           M 2 k p θ , where the column vector [k t ] corresponds to the element at the corresponding position in the column vector p θ , i.e., p θ (k t ). Note that k 1 = k − k 0 − n∆, k 2 = k − k 0 − ∆ are the two key points. Before the column [k 1 ], all elements in the matrix M 2 k are 1, and after the column [k 2 ], all elements are 0. So far, we make the two CDFs at the point k, P x (k) and P x+θ (k), related to the input distribution p x and the noise distribution p θ . The concrete information about point k is contained in the matrix M k , which only depends on k, where M k = M 1 k − M 2 k .(49) Then, we have W 0 = k \|P x (k) − P x+θ (k)\| (50) = k p T x M 1 k − M 2 k p θ = k p T x M k p θ = W 1 . Finally, we prove the constraints in the problem P 1 are equivalent to the ones in P 0 . • The DP constraint in (30b) Based on the necessary DP condition (9), we have that p θ (k) = 0. Then, we rewrite this constraint as an inequality constraint (detailed expression in (27) 0 · · · 0 1 0 · · · −e                   A k · p θ ≤              0 0 . . . 0 . . . 0              b ,(52) where k 1 = k − m, k 2 = k, k 3 = k + m. The three columns correspond to the elements p θ (k 1 ), p θ (k 2 ), p θ (k 3 ), respectively. Since for every point k, we have (2m + 1) -DP constrains, the row numbers of the matrix A k is 2m + 1. The column number relies on the length of the noise distribution p θ . Go through all the points k, and put A k together with the corresponding element k, we will have the equivalent form of the DP constraint: = log(c b ) ⇔ p θ (k − m 0 ) p θ (k) ≤ e ⇔ Ap θ ≤ b,(53) where A= · · · A k=0 A k=∆ A k=2∆ · · · T and b = 0 T . • The total sum constraint in (30c) k p θ (k) = 1 ⇔ k \|p θ (k)\| = 1 (54) ⇔ p θ 1 = 1 ⇔ \|p θ \| = 1. • The positive value constraint in (30d) ∀k, p θ (k) > 0 ⇔ · · · , p θ (∆) > 0, p θ (2∆) > 0, · · · T (55) ⇔ · · · , p θ (∆), p θ (2∆), · · · T 0 ⇔ p θ 0. By now, we have given the equivalent form of three constraints in problem P 0 with (53), (54) and (55), respectively. Thus, with (50), (53)-(55), we obtain the whole equivalent form of the original optimization problem P 0 : min p θ W 1 (x, x + θ) = k p T x M k p θ s.t. Ap θ ≤ b, \|p θ \| = 1, p θ 0, where the equivalence is proved by the objective variable, the objective function and the constraints, respectively. thus = log(c b ) = 0. Substituting and δ into (4), we have Pr {A (x) ∈ O} ≤ Pr {A (y) ∈ O}+ Fig. 1 . 1Discrete Gaussian noise distribution. Theorem 3 . 3The Gaussian mechanism A is ( , δ)-DP. Given an arbitrary large constant M , the two DP parameters and δ are estimated by = sup m 0 ∈[−m,m],m 0 ∈Z ∆ 1 2σ 2 (\|m 0 \|+∆) (2M +∆−m 0 −2µ) Fig. 2 . 0 20Discrete Laplacian noise distribution. precondition in(9). Then, based on the sufficient and necessary conditions in Theorem 1, we find that for any m 0 ∈ [−m, m] , m 0 ∈ Z ∆ , there existsp θ i +m 0 (k) p θ i (k) ≤e ∆/λ e − \|k−m Fig. 3 . 3Discrete Staircase-shaped noise distribution. Fig. 4 . 4Discrete Unifrom distributed noise distribution. Fig. 5 5 Fig. 5 . 5Discrete Exponential noise distribution. Fig. 6 . 6The Staircase-shaped optimal noise distribution. Fig. 7 7shows the optimal distributions with the discrete Gaussian input and the Poisson input, respectively, with the DP constraints set by = 2 and m = 15. Overall, the two optimal stair (a) Optimal distribution for the Gaussian input.(b) Optimal distribution for the Poisson input. Fig. 7 . 7Effect of input distributions on the optimal mechanism.(a) Input = Gaussian (µ = 0, σ = 10).(b) Input = Poisson (γ = 5). Fig. 8 . 8Effect of privacy costs on the mechanism utility with different inputs. ( a ) aInput = Gaussian (µ = 0, σ = 10). (b) Input = Poisson (γ = 5). Fig. 9 . 9Effect of adjacencies on the mechanism utility with different inputs. Fig. 10 . 10Statistic properties of the Wasserstein Distance. Fig . 10 shows a comparison of the utility for four -DP mechanisms, with different privacy level Proof. ⇐: We prove the necessity by contradiction. Assume that sup ∀m 0 ∈[−m,m],m 0 ∈Z ∆ ,∀i∈V a constant M ≥ m, for k ∈ [−M, M ], the DP parameter is bounded by = log(c b ) ≤ log (max {Q, R, S})(43)≤ max m 0 ∈[−m,m] 1 2σ 2 (m 0 +∆) (2M +∆−m 0 −2µ) = 1 2σ 2 (m + ∆) (2M + ∆ − m − 2µ) . k ≤ k 0 ; 1, k ≥ k 0 + n∆,due to the boundedness property of the CDF: ), i.e.,∀k, m 0 ∈ [−m, m] , p θ (k − m 0 ) p θ (k) ≤ c b = e(51)⇔ p θ (k−m 0 )−e ·p θ (k) ≤ 0. TABLE I ICOMPARISON OF WORKS ON MECHANISM PRIVACY AND UTILITYDifferential Privacy Properties Works [29] [30] This work Scenario Continuous Discrete Discrete Scope General Specific General Utility Properties Works [23] [24] This work Utility Metric The minimum noise magnitude/variance Wasserstein distance Optimal mechanism Geometric (Noises related) Staircase (Noises.) Staircase (Inputs.) TABLE II PRELIMINARIES IIOF THE DP PROPERTIES FOR MECHANISMSDimension Explanation Privacy Cost / Loss , δ Qualified by and e , respectively, allowing a small probability of error δ. Adjacency Property m Assume datasets have the same size, Table III IIIsummarizes several notations in this paper. TABLE III PRIMARY NOTATIONS IIINOTATIONSNotation Description ∆ The minimum discretization distance TABLE IV SUMMARY IVOF THE DP PARAMETERS ESTIMATION METHODSDP Property c b δ -DP Eq. (7) Eq. (8) δ = 0 ( , δ)-DP Eq. (11) Eq. (12) Eq. (10) TABLE V VDISCRETE DP PROPERTIES AND PRIVACY GUARANTEESMechanism Property δ Further, we transform (51) into a matrix form:                  [k 1 ] [k 2 ] [k 3 ] −e · · · 0 1 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . 0 · · · −e 1 0 · · · 0 0 · · · 0 1 −e · · · 0 . . . . . . . . . . . . . . . . . . . . . Differential private discrete noise adding mechanism: Conditions and properties. S Qin, J He, C Fang, J Lam, accepted by Proc. IEEE Amer. S. Qin, J. He, C. Fang, and J. Lam, "Differential private discrete noise adding mechanism: Conditions and properties," in accepted by Proc. IEEE Amer. Control Conf., 2022. 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Shanghai, ChinaShanghai Jiao Tong UniversityHer research interests include privacy and security in network systemsShuying Qin (S'22) is currently an undergraduate in the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. Her research interests include privacy and security in network systems. Jianping He (SM'19) is currently an associate professor in the Department of. Automation at Shanghai Jiao Tong UniversityJianping He (SM'19) is currently an associate professor in the Department of Automation at Shanghai Jiao Tong University. . Canada. He received the Ph.D. degree in control science and engineering from Zhejiang University ; Department of Electrical and Computer Engineering at University of VictoriaHe received the Ph.D. degree in control science and engineering from Zhejiang University, Hangzhou, China, in 2013, and had been a research fellow in the Department of Electrical and Computer Engineering at University of Victoria, Canada, from Dec. His research interests mainly include the distributed learning, control and optimization, security and privacy in network systems. to Mar. 2017. His research interests mainly include the distributed learning, control and optimization, security and privacy in network systems. He was the winner of Outstanding Thesis Award, Chinese Association of Automation. ; Ieee Dr, Ieee Tac, Tii, IEEE Open Journal of Vehicular Technology and KSII Trans. Internet and Information Systems. He serves as an Associate Editor for IEEE Trans. He received the best paper award from IEEE WCSP'17, the best conference paper award from IEEE PESGM'17, the finalist best student paper award from IEEE ICCA'17, and the finalist best conference paper award from IEEE VTC'20-FallDr. He serves as an Associate Editor for IEEE Trans. on Control of Network Systems, IEEE Open Journal of Vehicular Technology and KSII Trans. Internet and Information Systems. He was also a Guest Editor of IEEE TAC, IEEE TII, International Journal of Robust and Nonlinear Control, etc. He was the winner of Outstanding Thesis Award, Chinese Association of Automation, 2015. He received the best paper award from IEEE WCSP'17, the best conference paper award from IEEE PESGM'17, the finalist best student paper award from IEEE ICCA'17, and the finalist best conference paper award from IEEE VTC'20-Fall. He received the B.Sc. degree in automation and the Ph.D. degree in control science and engineering from. Shanghai, ChinaShanghai Jiao Tong UniversityChongrong Fang (M'21) is currently an Assistant Professor with the Department of AutomationChongrong Fang (M'21) is currently an Assistant Professor with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. He received the B.Sc. degree in automation and the Ph.D. degree in control science and engineering from His research interests include anomaly detection and diagnosis in cyber-physical systems and cloud networks. Hangzhou, ChinaZhejiang UniversityZhejiang University, Hangzhou, China, in 2015 and 2020, respectively. His research interests include anomaly detection and diagnosis in cyber-physical systems and cloud networks. and was awarded the Ashbury Scholarship, the A.H. Gibson Prize, and the H. Wright Baker Prize for his academic performance. He obtained the MPhil and Ph.D. degrees from the University of Cambridge. He is a Croucher Scholar, Croucher Fellow, and Distinguished Visiting Fellow of the. James Lam received a B.Sc. (1st Hons.) degree in Mechanical Engineering from the University of Manchester ; Royal Academy of Engineering, and Cheung Kong Chair Professor. Prior to joining the University ofJames Lam received a B.Sc. (1st Hons.) degree in Mechanical Engineering from the University of Manchester, and was awarded the Ashbury Scholarship, the A.H. Gibson Prize, and the H. Wright Baker Prize for his academic performance. He obtained the MPhil and Ph.D. degrees from the University of Cambridge. He is a Croucher Scholar, Croucher Fellow, and Distinguished Visiting Fellow of the Royal Academy of Engineering, and Cheung Kong Chair Professor. Prior to joining the University of Hong Kong in 1993 where he is now Chair Professor of Control Engineering, he was a faculty member at the. City University of Hong Kong and the University of MelbourneHong Kong in 1993 where he is now Chair Professor of Control Engineering, he was a faculty member at the City University of Hong Kong and the University of Melbourne. Fellow of Institute of Electrical and Electronic Engineers (FIEEE). Fellow of Institute of Mathematics and Its Applications (FIMA), Fellow of Institution of Mechanical Engineers (FIMechE), and Fellow of Hong Kong Institution of Engineers (FHKIE). Chartered Scientist (CSci)Professor Lam is a Chartered Mathematician (CMath), Chartered Scientist (CSci), Chartered Engineer (CEng), Fellow of Institute of Electrical and Electronic Engineers (FIEEE), Fellow of Institution of Engineering and Technology (FIET), Fellow of Institute of Mathematics and Its Applications (FIMA), Fellow of Institution of Mechanical Engineers (FIMechE), and Fellow of Hong Kong Institution of Engineers (FHKIE).	[]
[ "BASIC ANALOG OF FOURIER SERIES ON A q-QUADRATIC GRID", "BASIC ANALOG OF FOURIER SERIES ON A q-QUADRATIC GRID" ]	[ "Joaquin Bustoz ", "Sergei K Suslov " ]	[]	[]	We prove orthogonality relations for some analogs of trigonometric functions on a q-quadratic grid and introduce the corresponding q-Fourier series. We also discuss several other properties of this basic trigonometric system and the q-Fourier series.	10.4310/maa.1998.v5.n1.a1	[ "https://arxiv.org/pdf/math/9706216v1.pdf" ]	16,408,227	math/9706216	0f04ae30a76307359612d4f5e4c37e3efd74d496	BASIC ANALOG OF FOURIER SERIES ON A q-QUADRATIC GRID 4 Jun 1997 Joaquin Bustoz Sergei K Suslov BASIC ANALOG OF FOURIER SERIES ON A q-QUADRATIC GRID 4 Jun 1997 We prove orthogonality relations for some analogs of trigonometric functions on a q-quadratic grid and introduce the corresponding q-Fourier series. We also discuss several other properties of this basic trigonometric system and the q-Fourier series. Introduction For convergence conditions of (1.2) see, for example, [1], [28], and [30]. In the present paper we discuss a q-version of the Fourier series (1.1) with the aid of basic or q-analogs of trigonometric functions introduced recently in [15] (see also [5] and [24]). Our first main objective will be to establish analogs of the orthogonality relations (1.5)-(1.7) for q-trigonometric functions on a q-quadratic grid. There are several ways to prove the orthogonality relations (1.6)-(1.8) for trigonometric functions. The method based on the second order differential equation, u ′′ + ω 2 u = 0, (1.9) can be extended to the case of basic trigonometric functions. Consider, for example, two functions cos ωx and cos ω ′ x, which satisfy (1.9) with different eigenvalues ω and ω ′ . Then, (ω 2 − ω ′2 ) l −l cos ωx cos ω ′ x dx (1.10) = W (cos ωx, cos ω ′ x) \| l −l = cos ωx cos ω ′ x −ω sin ωx −ω ′ sin ω ′ x l −l . The right side of (1.10) vanishes when sin ωl = sin ω ′ l = 0, (1.11) which gives ω = π l n, ω ′ = π l m, (1.12) where n, m = 0, ±1, ±2, ±3, ... . In the same manner, one can prove (1.7). The last equation (1.8) is valid by symmetry. We shall extend this consideration to the case of the basic trigonometric functions in the present paper. This paper is organized as follows. In Section 2 we introduce the q-trigonometric functions. In the next section we derive a continuous orthogonality property of these functions, and then, in Section 4, we formally discuss the limit q → 1 of these new orthogonality relations. Section 5 is devoted to the investigation of some properties of zeros of the basic trigonometric functions and in Section 6 we evaluate the normalization constants in the orthogonality relations for these functions. In Section 7 we state the orthogonality relation for the corresponding q-exponential functions. Finally, we introduce basic analogs of Fourier series in Section 8, and in Sections 9-11 we give a proof of the completeness of the q-trigonometric system and establish some elementary facts about convergence of our q-Fourier series. Examples of these series are considered in Sections 12 and 14; we prove some basic trigonometric identities we heed in Section 13. Some miscellaneous results concerning q-trigonometric functions are discussed in Section 15. Analogs of Trigonometric Functions on a q-Quadratic Grid The following functions C(x) and S(x) given by C(x) = C q (x; ω) (2.1) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ × 2 ϕ 1 −qe 2iθ , −qe −2iθ q ; q 2 , −ω 2 and S(x) = S q (x; ω) (2.2) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ 2q 1/4 ω 1 − q cos θ × 2 ϕ 1 −q 2 e 2iθ , −q 2 e −2iθ q 3 ; q 2 , −ω 2 , were discussed recently [5], [15] and [24] as q-analogs of cos ωx and sin ωx on a qquadratic lattice x = cos θ. These functions are special cases y = 0 of more general basic trigonometric functions C(x, y) = C q (x, y; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ × 4 ϕ 3 −q 1/2 e iθ+iϕ ,−q 1/2 e iθ−iϕ ,−q 1/2 e iϕ−iθ ,−q 1/2 e −iθ−iϕ −q, q 1/2 , −q 1/2 ; q,−ω 2 (2.3) and S(x, y) = S q (x, y; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ 2q 1/4 ω 1 − q (cos θ + cos ϕ) × 4 ϕ 3 −qe iθ+iϕ , −qe iθ−iϕ , −qe iϕ−iθ , −q −iθ−iϕ −q, q 3/2 , −q 3/2 ; q, −ω 2 (2.4) which are q-analogs of cos ω(x + y) and sin ω(x + y), respectively (see [24]). Here x = cos θ and y = cos ϕ. Usually we shall drop q from the symbols C q (x; ω), S q (x; ω), C q (x, y; ω), and S q (x, y; ω) because the same base is used throughout the paper. The symbols 2 ϕ 1 and 4 ϕ 3 in (2.1)-(2.4) are, of course, special cases of basic hypergeometric functions, r ϕ s a 1 , a 2 , ... , a r b 1 , b 2 , ... , b s ; q, t (2.5) := ∞ n=0 (a 1 , a 2 , ... , a r ; q) n (q, b 1 , b 2 , ... , b s ; q) n (−1) n q n(n−1)/2 1+s−r t n . The standard notations for the q-shifted factorial are where n = 1, 2, ..., or ∞, when \|q\| < 1. See [7] for an excellent account of the theory of basic hypergeometric functions. Functions (2.1)-(2.4) are defined here for \|ω\| < 1 only. For an analytic continuation of these functions in a larger domain see [12], [15], and [24]. For example, C(x) = qω 2 e 2iθ , qω 2 e −2iθ ; q 2 ∞ (q, −qω 2 ; q 2 ) ∞ (2.8) × 2 ϕ 2 −ω 2 , −qω 2 qω 2 e 2iθ , qω 2 e −2iθ ; q 2 , q and S(x) = q 2 ω 2 e 2iθ , q 2 ω 2 e −2iθ ; q 2 ∞ (q 3 , −qω 2 ; q 2 ) ∞ 2q 1/4 ω 1 − q cos θ (2.9) × 2 ϕ 2 −ω 2 , −qω 2 q 2 ω 2 e 2iθ , q 2 ω 2 e −2iθ ; q 2 , q 3 . One can see from (2.8) and (2.9) that the basic trigonometric functions (2.1) and (2.2) are entire functions in z when e iθ = q z . Analytic continuation of q-trigonometric functions (2.3) and (2.4) can be obtained on the basis of the "addition" theorems, C (x, y) = C (x) C (y) − S (x) S (y) , S (x, y) = S (x) C (y) + C (x) S (y) , found in [24]. The basic trigonometric functions (2.1)-(2.4) are solutions of a difference analog of equation (1.9) on a q-quadratic lattice, [5], [15], [21], [23], and [24] for more details. Equation (2.10) can also be rewritten in a more symmetric form, σ ∆ ∇x 1 (z) ∇u(z) ∇x(z) + λ u(z) = 0, (2.10) where x(z) = 1 2 (q z + q −z ), q z = e iθ , x 1 (z) = x(z + 1/2), λ/σ = 4q 1/2 ω 2 / (1 − q) 2 , and ∆f (z) = ∇f (z + 1) = f (z + 1) − f (z). Seeδ δx(z) δu(z) δx(z) + 4q 1/2 (1 − q) 2 ω 2 u(z) = 0, (2.11) where δf (z) = f (z + 1/2) − f (z − 1/2). The q-trigonometric functions (2.1)-(2.4) satisfy the difference-differentiation formulas δ δx C(x, y) = − 2q 1/4 1 − q ω S(x, y) (2.12) and δ δx S(x, y) = 2q 1/4 1 − q ω C(x, y). (2.13) See [15] and [24]. Applying the operator δ/δx to the both sides of (2.12) or (2.13) we obtain equation (2.11) again. Equation (2.10) is a very special case of a general difference equation of hypergeometric type on nonuniform lattices (cf. [5], [21], [23], and [24]). The Askey-Wilson polynomials and their special and limiting cases [4], [17], and [21] are well-known as the simplest and the most important orthogonal solutions of this difference equation of hypergeometric type. Recently, Ismail, Masson, and Suslov [12], [13], [25], [26] have found another type of orthogonal solutions of this difference equation. In the present paper we shall discuss this new orthogonality property at the level of basic trigonometric functions. Continuous Orthogonality Property for q-Trigonometric Functions Our main objective in this paper is to find the orthogonality relations for qtrigonometric functions (2.1)-(2.2) similar to the orthogonality relations (1.5)-(1.7). Consider difference equations for the functions u(z) = C q (x(z); ω) and v(z) = C q (x(z); ω ′ ) in self-adjoint form, ∆ ∇x 1 (z) σρ(z) ∇u(z) ∇x(z) + λ ρ(z)u(z) = 0 (3.1) and ∆ ∇x 1 (z) σρ(z) ∇v(z) ∇x(z) + λ ′ ρ(z)v(z) = 0, (3.2) where the function ρ(z) satisfies the "Pearson equation" [21], [23], ρ(z + 1) ρ(z) = σ(−z) σ(z + 1) = 1 = q −4z−2 q 2 2z+1 , (3.3) and λ = 4q 1/2 σ (1 − q) 2 ω 2 , λ ′ = 4q 1/2 σ (1 − q) 2 ω ′2 . (3.4) One can easily check that ρ 0 (z + 1) ρ 0 (z) = q −4z−2 for ρ 0 (z) = (q 2z , q −2z ; q) ∞ q z − q −z (3.5) and ρ α (z + 1) ρ α (z) = q 4z+2 for ρ α (z) = q 2α+2z , q 2α−2z , q 2−2α+2z , q 2−2α−2z ; q 2 −1 ∞ (3.6) (cf. [12], [25], and [26]). Therefore, we can choose the following solution of (3.3), ρ(z) = (q 2z , q −2z ; q) ∞ (q z − q −z ) −1 (q 2α+2z , q 2α−2z , q 2−2α+2z , q 2−2α−2z ; q 2 ) ∞ , (3.7) where α is an arbitrary additional parameter. We shall see later that this solution satisfies the correct boundary conditions for our second order divided-difference Askey-Wilson operator (2.10) for certain values of this parameter α. Let us multiply (3.1) by v(z), (3.2) by u(z) and subtract the second equality from the first one. As a result we get (λ − λ ′ ) u(z) v(z) ρ(z)∇x 1 (z) = ∆ [σρ(z) W (u(z), v(z))] , (3.8) where W (u(z), v(z)) = u(z) v(z) ∇u(z) ∇x(z) ∇v(z) ∇x(z) (3.9) = u(z) ∇v(z) ∇x(z) − v(z) ∇u(z) ∇x(z) is the analog of the Wronskian [21]. Integrating (3.8) over the contour C indicated in the Figure on the next page; where z is such that z = iθ/ log q and −π/2 ≤ θ ≤ 3π/2; gives (λ − λ ′ ) C u(z) v(z) ρ(z)∇x 1 (z) dz (3.10) = C ∆ [σρ(z) W (u(z), v(z))] dz. As a function in z, the integrand in the right side of (3.10) has the natural purely imaginary period T = 2πi/ log q when 0 < q < 1, so this integral is equal to D σρ(z) W (u(z), v(z)) dz, (3.11) where D is the boundary of the rectangle on the Figure oriented counterclockwise. The basic trigonometric functions C(x) and S(x) are entire functions in the complex z-plane due to (2.8)-(2.9). Therefore, the poles of the integrand in (3.11) inside the rectangle in the Figure are the simple poles of ρ(z) at z = α, z = 1 − α and at z = α − iπ/ log q, z = 1 − α − iπ/ log q when 0 < Re α < 1/2. Hence, by Cauchy's theorem, 1 2πi D ρ W (u, v) dz (3.12) = Res f (z)\| z=α + Res f (z)\| z=1−α + Res f (z)\| z=α−iπ/ log q + Res f (z)\| z=1−α−iπ/ log q , r r r r Re z Im z α 1 − α α + iπ log q −1 1 − α + iπ log q −1 C 0 1 3iπ 2 log q −1 iπ 2 log q iπ 2 log q + 1 3iπ 2 log q −1 + 1 ✻ ✻ where f (z) = ρ(z) W (u(z), v(z)) (3.13) = q −z (q 2z , q 1−2z ; q) ∞ W (u(z), v(z)) (q 2α+2z , q 2α−2z , q 2−2α+2z , q 2−2α−2z ; q 2 ) ∞ . Evaluation of the residues at these simple poles gives Res f (z)\| z=α = lim z→α (z − α) f (z) (3.14) = − q −α (q 2α , q 1−2α ; q ) ∞ 2 log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × W (u(z), v(z))\| z=α , Res f (z)\| z=1−α = lim z→1−α (z − 1 + α) f (z) (3.15) = − q −α (q 2α , q 1−2α ; q ) ∞ 2 log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × W (u(z), v(z))\| z=1−α , Res f (z)\| z=α−iπ/ log q = lim z→α−iπ/ log q (z − α + iπ/ log q) f (z) (3.16) = q −α (q 2α , q 1−2α ; q ) ∞ 2 log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × W (u(z), v(z))\| z=α−iπ/ log q , and Res f (z)\| z=1−α−iπ/ log q = lim z→1−α (z − 1 + α + iπ/ log q) f (z) (3.17) = q −α (q 2α , q 1−2α ; q ) ∞ 2 log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × W (u(z), v(z))\| z=1−α−iπ/ log q . But W (u(z), v(z)) = v(z)u(z − 1) − u(z)v(z − 1) x(z) − x(z − 1) (3.18) by (3.9) and, therefore, W (u(z), v(z))\| z=α = W (u(z), v(z))\| z=1−α (3.19) = − W (u(z), v(z))\| z=α−iπ/ log q = − W (u(z), v(z))\| z=1−α−iπ/ log q due to the symmetries C(x) = C(−x), x(z) = x(−z), and x(z) = −x(z − iπ/ log q). Thus, the residues are equal and as a result we get q 1/2 (1 − q) 2 ω 2 − ω ′2 C u(z) v(z) ρ(z)∇x 1 (z) dz (3.20) = − πi q −α (q 2α , q 1−2α ; q ) ∞ log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × W (u(α), v(α)) , where 0 < Re α < 1/2. We have established our main equation (3.20) for the case u(z) = C (x(z); ω) and v(z) = C (x(z); ω ′ ). The same line of consideration shows that this equation is also true when u(z) = S (x(z); ω) and v(z) = S (x(z); ω ′ ). The corresponding analogs of the Wronskians in (3.20) can be written as W (C(x(z); ω), C(x(z); ω ′ )) (3.21) = 2q 1/4 1 − q [ω C (x(z); ω ′ ) S (x(z − 1/2); ω) − ω ′ C (x(z); ω) S (x(z − 1/2); ω ′ )] and W (S(x(z); ω), S(x(z); ω ′ )) (3.22) = 2q 1/4 1 − q [ω ′ S (x(z); ω) C (x(z − 1/2); ω ′ ) − ω S (x(z); ω ′ ) C (x(z − 1/2); ω)] by (2.12)-(2.13), respectively. One can see from (3.21) and (3.22) that the right side of (3.20) vanishes in both cases when eigenvalues ω and ω ′ are roots of the following equation S q (x(1/4); ω) = S q (x(1/4); ω ′ ) = 0. (3.23) This is a direct analog of (1.11) for basic trigonometric functions. In the last case, u(z) = C (x(z); ω) and v(z) = S (x(z); ω ′ ), the left side of (3.20) vanishes by symmetry. It is interesting to verify that by using our method as well. Equations (3.1) to (3.18) are valid again. But now W (u(z), v(z))\| z=α = W (u(z), v(z))\| z=1−α (3.24) = W (u(z), v(z))\| z=α−iπ/ log q = W (u(z), v(z))\| z=1−α−iπ/ log q due to the symmetries C(x) = C(−x), S(x) = −S(−x), x(z) = x(−z) , and x(z) = −x(z − iπ/ log q). Therefore, q 1/2 (1 − q) 2 ω 2 − ω ′2 C u(z) v(z) ρ(z)∇x 1 (z) dz (3.25) = − πi q −α (q 2α , q 1−2α ; q ) ∞ 2 log q −1 (q 2 , q 2 , q 4α , q 2−4α ; q 2 ) ∞ × [W (u(α), v(α)) − W (u(α), v(α))] ≡ 0, when 0 < Re α < 1/2. Combining all the above cases together, we finally arrive at the continuous orthogonality relations for basic trigonometric functions, π 0 C (cos θ; ω) C (cos θ; ω ′ ) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.26) =      0 if ω = ω ′ , π q 1/2 ; q 2 ∞ (q; q) 2 ∞ C (η; ω) ∂ ∂ω S (η; ω) if ω = ω ′ ; π 0 S (cos θ; ω) S (cos θ; ω ′ ) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.27) =      0 if ω = ω ′ , π q 1/2 ; q 2 ∞ (q; q) 2 ∞ C (η; ω) ∂ ∂ω S (η; ω) if ω = ω ′ ; and π 0 C (cos θ; ω) S (cos θ; ω ′ ) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = 0. (3.28) Here η := x(1/4) = q 1/4 + q −1/4 /2 and the eigenvalues ω and ω ′ satisfy the "boundary" condition (3.23). For arbitrary ω = ω ′ one gets from (3.20)- (3.22) π 0 C (cos θ; ω) C (cos θ; ω ′ ) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.29) = 2π ω 2 − ω ′2 q 1/2 ; q 2 ∞ (q; q) 2 ∞ × [ω C (η; ω ′ ) S (η; ω) − ω ′ C (η; ω) S (η; ω ′ )] and π 0 S (cos θ; ω) S (cos θ; ω ′ ) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.30) = 2π ω 2 − ω ′2 q 1/2 ; q 2 ∞ (q; q) 2 ∞ × [ω ′ S (η; ω) C (η; ω ′ ) − ω S (η; ω ′ ) C (η; ω)] . Also, in the limit ω → ω ′ , π 0 C 2 (cos θ; ω) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.31) = π q 1/2 ; q 2 ∞ ω (q; q) 2 ∞ ω C (η; ω) ∂ ∂ω S (η; ω) + C (η; ω) S (η; ω) − ω ∂ ∂ω C (η; ω) S (η; ω) and π 0 S 2 (cos θ; ω) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (3.32) = π q 1/2 ; q 2 ∞ ω (q; q) 2 ∞ ω C (η; ω) ∂ ∂ω S (η; ω) − C (η; ω) S (η; ω) − ω ∂ ∂ω C (η; ω) S (η; ω) . We remind the reader that η is defined by η = x(1/4) = q 1/4 + q −1/4 /2. This notation will be used throughout this work. Formal Limit q → 1 − In this section we formally obtain orthogonality of the trigonometric functions as limiting cases of our orthogonality relations (3.26)-(3.28) for basic trigonometric functions. According to [24], lim q→1 − C q x, y; 1 2 ω (1 − q) = cos ω (x + y) , (4.1) lim q→1 − S q x, y; 1 2 ω (1 − q) = sin ω (x + y) . (4.2) If ω = ω ′ we can rewrite (3.26) as π 0 C (cos θ; ω) C (cos θ; ω ′ ) e 2iθ , e −2iθ ; q 1/2 dθ = 0, (4.3) where (a; r) α := (a; r) ∞ (ar α ; r) ∞ . (4.4) Using the limiting relation [7] lim q→1 − (a; r) α = (1 − a) α , (4.5) one can see that e 2iθ , e −2iθ ; q 1/2 → 2 sin θ (4.6) as q → 1 − . Therefore, changing ω to (1 − q) ω/2 in (4.3) , with the help of (4.1) when y = 0 we obtain the orthogonality relation (1.6) with l = 1. The boundary condition (1.11) follows from (3.23) in the same limit. When ω = ω ′ we can rewrite (3.26) as π 0 C 2 (cos θ; ω) e 2iθ , e −2iθ ; q 2 1/2 dθ (4.7) = π (1 − q) Γ 2 q (1/2) C (η; ω) ∂ ∂ω S (η; ω) , where η = q 1/4 + q −1/4 /2 and Γ q (z) = (1 − q) 1−z (q; q) ∞ (q z ; q) ∞ (4.8) is a q-analog of Euler's gamma function Γ (z) (see, for example, [7]). Changing ω to (1 − q) ω/2 in (4.8), with the aid of lim q→1 − Γ q (z) = Γ (z) , (4.9) we get 2 1 −1 cos 2 πnx dx = 2π Γ 2 (1/2) cos 2 πn = 2, (4.10) where n = ±1, ±2, ... , in the limit q → 1 − . In a similar manner one can obtain (1.7) and (1.8) from (3.27) and (3.28), respectively. Some Properties of Zeros In Section 3 we have established the orthogonality relations for the basic trigonometric functions (3.26)-(3.28) under the boundary condition (3.23). Here we would like to discuss some properties of ω-zeros of the corresponding basic sine function, S (η; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ ω 1 − q 1/2 (5.1) × 2 ϕ 1 −q 3/2 , −q 5/2 q 3 ; q 2 , −ω 2 = q 3/2 ω 2 ; q ∞ (q 3 , −qω 2 ; q 2 ) ∞ ω 1 − q 1/2 × 2 ϕ 2 −ω 2 , −qω 2 q 3/2 ω 2 , q 5/2 ω 2 ; q 2 , q 3 , and the basic cosine function, C (η; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ (5.2) × 2 ϕ 1 −q 1/2 , −q 3/2 q ; q 2 , −ω 2 = q 1/2 ω 2 ; q ∞ (q, −qω 2 ; q 2 ) ∞ × 2 ϕ 2 −ω 2 , −qω 2 q 1/2 ω 2 , q 3/2 ω 2 ; q 2 , q . One can see that these functions have almost the same structure as the q-Bessel function discussed in [12], [13]. So we can apply a similar method to establish main properties of zeros of the functions (5.1)-(5.2). The first property is that the q-sine function S (η; ω) has an infinity of real ω-zeros. To prove that we can again consider the large ω-asymptotics of the function (5.1). The 2 ϕ 1 here can be transformed by (III.1) of [7], which gives S (η; ω) = −q 5/2 , q 3/2 ω 2 ; q 2 ∞ (q 3 , −qω 2 ; q 2 ) ∞ ω 1 − q 1/2 (5.3) × 2 ϕ 1 −q 1/2 , −ω 2 q 3/2 ω 2 ; q 2 , −q 5/2 . For large values of ω, such that ω 2 = q −3/2−2n where n = 0, 1, 2, ... , 2 ϕ 1 −q 1/2 , −ω 2 q 3/2 ω 2 ; q 2 , −q 5/2 (5.4) → 1 ϕ 0 −q 1/2 − ; q 2 , q = −q 3/2 ; q 2 ∞ (q; q 2 ) ∞ , by the q-binomial theorem. Therefore, as ω → ∞, S (η; ω) = −q 1/2 ; q ∞ (q; q 2 ) 2 ∞ (5.5) × ω q 3/2 ω 2 ; q 2 ∞ (−qω 2 ; q 2 ) ∞ [1 + o(1)] , by (5.3) and (5.4). But the function q 3/2 ω 2 ; q 2 ∞ oscillates and has an infinity of real zeros as ω approaches infinity. Indeed, consider the points ω = γ n , such that γ 2 n = β 2 q −2n , (5.6) where n = 0, 1, 2, ... and q 1/2 < β 2 < q −3/2 , as test points. Then, by using (I.9) of [7], S (η; γ n ) = −q 1/2 ; q ∞ (q; q 2 ) 2 ∞ β q 3/2 β 2 ; q 2 ∞ (−qβ 2 ; q 2 ) ∞ (5.7) × (−1) n q −n/2 q 1/2 /β 2 ; q 2 n (−q/β 2 ; q 2 ) n [1 + o(1)] , as n → ∞, and one can see that the right side of (5.7) changes sign infinitely many times at the test points ω = γ n as ω approaches infinity. In a similar manner, one can prove that the q-cosine function C (η; ω) has an infinity of real ω-zeros also. Thus we have established the following theorem. Theorem 5.1. The basic sine S (η; ω) and basic cosine C (η; ω) functions have an infinity of real ω-zeros when 0 < q < 1. Now we can prove our next result. Theorem 5.2. The basic sine S (η; ω) and basic cosine C (η; ω) functions have only real ω-zeros when 0 < q < 1. Proof. Suppose that ω 0 is a zero of the basic sine function (5.1) which is not real. It follows from (5.1) and (III.4) of [7] that S (η; ω) = q 5/2 ω 2 ; q 2 ∞ (−qω 2 ; q 2 ) ∞ ω 1 − q 1/2 (5.8) × 2 ϕ 2 −q 3/2 , −q 5/2 q 3 , q 5/2 ω 2 ; q 2 , q 3/2 ω 2 . Now we can see that ω 0 is not purely imaginary, because otherwise our function would be a multiple of a positive function. Let ω 1 be the complex number conjugate to ω 0 , so that ω 1 is also a zero of (5.1) because this function is a real function of ω. Since ω 2 0 = ω 2 1 the integral in the orthogonality relation (3.26) equals zero, but the integrand on the left is positive, and so we have obtained a contradiction. Hence a complex zero ω 0 cannot exist. One can consider the case of the basic cosine function in a similar fashion. Theorem 5.3. If 0 < q < 1, then the real ω-zeros of the basic sine S (η; ω) and basic cosine C (η; ω) functions are simple. Proof. This follows directly from the relations (3.31) and (3.32). Consider, for example, the case of the basic sine function. If ω = ω ′ , then the integral in the left side of (3.31) is positive, which means that ∂ ∂ω S (η; ω) = 0 when S (η; ω) = 0. The same is true for the zeros of the basic cosine function. Our next property is that the positive zeros of the basic sine function S (η; ω) are interlaced with those of the basic cosine function C (η; ω). Theorem 5.4. If ω 1 , ω 2 , ω 3 , ... are the positive zeros of S (η; ω) arranged in ascending order of magnitude, and ̟ 1 , ̟ 2 , ̟ 3 , ... are those of C (η; ω), then 0 = ω 0 < ̟ 1 < ω 1 < ̟ 2 < ω 3 < ̟ 3 < ... , (5.9) if 0 < q < 1. Proof. Suppose that ω k and ω k+1 are two successive zeros of S (η; ω). Then the derivative ∂ ∂ω S (η; ω) has different signs at ω = ω k and ω = ω k+1 . This means, in view of (3.32), that C (η; ω) changes its sign between ω k and ω k+1 and , therefore, has at least one zero on each interval (ω k , ω k+1 ). To complete the proof of the theorem, we should show that C (η; ω) changes its sign on each interval (ω k , ω k+1 ) only once. Suppose that C (η; ̟ k ) = C (η; ̟ k+1 ) = 0 and ω k < ̟ k < ̟ k+1 < ω k+1 . Then, by (3.32), the function S (η; ω) has different signs at ω = ̟ k and ω = ̟ k+1 and, therefore, this function has at least one more zero on (ω k , ω k+1 ). So, we have obtained a contradiction, and, therefore, the basic cosine function C (η; ω) has exactly one zero between any two successive zeros of the basic sine function S (η; ω). The proof of Theorem 5.1 has strongly indicated that asymptotically the large ω-zeros of the basic sine function S (η; ω) are ω n = ±κ n q −n , q 1/4 ≤ κ n < q −3/4 (5.10) as n → ∞. The same consideration as in [11] and [13] shows that S (η; ω) changes sign only once between any two successive test points ω = γ n and ω = γ n+1 determined by (5.6) for large values of n. We include details of this proof in Section 16 to make this work as self-contained as possible. Our next theorem provides a more accurate estimate for the distribution of the large zeros of this function. Theorem 5.5. If ω 1 , ω 2 , ω 3 , ... are the positive zeros of S (η; ω) arranged in ascending order of magnitude, then ω n = q 1/4−n + o (1) , (5.11) as n → ∞. Proof. In view of (5.1) and (III.32) of [7], S (η; ω) = ω 1 − q 1/2 −q 3/2 , −q 5/2 , q 3/2 ω 2 , q 1/2 /ω 2 ; q 2 ∞ (q, q 3 , −qω 2 , −q 2 /ω 2 ; q 2 ) ∞ (5.12) × 2 ϕ 1 −q 1/2 , −q 3/2 q ; q 2 , − q ω 2 + ω 1 − q 1/2 −q 1/2 , −q 3/2 , q 5/2 ω 2 , q −1/2 /ω 2 ; q 2 ∞ (q −1 , q 3 , −qω 2 , −q 2 /ω 2 ; q 2 ) ∞ × 2 ϕ 1 −q 3/2 , −q 5/2 q 3 ; q 2 , − q ω 2 , which gives the large ω-asymptotic of S (η; ω). When ω = q 1/4−n and n = 1, 2, 3, ..., the first term in (5.12) vanishes and we get S η; q 1/4−n = (−1) n q n/2−1/4 1 + q 1/2 1 − q 1/2 −q 3/2 ; q 2 n (−q 1/2 ; q 2 ) n (5.13) × 2 ϕ 1 −q 3/2 , −q 5/2 q 3 ; q 2 , −q 2n+1/2 with the help of (I.9) of [7], Thus, lim n→∞ S η; q 1/4−n = 0, (5.14) which proves our theorem. In a similar fashion, one can establish the following theorem. Theorem 5.6. If ̟ 1 , ̟ 2 , ̟ 3 , ... are the positive zeros of C (η; ω) arranged in ascending order of magnitude, then ̟ n = q 3/4−n + o (1) , (5.15) as n → ∞. The asymptotic formulas (5.11) and (5.15) for large ω-zeros of the basic sine S (η; ω) and basic cosine C (η; ω) functions confirm the interlacing property (5.9) from Theorem 5.4. Let us also discuss the large ω-asymptotics of the basic sine S (x; ω) and basic cosine C (x; ω) functions when x = cos θ belongs to the interval of orthogonality −1 < x < 1. From (2.1) and (2.2) one gets C (cos θ; ω) = 2 ϕ 1 −e 2iθ , −e −2iθ q ; q 2 , −qω 2 (5.16) = −e −2iθ , −qe −2iθ , qω 2 e 2iθ , e −2iθ /ω 2 ; q 2 ∞ (q, e −4iθ , −qω 2 , −q/ω 2 ; q 2 ) ∞ × 2 ϕ 1 −e 2iθ , −qe 2iθ q 2 e 4iθ ; q 2 , − q 2 ω 2 + −e 2iθ , −qe 2iθ , qω 2 e −2iθ , e 2iθ /ω 2 ; q 2 ∞ (q, e 4iθ , −qω 2 , −q/ω 2 ; q 2 ) ∞ × 2 ϕ 1 −e −2iθ , −qe −2iθ q 2 e −4iθ ; q 2 , − q 2 ω 2 and S (cos θ; ω) = 2q 1/4 ω 1 − q cos θ (5.17) × 2 ϕ 1 −qe 2iθ , −qe −2iθ q 3 ; q 2 , −qω 2 = 2q 1/4 ω 1 − q cos θ × −qe −2iθ , −q 2 e −2iθ , q 2 ω 2 e 2iθ , e −2iθ /ω 2 ; q 2 ∞ (q 3 , e −4iθ , −qω 2 , −q/ω 2 ; q 2 ) ∞ × 2 ϕ 1 −e 2iθ , −qe 2iθ q 2 e 4iθ ; q 2 , − q 2 ω 2 + −qe 2iθ , −q 2 e 2iθ , q 2 ω 2 e −2iθ , e 2iθ /ω 2 ; q 2 ∞ (q 3 , e 4iθ , −qω 2 , −q/ω 2 ; q 2 ) ∞ × 2 ϕ 1 −e −2iθ , −qe −2iθ q 2 e −4iθ ; q 2 , − q 2 ω 2 by (III.3) and (III.32) of [7]. For \|x\| < 1, \|q\| < 1 and large ω it is clear from (5.16) and (5.17) that the leading terms in the asymptotic expansions of C (cos θ; ω) and S (cos θ; ω) are given by C (cos θ; ω) ∼ −e −2iθ ; q ∞ (q, e −4iθ ; q 2 ) ∞ qω 2 e 2iθ ; q 2 ∞ (−qω 2 ; q 2 ) ∞ (5.18) + −e 2iθ ; q ∞ (q, e 4iθ ; q 2 ) ∞ qω 2 e −2iθ ; q 2 ∞ (−qω 2 ; q 2 ) ∞ and S (cos θ; ω) ∼ 2q 1/4 ω 1 − q cos θ (5.19) × −qe −2iθ ; q ∞ (q 3 , e −4iθ ; q 2 ) ∞ q 2 ω 2 e 2iθ ; q 2 ∞ (−qω 2 ; q 2 ) ∞ + −qe 2iθ ; q ∞ (q 3 , e 4iθ ; q 2 ) ∞ q 2 ω 2 e −2iθ ; q 2 ∞ (−qω 2 ; q 2 ) ∞ , respectively. In particular, when ω = ω n are large zeros of the basic sine function S (η; ω) we can estimate C (cos θ; ω n ) ∼ C cos θ; q 1/4−n , (5.20) S (cos θ; ω n ) ∼ S cos θ; q 1/4−n (5.21) due to (5.11) as n → ∞. Relations (5.18)-(5.21) lead to the following theorem. Theorem 5.7. For −1 < x = cos θ < 1 and \|q\| < 1 the leading term in the asymptotic expansion of C (cos θ; ω n ) as n → ∞ is given by For −1 < x = cos θ < 1 and \|q\| < 1 the leading term in the asymptotic expansion of S (cos θ; ω n ) as n → ∞ is given by C cos θ; q 1/4−n ∼ 2 q 1/2 ; q ∞ (q; q 2 ) 2 ∞ (5.22) × A e iθ cos ((2θ + π) n − χ) , where A e iθ = 1 − q 1/2 e 2iθ q 3/2 e −2iθ , q 5/2 e 2iθ ; q 2 ∞ (e 2iθ ; q) ∞ , (5.23) A e iθ −2 = e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ ,(5.S cos θ; q 1/4−n ∼ 2 q 1/2 ; q ∞ (q; q 2 ) 2 ∞ (5.26) × B e iθ cos ((2θ + π) (n − 1) − ψ) , where B e iθ = e iθ q 1/2 e −2iθ , q 3/2 e 2iθ ; q 2 It is worth mentioning also that the factor A e iθ −2 = B e iθ −2 coinsides with the weight function in our orthogonality relations (3.26)-(3.28) for the basic trigonometric functions. ∞ (e 2iθ ; q) ∞ , (5.27) B e iθ −2 = e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ ,(5. In a similar fashion, one can use the first lines in (5.16), (5.17), and Exercise 3.8 of [7] (see also the same line of reasonings in [8]) to establish complete asymptotic expansions of the basic sine and cosine functions for the large values of ω. Theorem 5.8. For −1 < x = cos θ < 1 and \|q\| < 1 complete asymptotic expansions of C (cos θ; ω) and S (cos θ; ω) as \|ω\| → ∞ are given by C (cos θ; ω) = qω 2 e 2iθ ; q 2 ∞ (e −2iθ ; q) ∞ (q, −qω 2 ; q 2 ) ∞ (5.31) × ∞ n=0 q 2n −e 2iθ ; q 2n (q 2 , q 2 e 4iθ ; q 2 ) n qω 2 e 2iθ ; q 2 −1 n + qω 2 e −2iθ ; q 2 ∞ (e 2iθ ; q) ∞ (q, −qω 2 ; q 2 ) ∞ × ∞ n=0 q 2n −e −2iθ ; q 2n (q 2 , q 2 e −4iθ ; q 2 ) n qω 2 e −2iθ ; q 2 −1 n and S (cos θ; ω) = e iθ q 2 ω 2 e 2iθ ; q 2 ∞ (e −2iθ ; q) ∞ (q, −qω 2 ; q 2 ) ∞ (5.32) × ∞ n=0 q 2n+1/4 −qe 2iθ ; q 2n (q 2 , q 2 e 4iθ ; q 2 ) n q 2 ω 2 e 2iθ ; q 2 −1 n +e −iθ q 2 ω 2 e −2iθ ; q 2 ∞ (e 2iθ ; q) ∞ (q, −qω 2 ; q 2 ) ∞ × ∞ n=0 q 2n+1/4 −qe −2iθ ; q 2n (q 2 , q 2 e −4iθ ; q 2 ) n qω 2 e −2iθ ; q 2 −1 n . The asymptotic expansions (5.31)-(5.32) are not in terms of the usual asymptotic sequence (xω) −n ∞ n=0 , but are sums of two complete asymptotic expansions in terms of the "inverse generalized powers" q 2 ω 2 e ±2iθ ; q 2 −1 n (cf. [8]). Remark 5.1. Mourad Ismail pointed out to our attention the following quadratic transformation formula 2 ϕ 1 −q ν+1 , −q ν+2 q 2ν+2 ; q 2 , −r 2 /4 (5.33) = (q; q) ∞ (q ν+1 ; q) ∞ (2/r) ν (−r 2 /4; q 2 ) ∞ J (2) ν (r; q) , where \|r\| < 2, relating the 2 ϕ 1 of a given structure with Jackson's basic Bessel functions J ν (r; q). A similar relation was earlier found by Rahman [22]. This transformation shows that our basic sine S (η; ω) and basic cosine C (η; ω) functions are just multiples of J 1/2 (2ω; q) and J (2) −1/2 (2ω; q), namely, S (η; ω) = (q; q) ∞ (q 1/2 ; q) ∞ ω 1/2 (−qω 2 ; q 2 ) ∞ J(2) 1/2 (2ω; q) , (5.34) C (η; ω) = (q; q) ∞ (q 1/2 ; q) ∞ ω 1/2 (−qω 2 ; q 2 ) ∞ J (2) −1/2 (2ω; q) . (5.35) The main properties of zeros of the q-Bessel functions J (2) ν (r; q) were established in Ismail's papers [9] and [10] by a different method. This gives independent proofs of our Theorems 5.1-5.4. Some monotonicity properties of zeros of J (2) ν (r; q) were discussed in [14]. Chen, Ismail, and Muttalib [8] have found a complete asymptotic expansion of J (2) ν (r; q) for the large argument, J (2) ν (r; q) = q 1/2 ; q ∞ 2 (q; q) ∞ r 2 ν (5.36) × i r 2 q (ν+1/2)/2 ; q 1/2 ∞ × ∞ n=0 q n/2 q ν+1/2 ; q n (q; q) n i r 2 q (ν+1/2)/2 ; q 1/2 −1 n + −i r 2 q (ν+1/2)/2 ; q 1/2 ∞ × ∞ n=0 q n/2 q ν+1/2 ; q n (q; q) n −i r 2 q (ν+1/2)/2 ; q 1/2 −1 n . This follows also from Exersises 3.15 and 3.8 of [7]. Equations (5.34)-(5.36) result in (5.11) and (5.15). Evaluation of Some Constants In this section we shall find explicitly the values of the normalization constants in the right sides of the orthogonality relations (3.26)-(3.27) for the basic sine and basic cosine functions. First, we evaluate the integral 2k (ω) = π 0 C 2 (cos θ; ω) + S 2 (cos θ; ω) (6.1) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = π 0 C (cos θ, − cos θ; ω) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ, where we have used the identity (4.14) of [24], C (x, −x; ω) = C 2 (x; ω) + S 2 (x; ω) . (6.2) In view of (2.3), for \|ω\| < 1 one can write 2 (−qω 2 ; q 2 ) ∞ (−ω 2 ; q 2 ) ∞ k (ω) = ∞ n=0 −ω 2 n q 1/2 ; q n (q, −q, −q 1/2 ; q) n (6.3) × π 0 e 2iθ , e −2iθ ; q ∞ (q n+1/2 e 2iθ , q n+1/2 e −2iθ ; q) ∞ dθ. The integral in the right side is a special case of the Askey-Wilson integral [4], π 0 e 2iθ , e −2iθ ; q ∞ (q n+1/2 e 2iθ , q n+1/2 e −2iθ ; q) ∞ dθ (6.4) = 2π (q 2n+2 ; q) ∞ (q, −q n+1/2 , q n+1 , −q n+1 , q n+1 , −q n+1 , −q n+3/2 ; q) ∞ . Therefore, 2 (−qω 2 ; q 2 ) ∞ (−ω 2 ; q 2 ) ∞ k (ω) = 2π q 1/2 ; q ∞ (q, q, −q, −q 1/2 ; q) ∞ (6.5) × ∞ n=0 (−ω 2 ) n 1 − q n+1/2 , where we have used the identity q 2n+2 ; q ∞ = q n+1 , −q n+1 , q n+3/2 , −q n+3/2 ; q ∞ . But, ∞ n=0 (−ω 2 ) n 1 − q n+1/2 = 1 1 − q 1/2 2 ϕ 1 q, q 1/2 q 3/2 ; q, −ω 2 = q, −q 1/2 ω 2 ; q ∞ (q 1/2 , −ω 2 ; q) ∞ 2 ϕ 1 q 1/2 , −ω 2 −q 1/2 ω 2 ; q, q by (III.1) of [7]. The last line provides an analytic continuation of this sum in the complex ω-plane. Finally, we obtain k (ω) = 1 2 π 0 C 2 (cos θ; ω) + S 2 (cos θ; ω) (6.6) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = π q 1/2 , −q 1/2 ω 2 ; q ∞ (q, −ω 2 ; q) ∞ (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ (6.7) × 2 ϕ 1 q 1/2 , −ω 2 −q 1/2 ω 2 ; q, q . The second line gives the large asymptotic of the function k (ω), k (ω) = π −q 1/2 ω 2 ; q ∞ (−ω 2 ; q) ∞ (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ 1 + o ω −2 , (6.8) as ω 2 → ∞ but ω 2 = −q −n−1/2 for a positive integer n. In particular, when ω = ω n are large zeros of the basic sine function S (η; ω) one gets as n → ∞ k (ω n ) ∼ k q 1/4−n ∼ 2π (−q; q) 2 ∞ (−q 1/2 ; q) 2 ∞ (6.9) by (5.11) and (I.9) of [7]. With the aid of (6.6)-(6.7) one can now rewrite (3.31) and (3.32) in more explicit form, π 0 C 2 (cos θ; ω) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (6.10) = k (ω) + π q 1/2 ; q 2 ∞ ω (q; q) 2 ∞ C (η; ω) S (η; ω) and π 0 S 2 (cos θ; ω) e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ (6.11) = k (ω) − π q 1/2 ; q 2 ∞ ω (q; q) 2 ∞ C (η; ω) S (η; ω) . These basic integrals are, obviously, q-extensions of the following elementary integrals When ω satisfies the boundary condition (3.32) the last terms in the right sides of (6.10) and (6.11) vanish and we obtain the values of the normalization constants in the orthogonality relations (3.26)-(3.28) in terms of the function k (ω) defined by (6.7). Orthogonality Relations for q-Exponential Functions Euler's formula, e iωx = cos ωx + i sin ωx, The q-analog of Euler's formula (7.1) is E q (x; iω) = C q (x; ω) + iS q (x; ω) , (7.4) where E q (x; α) with α = iω is the q-exponential function introduced in [15] (see also [5] and [24], we shall use the same notations as in [24]); C q (x; ω) and S q (x; ω) are basic cosine and sine functions defined by (2.1) and (2.2), respectively. Our orthogonality relations for the basic trigonometric functions (3.26)-(3.28) result in the following orthogonality property for the q-exponential function 1 2k (ω n ) π 0 E q (cos θ; iω m ) E q (cos θ; −iω n ) (7.5) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = δ mn , where ω m , ω n = 0, ±ω 1 , ±ω 2 , ±ω 3 , ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude; the normalization constants k (ω n ) are defined by (6.7). A basic analog of e iω(x+y) = cos ω (x + y) + i sin ω (x + y) (7.6) is E q (x, y; iω) = C q (x, y; ω) + iS q (x, y; ω) , (7.7) see [15] and [24]. The general exponential function on a q-quadratic grid E q (x, y; iω) has the following orthogonality property. Theorem 7.1. π 0 E q (cos θ, cos ϕ; iω m ) E q (cos θ, cos ϕ ′ ; −iω n ) (7.8) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = 2k (ω n ) E q (cos ϕ; iω n ) E q (cos ϕ ′ ; −iω n ) δ mn , where ω m , ω n = 0, ±ω 1 , ±ω 2 , ±ω 3 , ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude; the normalization constants k (ω n ) are defined by (6.7). Proof. Using of the "addition" theorem for basic exponential functions [24], E q (x, y; iω) = E q (x; iω) E q (y; iω) , (7.9) and the orthogonality relation (7.5) one gets (7.8). In a similar fashion, we can establish the following results. Theorem 7.2. π 0 C (cos θ, cos ϕ; ω m ) C (cos θ, cos ϕ ′ ; ω n ) (7.10) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = 0 if m = n, k (ω n ) C (cos ϕ, − cos ϕ ′ ; ω n ) if m = n; π 0 S (cos θ, cos ϕ; ω m ) S (cos θ, cos ϕ ′ ; ω n ) (7.11) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = 0 if m = n, k (ω n ) C (cos ϕ, − cos ϕ ′ ; ω n ) if m = n; and π 0 C (cos θ, cos ϕ; ω m ) S (cos θ, cos ϕ ′ ; ω n ) (7.12) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ = 0 if m = n, k (ω n ) S (cos ϕ, − cos ϕ ′ ; ω n ) if m = n; where ω m , ω n = ω 1 , ω 2 , ω 3 , ..., are positive zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude; the normalization constants k (ω n ) are defined by (6.7). Proof. Use the "addition" theorem for the basic trigonometric functions [24] and the orthogonality relations (3.26)-(3.28). Basic Fourier Series By analogy with (1.2) we can now introduce a q-version of Fourier series, f (cos θ) = a 0 + ∞ n=1 (a n C q (cos θ; ω n ) + b n S q (cos θ; ω n )) , (8.1) where ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude, and a 0 = 1 2k (0) π 0 f (cos θ) (8.2) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ, a n = 1 k (ω n ) π 0 f (cos θ) C q (cos θ; ω n ) (8.3) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ, b n = 1 k (ω n ) π 0 f (cos θ) S q (cos θ; ω n ) (8.4) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ. The complex form of the basic Fourier series (8.1) is f (cos θ) = ∞ n=−∞ c n E q (cos θ; iω n ) (8.5) with c n = 1 2k (ω n ) π 0 f (cos θ) E q (cos θ; −iω n ) (8.6) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ where ω n = 0, ±ω 1 , ±ω 2 , ±ω 3 , ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude; the normalization constants k (ω n ) are defined by (6.7). These expressions, of course, merely indicate how the coefficients of our basic Fourier series are to be determined on the hypothesis that the expansion exists and is uniformly convergent. We shall study the question of convergence of the series (8.1) and (8.5) in the next sections. The q-Fourier series of f in either of the forms (8.1) and (8.5) will be denoted in a usual manner by S [f ]. Completeness of the q-Trigonometric System Completeness of the trigonometric system {e iπnx } ∞ n=−∞ on the interval (−1, 1) is one of the fundamental facts in the theory of trigonometric series (see, for example, [1], [18], [20], [19] and [30]). In this section we shall prove a similar property for the system of basic trigonometric function {E q (x; iω n )}, where ω n = 0, ±ω 1 , ±ω 2 , ±ω 3 , ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude. But first we need to discuss connections between the basic trigonometric functions and the continuous q-Hermite polynomials. The continuous q-Hermite polynomials, H n (cos θ\|q) = n k=0 (q; q) n (q; q) k (q; q) n−k e i(n−2k)θ , (9.1) have two generating functions, ∞ n=0 r n (q; q) n H n (cos θ\|q) = 1 (re iθ , re −iθ ; q) ∞ , (9.2) when \|r\| < 1 and ∞ n=0 q n 2 /4 (q; q) n α n H n (cos θ\|q) = qα 2 ; q 2 ∞ E q (cos θ; α) (9.3) (see, for example, [7], [15], and [24]). Lemma 9.1. The following functions with h n = h n (x) = q n 2 /4 (q; q) n H n (x\|q) . Thus, e (x, α) is an entire function in α. The order of this entire function is [19] lim n→∞ n log n − log \|h n \| (9.10) = lim n→∞ n log n − log \|q n 2 /4 H n (x\|q) / (q; q) n \| = 0. e (x, α) = qα 2 ; q 2 ∞ E q (x; α) , (9.4) s(ω) = −qω 2 ; q 2 S (η; ω) (9.5) = 1 2i (e(x, iω) − e(x, −iω)) , Functions (9.5) and (9.6) are just a sum or difference of two functions of type (9.4), so they are also entire functions of order zero. This proves the lemma. The next step is to establish the following inequalities. (q; q) n (q; q) k (q; q) n−k cosh (n − 2k) τ (9.14) = H n (cosh τ \|q) . Lemma 9.2. Let − cosh τ ≤ −1 ≤ x ≤ 1 ≤ cosh τ, where x = cos θ, 0 ≤ θ ≤ π, Estimating both sides of (9.3) gives qα 2 ; q 2 ∞ E q (cos θ; α) ≤ ∞ n=0 q n 2 /4 (q; q) n \|α\| n \|H n (cos θ\|q)\| ≤ ∞ n=0 q n 2 /4 (q; q) n \|α\| n H n (cosh τ \|q) = q \|α\| 2 ; q 2 ∞ E q (cosh τ ; \|α\|) by (9.14) and (9.3). This proves (9.11). The monotonicity property (9.12) follows from the monotonicity of the hyperbolic cosine function. It is clear that the system {E q (x; iω n )} ∞ n=−∞ is complete if the equivalent system {e (x, iω n )} ∞ n=−∞ is closed. Suppose that the system {e (x, iω n )} ∞ n=−∞ is not closed on (−1, 1). This means that there exists at least one function φ(x), not identically zero, such that ±1, ±2, ..., (9.15) where ρ(x) is the absolutely continuous measure in the orthogonality relation (7.5). 1 −1 φ(x) e (x, iω n ) ρ(x) dx = 0, n = 0, Then, the function f (ω) = 1 −1 φ(x) e (x, iω n ) ρ(x) dx (9.16) is an entire function of order zero and f (ω n ) = 0 for all n = 0, ±1, ±2, .... Thus the study of closure amounts to the study of zeros of a certain entire function. Suppose that φ(x) is integrable on (−1, 1), 1 −1 \|φ(x)\| ρ(x) dx = A < ∞. (9.17) Then \|f (ω)\| ≤ 1 −1 \|φ(x) e (x, iω)\| ρ(x) dx (9.18) ≤ e (cosh τ, \|ω\|) × 1 −1 \|φ(x)\| ρ(x) dx = A e (cosh τ, \|ω\|) by (9.11) and (9.17). Consider the quotient g(ω) = f (ω) s(ω) (9.19) of two entire functions, f (ω) and s(ω) defined by (9.16) and (9.5), respestively. The functions f (ω) and s(ω) have the same zeros, so g(ω) is an entire function. The order of this entire function is zero because both f (ω) and s(ω) are of order zero (see [19], Corollary of Theorem 12 on p. 24). Moreover, this function g(ω) is bounded on a straight line parallel to the imaginary axis. Indeed, let ω = γ + iδ. Using the same arguments as in Section 5 one can see that lim \|δ\|→∞ s(iδ) e(η, \|δ\|) < ∞. (9.20) From this condition and the inequality (9.18), it follows that the entire function g(ω) is bounded on the imaginary axis. But an entire function of order zero bounded on a line must be a constant (see and Corollary on pp. 49-51 of [19]). Then, f (ω) = c s(ω) (9.21) and, therefore, \|c\| = 1 −1 φ(x) e (x, iω) s(ω) ρ(x) dx (9.22) ≤ 1 −1 φ(x) e (x, iω) s(ω) ρ(x) dx ≤ A E q (cosh τ ; \|ω\|) S(cosh τ 1 ; \|ω\|) → 0 as \|ω\| → ∞ and τ < τ 1 . Thus, f (ω) is identically zero and the function φ(x) does not exist. We have established the following theorem. Theorem 9.3. The system of the basic trigonometric function {E q (x; iω n )}, where n = 0, ±1, ±2, ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude, is complete on (−1, 1). As corollaries we have the following results. Theorem 9.4. If f (x) and g(x) have the same q-Fourier series, then f ≡ g. Proof. The q-Fourier coefficients of f − g all vanish, so that f − g ≡ 0. Bessel's inequality for the q-trigonometric system {E q (x; iω n )} ∞ n=−∞ , where ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude, takes the form 1), which means that \|f (x)\| 2 is integrable on (−1, 1) with respect to the weight function ρ(x) in the orthogonality relation (7.5). Here c n are the q-Fourier coefficients of f (x) defined by (8.6). When N → ∞ we get Parseval's formula (9.24) due to the completeness of the q-trigonometric system {E q (x; iω n )} ∞ n=−∞ and the space L 2 ρ (−1, 1) [1], [18]. It follows that the q-Fourier coefficients c n tend to zero if f ∈ L 2 ρ (−1, 1). N n=−N \|c n \| 2 ≤ 1 −1 \|f (x)\| 2 ρ(x) dx (9.23) provided f ∈ L 2 ρ (−1,∞ n=−∞ \|c n \| 2 = 1 −1 \|f (x)\| 2 ρ(x) dx Bilinear Generating Function In this section we shall derive the following bilinear generating relation, ∞ n=−∞ (−qr 2 ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ k −1 (ω n ) (10.1) × E q (cos θ; iω n ) E q (cos ϕ; −irω n ) = q, r 2 , q 1/2 e 2iθ , q 1/2 e −2iθ ; q ∞ π (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ , for the basic exponential functions. Here as before ω n = 0, ±ω 1 , ±ω 2 , ±ω 3 , ... and ω 0 = 0, ω 1 , ω 2 , ω 3 , ... are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude. We shall use this generating function for a further investigation of the convergence of the basic Fourier series (8.5) in the subsequent section. Let us establish a connecting relation of the form, (qα 2 r 2 ; q 2 ) ∞ (qα 2 ; q 2 ) ∞ E q (cos θ; αr) (10.2) = 1 2π π 0 (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ × E q (cos ϕ; α) dϕ, where \|r\| < 1. One can easily see that if we could prove the uniform convergence in the variable x = cos θ of the series in the left side of (10.1), than the integral in (10.2) gives the correct values of the basic Fourier coefficients (see (8.5)-(8.6)), which verifies the generating relation (10.1) by Theorem 9.5. So, one needs to give a prove of (10.2) first. The continuous q-Hermite polynomials have the following bilinear generating function (the Poisson kernel), ∞ n=0 r n (q; q) n H n (cos θ\|q) H n (cos ϕ\|q) (10.3) = (r 2 ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ , where \|r\| < 1. The orthogonality relation for these polynomials is (q; q) n (q; q) ∞ δ mn (see, for example, [7]). Expanding E q (cos ϕ; α) in the right side of (10.2) in the uniformly convergent series of the q-Hermite polynomials with the aid of (9.3), we get 1 2π π 0 (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ (10.5) × qα 2 ; q 2 ∞ E q (cos ϕ; α) dϕ = ∞ n=0 q n 2 /4 (q; q) n α n × 1 2π π 0 (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ ×H n (cos ϕ\|q) dϕ. The series in (10.3) converges uniformly when \|r\| < 1. Then, using (10.4), 1 2π π 0 H n (cos ϕ\|q) (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ dϕ (10.6) = r n H n (cos θ\|q) . Uniform convergence of the series in (10.1) can be justified with the help of the inequality (9.11) and the corresponding asymptotic expressions. This proves (10.1) by Theorem 9.5. It is worth mentioning a few special cases of (10.1). When r = 0 we obtain the following generating function, ∞ n=−∞ 1 (−qω 2 n ; q 2 ) ∞ k (ω n ) (10.7) × E q (cos θ; iω n ) = q, q 1/2 e 2iθ , q 1/2 e −2iθ ; q ∞ , for E q (x; iω n ). If ϕ = π/2, one gets ∞ n=−∞ (−qr 2 ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ k −1 (ω n ) (10.8) × E q (cos θ; iω n ) = q, r 2 , q 1/2 e 2iθ , q 1/2 e −2iθ ; q ∞ π (−r 2 e 2iθ , −r 2 e −2iθ ; q 2 ) ∞ . A terminating case of this generating relation appears when r 2 = −1/qω 2 m for an integer m = 0, \|m\| n=−\|m\| (ω 2 n /ω 2 m ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ k −1 (ω n ) (10.9) × E q (cos θ; iω n ) = q, r 2 , q 1/2 e 2iθ , q 1/2 e −2iθ ; q ∞ π (e 2iθ /qω 2 m , e −2iθ /qω 2 m ; q 2 ) ∞ . Here m = ±1, ±2, ±3, .... Method of Summation of Basic Fourier Series According to Theorem 9.5, for a continuous function f (x) the basic Fourier series S [f ] converges to f (x) if it converges uniformly. In this section we shall discuss another method of summation of basic Fourier series. Let f (x) be a bounded function that is continuous on (−1, 1) and let S [f ] be its q-Fourier series defined by the right side of (8.5). Replace this series by S r [f ] = ∞ n=−∞ c n (r) E q (cos θ; iω n ) , (11.1) where c n (r) = (−qr 2 ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ (11.2) × 1 2k (ω n ) π 0 f (cos θ) E q (cos θ; −irω n ) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ provided that 0 < r < 1. Comparing (11.2) and (8.6), lim r→1 − c n (r) = c n , (11.3) where c n are the regular q-Fourier coefficients of f (x). Suppose that the series S r [f ] converges uniformly with respect to the parameter r when 0 < r < 1. Then, lim r→1 − S r [f ] = S [f ] . (11.4) On the other hand, from (11.1)-(11.2) one gets S r [f ] = ∞ n=−∞ (−qr 2 ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ E q (cos θ; iω n ) (11.5) × 1 2k (ω n ) π 0 f (cos ϕ) E q (cos ϕ; −irω n ) × (e 2iϕ , e −2iϕ ; q) ∞ (q 1/2 e 2iϕ , q 1/2 e −2iϕ ; q) ∞ dϕ. Using the uniform convergence of the series in the bilinear generating function (10.1), we finally obtain S r [f ] = 1 2π π 0 f (cos ϕ) (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ dϕ. (11.6) It has been shown in [3] (see also [29]) that lim r→1 − 1 2π π 0 f (cos ϕ) (q, r 2 , e 2iϕ , e −2iϕ ; q) ∞ (re iθ+iϕ , re iθ−iϕ , re iϕ−iθ , re −iθ−iϕ ; q) ∞ dϕ = f (cos θ) (11.7) for every bounded function f (cos θ) that is continuous on 0 < θ < π. As a result we have proved the following theorem. Theorem 11.1. Let f (x) be a bounded function that is continuous on (−1, 1) and let S r [f ] be the series defined by (11.1)- (11.2). If S r [f ] converges uniformly with respect to the parameter r when 0 < r < 1, then lim r→1 − S r [f ] = S [f ] = f (x). 12. Relation Between q-Trigonometric System and q-Legendre Polynomials The trigonometric system {e iπnx } ∞ n=−∞ and the system of the Legendre polynomials {P m (x)} ∞ m=0 are two complete systems in L 2 (−1, 1). The corresponding unitary transformation between these two orthogonal basises and its inverse are e iπnx = 2 πn 1/2 ∞ m=0 i m (m + 1/2) J m+1/2 (πn) P m (x) (12.1) and P m (x) = ∞ n=−∞ (−i) m 1 2πn 1/2 J m+1/2 (πn) e iπnx ,(12.2) respectively. Relation (12.1) is a special case of a more general expansion, e irx = 2 r ν Γ (ν) ∞ m=0 i m (ν + m) J ν+m (r) C ν m (x) , (12.3) where C ν m (x) are ultraspherical polynomials and J ν+m (r) are Bessel functions [27]. Expansion (12.2) is the Fourier series of the Legendre polynomials on (−1, 1). Orthogonality properties of the trigonometric system and Legendre polynomials lead to the orthogonality relations, (12.5) for the corresponding Bessel functions. The basic trigonometric system {E q (x; iω n )} ∞ n=−∞ and the system of the continuous q-ultraspherical polynomials {C m (x; β\| q)} ∞ m=0 with β = q 1/2 , which are the basic analogs of the Legendre polynomials, are two complete orthogonal systems in L 2 ρ (−1, 1), where ρ is the weight function in the orthogonality relation (7.5). Therefore, there exists a q-version of the unitary transformation (12.1)-(12.2). ∞ m=0 m + 1/2 πn J m+1/2 (πn) J m+1/2 (πl) = δ nl , (12.4) ∞ n=−∞ m + 1/2 πn J m+1/2 (πn) J p+1/2 (πn) = δ mp , Ismail and Zhang [15] have found the following q-analog of (12.3), (12.6) where J (2) ν+m (2ω; q) is Jackson's q-Bessel function (see, for example, [7]). Special case ν = 1/2 gives the basic analog of the expansion (12.1), E q (x; iω) = (q; q) ∞ ω −ν (−qω 2 ; q 2 ) ∞ (q ν ; q) ∞ × ∞ m=0 i m 1 − q ν+m q m 2 /4 J (2) ν+m (2ω; q) C m (x; q ν \| q) ,E q (x; iω n ) = (q; q) ∞ ω −1/2 n (−qω 2 n ; q 2 ) ∞ (q 1/2 ; q) ∞ × ∞ m=0 i m 1 − q m+1/2 q m 2 /4 J (2) m+1/2 (2ω n ; q) C m x; q 1/2 q , (12.7) where ω −n = −ω n and ω 0 = 0, ω 1 , ω 2 , ω 3 , ..., are nonnegative zeros of the basic sine function S (η; ω) arranged in ascending order of magnitude. On the other hand, the continuous q-ultraspherical polynomials C m x; q 1/2 q can be expanded in the q-Fourier series as C m x; q 1/2 q = π q 1/2 ; q ∞ (q; q) ∞ × ∞ n=−∞ (−i) m q m 2 /4 ω −1/2 n k (ω n ) (−qω 2 n ; q 2 ) ∞ J (2) m+1/2 (2ω n ; q) (12.8) × E q (x; iω n ) . Indeed, by (8.5)-(8.6), C m x; q 1/2 q = ∞ n=−∞ c n E q (x; iω n ) ,(12.9) where c n = 1 2k (ω n ) π 0 C m cos θ; q 1/2 q E q (cos θ; −iω n ) (12.10) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ. Using (12.7), where the series on the right converge uniformly in x for any ω, and the orthogonality relation π 0 C m cos θ; q 1/2 q C p cos θ; q 1/2 q (12.11) × e 2iθ , e −2iθ ; q ∞ (q 1/2 e 2iθ , q 1/2 e −2iθ ; q) ∞ dθ × J (2) m+1/2 (2ω n ; q) J (2) m+1/2 (2ω l ; q) = δ nl and ∞ n=−∞ π 1 − q m+1/2 ω n k(ω n ) (−qω 2 n ; q 2 ) 2 ∞ q m 2 /2 (12.17) × J (2) m+1/2 (2ω n ; q) J (2) p+1/2 (2ω n ; q) = δ mp for the corresponding Jackson's q-Bessel function. These relations are, clearly, qanalogs of (12.4)-(12.5). Some Basic Trigonometric Identities One of the most important formulas for the trigonometric functions is the main trigonometric identity, cos 2 ωx + sin 2 ωx = 1. (13.1) It follows from the Pythagorean Theorem or from the addition formulas for the trigonometric functions, but one can also prove this identity on the base of the differential equation. The functions cos ωx and sin ωx are two solutions of (1.9) corresponding to the same eigenvalue ω. Therefore, One can extend this consideration to the case of the basic trigonometric functions. Consider equation (3.8) with u(z) = C q (x(z); ω), v(z) = S q (x(z); ω), and ρ (z) = 1, ∆ [W (u(z), v(z))] = 0, (13.4) where W (u, v) = W (C (x; ω) , S (x; ω)) (13.5) = 2q 1/4 ω 1 − q [C (x(z); ω) C (x(z − 1/2); ω) + S (x(z); ω) S (x(z − 1/2); ω)] is the analog of the Wronskian (3.9) and we also used (2.12)-(2.13). One can easily see that W (u, v) here is a doubly periodic function in z without poles in the rectangle on the Figure. Therefore, this function is just a constant by Liouville's theorem, C (x (z)) C (x (z − 1/2)) + S (x (z)) S (x (z − 1/2)) = C. The value of this constant C can be found by choosing x = 0, which gives C = (−ω 2 ; q 2 ) 2 ∞ (−qω 2 ; q 2 ) 2 ∞ × 2 ϕ 1 q, q q ; q 2 , −ω 2 × 2 ϕ 1 1, q q ; q 2 , −ω 2 = (−ω 2 ; q 2 ) 2 ∞ (−qω 2 ; q 2 ) 2 ∞ × 1 ϕ 0 q − ; q 2 , −ω 2 = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ by the q-binomial theorem. As a result one gets C q (cos θ; ω) C q (cos (θ + i log q/2) ; ω) (13.6) + S q (cos θ; ω) S q (cos (θ + i log q/2) ; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ as a q-extension of the main identity (13.1). The special case z = 1/4, when η = x(1/4), of (13.6) has the simplest form C 2 (η; ω) + S 2 (η; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ . (13.7) Our identity (13.6) can also be derived as a special case of the "addition" theorem for the basic trigonometric functions established in [24]. In a similar fashion, we can find an analog of the identity cos 2 ω (x + y) + sin 2 ω (x + y) = 1 (13.8) considering more general basic sine and cosine functions, C (x, y; ω) and S (x, y; ω), as two solutions of equation (2.10). The result is C q (cos θ, cos ϕ; ω) C q (cos (θ + i log q/2) , cos ϕ; ω) (13.9) + S q (cos θ, cos ϕ; ω) S q (cos (θ + i log q/2) , cos ϕ; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ C 2 q (cos ϕ; ω) + S 2 q (cos ϕ; ω) = (−ω 2 ; q 2 ) ∞ (−qω 2 ; q 2 ) ∞ C q (cos ϕ, − cos ϕ; ω) . We have used (6.2) here. This identity can also be verified with the aid of the "addition" theorems for the basic trigonometric functions. Identity (13.7) gives the values of the basic cosine function C (η; ω) at the zeros of the basic sine function S (η; ω), C (η; ω n ) = (−1) n (−ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ , (13.10) and vice versa, (13.11) with the aid of Theorem 5.4. S (η; ̟ n ) = (−1) n (−̟ 2 n ; q 2 ) ∞ (−q̟ 2 n ; q 2 ) ∞ , Example Let us consider a periodic function p 1 (x) which is defined in the interval (−1, 1) by p 1 (x) = x. Its Fourier coefficients are c 0 = 0; c n = 1 2 1 −1 xe −iπnx dx = (−1) n−1 iπn , n = 0. Therefore, x = ∞ n=−∞ ′ (−1) n−1 2iπn e iπnx (14.1) = 2 ∞ n=1 (−1) n−1 sin πnx πn . The special case m = 1 of (12.8), C 1 x; q 1/2 q = −iπ q 1/2 ; q ∞ (q; q) ∞ q 1/4 (14.2) × ∞ n=−∞ ω −1/2 n k (ω n ) (−qω 2 n ; q 2 ) ∞ J(2) 3/2 (2ω n ; q) × E q (x; iω n ) , gives us a possibility to establish the q-analog of (14.1). Let us first simplify the right side of (14.2). Using the three-term recurrence relation for the q-Bessel functions (see Exercise 1.25 of [7]) and (5.32)-(5.33), one gets J (2) 3/2 (2ω n ; q) = −q −1/2 J (2) −1/2 (2ω n ; q) (14.3) = − q 1/2 ; q ∞ (q; q) ∞ (−qω 2 n ; q 2 ) ∞ (qω n ) 1/2 C (η; ω n ) . On the other hand, C 1 x; q 1/2 q = 2 1 + q 1/2 x. (−1) n−1 k (ω n ) ω n (−ω 2 n ; q 2 ) ∞ (−qω 2 n ; q 2 ) ∞ × S q (x; ω n ) . These equations are, clearly, q-analogs of (14.1). Miscellaneous Results Under certain restrictions a function f (z) analytic in the entire complex plane and having zeros at the points a 1 , a 2 , a 3 , ... (these are the only zeros of f (z)), where lim n→∞ \|a n \| is infinite, can be represented as an infinite product f (z) = f (0) e zf ′ (0)/f (0) ∞ n=1 1 − z a n e z/an (15.1) see, for example, [28], [19]. Consider the entire function between the zeros of the basic sine S(η; ω) and basic cosine C(η; ω) functions. 16. Appendix: Estimate of Number of Zeros of S(η; ω) In this section we give an estimate for number of zeros of the basic sine function S(η; ω) on the basis of Jensen's theorem (see, for example, [6] and [19]). We shall apply the method proposed by Mourad Ismail at the level of the third Jackson q-Bessel functions [11](see also [13] for an extension of his idea to q-Bessel functions on a q-quadratic grid). Let us consider the entire function f (ω) defined in (15.2) again and let n f (r) be the number of of zeros of f (ω) in the circle \|ω\| < r. Consider also circles of radius R = R n = κq −n , q 1/4 ≤ κ < q −3/4 with n = 1, 2, 3, ... in the complex ω-plane. Since n f (r) is nondecreasing with r one can write n f (R n ) ≤ n f (r) ≤ n f (R n+1 ) (16.1) if R n ≤ r ≤ R n+1 , and, therefore, and, finally, one gets n f (R n ) Rlog q −1 n f (R n ) ≤ R n+1 Rn n f (r) r dr ≤ log q −1 n f (R n+1 ) . For large values of n in view of (5.5), f κq −n−1 e iϑ f (κq −n e iϑ ) ∼ q 3/2 κ 2 q −2n−2 e 2iϑ ; q 2 ∞ (q 3/2 κ 2 q −2n e 2iϑ ; q 2 ) ∞ = 1 − q 3/2 κ 2 q −2n−2 e 2iϑ , and log f κq −n−1 e iϑ f (κq −n e iϑ ) ∼ 2n log q −1 + log α, where α = κ 2 q −1/2 . Therefore, n f (R n ) ≤ 2n + log α/ log q −1 < n f (R n+1 ) (16.9) or n f (R n ) < 2n + log α/ log q −1 ≤ n f (R n+1 ) , (16.10) which gives n f (R n+1 ) − n f (R n ) < 2n + 2 + log α/ log q −1 − 2n − 2 + log α/ log q −1 = 4 Thus, we have established that n f (R n+1 ) − n f (R n ) < 4 (16.11) as n → ∞. Due to the symmetry f (ω) = f (−ω) the last inequality implies that there is only one positive root of S(η; ω) between the test points ω = γ n defined by (5.6) for large values of n. Acknowledgments We wish to thank Dick Askey, Mourad Ismail, John McDonald, and Mizan Rahman for valuable discussions and comments. One of us (S. S.) gratefully acknowledge the hospitality of the Department of Mathematics at Arizona State University were this work was done. The formulas (1.3)-(1.5) for the coefficients of the Fourier series are consequences of the orthogonality relations for trigonometric functions 1 , a 2 , ..., a m ; q) 5.23) and(5.27),A e iθ = e iθ e −2iθ ; q ∞ (e 2iθ ; q) ∞ B e −iθ .(5.30) and c(ω) = −qω 2 ; q 2 C (η; ω) x, iω) + e(x, −iω)) are entire functions in α and ω, respectively, of order zero for all real values of x.Proof. The generating function (9.3) gives a power series expansion for the function q n 2 /4 (q; q) n H n (x\|q) and τ ≥ 0 . 0Then \|e (cos θ; α)\| ≤ e (cosh τ ; \|α\|) (9.11) and e (cosh τ ; \|α\|) ≤ e (cosh τ 1 ; \|α\|) (9.12)if τ < τ 1 .Proof. One can rewrite (9.1) as H n (cos θ\|q) ; q) n (q; q) k (q; q) n−k cos (n − 2k) θ. Theorem 9. 5 . 5If f (x) is continuous and S [f ], the q-Fourier series of function f , converges uniformly, then its sum is f (x). Proof. Let g(x) denote the sum of S [f ], the q-Fourier series in the right side of (8.5). Then the coefficients of S [f ] are q-Fourier coefficients of g. Hence, S [f ] = S [g], so that f ≡ g and, f and g being continuous, f (x) ≡ g(x). π 0 H 0m (cos θ\|q) H n (cos θ\|q) e 2iθ , e −2iθ ; q ∞ dθ From (10.5), (10.6), and (9.3) we finally arrive at the connecting relation (10.2). d dx [W (cos ωx, sin ωx)] ωx + sin 2 ωx = constant. (13.3) Substituting x = 0, one verifies (13.1). . simple real zeros at ω = ±ω n by Theorems 5.As a result we arrive at the infinite product representation for the basic sine function,S(η; ω) = 1 1 − q 1/2 ω (−qω 2 ; q 2 ) ∞ (15.3)In a similar manner, one can obtain an infinite product representation for the basic cosine function, proof of Theorem 5.1 we have established the fact that for sufficiently large n there are at least two roots of f (ω) between the circles \|ω\| = R n and \|ω\| = R n+1 . Thus, for sufficiently large n the inequality (16.3) should really have one of the following formslog q −1 n f (R n ) ≤ R n+1 Rn n f (r) r dr < log q −1 n f (R n+1 ) , ≤ log q −1 n f (R n+1 ) . (16.5)Our next step is to estimate the integral in (16.4)-(16.5). By Jensen's theorem[6],[19] κq −n−1 e iϑ f (κq −n e iϑ )dϑ. From (16.3) and (16.7),1 + log α/ log q −1 2n − 1 n ≤ n f (R n ) 2n ≤ 1 + log α/ log q n+1Rn dr r ≤ R n+1 Rn n f (r) r dr ≤ n f (R n+1 ) R n+1 Rn dr r . (16.2) But R n+1 Rn dr r = log r R n+1 Rn = log q −1 = 2π q 1/2 ; q 2 ∞ (q; q) 2 ∞ 1 − q m+1/2 −1 δ mp (see, for example,[7]), one gets c n = π q 1/2 ; q ∞ (q; q) ∞ (−i) m q m 2 /4 (12.12)m+1/2 (2ω n ; q) , or,14) by (5.31) and (III.32) of[7], respectively. The last equation gives the large ωasymptotic of the basic Fourier coefficients. With the aid of (5.11), (6.9), and (I.9) of[7], we finally obtain \|c n \| ∼ D q n/2 → 0 (12.15) as n → ∞, where D is some constant. 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Sergei K Suslov, Kurchatov Institute, Moscow, 123182, Russia Current; Tempe, Arizona; U.SDepartment of Mathematics, Arizona State UniversityA. E-mail address: [email protected] K. Suslov, Kurchatov Institute, Moscow, 123182, Russia Current address: Department of Mathematics, Arizona State University, Tempe, Arizona 85287- 1804, U.S.A. E-mail address: [email protected]	[]
[]	[ "Agnid Banerjee ", "Bernd Kawohl " ]	[]	[ "Mathematics Subject Classification" ]	We extend the symmetry result of Serrin[21]and Weinberger [24] from the Laplacian operator to the highly degenerate game-theoretic p-Laplacian operator and show that viscosity solutions of −∆ N p u = 1 in Ω, u = 0 and ∂u ∂ν = −c = 0 on ∂Ω can only exist on a bounded domain Ω if Ω is a ball.	10.1090/bproc/33	[ "https://arxiv.org/pdf/1711.08696v2.pdf" ]	13,672,097	1711.08696	00929f3efd06c0973f9e6282664a9254c38fa53a	2010 Agnid Banerjee Bernd Kawohl Mathematics Subject Classification 35252010Overdetermined problems for the normalized p-Laplacianoverdetermined boundary value problemgame-theoretic p-Laplacianviscosity solution We extend the symmetry result of Serrin[21]and Weinberger [24] from the Laplacian operator to the highly degenerate game-theoretic p-Laplacian operator and show that viscosity solutions of −∆ N p u = 1 in Ω, u = 0 and ∂u ∂ν = −c = 0 on ∂Ω can only exist on a bounded domain Ω if Ω is a ball. Introduction In a seminal paper [21] Serrin showed that the following overdetermined boundary problem can only have a solution u ∈ C 2 (Ω) if Ω is a ball. − ∆u = 1 in Ω, u = 0 and ∂u ∂ν = c < 0 on ∂Ω (1.1) Here c is constant and Ω ⊂ R n is a bounded connected domain with boundary of class C 2 . Serrin used Alexandrov's moving plane method for his proof, while Weinberger [24] found a proof using Rellich's identity and the fact that a related function P (x) = \|∇u\| 2 + 2 n u is constant in Ω. Only the second method of proof has been adapted to a situation where the Laplacian operator is replaced by the p-Laplacian in [9] and [13]. In this paper we treat the case that the Laplacian is replaced by the normalized or gametheoretic p-Laplacian ∆ N p which is defined for any p ∈ (1, ∞) by Note that this operator is not in divergence form. Therefore it resists attempts to treat it with variational methods. On the other hand it is quite benign, because its coefficient matrix is bounded from below by min{ 1 p , p−1 p }I and from above by max{ 1 p , p−1 p }I. Therefore the moving plane method seems more appropriate in this context. Note also, that the above definition of the normalized p-Laplacian needs further explanation when ∇u = 0. The definition of and a weak comparison principle for continuous viscosity solutions are given below. These and an existence and uniqueness result can be found for instance in [18] or [17]. Our main result answers an open problem from [14]. ∆ N p u := 1 p \|∇u\| 2−p div \|∇u\| p−2 ∇u = 1 p ∆ N 1 u + p − 1 p ∆ N ∞ u,(1.Theorem 1.1. For p ∈ (1, ∞) let u ∈ C(Ω) be a viscosity solution to the overdetermined boundary value problem − ∆ N p u = 1 in Ω, u = 0 and ∂u ∂ν = c < 0 on ∂Ω (1.4) on a connected bounded domain Ω with with boundary of class C 2 . Then Ω must be a ball. Remark 1.2. We note that the Neumann condition ∂u ∂ν = c < 0 on ∂Ω is interpreted in the following sense: Any C 2 function ϕ such that ϕ − u has a minimum at a point x ∈ ∂Ω satisfies ∂ϕ ∂ν (x) ≤ c at x. Similarly, any C 2 function ψ such that ψ − u has a maximum at a point y ∈ ∂Ω satisfies ∂ψ ∂ν (y) ≥ c. Remark 1.3. It was pointed out in [14] that Theorem 1.1 remains true for p = 1, while for p = ∞ it is generally false. In fact for p = 1 the equation can be rewritten as − n−1 p H(x)u ν (x) =, where H(x) denotes mean curvature of the level set passing through x, and in view of the constant Neumann data this means that ∂Ω has constant mean curvature. Therefore Ω is a ball of radius 1−n p c. As explained in [5], for p = ∞ the right P -function is \|∇u\| 2 + 2u, and annuli are cases in which the overdetermined problem has viscosity solutions of class C 1 . The case p = ∞ was also studied in great detail in a series of papers by Crasta and Fragalá, who relaxed the C 2 smoothness of the boundary, see e.g. [7]. The normalized p-Laplacian has also been studied in the context of evolution equations in a number of papers, see [11,8,4,3,20,12,10]. Definitions and Comparison Result In the notation of the theory of viscosity solutions we study the equation F p (∇u, ∇ 2 u) = − p − 2 2 \|∇u\| −2 ∇ 2 u∇u, ∇u − 1 p trace∇ 2 u − 1 = 0. (2.1) Definition 2.1. Following [6], u ∈ C(Ω) is a viscosity solution of the equation F (∇u, ∇ 2 u) = 0, if it is both a viscosity subsolution and a viscosity supersolution. u is a viscosity subsolution of F (∇u, ∇ 2 u) = 0, if for every x ∈ Ω and ϕ ∈ C 2 such that ϕ − u has a minimum at x, the inequality F * (∇ϕ, ∇ 2 ϕ) ≤ 0 holds. Here F * is the lower semicontinuous hull of F . u is a viscosity supersolution of F (∇u, ∇ 2 u) = 0, if for every x ∈ Ω and ψ ∈ C 2 such that ψ − u has a maximum at x, the inequality F * (∇ψ, ∇ 2 ψ) ≥ 0 holds. Here F * is the upper semicontinuous hull of F . If X denotes a symmetric real valued matrix, we denote its eigenvalues by λ min = λ 1 ≤ λ 2 ≤ . . . ≤ λ n = λ max . Using this notation, it is a simple exercise to find out that F * (q, X) = F (q, X) if q = 0, inf a∈R n \{0} F (a, X) if q = 0, (2.2) so F * (0, X) = − p−1 p λ 1 − 1 p n i=2 λ i − 1 for p ∈ (1, 2], − p−1 p λ n − 1 p n−1 i=1 λ i − 1 for p ∈ [2, ∞), (2.3) while F * (q, X) = F (q, X) if q = 0, sup a∈R n \{0} F (a, X) if q = 0, (2.4) that is F * (0, X) = − p−1 p λ n − 1 p n−1 i=1 λ i − 1 for p ∈ (1, 2], − p−1 p λ 1 − 1 p n i=2 λ i − 1 for p ∈ [2, ∞). (2.5) The following comparison principle has been derived in [18,17]. It can be used to show the positivity of u and a Hopf Lemma. Lemma 2.3. Suppose Ω satisfies a uniform interior sphere condition and u ∈ C(Ω) is a viscosity solution of F p (∇u, ∇ 2 u) = 0 in Ω such that u = 0 on ∂Ω. Then u is positive in Ω and there exists a number a > 0 such that for all y ∈ ∂Ω lim sup t→0 + u(y) − u(y − tν(y)) t ≤ −a < 0. Here ν(y) denotes the outward unit normal at y. In fact, one can compare u to a radially symmetric and radially decreasing solution v on the interior of the sphere. On a ball solutions v of the Dirichlet problem for F p = 0 are unique by Proposition 2.2, so they are necessarily radial. In polar coordinates F p = 0 turns into the tractable ODE [15] − p−1 p v rr − n−1 pr v r = 1 in (0, R) with v r (0) = 0 = v(R) as boundary conditions, and this boundary value problem has the explicit solution v(r) = p 2(p + n − 2) (R 2 − r 2 ), so that u is positive in every ball with radius R contained in Ω. Moreover, Lemma 2.3 holds with a = Rp p+n−2 . Proof of Main Result To prove Theorem 1.1 we follow an idea developed in [1]. We first note that from the regularity result stated in Theorem 4.2 in the Appendix, we have that u is in C 1,β (Ω) for some β = β(n, p, Ω) and therefore the Neumann condition is realized in the classical pointwise sense. Now because by assumption \|∇u\| = −c > 0 on ∂Ω and u ∈ C 1,β (Ω), we have that \|∇u\| > 0 in an ε-neighborhood S ε of ∂Ω inside Ω defined by S ε := {x ∈ Ω ; d(x, ∂Ω) < ε}. Therefore the operator F p is well-defined in the classical sense in S ε . Moreover in S ε , since \|∇u\| > 0, we have that u solves Σ n i,j=1 a ij (x)u x i x j = −1 where a ij (x) = 1 p (δ ij + (p − 2) u x i u x j \|∇u\| 2 ) is uniformly elliptic and is in C β (S ε ). Consequently, by the classical Schauder theory we can assert that u is of class C 2,β loc in S ε . We now move a hyperplane, say T λ := {x ∈ R n \| x 1 = λ} from the left by the amount ε/2 in x 1 -direction into Ω and compare the original solution u(x) to the reflected one v(x) = u(x − 2λe 1 ) in the reflected cap. By the weak comparison principle, Proposition 2.2, we know that u ≥ v in the reflected cap Σ ′ λ . Moreover, since \|∇u\|, \|∇v\| > 0 in S ε , we have that u, v solve an equation in S ε of the form F (∇h, ∇ 2 h) = 0 (3.1) whereF is uniformly elliptic and smooth in its arguments. Therefore w = u − v solves the following equation linearized equation in S ε , Σ n i,j=1 c ij w x i x j + b, ∇w = 0,(3.2) where c ij = 1 0 ∂F ∂m ij (t∇u + (1 − t)∇v, t∇ 2 u + (1 − t)∇ 2 v)dt and b i = 1 0 ∂F ∂p i (t∇u + (1 − t)∇v, t∇ 2 u + (1 − t)∇ 2 v)dt. Moreover [c ij ] is uniformly elliptic and the first order coefficients b i are bounded in S ε . Note that over here, we think ofF : R n × R n 2 → R as a function of the matrix [m ij ] ∈ R n 2 and the vector p = (p 1 , ....., p n ) ∈ R n . Therefore, ∂F ∂m ij is to be thought of as the partial derivative ofF with respect to the coordinate m ij in R n 2 and ∂F ∂p i is the partial derivative with respect to the coordinate p i in R n . Since w solves the uniformly elliptic PDE (3.2) in Σ ′ λ ∩ S ε , by the classical strong maximum principle applied to w, we get that u > v in Σ ′ λ ∩ S ε and w x 1 > 0 on the plane T λ ∩ S ε/2 . The latter inequality follows from the classical Hopf Lemma applied to w. We continue to move the hyperplane across. Even if the operator might become degenerate because we pass a critical point of u, the weak comparison principle continues to hold, so that u ≥ v in the reflected cap, until one of the following cases occurs. i) The hyperplane and ∂Ω meet under a right angle in a point P . ii) The reflected cap touches ∂Ω from the inside of Ω in a point Q. In case i) we can apply the strong maximum principle again and conclude that either u > v in the reflected cap intersected with an ε/2 neighborhood of the point P , or u ≡ v there. But by Serrin's corner lemma applied to w = u − v which solves (3.2), the first case w > 0 is ruled out. In fact ∇u(P ) = ∇v(P ) because the normal derivatives coincide there and the tangential ones vanish. So a partial derivative of w in any direction η must vanish there. Let us see what happens to second partial derivatives in direction η = αν + βτ , where τ is a unit vector tangent to ∂Ω at P . We claim that u ηη (P ) = α 2 u νν (P ) + αβu ντ (P ) + αβu τ ν (P ) + β 2 u τ τ (P ) = v ηη (P ). (3.3) In fact in P one can rewrite the differential equations for u and v as (see [14]) p−1 p u νν (P ) + n−1 p H(P )u ν (P ) = −1 = p−1 p v νν (P ) + n−1 p H(P )v ν (P ) (3.4) where H is the mean curvature of the C 2 boundary. Since u ν (P ) = c = v ν (P ) we conclude that u νν (P ) = v νν (P ). For the same reason (u ν ) τ = 0 = (v ν ) τ . Now (u τ ) ν = u ν τ − κu ν , where κ denotes the curvature of ∂Ω in direction τ , so that (u τ ) ν (P ) = (v τ ) ν (P ). Finally one can observe that u τ τ (P ) = u ss (P ) + κ(P )u ν (P ), where s denotes arclength along the curve that is cut out of ∂Ω by the plane spanned by τ and ν. Since u and v are constant on ∂Ω, we have u ss (P ) = v ss (P ), and since ∇u(P ) = ∇v(P ), this completes the proof of (3.3). Therefore we can conclude that w η (P ) = w ηη (P ) = 0 and at this point, Serrin's corner lemma implies that w ≡ 0 in Σ ′ λ ∩ S ε . In case ii) we can also conclude that either u > v in the reflected cap intersected with an ε/2 neighborhood of the point Q or u ≡ v. Now in former case, i.e. when u > v in the reflected cap, we recall again that w solves the uniformly elliptic PDE (3.2) in S ε .But then by the Hopf's Lemma applied to w, we get that ∂w/∂ν < 0 at Q, which contradicts the fact that u satisfies constant Neumann data. Consequently, we have that w ≡ 0 in Σ ′ λ ∩ S ε . In both cases ∂Ω and u are locally Steiner-symmetric in direction x 1 . To see that they are also globally symmetric, one can argue as follows. For reasons of continuity the set of points in which the reflected boundary coincides with the original boundary is closed in ∂Ω. But it is also open in ∂Ω. In fact in any point that belongs to the boundary of this intersection one can apply the corner lemma again to see that a whole neighborhood still belongs to it. Since this Steiner symmetry happens in any direction, we can conclude that Ω is a ball and u is radial and radially decreasing. Appendix In this section, we state and prove a basic regularity result which has been referred to in the proof of Theorem 1.1 in the previous section. In order to do so, we first introduce the relevant notion of extremal Pucci type operators. Let M + and M − denote the maximal and minimal Pucci operators corresponding to λ, Λ, i.e., for every M ∈ S n we have M + (M) = M + (M, λ, Λ) = Λ e i >0 e i + λ e i <0 e i , M − (M) = M − (M, λ, Λ) = λ e i >0 e i + Λ e i <0 e i , where e i = e i (M) indicate the eigenvalues of M. Hereafter, the dependence of M + and M − on λ, Λ will be suppressed. As is well known, M + and M − are uniformly elliptic fully nonlinear operators. We now state the following important lemma which connects the normalized p Laplacian operator ∆ N p to appropriate maximal and minimal Pucci operators. Lemma 4.1. Let u ∈ C(Ω) be a viscosity solution to ∆ N p u = f in Ω (4.1) and let f be bounded. Then u satisfies the following differential inequalities in the viscosity sense M + (∇ 2 u) + K ≥ 0 ≥ M − (∇ 2 u) − K (4.2) where M + , M − are the pair of extremal Pucci operators corresponding to λ = min{ 1 p , p−1 p } and Λ = max{ 1 p , p−1 p } and K = \|\|f \|\| L ∞ (Ω) . Proof. The proof is a straightforward consequence of the fact that the coefficient matrix corresponding to ∆ N p is bounded from below by min{ 1 p , p−1 p }I and from above by max{ 1 p , p−1 p }I. We now state our main result in this section concerning the regularity of viscosity solutions to the equation in Theorem 1.1 up to the boundary. Theorem 4.2. Let Ω be a C 2 domain and x 0 ∈ ∂Ω. Let u be a viscosity solution to −∆ N p u = f in Ω ∩ B 2r (x 0 ) u = g on ∂Ω ∩ B 2r (x 0 ) (4.3) where f ∈ C(B 2r (x 0 ) ∩ Ω) and g ∈ C 1,α (∂Ω ∩ B 2r (x 0 )). Then u ∈ C 1,β (Ω ∩ B r (x 0 )) where β = β(α, Ω, f, g). Proof. We first note that by Lemma 4.1, u satisfies the differential inequalities (4.2) in the viscosity sense. Therefore we may apply Theorem 1.1 in [22] and can assert that for some β = β(α, n, p), there exists G ∈ C β 1 (∂Ω ∩ B r (x 0 )) which is the "gradient" of u at the boundary such that \|u(x) − u(x 1 ) − G(x 1 ), x − x 1 \| ≤ C\|x − x 1 \| 1+β 1 for all x ∈ Ω ∩ B r (x 0 ). (4.4) Here C depends also on the C 2 character of the domain Ω. Therefore, (4.4) expresses the fact that u is C 1,β at the boundary. Now the fact that u ∈ C 1,β 2 loc (Ω ∩ B 2r (x 0 )) for some β 2 depending on n, p follows from the interior regularity result established in the recent paper [2]. At this point, by taking β = min(β 1 , β 2 ), we can argue as in the proof of Proposition 2.4 in [19] to conclude that u ∈ C 1,β (Ω ∩ B r (x 0 )). j=1 u x i u x i x j u x j \|∇u\| 2 . (1.3) Proposition 2. 2 . 2Suppose u and v are in C(D) and are viscosity super-resp. subsolutions of F p = 0 on a domain D and u ≥ v on ∂D. Then u ≥ v in D. Acknowledgements: This research was essentially done during a visit of the second author to TIFR CAM Bangalore. B.K. thanks the Institute and in particular Agnid Banerjee for their hospitality and support. Both authors thank the anonymous referee for carefully reading the manuscript and for his/her helpful comments and suggestions. 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[ "Decays of the B c Meson in a Relativistic Quark-Meson Model", "Decays of the B c Meson in a Relativistic Quark-Meson Model" ]	[ "Matthew A Nobes \nDepartment of Physics\nV5A 1S6 and R. M. Woloshyn TRIUMF\nSimon Fraser University Burnaby\n4004 Wesbrook MallV6T 2A3VancouverBC, BCCanada, Canada\n" ]	[ "Department of Physics\nV5A 1S6 and R. M. Woloshyn TRIUMF\nSimon Fraser University Burnaby\n4004 Wesbrook MallV6T 2A3VancouverBC, BCCanada, Canada" ]	[]	The semileptonic decay form factors of the double heavy B c meson provide a unique opportunity to study the strong interactions between two heavy quarks. A fully relativistic model, with effective non-local quark-meson interactions, is used to compute semileptonic decay form factors, for both the B c and a wide range of other heavy-light mesons.Using these form factors predictions for decay rates and branching ratios are obtained. The results are compared to other theoretical approaches and, where available, to experimental results. In addition the radiative decay of B * c is discussed.	10.1088/0954-3899/26/7/308	[ "https://arxiv.org/pdf/hep-ph/0005056v2.pdf" ]	6,440,002	hep-ph/0005056	4d998d6fc4623716b8d62167312e14a35a49e734	Decays of the B c Meson in a Relativistic Quark-Meson Model 23 Jun 2000 Matthew A Nobes Department of Physics V5A 1S6 and R. M. Woloshyn TRIUMF Simon Fraser University Burnaby 4004 Wesbrook MallV6T 2A3VancouverBC, BCCanada, Canada Decays of the B c Meson in a Relativistic Quark-Meson Model 23 Jun 2000 The semileptonic decay form factors of the double heavy B c meson provide a unique opportunity to study the strong interactions between two heavy quarks. A fully relativistic model, with effective non-local quark-meson interactions, is used to compute semileptonic decay form factors, for both the B c and a wide range of other heavy-light mesons.Using these form factors predictions for decay rates and branching ratios are obtained. The results are compared to other theoretical approaches and, where available, to experimental results. In addition the radiative decay of B * c is discussed. Introduction A primary goal in the study of semileptonic decays of heavy mesons is to extract the values of the CKM matrix elements. The great virtue of semileptonic decays is that the effects of the strong interaction can be separated from the effects of the weak interaction into a set of Lorentz invariant form factors [1]. Thus the theoretical problem associated with analysing semileptonic decays is essentially that of calculating the form factors. The focus of this work is the decay of the B c meson (for a review of the properties of this system see [2]). This system is unique among mesons made up of heavy (charm or bottom) quarks, it is the only one which is stable with respect to strong and electromagnetic interactions. Therefore, the B c system is the only heavy meson for which form factors (albiet transition form factors rather than elastic) can be measured. These form factors then provide a unique probe of the dynamics of heavy quark systems. There are many approaches to the calculation of decay form factors, for example, lattice QCD [3], QCD sum rules [4], and phenomological modelling [5]. In this work a particular model with an effective quark-meson coupling is adopted. There are many models of this type [6,7,8,9,10]. The one used here has its genesis in the QCD version of the Nambu-Jona-Lasinio model [11] extended to heavy quarks [6] and is most closely related to the model used recently by Ivanov and Santorelli in a their study of pseudoscalar meson decays [12]. The advantage of this approach is that it is fully relativistic and very versatile. Quarks and mesons for all masses are treated within the same framework. For light quarks the model has the features of spontaneous chiral symmetry breaking and in the single heavy quark limit the form factor constraints of heavy quark symmetry are obtained. Our work differs from Ivanov and Santorelli in the choice of the quarkmeson vertex function and in the way that parameters are fixed. A number of heavy mesons decays not calculated in Ref. [12] are treated here. The Quark-Meson Coupling The particular quark-meson coupling used in this work is based on an effective Lagrangian which models the interaction between mesons and quarks with a non-local interaction vertex [6,12]. The interaction Lagrangian has the form L int (x) = g M M (x) dx 1 dx 2 δ x − 1 2 m 1 x 1 + m 1 x 2 m 1 + m 2 × f [(x 1 − x 2 ) 2 ]q 1 (x 1 )Γ M q 2 (x 2 ),(1) where Γ M is the Dirac matrix appropriate to the meson field M, f [(x 1 −x 2 ) 2 ] is a non-local vertex function, which simulates the finite size of the meson, and q 1 and q 2 are the quark fields. A condition imposed on the vertex function is that it should render all loop diagrams UV finite. The coupling constant g M is determined by the compositeness condition, which is the requirement that the renormalization constant of the meson fields be zero, i.e. Z M = 1 − g 2 M 2 dΠ M (p 2 ) dp 2 p 2 =M 2 M = 0.(2) Here Π M (p 2 ) is the self energy of the meson field, given by Π M (p 2 ) = 2N c d 4 k (2π) 4 i f 2 (Q 2 ) tr Γ M 1 m 1 − (/ k + / p) Γ M 1 m 2 − / k ,(3) where m 1 and m 2 are the masses of the quarks in the loop and Q is a relative momentum chosen to be Q = k + αp with α = m 2 m 1 +m 2 . The constituent quark masses in (3) are free parameters. As well, the vertex function will contain a free parameter which reflects the size of the meson. These parameters will be different for the different mesons. The use of free constituent quark propagators in expressions like (3) can lead to a problem which reflects the lack of quark confinement in the model. If the meson mass M M is greater than the sum of its constituent quark masses loop integrals will develop imaginary parts. This indicates a nonzero amplitude for the creation of a free quark-antiquark pair. There have been some various attempts to obviate this problem within quark-meson effective theories [7,8,9,10]. Here we adopt the approach of Ref. [12] and use free propagators. The constituent quark masses are then fit to allow for the inclusion of as many mesons as possible. In order to carry out calculations a choice must be made for the vertex function f (q 2 ). The function that was used in this analysis was the dipole f (Q 2 ) = Λ 4 [Λ 2 − Q 2 ] 2 . This choice was made for two reasons; first the form of the dipole vertex function is the same as a propagator, allowing standard Feynman parameter techniques to be used in evaluating loop integrals. Second, the vector decay constant f V would diverge if only a monopole vertex function was used. Since one of the primary criteria for the vertex functions is that they should render all diagrams UV finite, a function with UV fall-off as least as fast as a dipole is needed. The parameter Λ characterizes the finite size of the meson, and will be different for different mesons. To account for this the various values of Λ will be distinguished by subscripts which reflect either the meson type or the quark content, e.g. Λ Bc and Λ bc will be used interchangeably. Further, in expressions involving the vertex form factor, the same comvention will be used. Note that the calculations of Ref. [12] used a Gaussian vertex function so that the parameters used there can not be compared directly with ours. The parameters of the model were fit to the leptonic decay constants, f P and f V . These quantities are defined by 0\| − iγ µ γ 5 \|P = if P p µ ,(4)0\|iγ µ \|V, ǫ = M 2 V f V ǫ µ (5) where M V is the vector meson mass. The pseudoscalar decay constant is given by the one-loop expression f P p µ = N c d 4 k (2π) 4 i g P f (Q 2 ) tr −γ µ γ 5 [m 1 + / k + / p]γ 5 [m 2 + / k] m 2 1 − (k + p) 2 m 2 2 − k 2 .(6) Here m 1 and m 2 refer to the masses of the quarks in the loop, this convention will be used throughout this paper. Using the dipole vertex function, combining the denominators using Feynman parameters, and performing the integration over k yields f P = g P 3Λ 4 P 4π 2 D x x 1 [m 2 (1 − σ) + m 1 σ] ∆ 2 ,(7) with σ = αx 1 + x 2 , η = α 2 x 1 + x 2 , ∆ = Λ 2 P x 1 + m 2 1 x 2 + m 2 2 x 3 + (σ 2 − η)M 2 P , ℓ = k + σp, D x = 1 0 3 i=1 dx i δ 3 i=1 x i − 1 , where M P is the mass of the pseudoscalar meson. Likewise the expression for the vector decay constant is f V = g V 3Λ 4 V 4π 2 M 2 V D x x 1 m 1 m 2 + ∆ + σ(1 − σ)M 2 V ∆ 2 ,(8) where the same defintions have been used, with the obvious change of M P to M V and Λ P to Λ V in the expression for ∆. To compute the coupling constants g P and g V , the self energies and their derivatives must be computed. Then the compositeness condition (2) can be used to find the couplings. The self energy for a pseudoscalar meson is given by Π P (p 2 ) = 3Λ 8 P 2π 2 D x x 3 1 m 1 m 2 + p 2 σ(1 − σ) + 2 3∆ ∆ 4 ,(9) where∆ = Λ 2 P x 1 + m 2 1 x 2 + m 2 2 x 3 + (σ 2 − η)p 2 and all the other quantities are the same as the ones defined above. The self energy for a pseudoscalar meson is given by the tensor Π µν V which can be expressed as Π µν V (p) = Π V (p 2 )g µν +Π V (p 2 ) p µ p ν p 2 .(10) Unfortunately Π V =Π V , so this does not have the proper structure for a vector propagator. This problem was solved (following [13]) by simply dropping theΠ V term, which would cancel out of any calculation of a physical process at one-loop order (since ǫ · p = 0). The relevant part of the vector meson self energy is given by Π V (p 2 ) = 3Λ 8 V 2π 2 D x x 3 1 m 1 m 2 + 1 3∆ + σ(1 − σ)p 2 ∆ 4 ,(11) where all the quantities appearing have been defined previously. The free parameters of the model are fit to the six values of f P and the measured values f J/ψ = 0.1309 and f Υ = 0.075012 [14]. These data, which are displayed in Table 1, fix eight free parameters. In order to reduce the number of free parameters to match the available data the value of the strange quark mass was fixed at 450 MeV and the vertex parameter for a vertex containing only u and d quarks Λ π was taken (following [6]) to be 1 GeV. In addition the following further simplifying assumptions were made Λ us = Λ ds = Λ ss = Λ K , Λ uc = Λ dc = Λ sc = Λ D , Λ ub = Λ db = Λ sb = Λ B . This leaves the following parameters to be fit, m q , m c , m b , Λ K , Λ D , Λ B , Λ cc , Λ bb , and Λ bc . The parameter Λ bc could only be fit to a value for f Bc which is not in the values listed in Table 1 The values for the self energies, coupling constants, and leptonic decay constants arising from these parameters are displayed in Tables 2 and 3. In order to fix Λ bc a value of f Bc must be given. There is no experimental value for this quantity and theoretical estimates tend to fall in the range 400 MeV f Bc 500 MeV (see, for example, [2,16,17,18,19]). A further complication is that the mass of M Bc is also not yet known very well. The current measurement [20] is M CDF Bc = 6.4±0.39(stat)±0.13(sys) GeV c 2 , which comes from the few confirmed B c events at the Tevatron. Theoretical results tend to lie within this range, so following the potential model prediction of [2] the mass of the B c was chosen to be 6.25 GeV. One general argument guides the selection of Λ bc , it should lie between Λ B and Λ bb . With this in mind, and using the value for M Bc above, a number of values of Λ bc were tried, spanning the possible range. Fig. 1 shows the value of f Bc as a function of Λ bc . The value selected selected for use in this work was Λ bc = 2.3 GeV, which gives f Bc = 450 MeV, a value in the middle of the range of the theoretical predictions. Semileptonic Decays of K, D, and B Mesons The model used in this work is phenomenological but having fixed its parameters, the results for semileptonic decays are predictions. Before proceeding to decays of B c it is important to test the model against experimental results where they are available. Therefore several semileptonic decays of K, D, and B meson are calculated. The formalism for these calculations, presented in this section, extends directly also to the calculation of B c decay. Some of the decays considered here have already been treated by Ivanov and Santorelli [12]. However, that work does not demonstrate the full applicablity of the approach. Apart from decays to light vector mesons, the model is capable of treating virtually any semileptonic decay (with the restriction that a value for the meson mass must be supplied as input). The amplitude A for a semileptonic decay is given by, A = G F √ 2 V QQ ′ L µ H µ .(12) Here G F is the Fermi constant, V QQ ′ is the relevant CKM matrix element, L µ is the lepton current L µ =ū ν ℓ γ µ (1 − γ 5 )v ℓ , and H µ is the hadron current H µ = k, ǫ\|(V µ − A µ )\|P ,(13) where P is the momentum of the parent meson, k is the momentum of the daughter meson, and ǫ is the polarization, if the daughter meson is a vector. The two currents in (13) are the vector V µ and axial A µ . If the final state is a pseudoscalar the hadron current can be decomposed as follows, k\|A µ \|P = 0, k\|V µ \|P = f + (q 2 )(P + k) µ + f − (q 2 )(P − k) µ , where f + (q 2 ) and f − (q 2 ) are Lorentz invariant form factors. Likewise, if the final state is a vector meson, k, ǫ\|A µ \|P = f (q 2 )ǫ * µ + a + (q 2 )(ǫ * · P )(P + k) µ + a − (q 2 )(ǫ * · P )(P − k) µ , k, ǫ\|V µ \|P = ig(q 2 )ǫ µνρσ ǫ * ν (P + k) ρ (P − k) σ , where the form factors are g, f , a + , and a − . In each of these expressions q = (P − k) is the momentum transfer. For a decay to a pseudoscalar meson (with mass denoted by M P ′ ) the differential decay rate can be reduced to [1] dΓ dq = G 2 F \|V QQ ′ \| 2 M 2 P K 3 24π 3 \|f + (q 2 )\| 2 .(14) where, K = M P 2 1 − M 2 P ′ M 2 P − y 2 − 4 M 2 P ′ M 2 P y.(15) The lepton spectrum is given by, dΓ dx = G 2 F \|V QQ ′ \| 2 M 5 P 16π 3 (1 − 2x) ymax(x) 0 [y max (x) − y]\|f + (q 2 )\| 2 dy,(16) where y max (x) = 4x(xmax−x) 1−2x with x max = M 2 P −M 2 P ′ 2M 2 P . If the final state is a vector meson (with mass M V )the corresponding differential decay rate is, dΓ dy = G 2 F \|V QQ ′ \| 2 KM 2 P y 96π 3 \|H + \| 2 + \|H − \| 2 + \|H 0 \| 2 ,(17) whereH ± = f (q 2 ) ∓ 2M P Kg(q 2 ), H 0 = M P 2M V √ y 1 − M 2 V M 2 P − y f (q 2 ) + 4K 2 a + (q 2 ) , and the final mass M V should be subsititued for M P ′ in (15). The expression for the lepton spectrum is given by dΓ dx = G 2 F \|V QQ ′ \| 2 M 5 P 32π 3 ymax(x) 0 α(y) M 2 P y+ 2(1 − 2x)[y max (x) − y]β ++ (y) + γ(y)y[2x max − 4x + y]} ,(18) where the following definitions were made α(q 2 ) = \|f (q 2 )\| 2 + λ\|g(q 2 )\| 2 , γ(q 2 ) = 2f (q 2 )g(q 2 ), β ++ (q 2 ) = 1 4M V \|f (q 2 )\| 2 + λ\|a + (q 2 )\| 2 − 4M 2 V q 2 \|g(q 2 )\| 2 +2(M 2 P − M 2 V − q 2 )f (q 2 )a + (q 2 ) , λ(q 2 ) = (M 2 P − M 2 V − q 2 ) 2 − 4M 2 V q 2 , x max = M 2 P − M 2 V 2M 2 P Note that all of these expressions assume that lepton mass m ℓ is zero. The form factors for decay to a pseudoscalar meson are f + (q 2 ) = g P g P ′ 9Λ 4 P Λ 4 P ′ π 2 D x x 1 x 2 χ + − 3 4∆ (σ 1 + σ 2 ) +∆ ∆ 5 ,(19)f − (q 2 ) = g P g P ′ 9Λ 4 P Λ 4 P ′ π 2 D x x 1 x 2 χ − − 3 4∆ (σ 1 − σ 2 ) ∆ 5 .(20) The following definitions were made to simplify the expressions D x = 1 0 5 i=1 dx i δ 5 i=1 x i − 1 , ∆ = Λ 2 P ′ x 1 + Λ 2 P x 2 + m 2 3 x 3 + m 2 1 x 4 + m 2 x 5 , µ ij = m i m i + m j , σ 1, (2) = x 4, (3) + µ 12,(23) x 2, (1) , η 1, (2) = x 4, (3) + µ 2 12,(23) x 2, (1) , ∆ = ∆ + (σ 2 1 − η 1 + σ 1 σ 2 )M 2 P +(σ 2 − η 2 + σ 1 σ 2 )M 2 P ′ − σ 1 σ 2 q 2 , κ = m 3 (m 2 − m 1 ) + m 1 m 2 + 1 2 (M 2 P + M 2 P ′ − q 2 ), ǫ = (σ 1 + σ 2 )σ 1 M 2 P + (σ 1 + σ 2 )σ 2 M 2 P ′ − σ 1 σ 2 q 2 , ζ 1 = m 1 m 2 − σ 1 (σ 1 − 1) + σ 2 σ 1 − 1 2 M 2 P −σ 2 σ 1 + σ 2 − 1 2 M 2 P ′ + σ 2 σ 1 − 1 2 q 2 , ζ 2 = m 2 m 3 − σ 1 σ 1 + σ 2 − 1 2 M 2 P − σ 2 (σ 2 − 1) + σ 1 σ 2 − 1 2 M 2 P ′ +σ 1 σ 2 − 1 2 q 2 , χ ± = (ǫ − κ)(σ 1 ± σ 2 ) ± ζ 1 + ζ 2 . These definitions (in addition to α and µ) will be used throughout the rest of this paper, with the obvious substitution of M V and Λ V for M P ′ and Λ P ′ when the final state is a vector meson. The form factors for decays to vector mesons are given by g(q 2 ) = g M P g M V 9Λ 4 P Λ 4 V π 2 D xx 1 x 2 σ 2 (m 2 − m 3 ) + σ 1 (m 2 − m 1 ) − m 2 ∆ 5 ,(21)f (q 2 ) = g M P g M V 18Λ 4 P Λ 4 V π 2 D xx 1 x 2 × 1 ∆ 5 m 1 m 2 m 3 − 1 4 (m 2 − m 1 − 2m 3 )∆ +[ξ 1 + ξ 3 ]M 2 P + [ξ 2 + ξ 3 ]M 2 V − ξ 3 q 2 ,(22)a ± (q 2 ) = g M P g M V 18Λ 4 P Λ 4 V π 2 D xx 1 x 2 1 2 (β 1 ± β 2 ∓ σ 2 m 3 ) ∆ 5 .(23) The following further definitions have been made, ξ 1 = σ 1 (m 3 − m 2 )(1 − σ 1 ) − m 1 σ 2 1 , ξ 2 = σ 2 (m 1 − m 2 )(1 − σ 2 ) − m 3 σ 2 2 , ξ 3 = m 2 1 2 − 1 2 (σ 1 + σ 2 ) + σ 1 σ 2 + m 1 σ 1 1 2 − σ 2 +m 3 σ 2 1 2 − σ 1 , β 1 = 2σ 1 [m 1 σ 1 + m 2 (1 − σ 1 )], β 2 = m 2 (σ 1 + σ 2 − 1 − 2σ 1 σ 2 ) − m 1 σ 1 (1 − 2σ 2 ). Excluding the B c decays a total of sixteen pseudoscalar to pseudoscalar decays were considered. Due to the difficulty with confinement the corresponding number of pseudoscalar to vector decays that could be treated was only four. Table 4 shows the predictions for the decay rates and branching ratios for all of the decays considered. The values of the CKM matrix elements, and the necessary lifetimes were taken from [14]. Many of the decay rates treated in this section have been measured, hence most of the predictions can be compared to observed quantities. Table 4 shows the predicted and measured results for the branching ratios. The experimental results are taken from [14] and the errors in the predictions represent the uncertainties in the CKM matrix elements. Overall, the agreement with experiment is reasonable which increases the level of confidence in the areas where direct comparison with experiment is not possible. Table 5 shows values of f + (0) as computed in this work and in various other theoretical approaches. The other approaches are widely varied: [21] uses the ISGW model, [5] uses the WBS model, [22] gives results from a bag model, [23] uses a Dyson-Schwinger equation approach and [3] gives lattice QCD results. Of particular interest is the work of Ivanov et al. [24] which uses the quark confinement model. This quark-meson model is based on similar considerations to the model used in this work so its predictions should be close to ours. As well, the decay B → D+ℓ + +ν ℓ can be treated in a model independent way using the HQET. Ivanov et al. have shown in several papers [10,12,25] that quark-meson models of the type used here give the correct tree level HQET relations in the infinite mass limit. Nevertheless a direct check with finite quark mass is useful. The HQET gives the prediction [26] f HQET + (q 2 max ) = m B 0 + m D − 2 √ m B 0 m D − = 1.138, which compares well with our value f + (q 2 max ) = 1.133. The most important comparison that can be made is with Ref. [12]. This paper uses a different vertex function to treat B and D decays. This serves as a check on the dependence of the model on the choice of vertex function. Apart from the case B → π + ℓ + ν agreement with [12] is very good. In addition [12] presents the values of f + (q 2 ) over the full range of q 2 . Overall agreement is good between the two calculations, Fig. 2 illustrates the agreement in the case D 0 → K − + ℓ + + ν ℓ . Fig. 3 shows the case B 0 → π − + ℓ + + ν ℓ , for which the agreement is better over the whole range than indicated in Table 5. Due to lack of confinement very few pseudoscalar to vector decays can be calculated. Of the few decays treated in this work only the decay B → D * + ℓ + ν ℓ has been studied extensively. Table 6 compares our predictions with some other calculations. Overall the agreement is reasonable. Semileptonic Decays of the B c Meson The methods of the previous section can be directly applied to the semileptonic B c decays. Using the procdeure outline above, decay rates, lepton spectra, and branching ratios can be computed. In this work the lifetime of the B c was taken to be 0.5 ps, which agrees with the CDF value of τ CDF Bc = 0.46 +0. 18 −0.16 ± 0.03 ps [20]. Table 7 shows f + (0), f + (q 2 max ), the total decay rate Γ and the branching ratio for the four pseudoscalar decays. For the decays to vector mesons, values of the form factors at q 2 = 0 as well as total decay rates and branching ratios are displayed in Tables 8 and 9. There are a number of other calculations of the semileptonic decays of disagreement not only over the values of the branching ratios but also as to which decay will be favoured. For example, the quark-meson model used in this work predicts the decay to the J/ψ to be slightly favoured over the decay to the B * s while the heavy quark approach used in [28] and [30] predicts the decay to B * s to dominate. This divergence of predictions may be expected; the heavy-heavy quark content of the B c poses a challenge for models. Light-quark mesons may be constrained by chiral symmetry and heavy-light mesons by heavy quark symmetry. On the other hand the physics of heavy-heavy systems is less constrained by symmetries so extending models into this domain provides a severe test. Electromagnetic Decays V → P + γ In addition to semileptonic decays the electromagnetic decays of vector mesons can be treated within our effective quark-meson coupling model. Since the amplitude involves the matrix element V \|V µ \|P it is clear this process will be related to the form factor g(q 2 ). The the amplitude for this process is A = −2eǫ µναβǫ µ ǫ ν p V α p P β [Q 1 g 1 (0) + Q 2 g 2 (0)],(24) where Q 1, (2) is the charge of q 1, (2) , and p P, V are the momenta of the pseudoscalar and vector mesons. The functions g i (0) are the form factors given by (21), with the appropriate masses inserted, and with q 2 = 0. The appropriate masses in these functions are given by the interchange of M P and M V and the subscript which denotes which of the quark lines the gauge field is coupled to (i.e. for g 1 (0) the appropriate expression sets m 3 =m 1 ). Defining g V P γ = 2[Q 1 g 1 + Q 2 g 2 ], and summing over initial and final polarizations gives \|A\| 2 = 2απ 3 M 4 V 1 − M 2 P M 2 V 2 g 2 V P γ ,(25) where α = 1 137 is the fine structure constant. Standard techniques [31] yield the total rate Γ V P γ = α 24 M 3 V g 2 V P γ 1 − M 2 P M 2 V 3 .(26) Electromagetic decays have been the subject of several theoretical studies. As well the decay J/ψ → η c + γ has been measured. Table 11 shows our predictions for g V P γ along with the single experimental result and the predictions of some other models. In [32] and [33] two different heavy quark approaches were used. The quark confinement model [13], which has some similarity to the quark-meson model used in this work, gives results which are quite close to ours. There are measured branching ratios for the D * decays, however no lifetime measurement has been made. Therefore our predictions (which do not include the lifetime) cannot be compared directly with experiment. In order to obtain branching ratios a theoretical estimate of the lifetime must be used. The quark confinement model is the ideal choice, since its predictions are closest to our work. Using the results from [13] and our predictions for the total rates (obtained from (26)) the following branching ratios are obtained: BR (D * ) 0 → D 0 + γ = 33.0%, BR (D * ) + → D + + γ = 1.43%. These compare well with the experimental values [14] BR expt. (D * ) 0 → D 0 + γ = 38.1%, BR expt. (D * ) + → D + + γ = 1.1%. In order to treat the electromagnetic decay B * c → B c + γ the mass of the B * c meson must be specified. Theoretical estimates [2] indicate that the mass difference should be small; M B * c − M Bc < 100 MeV. To examine the effect of a small change in the B * c mass, the self energy and coupling constant were calculated over a range of masses. These results were used to calculate g B * c Bcγ and are displayed in Table 12. The decay rate is shown in Fig. 4. The radiative decay of B * c has not been studied extensively. A QCD sum rule approach [34], using M B * c = 6.6 GeV and M Bc = 6.3 GeV, gives the result g SR B * c Bcγ = 0.270 ± 0.095GeV −1 . Using these masses our prediction is g B * c Bcγ = 0.2196GeV −1 . The two values are in agreement. Conclusion A Lagrangian which models mesons in terms of an effective non-local quarkmeson interaction vertex [6,12] was extended in this paper to describe mesons such as B * c composed of two heavy quarks. The model has the advantage of treating all quarks (heavy and light) on the same footing, thereby permitting a unified investigation of heavy → heavy, and heavy This is in contrast to the situation in K, D and B meson decays where much smaller differences between different models are found. As a further illustration of the versatility of the model, the electromagnetic decays V → P + γ were investigated. A reasonable description of J/ψ and D * radiative decay was found and the rate for B * c → B c + γ was calculated for a range of B * c masses. There is room for further analysis within the model considered in this work. Hadronic decays, such as B + c → J/ψ + π + can also be treated. A detailed analysis of all of these decays, combined with the results for the leptonic and semileptonic decays, could be used to make a prediction for the lifetime τ Bc . A further area that needs work is the difficulty with confinement. The natural solution to this problem appears to be provided by the quark confinement model [10]. Recently Ivanov et al. have proposed a modification of the quark confinement model which may aid its application to heavy mesons [35]. Table 9: Predictions for the form factors at q 2 = 0 for B c → V decays. main new results are the extension of the model to include doubly heavy mesons, the calculation of B c semileptonic decays and the electromagnetic vector to pseudoscalar transitions. This paper is organized as follows: the next section introduces the model, discusses the general method of calculation, and fits the models free parameters. Sect. 3 presents the calculation of the form factors and decay rates for the semileptonic decays of a wide varity of heavy-light pseudoscalar mesons. These calculations are compared with both measured results, and other theoretical approaches. Sect. 4 presents the same set of calculations for the eight primary semileptonic decays of the B c meson. The predictions are compared with other theoretical work, in order to highlight the differences that exist between various approaches. Sect. 5 briefly discusses the electromagnetic decays V → P + γ for a number of vector mesons, including the B * c . Sect. 6 gives conclusions and directions for future work. , hence it is retained as a free parameter, leaving eight to be fit. The fit to the remaining eight parameters is given by (all values in MeV) Our model can be used to calculate all of these form factors. In all the following expressions the inital meson is composed of quarks with masses m 1 and m 2 and the final meson is composed of quarks with masses m 3 and m 2 (i.e. m 1 is the mass of the quark which decays to a new quark with mass m 3 , and m 2 is the mass of the spectator). B c . A comparison of some results for the dominant decay modes is given in Table 10. In contrast to the situation in Sect. 3 where our quark-meson model predictions, for the most part, agreed with other models and the various other models agreed with each other, there are substantial differences between calculations of B c decays. The clearest examples of this are the predictions for the decays to the B * s and J/ψ. These two decays are expected to be the most important semileptonic decay channels However there is → light quark decays. The model does not provide a complete dynamical description of quark interactions, meson masses can not be calculated and must be introduced as input parameters. Quark masses and the parameters associated with the quark-meson vertex were determined by fitting the pseudoscalar and vector meson decay constants f P and f V . Due to lack of confinement in this approach some light vector mesons had to be excluded from the analysis.To both test the model, and demonstrate its versatility, a large number of semileptonic decays of K, D, and B mesons were analysed. Agreement with measured results and other theoretical approaches was good.The main focus of this work was on the analysis of the semileptonic decays of the doubly-heavy B c meson. The form factors characterizing the strong interactions of the B c system were computed over the entire available range of momentum transfer. Using these results, decay rates and branching ratios were computed for all eight decay going to mesonic ground states.A comparison with some other approaches highlighted the significant differences among various model predictions concerning the B c . On very general grounds one would expect the decays to J/ψ and (B * s ) 0 states to be the most important. However, there is no agreement from models which of the channels dominates and absolute rates differ by a factor of 3 to 4. MFigure 2 : 2B * c (GeV ) Π (M B * c ) 2 (GeV 2 ) g B * Formfactor for the decay D 0 → K − calculated in this work (dashed line) and in Ref.[12] (solid line). Figure 3 :Figure 4 : 34Form factor for the decay B 0 → π − calculated in this work (dashed line) and in Ref. [12] (solid line). The decay rate Γ [B * c → B c + γ] vs. M B * c . Table 1 : 1Pseudoscalar Decay ConstantsMeson (P) f P (MeV) Source π + 130.7 ± 0.5 Expt. [14] K + 159.8 ± 1.8 Expt. [14] D + 183 +12+41+9 −13−0−25 Lattice QCD [15] D + s 229 +10+51+3 −11−0−19 Lattice QCD [15] B + 156 +12+29+9 −14−0−9 Lattice QCD [15] B 0 s 177 +11+39+13 −12−0−11 Lattice QCD [15] Table 2 : 2Fitted Values of Pseudoscalar Meson PropertiesMeson Π(M 2 ) (GeV 2 ) g P f P (MeV) π + 0.038588 5.14916 131.06 K + 0.068760 5.16139 160.85 D + 0.083122 6.26671 182.80 D + s 0.087025 6.82039 223.81 B + 0.134842 6.44510 142.21 B 0 s 0.136072 7.75408 187.42 Table 3 : 3Fitted Values of Vector Meson PropertiesMeson Π(M 2 ) (GeV 2 ) g V f V J/ψ 0.095419 8.81505 0.131227 Υ 0.621464 7.54393 0.074796 Table 4 : 4Predictions for Decay Rates and Branching RatiosParent Daughter Γ (ps −1 ) BR(%) BR expt. (%) K + π 0 3.589 × 10 −6 4.45 3.18 ± 0.08 K 0 S π − 7.274 × 10 −6 0.0650 0.0670 ± 0.0007 K 0 L π − 7.274 × 10 −6 37.6 38.78 ± 0.27 D 0 π − 5.488 × 10 −3 0.228 0.37 ± 0.06 D 0 K − 8.476 × 10 −2 3.52 3.50 ± 0.17 D + π 0 2.790 × 10 −3 0.295 0.31 ± 0.15 D + K 0 8.515 × 10 −2 9.000 6.8 ± 0.8 D + s K 0 4.184 × 10 −3 0.195 D + s D 0 4.786 × 10 −8 2.24 × 10 −6 B 0 π − 6.455 × 10 −5 0.0101 0.018 ± 0.006 B 0 D − 1.716 × 10 −2 2.68 2.00 ± 0.006 B 0 (D * ) − 3.983 × 10 −2 6.21 4.60 ± 0.27 B + π 0 3.457 × 10 −5 5.70 × 10 −3 < 0.22 B + D 0 1.726 × 10 −2 2.85 1.86 ± 0.33 B + (D * ) 0 4.059 × 10 −2 6.70 5.3 ± 0.8 B 0 s K − 6.118 × 10 −5 9.42 × 10 −3 B 0 s D − s 1.642 × 10 −2 2.53 B 0 s (D * s ) − 4.185 × 10 −2 6.45 B 0 s B − 2.619 × 10 −8 4.03 × 10 −6 B 0 s (B * ) − 8.032 × 10 −10 1.24 × 10 −7 Table 5 : 5Comparison of this work with other approaches for the form factor f + (0). Here P is the parent meson and D is the DaughterP D This work [5] [21] [22] [24] [12] [3] [23] K + π 0 0.9617 0.98 Table 6 : 6Vector Form Factors for the DecayB → D * + ℓ + + ν ℓ Reference g(0) (GeV −1 ) a + (0) (GeV −1 ) f (0) (GeV )This Work -0.10391 -0.09240 5.286 [5] -0.09745 -0.09471 4.736 [27] -0.16 -0.15 6.863 [23] -0.09054 -0.07271 3.863 Table 7 : 7Predictions for B c → P DecaysP f + (0) f + (q 2 max ) Γ (ps −1 ) BR (%) B 0 0.4504 0.6816 9.7001 × 10 −4 0.049 B 0 s 0.5917 0.8075 1.8774 × 10 −2 0.94 D 0 0.1446 1.017 2.8244 × 10 −5 0.0014 η c 0.5359 1.034 1.0355 × 10 −2 0.52 Table 8 : 8Predictions for B c → V DecaysV Γ (ps −1 ) BR(%) (B * ) 0 1.048 × 10 −3 0.052 (B * s ) 0 2.872 × 10 −2 1.44 (D * ) 0 4.739 × 10 −5 0.0024 J/ψ 2.943 × 10 −2 1.47 Table 10 : 10Branching Ratios for the Semileptonic Decays of the B c .Decay Meson This Work [4] [28] [29] [30] B 0 s 0.94% (0.68 ± 0.23)% 1.35% 0.80% (B * s ) 0 1.44% (0.68 ± 0.23)% 3.22% 2.3% η c 0.52% (0.57 ± 0.17)% 0.553% 0.836% 0.15% J/ψ 1.47% (0.68 ± 0.17)% 2.32% 2.13% 1.5% Table 11 : 11The decay constant g V P γ in GeV −1 .V This Work Expt. [14] [23] [32] [33] [13] (D * ) 0 2.0321 1.043 1.079 1.598 1.939 (D * ) + 0.5224 0.1535 0.0702 0.2886 0.3950 (D * s ) + 0.2369 0.1917 0.2598 J/ψ 0.7419 0.5538 (B * ) 0 0.9770 0.3098 0.4177 0.5720 0.9104 (B * ) + 1.7627 0.3461 0.6540 0.9701 1.618 (B * s ) 0 0.6417 Υ 0.1314 Table 12 : 12Calculations of B * c Properties and g B * c Bcγ . Acknowledgements. We would like to thank M.A. Ivanov and P. Santorelli for a helpful communication. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. . F J Gilman, R L Singleton, Phys. Rev. D. 41142Gilman F J and Singleton R L 1990 Phys. Rev. D 41 142 . S S Gershtein, V V Kiselev, A K Likhoded, A V Tkabladze, A Berezhnoy, A I Onishchenko, hep- ph/9803433Theoretical Status of the B c Meson. Gershtein S S, Kiselev V V, Likhoded A K, Tkabladze A V, Berezhnoy A V and A.I. Onishchenko "Theoretical Status of the B c Meson," hep- ph/9803433. Heavy Quark Physics From Lattice QCD. 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M A Ivanov, P Santorelli, N Tancredi, hep. Ivanov M A, Santorelli P and Tancredi N "The Semileptonic Form Factors of B and D Mesons in the Quark Confinement Model," hep- Figure 1: f Bc vs. Λ bc. Figure 1: f Bc vs. Λ bc	[]
[ "Semi-metallicity and electron-hole liquid in two-dimensional C and BN based compounds", "Semi-metallicity and electron-hole liquid in two-dimensional C and BN based compounds" ]	[ "Alejandro Lopez-Bezanilla ", "Peter B Littlewood \nArgonne National Laboratory\n9700 S. Cass Avenue60439LemontIllinoisUnited States\n\nJames Franck Institute\nUniversity of Chicago\n60637ChicagoIllinoisUnited States\n", "\nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n" ]	[ "Argonne National Laboratory\n9700 S. Cass Avenue60439LemontIllinoisUnited States", "James Franck Institute\nUniversity of Chicago\n60637ChicagoIllinoisUnited States", "Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA" ]	[]	Insulating-metallic transition mediated by substitutional atoms is predicted in a series of twodimensional carbon-based structures. Introducing Si atoms in selected sites of tetrahexcarbon [Carbon 137 (2018) 266] according to rational chemical rules, metallicity by trivial band inversion without band gap opening is induced. Additional substitution of remaining C atoms by BN dimers introduces no changes in the metallic properties. A series of isomorphous two-dimensional materials with isoelectronic structures derived by exchanging group IV elements exhibiting various band gaps is obtained. Dynamical stability is verified with phonon analysis and beyond the harmonic approximation with molecular dynamics up to room temperature. The semi-metallic compounds have well-nested pockets of carriers and are good candidates for the formation of an excitonic insulator.	10.1103/physrevmaterials.5.014006	[ "https://arxiv.org/pdf/2202.07618v1.pdf" ]	234,090,387	2202.07618	b591d478329d0dec34fd77f4d65509d1cc68873d	Semi-metallicity and electron-hole liquid in two-dimensional C and BN based compounds Alejandro Lopez-Bezanilla Peter B Littlewood Argonne National Laboratory 9700 S. Cass Avenue60439LemontIllinoisUnited States James Franck Institute University of Chicago 60637ChicagoIllinoisUnited States Theoretical Division Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Semi-metallicity and electron-hole liquid in two-dimensional C and BN based compounds (Dated: February 16, 2022)PACS numbers: Valid PACS appear here Insulating-metallic transition mediated by substitutional atoms is predicted in a series of twodimensional carbon-based structures. Introducing Si atoms in selected sites of tetrahexcarbon [Carbon 137 (2018) 266] according to rational chemical rules, metallicity by trivial band inversion without band gap opening is induced. Additional substitution of remaining C atoms by BN dimers introduces no changes in the metallic properties. A series of isomorphous two-dimensional materials with isoelectronic structures derived by exchanging group IV elements exhibiting various band gaps is obtained. Dynamical stability is verified with phonon analysis and beyond the harmonic approximation with molecular dynamics up to room temperature. The semi-metallic compounds have well-nested pockets of carriers and are good candidates for the formation of an excitonic insulator. INTRODUCTION The successful synthesis of graphene[1] and numerous other two-dimensional (2D) materials marked the beginning of an acclaimed revolution in materials science. Much of the scientific and technological effort deployed on the field of two-dimensional (2D) materials is to ensure the creation of new compounds with outstanding electronic properties compatible with current synthesis techniques as in graphene. However, chemical modification poses a design challenge in the design of new materials. For example, the conjugated π-network of graphene lattice allows it to exhibit metallic properties, relativistic dispersion of charge carriers, and ballistic transport [2]. These unique electronic features may be altered by modifying the graphene composition, even when its geometry and number of valence electrons are preserved. This is the case of massive graphene BN codoping [3], where polar bonds create electrostatic fields and potential variations at the atomic scale that disrupt graphene hyperconjugation. Richness or disorder in the composition may entail disruption of original properties [4] without necessarily obtaining new interesting properties. As the intricacy of the novel compounds becomes more complex, a careful inspection of intermediate materials obtained in the process of discovery is revealed as a fundamental requirement to infer new material properties. Methodologies consisting in introducing modifications on existing structures guided by the intuition of accumulated experience are gaining relevance [5]. Indeed, examining the physicochemical properties of the materials' constituents provides detailed insight of the interplay between elements in the arising of unexpected properties. Under this backdrop, accurate density functional theory (DFT) based calculations have enabled the prediction of multielement nanostructures with specific functionalities [6], facilitating the analysis of their properties before the actual synthesis. In a previous paper [7], we presented a theoretical study of a new 2D material exhibiting a pentagonal arrangement of C and Si atoms that was obtained upon analysis of the atomic bonding of penta-graphene [8]. In the buckled pentagonal structure of penta-graphene, some C atoms are fourfold coordinated with their atomic orbitals in sp 3 hybridization. Substituting those C atoms by Si atoms, the final nanostructure exhibits enhanced stability and new electronic properties. This is due to the ability of Si valence atomic orbitals to rehybridize in the sp 3 configuration, as the existence of silicene demonstrates [9]. A different hybridization is sp 2 d, where interspersed fourfold coordinated Si atoms in graphene form stable structures [10,11]. Following the same strategy, in this paper we report a series of buckled 2D compounds composed of four-and six-atoms rings obtained by chemical modification of tetrahexcarbon [12,13] (thC), a carbon allotrope obtained by applying a Stone-Wales transformation to penta-graphene [14]. Substitution of sp 3 hybridized C atoms by group IV atoms introduces a stress release in the nanostructure that expands the in-plane unit cell vectors, increases the buckling, and turns the insulating carbon allotrope into several forms of metallic and semiconducting materials. Additional substitution of C dimers by BN dimers yields tetrahexSiBN (thSiBN), a metallic 2D material. Table I. Lower panel displays the diamond and hexagon arrangement of tetrahexSiBN. Grey, green, yellow, and blue spheres represent C, Si, B, and N atoms respectively. TABLE I. Unit cell vectors (nx, ny) and angles between central (c), lower (l) and two upper (u1,u2) atoms (as shown in Figure 1 of all the nanostructures. The maximum vertical separation between atoms is provided as the thickness of the buckled nanostructures. Wales transformation) to penta-graphene [14]. The resulting structure is composed of a periodic arrangement of buckled diamond and hexagons exhibiting dynamical and thermal stability. Two types of C atoms can be differentiated according to the hybridization of their orbitals, namely sp 2 and sp 3 atoms. Whereas the former are threefold coordinated and their hybridized orbitals form angles of ∼120 • , similar to graphene, the latter are fourfold coordinated and tend to adopt a tetrahedral geometry (similar to diamond) which is frustrated by the nearly planar configuration of the layer. This leads to a nanostructure that is higher in energy than that of graphene [12]. A natural way to alleviate the stress imposed by the frustrated tetrahedral geometry is to substitute the C sp 3 -hybridized atoms by an atom whose orbitals can ei-ther combine yielding a flat configuration or simply be more prone to hybridizing in the sp 3 configuration than a C atom. Si exhibits sp 3 hybridization in a 2D buckled structure (silicene) [9,15], suggesting that a closerto-tetrahedral arrangement of the Si atomic orbitals is preferred. Also, experimental observations of inclusions of Si atoms in graphene layers [10,11] were reported and an sp 2 d atomic hybridization proposed to explain the flat fourfold Si coordination. Figure 1 shows the geometry of tetrahexSiC 2 (thSiC 2 ) where each Si atom is bonded at first neighbours to four C dimers. This buckled structure exhibits a formation energy 2.15 eV lower than that of the completely flat geometry, suggesting that the sp 3 atomic hybridization is preferred over the sp 2 d. Although the shape of the unit cell and atomic arrangements are the same than in thC, cell vectors increase substantially (18.8% in x and 24.4% in y ) to accommodate the Si atoms, as shown in Table I. Angles change accordingly as shown in Table I. As discussed in section , in addition to Si atom other larger size atoms of group IV are considered in substitution of sp 3 hybridized atoms, which yields larger unit cells ELECTRONIC PROPERTIES The electronic properties of thC are dominated by the alternating arrangement of hexagons and diamonds that prevents thC from being hyperconjugated. Within the LDA approximation, the band diagram exhibits an electronic band gap at the Fermi level of 1.63 eV, as shown in Figure 2. The exact gap value may depend on the exchange-correlation functional used in the mean-field calculation [12]. A detailed inspection of the contribution of each atomic orbital to the band diagram is conducted with color-weighted bands as plotted in Figure 3. Whereas the fourfold coordinated C atoms barely participate in the low-energy spectrum of thC, the C atoms in sp 2 hybridization are responsible for creating the electronic states in the vicinity of the Fermi level. Orbitals contribute differently depending on the their orientation: perpendicular to the plane p ⊥ orbitals contribute more than the in-plane orbitals (p \|\| ) to both conduction and valence bands. Dimers of p ⊥ orbitals in one unit cell are inefficient in forming a chemical bond with neighbouring p ⊥ orbitals in the next cell due to the intervening sp 3 hybridized C atoms; this results in low-dispersive states at the Fermi level and an insulating band gap. Note an isolated set of four empty conduction bands formed by the lateral overlap of p ⊥ orbitals of C sp 2 hybridized dimers. Separated from the rest of conduction bands by a 0.9 eV band gap, these bands exhibit a dispersion of ∼4 eV. As explained below, the shift in energy of these four bands as a result of chemical modification is the key factor that determines the metallic or insulating character of the modified 4 3 2 1 0 1 2 3 4 5 6 E E Fermi (eV) X M Y M thC X M Y M thSiC 2 X M Y M thGeC 2 X M Y M thSnC 2 X M Y M thCBN X M Y M thSiBN FIG. 2. Electronic band diagrams of tetrahexC, tetrahexSiC2, tetrahexGeC2, tetrahexSnC2, tetrahexCBN, and tetrahexSiBN. The insulating behavior of the carbon allotrope becomes metallic upon Si substitution of the sp 3 hybridized C atoms. Substitution of C atoms by BN dimers introduces little modification in the band structure. nanostructures. Si is one position below C in the group IV of the periodic table, namely both atoms share many characteristics such as the same number of valence electrons in a p-orbital: 2p in C atoms and 3p in the Si atoms. Going down in the group IV, heavy metal atoms such Ge and Sn also accommodate four electrons in their outermost 4p and 5p-orbitals respectively. As a result of the increasing difference between the s and p valence orbitals as the size of the atoms increase, σ-orbitals and π-orbitals exhibit a non-negligible mixing in two-dimensional structures based on Si [9,16], Ge [17], and Sn [18]. This leads to honeycombed structures where sp 3 hybridization is a configuration energetically more stable with respect to all-sp 2 graphene which favors the puckering of the layers. With the sp 3 hybridization being highly favorable, chemical substitution of sp 3 hybridized C atoms by Si, Ge, and Sn atoms is considered in the following as alternative materials. where the Si atoms are bonded to four C atoms. The same schematic representation is valid for Ge and Sn substituting atoms, although both the unit cell vectors and the bonding distances increase with increasing group IV atom size. The electronic band diagram of thSiC 2 in Figure 2 shows that atomic substitution renders the nanostructure metallic, reduces the dispersion of the conduction bands, and leads to a trivial band inversion by which two of the valence bands become partially empty at the expense of filing partially two conduction bands. The insulating-metallic transition is driven by a shift down in energy of the conduction bands (with respect to thC), that increase their gap to ∼2 eV with the higher bands, and by a shift up in energy of a set of two occupied bands. An inspection of the color-weighted bands plotted in Figure 3 shows that the cone-shape band that in thC was totally filled and formed by the in-plane p-orbitals shifts in energy and becomes partially empty in thSiC 2 . The cone-shape conduction band formed almost exclusively by p ⊥ -orbitals compensates and produces an electron pocket. Near the zone center two bands play a similar role in the metallicity of the this structure. A charge transfer from p \|\| to p ⊥ orbitals occurs as a result of introducing the Si atom in the nanostructure. Calculations including spin-orbit coupling leave the non-relativistic results unaltered. Si substitution by Ge atoms leads to thGeC 2 . A major change introduced in the electronic band diagram when compared to the one of thSiC 2 is the semiconducting band gap of 400 meV at the Γ point. Both the shape and dispersion of the conduction bands remain the same. It is interesting to point out that substitution of Si by Ge with no relaxation of the structure leads to a band gap of only 14 meV. (Similarly, substitution of C by Si or Sn shifts conduction bands away from valence bands without opening a band gap). Therefore, the combined effect of atomic substitution and structural relaxation adjust the level of the electronic states and the subsequent creation or disappearance of a band gap. Substitution of sp 3 atoms of thC by Sn atoms yields thSnC 2 . As in thSiC 2 the gap vanishes and a small band inversion at Γ point renders the nanostructure metallic. The electronic gap above the set of four conduction bands also closes and no major variation of the electronic band dispersion is observed. The non-monotonic trend of gap closing and opening in the sequence C-Si-Ge-Sn is on the first sight surprising, because it runs counter to the usual trend of gap reduction and closing in tetrahedrally coordinated semiconductors as the s-p splitting increases on moving down the periodic table. That trend is in fact present in the manifold of σ * antibonding orbitals which lies above the almost isolated sp 2 ⊥ . In the most simplified view the sp 2 ⊥ band from three-fold coordinated atoms is sitting rather like an "impurity band" of dimers in a wider gap insulator. The bonding-antibonding σ σ * manifold is constructed from hybridised four-fold sp 3 and three-fold sp 2 \|\| ; that gap shrinks monotonously from 6 eV in thC to around 2 eV in thSnC 2 . The p ⊥ band sits at a relatively fixed energy as the larger gap collapses around it. A common characteristic of the four structures presented above is the isoelectronic character of their constituents. The number of valence electron remain constant and hence the band structures retain a similarity across the substitution series. An additional isoelectronic substitution in thC could change the C atoms in sp 2 hybridization. Given that 2D BN is an isomorph of graphene [19] able to integrate the layered structure in codoping, the C dimers of the four nanostructures could in principle be substituted by BN dimers. It is worth remembering that BN-based structures are wide-gap insulators due to the polar and ionic character of the B-N bond. 2D hexagonal BN was observed to show metallic behavior when cut along zigzag nanoribbons, which was ascribed to half-metallicity derived from the edge states [19]. In Figure 2 the band diagram of tetrahexCBN (thCBN) shows two sets of isolated four bands at both sides of the Fermi level separated by a 1.95 eV wide band gap. The most appealing BN substitution occurs in the dimers of thSiC 2 , which yields tetrahexSiBN (thSiBN). Figure 2 shows that thSiBN is metallic with a compensated electron-hole pocket similar to that of thSiC 2 . The dispersive band in the Y − Γ region vanishes, although the cone-shape band remains yielding band inversion in the zone center. An analysis of the weighted bands in Figure 3 shows that the combined p ⊥ orbitals of both B and N atoms in the dimers contribute to the conduction bands more than the p \|\| orbitals, and vice versa for the valence bands. As in the previous structures, the weight of sp 3 hybridized atoms in the formation of electronic states in the vicinity of the Fermi level is negligible. A qualitative Mulliken charge analysis reveals a strong ionic character in multi-atom compounds, specially both thSiC 2 and thSiBN. Whereas Si atoms in the former accept ∼0.1 electrons from the C atoms, in the latter each Si atom behaves as a donor of ∼0.3 electrons to each BN dimer. DYNAMICAL STABILITY The first report on thC [12] demonstrated the stability of the monolayer based on a phonon spectrum analysis. The absence of imaginary frequencies indicates dynamical stability since only restoring forces act on an atom that is perturbed a short distance from its equilibrium position. Tiny deviations observed in all structures were minimized within the LDA approximation and very tight convergence criteria in the first-principles calculations. Phonon calculations displayed in Figure 4 shows deviations of ∼0.75 THz towards imaginary values for one of the acoustic branches starting at Γ point. Whereas a deviation in between two points or at a point of the zone edge could indicate a structural phase transition, phonon softening at the Γ point is indicative of some numerical inaccuracy, and does not demonstrate a structural instability. All compounds presented are dynamically stable and restoring forces are efficient against small perturbation and maintain the structural integrity. Structures formed by substitution of C and Si atoms by Ge and Sn atoms in BN-based structures exhibit instabilities and were discarded in this study. A total of 32 acoustic and optic modes extend over a frequency range of ∼40 THz. Separated by a ∼8 THz gap, four additional optical modes corresponding to in-plane three-fold coordinated atom vibration, lay at ∼50 THz. This gap increases when the difference of the atomic masses becomes more pronounced in thSiC 2 and thSiBN. Also the phonon dispersion lowers with the 32 branches spanning a range of ∼25 THz. The structural integrity of the layered structures is further verified with molecular dynamic calculations at finite temperature. Starting from the ground-state at T = 150 K and T = 300 K, 5 ps long runs were performed after an initial equilibration time of 1 ps with a 1 fs time-step. From heating at constant temperature the two-dimensional slabs composed of 192 atoms for 5 ps, no structural rearrangement of the atoms was observed. Figure 5 shows relative interatomic distance variations in each orthogonal direction, of nearest Si-Si atoms, and Si-C and C-C dimers. The longer distance variation between atoms is the perpendicular-to-the-plane direction (up to 1Å), whereas atoms modify their relative distances in the structure plane in a much smaller range. The collective motion of the atoms is a breathing vibration of the nanostructure that expands and compresses, with no large variation of the bonding distance between pairs of atoms. Similar structural stability was observed for the rest of the compounds at finite temperatures. This demonstrates that the structural integrity of the nanostructures is preserved beyond the harmonic approximation, confirming that the compounds are stable up to room temperature. An alternative arrangements were considered corresponding to a BN dimer followed by a NB dimer along the x-axis after the C bond, and then the opposite arrangement from the parallel BN-C-NB line. That yields two N and two B atoms in front of each other at each side of a C atom, which reduces the distance between two atoms of the same kind, increasing the dipole and destabilizing the structure. The B-C-B and N-C-N distances would be 1.96Åand 2.06Å, as opposed to 2.1Åof B-C-N. In the structures studied here, the shortest distance between two B or two N atoms is 2.5Å, which helps to balance the negative repulsion between electronegative N atoms. In addition, phonon calculations reveal important softening of some modes as a result of the atomic repulsion. In a layered structure, the z-axis acoustic modes are relatively soft, and the z-axis lattice constant is potentially temperature-dependent and of course difficult to estimate from numerics with high accuracy. This raises the question about whether the semimetallicity might be dependent on the c-axis lattice constant. The chemical trends elucidated in section III around the closing of the gap between σ and σ * orbitals is robust. The overlap of the bands is determined by the bandwidth of the dimer band relative to the bonding-antibonding gap. Both of these parameters are established by local in-plane atomic correlations which are not strongly affected by the interlayer distance in the simulation. CONCLUSIONS To summarize, a series of 2D materials composed of C and BN dimers woven in a hexagonal-square motif with group IV elements has been studied. Despite the isomorphism and isoelectronic features of the structures, a wide range of electronic behaviors are found. Exchanging C by BN dimers and substituting fourfold coordinated C atoms by isoelectronic Si, Ge, and Sn atoms, insulatingmetallic transitions are observed. Whilst the carbon allotrope exhibits an electronic band gap, substitutional of sp 3 hybridized C atoms by Si or Sn atoms brings together conduction and valence bands leading to band inversion. Ge atoms yield a similar band structure to the one composed with the previous atoms but exhibit a small band gap. It is remarkable that a 2D nanostructure composed of three different elements exhibits a metallic band structure, since the polar bonds typically lead to gaped electronic band diagrams. Finally we remark that the electron-hole pockets that form at the fermi surface are very well nested, and the electron gas parameter r s = 1/(k F a B ) is about 5 (here k F is the Fermi momentum and a B the Bohr radius. Another way to put this is that the Fermi energy of the electron (hole) pockets is around 0.1 eV, which is considerably smaller than the (unscreened) Coulomb energy e 2 /r s of order 2 eV. Since the charge transfer from the in plane (hole pocket) to out of plane (electron pocket) will generate a Coulomb interaction that is relatively weakly screened, these semimetals will be good candidates for an excitonic insulator instability where the electron-hole pairs condense. [20] COMPUTATIONAL METHODS Self-consistent density functional theory (DFT) based calculations were performed within local density approximation (LDA) approach [21] for the exchange-correlation functional was used as implemented in the SIESTA code [22][23][24]. A double-ζ polarized basis set was used to relax atomic coordinates and compute phonon spectra. Norm-conserving pseudopotentials as implemented in the SIESTA code were employed to represent the electronion interaction, so that the pseudopotentials account for the effect of the Coulomb potential created by the nuclei and core electrons creating an effective ionic potential acting on the valence electrons [25]. Atomic positions of the layered materials were fully relaxed with a force tolerance of 0.01 eV/Å. The integration over the Brillouin zone (BZ) was performed using a Monkhorst sampling of 38 × 42 × 1 k-points. The radial extension of the orbitals had a finite range with a kinetic energy cutoff of 50 meV. A vertical separation of 35Å in the simulation box prevents virtual periodic parallel layers from interacting. The force-constant method and the PHONOPY package [26] were employed for computing phonon spectra. DFT-based molecular dynamic simulations with a Nosé thermostat [27] were performed in the NVT ensemble for a 4×4×1 unit supercells using only the Γ point. FIG. 1 . 1Top and side view of the schematic representation of the tetrahexC, tetrahexSiC2, and tetrahexSiBN nanostructures. Angles between the central (c), lower (l) and two upper (u1,u2) atoms for each isomorph structure are given in FIG. 3 . 3Color-weighted band diagrams of threefold coordinated atoms showing the independent contribution of perpendicular to the plane p ⊥ and in-plane p \|\| orbitals of tetrahexC, tetrahexSiC2, and tetrahexSiBN. Fourfold coordinated atoms barely contribute to the formation of electronic states in the vicinity of the Fermi level. The color intensity of the lines is proportional to the contribution. Figure 1 FIG. 4 . 14shows the unit cell of tetrahexSiC 2 (thSiC 2 ) Phonon spectra of tetrahexC, tetrahexSiC2, tetrahexGeC2, tetrahexSnC2, tetrahexCBN, and tetrahexSiBN. The frequency of each phonon mode is plotted versus its propagation direction along the high-symmetry lines in the Brillouin zone, as depicted in the inset. FIG. 5 . 5Molecular dynamic simulations at 150K and 300K demonstrate structural stability of thSiC2 and thSiBN beyond the harmonic approximation at 0K. Variations of distances between atoms in the three perpendicular directions between pairs of atoms remain within a 1Å limit for the out-of-plane direction throughout 5 ps after an time of 1 ps. Blue lines indicate out of plane displacement while red and green inplane x and y displacement respectively. 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[ "ASTRONOMY AND ASTROPHYSICS Spallative Nucleosynthesis in Supernova Remnants II. Time-dependent numerical results", "ASTRONOMY AND ASTROPHYSICS Spallative Nucleosynthesis in Supernova Remnants II. Time-dependent numerical results" ]	[ "Etienne Parizot [email protected] \nDublin Institute for Advanced Studies\n5 Merrion SquareDublin 2Ireland\n", "Luke Drury \nDublin Institute for Advanced Studies\n5 Merrion SquareDublin 2Ireland\n" ]	[ "Dublin Institute for Advanced Studies\n5 Merrion SquareDublin 2Ireland", "Dublin Institute for Advanced Studies\n5 Merrion SquareDublin 2Ireland" ]	[]	We calculate the spallative production of light elements associated with the explosion of an isolated supernova in the interstellar medium, using a timedependent model taking into account the dilution of the ejected enriched material and the adiabatic energy losses. We first derive the injection function of energetic particles (EPs) accelerated at both the forward and the reverse shock, as a function of time. Then we calculate the Be yields obtained in both cases and compare them to the value implied by the observational data for metalpoor stars in the halo of our Galaxy, using both O and Fe data. We find that none of the processes investigated here can account for the amount of Be found in these stars, which confirms the analytical results ofParizot and Drury (1999). We finally analyze the consequences of these results for Galactic chemical evolution, and suggest that a model involving superbubbles might alleviate the energetics problem in a quite natural way.	null	[ "https://arxiv.org/pdf/astro-ph/9903358v1.pdf" ]	2,777,110	astro-ph/9903358	54222e70eccc32ecbeef5b83696518fbf3e27e83	ASTRONOMY AND ASTROPHYSICS Spallative Nucleosynthesis in Supernova Remnants II. Time-dependent numerical results 9903358v1 23 Mar 1999 Etienne Parizot [email protected] Dublin Institute for Advanced Studies 5 Merrion SquareDublin 2Ireland Luke Drury Dublin Institute for Advanced Studies 5 Merrion SquareDublin 2Ireland ASTRONOMY AND ASTROPHYSICS Spallative Nucleosynthesis in Supernova Remnants II. Time-dependent numerical results 9903358v1 23 Mar 1999arXiv:astro-ph/ A&A manuscript no. (will be inserted by hand later) Your thesaurus codes are: 12 (accepted for publication in A&A)Acceleration of particlesNuclear reactions, nucleosynthesisISM: supernova remnantsGalaxy: abun- dances We calculate the spallative production of light elements associated with the explosion of an isolated supernova in the interstellar medium, using a timedependent model taking into account the dilution of the ejected enriched material and the adiabatic energy losses. We first derive the injection function of energetic particles (EPs) accelerated at both the forward and the reverse shock, as a function of time. Then we calculate the Be yields obtained in both cases and compare them to the value implied by the observational data for metalpoor stars in the halo of our Galaxy, using both O and Fe data. We find that none of the processes investigated here can account for the amount of Be found in these stars, which confirms the analytical results ofParizot and Drury (1999). We finally analyze the consequences of these results for Galactic chemical evolution, and suggest that a model involving superbubbles might alleviate the energetics problem in a quite natural way. Introduction The class of light elements, namely Li, Be and B, sets itself apart from any other by its interstellar origin (except for part of the 7 Li, produced in the Big Bang ages, and perhaps part of the 11 B, produced in supernova (SN) explosions by neutrino-spallation). Concentrating on the most representative isotope, the abundance of 9 Be in stars of increasing metallicity can be regarded as the witness and tracer of the nuclear spallation efficiency during Galactic chemical evolution. Indeed, virtually every atom of Be observed in the atmosphere of stars must have been produced by the spallation of a larger nucleus, most probably C or O, induced by the interaction of energetic particles (EPs) with the interstellar medium (ISM). Send offprint requests to: E. Parizot Since the first measurement of Be in a very metal-poor star at the beginning of the decade (Gilmore et al. 1991), increasing evidence has been gathered showing that the abundance of Be and B in the early Galaxy (until the ambient metallicity is 10% that of the sun, say) kept increasing jointly and linearly with ordinary metallicity tracers, such as Fe or O, as if they were actually primary elements (Duncan et al. 1992(Duncan et al. ,1997Edvardsson et al. 1994;Gilmore et al. 1992;Kiselman & Carlsson 1996;Molaro et al. 1997;Ryan et al. 1994). Now they are not, since as we just recalled C and O nuclei have to be produced first in order that they can be spalled by EPs into light elements. The observations therefore suggest that some process must act to ensure that, on average, an equal amount of Be is synthesized each time a given mass of Fe or O is ejected into the ISM. It should be clear, however, that this statement relies on the assumption that the abundances of O and Fe are proportional to one another, at least during the early evolution stages in which we are interested here. This assumption has long been used with high confidence level based on both theoretical and observational arguments, but new observations seem to contradict it dramatically (Israelian et al. 1998;Boesgaard et al. 1998). Although an independent confirmation of these observations would be welcome, they have recently been used to reappraise the alleged 'primary behavior' of 6 LiBeB Galactic evolution (Fields and Olive, 1999). Indeed, if the O/Fe abundance ratio is not constant but actually decreases with metallicity, then the observed approximate constancy of the Be/Fe ratio implies an increasing Be/O ratio. Fields and Olive (1999) find a Be-O logarithmic slope in the range 1.3-1.8, which seems to contradict both the primary scenario (slope 1) and the secondary scenario (slope 2), in which the spallation reactions producing the light elements are induced by standard Galactic cosmic rays (GCRs) accelerated out of the ISM. However, the current lack of Be and O abundance measurements in the same very metal-poor stars (with [O/H] = 10 −3 , say) makes the data marginally compatible, within error bars, with both scenarii. While the situation should be soon clarified, notably by the accumulation of data at lower metallicity and independent measurements of Be, B, O and Fe in the same set of halo stars, we (Parizot and Drury, 1999; Paper I) choose to investigate the Be production in the ISM from the other direction, i.e calculate the Be yield associated with the explosion of an isolated supernova (SN) in the ISM, according to current knowledge about supernova remnant (SNR) evolution and standard shock acceleration, and compare this Be yield with the value required to explain the observed Be/Fe ratio in metal-poor stars. We identified two different mechanisms leading naturally to a primary evolution of Be in the early Galaxy. In the first mechanism, particles from the ambient ISM (i.e. metal-poor) are accelerated at the forward shock of the SN and confined within the SNR until the end of the Sedov-like evolution phase. There, they interact with the freshly synthesized C and O nuclei, and therefore produce Be by spallation at a much higher rate than in the (secondary) GCR nucleosynthesis scenario in which they merely interact with the ambient, metal-poor ISM. In the second mechanism, particles from the enriched SN ejecta are accelerated at the reverse shock and again confined within the SNR during Sedov-like phase, where they suffer adiabatic losses through which they lose between 30% and 70% of their initial energy, depending on the ambient density. After the end of the Sedov-like phase, these particles diffuse out in the ISM where the energetic C and O nuclei can be spalled by the H and He atoms at rest in the Galaxy. We have shown in Paper I, through approximate analytical calculations, that the total Be yield obtained by processes 1 and 2 depends on the ambient density, and that this third mechanism is actually the most efficient (for light element production) in most cases, though not efficient enough to account for the observed Be/Fe ratio of ∼ 1.6 10 −6 . If each SN ejects on average 0.1 M ⊙ of Fe in the ISM, then the average Be yield per SN must be ∼ 4 10 48 atoms (cf. Ramaty et al. 1997), which exceeds even our most optimistic calculated yields by about one order of magnitude. We concluded that another mechanism or source of energy should be invoked, and argued that a model based on superbubble acceleration (involving the collective effect of SNe rather than individual SN shock acceleration) is a quite natural and promising candidate. In this paper, we confirm the results of Paper I by performing time-dependent numerical calculations, and discuss in more details their implications for Galactic chemical evolution scenarii. The reader is referred to Paper I for a more detailed description of the mechanisms considered here, and a discussion of their motivation and theoretical justification. Why we need to do time-dependent calculations We intend to calculate the Li, Be and B (LiBeB) production induced by the interaction of energetic particles within a SNR. We shall first consider the fate of the particles accelerated out of the ambient, zero metallicity ISM entering the forward shock created by a SN explosion (process 1), and then turn to the acceleration of particles from the SN ejecta at the reverse shock, on a very short time scale around the so-called sweep-up time, t SW (process 2). It turns out that both of these processes are highly nonstationary, for a number of reasons which we now review. EPs accelerated at the forward shock Considering first process 1, we expect that the particle injection power be more or less proportional to the power of the shock, which is a decreasing function of time as the SNR evolves. Therefore the injection rate of the EPs is not constant, and no steady-state distribution function of the EPs within the SNR is ever reached. If everything else was constant in the problem, we could however calculate the total energy injected in the form of EPs during the whole process, and multiply it by the steady-state spallation efficiency (defined as the 'number of nuclei synthesized per erg injected'), evaluated from standard steady-state calculations. This would provide us with the total spallation yields (i.e. the time integral of the spallation rates), which are the only observationally relevant quantities. This, however, cannot be done in the case we are considering, because the chemical composition of the target, namely the interior of the SNR, is also evolving during the expansion. Indeed, as more and more metal-poor material is swept-up from the ISM by the shock, the metal-rich SN ejecta suffer stronger and stronger dilution, which makes the spallation of C and O less and less efficient. As a consequence, even though we can evaluate the total energy eventually imparted to EPs, we cannot deduce the spallation yields from it because we don't know what composition to choose for the target. Again, if this was the only non stationary feature in the process, we could still calculate the average target composition and compute the spallation yields from it. But since both the EP injection rate and the target composition are functions of time, steadystate models cannot be used in any consistent way, and a fully time-dependent calculation is required. Qualitatively, it is easy to show that the yields which we obtain by integrating the time-dependent spallation rates must be appreciably higher than those derived from steady-state estimates using the total energy injected in the form of EPs and the (constant) mean target composition. Indeed, the latter amounts to assuming that the injection rate is also constant and equal to the average power of the EPs (i.e. the total energy divided by the duration of the process). However, in the time-dependent model, we take advantage of the fact that the EP power is higher at the beginning, when the target composition is richer in C and O. In other words, the spallation efficiency is higher when the EP fluxes are higher too, and conversely, less energy is imparted to the less efficient EPs accelerated towards the end of the process. In addition to the sources of non-stationarity just mentioned (time-dependent injection and dilution of the ejecta), we also have to take into account the adiabatic losses suffered by the EPs as they wander inside the expanding volume of the SNR. Now these adiabatic losses are essentially function of time, becoming smaller and smaller as the SNR expands and the shock velocity gets lower. This again can only be taken into account in a timedependent model. EPs accelerated at the reverse shock Coming now to the case of process 2, where particles from the enriched material ejected by the supernova are accelerated at the reverse shock, it is clear that the dilution effect mentioned above does not have any significant influence anymore. Indeed, the light element production is now dominated by the spallation of energetic C and O nuclei interacting with ambient H and He, instead of ambient C and O interacting with energetic H and He nuclei in the case of process 1 (see Figs. 1 and 4). The abundance of (non energetic) C and O in the target has therefore only a negligible influence, since these nuclei hardly contribute to the spallation yields. Nevertheless, the time dependence of the EP injection and the adiabatic losses still have to be taken into account, which is enough to make time-dependent calculations indispensable. As has been argued in Paper I, the curve representing the power in the reverse shock, P in , as a function of time strongly peaks around the sweep-up time, t SW , which is defined as usual as the time at which the swept-up mass is equal to the ejected mass. This also approximately marks the end of the free expansion phase and the beginning of the adiabatic (or Sedov-like) phase. In the absence of a motivated prescription for the reverse shock power function, P in (t), and on the understanding that its time scale is short as compared to the energy loss time scale, we shall consider below the injection of EPs as instantaneous in the case of process 2. We thus just could not be further away from a steady-state. However, as above, were this the only non-stationary feature of the process, we could still obtain the integrated spallation yields from steady-state calculations by merely multiplying the total energy injected in the form of EPs by the spallation efficiency, which in this case is almost independent of the target composition. Unfortunately, as already indicated, the adiabatic losses are also a function of time, and will therefore cause the aforementioned spallation efficiency to vary as the process goes on. On the other hand, once the EPs leave the SNR at the end of the Sedov-like phase to interact with the surrounding ISM, they will only suffer the usual Coulombian losses, which are essentially independent of time. The above argument therefore does not apply anymore and this final part of process 2 (occuring outside the remnant) could be worked out with purely steady-state machinery. This is in fact what we did in Paper I (see its Sect. 4), in our study of what we then called process 3. Here, however, we shall not distinguish between the part of the process occuring inside the SNR, and the part occuring outside (former process 3), because our time-dependent numerical model allows us to treat both on the same footing. In particular, we obtain in this way not only the total LiBeB yields, but also their production rates as a function of time, whose time integral can be sucessfully checked to be equal to the steady-state yields. Description of the theoretical and numerical model It has been shown in the previous section that the spallative production of light elements associated with the explosion of a SN in the ISM is essentially a dynamical process, and therefore requires non-stationary calculations. A general time-dependent model for the interaction of EPs in the ISM has been developed and presented in Parizot (1999), so we shall use it here extensively, recalling only the results relevant to our specific problem and calculating the required inputs for processes 1 and 2. The mathematical formalism and the physical ingredients In each case, we separate the acceleration of the energetic particles (EPs) from their propagation and interaction within the SNR. This is legitimate because the time scale for acceleration up to the energies we are concerned with is very much smaller than any other time scale in the problem, whether dynamical (SNR evolution) or physical (energy loss rate, spallation rates). Consequently, our calculations apply to the EPs once they have been 'injected' inside the SNR (from the region close to the shock). Let us assume for the moment that we have determined the so-called injection function, Q i (E, t), which we define as the number of particles of species i introduced at energy E and time t, per unit energy and time (in (MeV/n) −1 s −1 ). The EP distribution function, N i (E, t) then satisfies the usual propagation equation (see Parizot 1999) : ∂ ∂t N i (E, t) + ∂ ∂E (Ė i (E)N i (E, t)) = Q i (E, t) + Q ′ i (E, t) − N i (E, t) τ tot i (E) ,(1) whereĖ i (E) is the energy loss rate for the nuclei of species i at energy E (in (MeV/n)s −1 ), Q ′ i (E, t) is the production rate of nuclei i as secondary particles, and τ tot i (E) is the time scale for catastrophic losses, such as nuclear destruction or escape from the region under study. Since we are concerned with spallation reactions involving the nuclei of the CNO group, we can neglect the two-step processes such as 16 O + p → 12 C followed by 12 C + p → 9 Be. We indeed found, using a steady-state model, that the omission of the two-step processes leads too an error of at most ∼ 10%, in good agreement with Ramaty et al. (1997) calculations. Since this is smaller than the other observational and theoretical uncertainties, and their implementation in a time-dependent model greatly complicates the situation, we shall neglect them here (Note that in any case, even if there were no other uncertainty in the problem, it is much more accurate to do time-dependent calculations without two-step processes than steady-state calculations including two-step processes, as our simulations have shown). To state this in a more physical way, we can claim that the spallative production of carbon amounts to at most a few percents of the initial CNO supply from the supernova explosion. To the level of precision of the SN models, to mention that only, this correction is of no significance, so we shall simply drop Q ′ i (E, t) in Eq. (1). Concerning the catastrophic loss time, τ tot i , it is obtained for stable nuclei as : 1 τ tot i (E, t) = 1 τ esc i (E, t) + 1 τ D i (E, t) ,(2) where τ esc i is the escape time, and τ D i is the destruction time. The latter is derived from semi-empirical formulas giving the total inelastic cross sections σ i,j for a projectile i in a target of species j (Silberberg & Tsao 1990), according to : 1 τ D i (E, t) = [ j σ i,j (E)n j (t)]v(E),(3) where v(E) is the velocity of the energetic particle and n j (t) is the number density of target species j at time t. Following the above qualitative analysis (see Paper I for more details), we assume that the time of escape out of the SNR is infinite during the Sedov-like phase of the SNR expansion, and 'zero' afterwards. This merely translates the fact that the EPs are confined within the SNR during the adiabatic phase (at least those of lowest energy, which produce most of the spallative LiBeB), and then leak out on a very short time scale. Once the EPs have escaped from the SNR, we need to distinguish between our two processes. In the first case (acceleration at the forward shock), the EPs are deprived of CNO and will not give rise to enough spallation reactions out of the SNR to raise the LiBeB production in any significant way. This is due to the very low ambient metallicity. In the second case, however, the EPs are made of the supernova ejecta themselves and are thus rich in CNO. As a consequence, as far as LiBeB production is concerned, there is no difference whether they interact within or outside the SNR, as interactions with H and He nuclei dominate anyway. We must therefore follow these accelerated nuclei after the end of the adiabatic phase, and compute the corresponding contribution to the total production of light elements. Concerning the energy loss rate,Ė i (E), we need to take into account both ionisation (Coulombian) and adiabatic losses. The former are very common and just cannot be avoided as soon as energetic particles are to be interacting in the ISM. The latter, however, must be included here because the EPs are confined within the SNR where their velocities are randomized. As a consequence, they do participate to the internal pressure which drives the remnant during the Sedov-like phase, and suffer the adiabatic losses like any other particle working outward when reflected at the expanding shell. Quantitatively, these adiabatic losses have been calculated in Paper I. They are given by Eq. (14) there, namely : p p = − 3 4Ṙ R ,(4) where p is the momentum of the particle and R(t) is the radius of the shock. Assuming the Sedov-like expansion law (R(t) ∝ t 2/5 ) and writing the loss rate in terms of energy, we obtain : E ad (E, t) = − 3 10 E t E + 2m p c 2 E + m p c 2(5) This energy loss rate does not depend on the EP species, but is clearly a function of time. On the other hand, the ionisation losses,Ė ion (E), do depend on the nuclear species, as well as on time, indirectly, through the density and composition of the ambient medium. Indeed, it has to be realised that the medium in which the EPs are 'propagating', namely the interior of the SNR, is initially very rich in freshly synthesized CNO nuclei, and then gets poorer and poorer in metals as the ejecta are being diluted in the ambient, metal-poor, swept up material. This dilution effect is most important for the calculation of the total LiBeB production through our first mechanism (acceleration of the ISM at the forward shock). Indeed, the instantaneous production rates are directly proportional to the density of CNO within the remnant at time t, which goes like R −3 , i.e. t −6/5 . Quantitatively, the LiBeB production rates are obtained by integrating the spallation cross sections over the EP distribution functions : dN k dt = i,j ∞ 0 dE ′ N i (E ′ , t)n j (t)σ i,j;k (E ′ )v i (E ′ ),(6) where σ i,j;k is the cross section for the reaction i + j → k, and n j is the number density of nuclei j in the target (here, the interior of the SNR). Etienne Parizot & Luke Drury: Spallative Nucleosynthesis in Supernova Remnants 5 The total LiBeB production is then obtained for the first mechanism by integrating these production rates from t SW to t end , which marks the end of the Sedov-like phase as well as the end of the confinement of the EPs inside the SNR. For the second mechanism, we need to integrate from t SW to the confinement time of the cosmic rays within the Galaxy. As we shall see below, integrating up to infinity only leads to a small overestimate of the total LiBeB production, since the low energy cosmic rays responsible for most of that production have anyway a short lifetime above the spallation thresholds. The sweep-up time, t SW , is obtained straightforwardly from its definition as a function of the SN parameters and the ambient number density, n 0 : t SW = (1.4 10 3 yr) M ej 10M ⊙ 5 6 E SN 10 51 erg − 1 2 n 0 1cm −3 − 1 3 .(7) The determination of t end is more difficult and somewhat arbitrary, even in the approximation of a perfectly homogeneous circumstellar medium. We argued above and in Paper 1 that t end should more or less coincide with the end of the Sedov-like phase, when the shock induced by the SN explosion becomes radiative, that is when the cooling time of the post-shock gas becomes of the same order as the dynamical time. This depends on the cooling function which in turn depends on the density and metallicity of the post-shock gas. Such details and their influence on t end have been considered in Paper I. Here, we only give the asymptotic result, valid in the limit of large ambient densities, n 0 : t end = (1.1 10 5 yr) E SN 10 51 erg 1/8 n 0 1cm −3 −3/4 . (8) Comparing the dependence of t SW and t end on density, we find that the Sedov-like phase gets shorter when n 0 is increased, and thus the duration of process 1 decreases. The injection function at the forward shock We now turn to the determination of the injection function, Q i (E, t), in the case of our first mechanism. As suggested by shock acceleration calculations, we assume that the distribution function of the accelerated particles is f (p) ∝ p −4 , so that the number of protons injected inside the SNR per unit time between momenta p and p + dp, irrespective of their direction, is : Q(p)dp = Q 0 dp p 2 ,(9) from thermal values up to ∼ 10 14 eV/c. This leaves us only with the calculation of the normalisation, Q 0 , as a function of time. Following again the most widely accepted theoretical ideas, we assume that the total energy injected per unit time in the form of energetic particles at time t is equal to a constant fraction, θ 1 , of the power, P in , flowing through the shock at that time (recalling that the acceleration time scale is small as compared to the dynamical one). Mathematically, this normalisation condition reads : pmax pmin Q(p)E(p)dp = θ 1 P in ,(10) where E(p) = p 2 c 2 + m 2 c 4 − mc 2 is the energy of a proton of impulsion p. Integrating the left hand side (LHS) of Eq. (10), one finds : LHS = Q 0 c pmax pmin mc p 1 + (p/mc) 2 − 1 dp p = Q 0 c umax umin √ 1 + u 2 − 1 u 2 du ≡ Q 0 cκ,(11) where u = p/mc and κ is the number κ = 1 − √ 1 + u 2 u + ln(u + 1 + u 2 ) umax umin(12) Typical values for u min and u max are u max = p max /mc ∼ 10 14 /2 10 9 ∼ 5 10 4 , and u min = 2E min /m p c 2 < ∼ 10 −2 . Then, to first order : κ = ln u max − 1 + ln 2 + O(u min + O(1/u max ) ∼ 10.5(13) depending on p max only logarithmically. Combining Eqs. (10) and (11), we obtain the injection function at the forward shock as : Q(p) = θ 1 P in κp 2 c ,(14) or in terms of energy : Q(E) = Q(p) dp dE = θ 1 P in κ 1 E 3/2 E + mc 2 (E + 2mc 2 ) 3/2(15) The asymptotical behavior is thus : Q(E) ∝ E −1.5 for E ≪ m p c 2 , and Q(E) ∝ E −2 for E ≫ m p c 2 . It should be clear that the above injection function is indeed a function of time, through the incoming power P in . To evaluate it, one can make use of the well known formulas giving the time evolution of the shock radius, R s , and velocity, V s , during the Sedov-like phase, and calculate P in = 1 2 ρ 0 V 2 s × 4πR 2 s × V s . However, since the Sedov phase is a similarity solution, we know that the result will be nothing else but P in (t) ≃ E SN /t, where E SN is the explosion energy. The time-dependent injection function is then finally : Q(E, t) = θ 1 κ E SN t 1 E 3/2 E + mc 2 (E + 2mc 2 ) 3/2 .(16) As can be seen, the power injected in the form of energetic particles decreases as t −1 as the SNR expands. This is not a futile result, since it happens that the earliest times are also the most favourable to the spallative production of light elements in a SNR. Indeed, as was discussed in Sect. 2, the CNO nuclei suffer a rapid dilution as the remnant expands, lowering the spallation rates. Ignoring the enhancement of the EPs when the SNR is still rather small would thus leads one to significantly underestimate the LiBeB production. In the above derivation, we did not worry about the chemical composition of the EPs. Clearly the injection function still has to be weighted by the relative abundance of each nuclear species present in the ISM swept up by the SNR. As already mentioned, we are interested only in the LiBeB production at low ambient metallicity, since this is when the observed proportionality between Be and Fe abundances is the most striking and unexpected. According to the assumption that we are testing here, each supernova leads to the same amount of 9 Be production, whatever the ambient metallicity. Therefore, all our calculations are made with a zero ambient metallicity. The EPs accelerated out of the ISM are thus made of H and He only, with their primordial relative abundances. The injection function at the reverse shock In the case of the acceleration of the supernova ejecta through the reverse shock, the injection function can be written straightforwardly as : Q i (E, t) = n iQ (E)δ(t − t SW ),(17) where it is assumed that the acceleration takes place instantaneously at t SW . This may be justified by noting that the genuine acceleration and reverse shock evolution time scales are certainly smaller than EP evolution time scales (nuclear interactions and energy losses). The relative abundance of the different nuclei in the accelerated particles just reflects that of the supernova ejecta, n i , and the shape ofQ(E) is the same as above. This time, however, the injection function has to be normalised to : i dt Q i (E, t)EdE = θ 2 E SN ,(18) where θ 2 is the fraction of the explosion energy which goes into the EPs accelerated at the reverse shock. This can be phenomenologically expressed as the product of two coefficient : θ 2 = θ acc ×θ rev , where θ acc is the fraction of the shock energy imparted to the EPs (i.e. θ acc ≈ θ 1 , defined above), and θ rev is the fraction of the explosion energy which goes into the reverse shock. In our calculation, we adopt the 'canonical values' of θ 1 = 0.1 and θ rev = 0.1, and thus θ 2 = 0.01. It should be clear, however, that these values are only indicative, and that the results simply scale proportionally to θ 1 and θ 2 . The time integration in Eq. (18) is straightforward, and with i n i = 1, we get : Q(E) = θ 2 E SN κ 1 E 3/2 E + mc 2 (E + 2mc 2 ) 3/2 ,(19) where κ has been given in Eq. (12) and (13). Note that the mass m appearing in the above expressions is always the proton mass, and that correlatively the energies are expressed in MeV/n for all the nuclear species. The formal solution for the EP distribution function The formal solution of the time-dependent propagation equation (1) is (Parizot 1999) : N i (E, t) = 1 \|Ė i (E)\| +∞ E Q i (E 0 , t − τ i (E 0 , E)) × exp − E E0 dE ′ E i (E ′ )τ tot,i (E ′ ) dE 0 ,(20) where τ i (E 0 , E) = E E0 dE ′ E i (E ′ ) .(21) This solution, however, only considers the timedependence of the injection function, Q(E, t), and not that of the conditions of propagation, namely the energy losses and the destruction time. Now it is clear that the adiabatic losses do depend on time as well as the ionisation losses and the nuclear destruction time, through the chemical composition within the SNR. One then needs to divide the whole process into sufficiently short phases so that these parameters stay approximately constant during each phase, and put together the solutions (20) for each phase in a proper way (for details, see Parizot, 1999). For the present calculations, it proved sufficient to divide the Sedov-like phase into 15 successive phases. In the case of our second injection function, Eq. (17), corresponding to the reverse shock acceleration, the time delta function allows us to integrate Eq. (20) to obtain : N i (E, t) = \|Ė i (E in )\| \|Ė i (E)\| n iQ (E in ) exp − E Ein dE ′ E i (E ′ )τ tot,i (E ′ ) ,(22) where E in (i, E, t) is the solution of : E Ein dE ′ E i (E ′ ) = t − t SW .(23) In other words, E in is the energy at which a particle of species i must have been accelerated at time t SW in order to have slowed down to energy E at time t. Similarly, the Woosley and Weaver, 1995), and the ambient density is n 0 = 10 cm −3 . exponential factor in Eqs (20) and (22) is nothing but the survival probability of a particle i from its injection at energy E 0 (or E in ) to the current energy, E. The above solutions allow us to calculate the EP distribution function for both of our injection functions, Eqs (16) and (17)-(19). Equation (6) can then be used to compute the LiBeB production rates at any time after the beginning of acceleration, at t SW . The results are presented in the following section. The results LiBeB production by the EPs from the forward shock The results we show in this section are obtained with the SN explosion models calculated by Woosley and Weaver (1995). We use their models Z, U and T, corresponding to stars with initial metallicity Z = 0, 10 −4 Z ⊙ , and 10 −2 Z ⊙ , respectively, and keep the same labels as the authors to refer to specific models (e.g. model U15A corresponds to a star of 15 M ⊙ with 10 −4 Z ⊙ initial metallicity and a standard explosion energy of ≈ 1.2 10 51 erg). We adopt the value θ 1 = 0.1 throughout, on the understanding that all the spallation rates are merely proportional to this parameter. In Fig. 1, we show the typical evolution of the spallation rates for Be production as a function of time, for a SN exploding in a medium with mean density n 0 = 10 cm −3 . The main contribution is seen to come from reaction p+ 16 O, which is due to the low C/O abundance ratio in the SN ejecta. For reactions involving alpha particles, this deficiency of carbon as compared to oxygen is compensated by a greater spallation efficiency. The general shape of the curves is easily understood if one refers to Eq. (6) and to the analysis of the preceding section. Indeed, the spallation rates are basically the product of the relevant cross section by the spectral density of energetic protons, N p (t), and the number density of Oxygen within the SNR. Now the latter is subject to dilution by the swept-up metal-free gas, and therefore decreases as R −3 , or R −3 SW (t/t SW ) −6/5 , while N p (t) is merely the time integral of the injection function, Eq. (16) (at least as long as one can neglect the energy losses). We thus find N p (t) ∝ ln(t/t SW ), and the spallation rates : dN Be dt ∝ R −3 SW (t/t SW ) −6/5 ln t t SW ,(24) which fits very well the curves in Fig. 1. Differentiating the above expression, we find the maximum production rates to occur at t = e 5/6 t SW ≈ 2.3 t SW , which expresses the best compromise between Oxygen dilution in the SNR and a sufficient injection of EPs since the onset of the acceleration process. This behavior can be further observed on Fig. 2 where we plot the total production rates of Be as a function of time after explosion, for different values of the ambient density, ranging from 1 to 10 4 cm −3 . The shortening of the Sedov-like phase already mentioned is clearly apparent on the figure, as is the behavior of t SW ∝ n −1/3 0 and t end ∝ n −3/4 0 . The calculations also confirm that the position of the maximum is always at t max ≈ 2.3 t SW , although at the highest densities, this is very close indeed to the end of the adiabatic expansion phase, when the confinement of the EPs ceases and the whole process stops. The position of the maximum then varies as n −1/3 0 , while its height, obtained by replacing t by t nax in Eq. (24), is proportional to R −3 SW , and thus n 0 . In Fig. 3 we show the evolution of the production rates for the five light element isotopes, either taking and not taking the adiabatic losses into account. The behavior of 6 Li and 7 Li is different from that of the other isotopes, because lithium is mainly produced through α + α reactions, as shown in Fig. 4, and these reactions are not sensitive to the dilution of the SN ejecta by the ambient material. The evolution of Li production rates therefore reflects directly the evolution of the EP fluxes. As just stated, this would be a pure logarithm if one could neglect the energy losses. It turns out that the adiabatic losses dominate the Coulombian losses for any reasonable ambient density. To see how they influence the EP fluxes, let us re-write Eq. (1) in the form : ∂ ∂t N (E, t) = Q(E, t) − ∂ ∂E (Ė ad (E, t)N (E, t)),(25) where we dropped the destruction and second order terms. At energies of a few tens of MeV/n, Eq. (5) simplifies to give the expression for adiabatic losses : E ad (E, t) = − 6 10 E t(26) Replacing in Eq. (25), we obtain : ∂ ∂t N = Q + 6 10 t ∂ ∂E (EN ) = Q − 6(α − 1) 10 t N,(27) where we recognized that a power-law for the injection function Q, with spectral index −α (Q = Q 0 E −α /t), translates into a power-law for the EP spectral density N with the same index : N = N 0 (t)E −α . This is a consequence of the proportionality between the energy loss rate and the energy itself. The equation for N 0 is then straightforward : ∂ ∂t N 0 = Q 0 t − 6(α − 1) 10 t N 0 ,(28) from where we see that instead of the logaritmic increase N (E, t) = Q 0 E −α ln(t/t SW ) prevailing in the absence of energy losses, a steady-state value should be reached (if the Sedov-like phase last long enough) with : N 0 = 10Q 0 6(α − 1) .(29) So the adiabatic losses are important when both terms in the right hand side of Eq. (28) are of the same order, that is (evaluating the second term from its 'no-loss value', and using α = 1.5 for the low-energy part of the spectrum) : ln t t SW ≈ 6 10 (α − 1),(30) or t < ∼ t SW e 6 10 (α−1) ≈ 1.35 t SW . This result is in very good agreement with the numerical results shown in Fig. 3. Likewise, the gap between the calculations with adiabatic losses turned on or off is increasing only logarithmically with time, so that the difference is rather small, even at the end of the Sedov-like phase. We find total Be production only a few tens of percent higher if we drop the adiabatic losses, and the difference even falls to zero when higher ambient densities are considered. This is of course because the Sedov phase is then considerably shortened. Although Fig. 1, 2 and 3 help us to clarify the dynamics of the process and understand the role of the different parameters, only the total, integrated light elements production is actually relevant to the Galactic chemical evolution. We show in Fig. 5 the results of the integration of the Be production rates over the whole Sedov-like phase, for different SN explosion models, as a function of the ambient density. Except for the case of the Z30A model, we find that for a given mass of the progenitor the total Be yield is independent of the initial metallicity of the star (zero, 10 −4 or 10 −2 times solar). The very small production of Be obtained with the Z30A model is in fact due to a very small amount of Oxygen expelled by the supernova. A model with a higher explosion energy (Z30B) gives results closer to those of T30A and U30A. Although yields significantly different are obtained for different masses of the progenitor, due to different compositions and masses of the ejecta, it is clear from Fig. 5 that the total amount of Be produced by process 1 (forward shock) is much too low to account for the Be observed in metal-poor star. Indeed, the results obtained for a 15 M ⊙ star with ambient density n 0 = 1 cm −3 are about three orders of magnitude too low, for our choice of θ 1 = 0.1. This is in very good agreement with the analytical estimates presented in Paper I. Concerning the density dependence of the Be yields, the numerical results shown in Fig. 2 are also in good agreement with the analytical calculations. In particular, the yields increase with ambient density and reach a maximum at about a few 10 3 cm −3 , above which the Sedov-like phase becomes extremely short, and even vanishes for high mass progenitors (implying large ejected masses). Using Eq. (7) and (8) we can write this limiting density as : n lim ≃ (4 10 4 cm −3 ) M ej 10 M ⊙ −2 E SN 10 51 erg 3/2 . (32) LiBeB production by the EPs from the reverse shock We now turn to the results obtained for the second mechanism, in which the SN ejecta are accelerated at the reverse shock at the onset of the Sedov-like phase. The 6 Li and 9 Be production rates are shown on Fig. 6 as a function of time, with and without adiabatic losses, for an ambient density of 10 cm −3 and a progenitor corresponding to the U15A model of SN. As can be seen, the Be production rates are strongly dominated by inverse spallation reactions, i.e. reactions in which the projectile is the heavier nuclei. Moreover, since the abundance of C and O in the target suffers from dilution by ISM gas, the direct-to-inverse spallation efficiency ratio keeps decreasing during the Sedov-like phase. At the end of it, as already discussed, the direct reactions stop, while inverse ones are not affected. In Figs. 6b and 6d, the adiabatic losses have not been taken into account. The decrease of the direct spallation rates is thus due only to dilution, and we obtain the expected power law in R −3 , or t −6/5 . In the meanwhile, the inverse spallation rates are almost constant, as the Sedov-like phase is much shorter than the time-scale for coulombian losses. This time-scale can literally be read from the figure. It is of order a few times 10 5 years for this set of parameters. Note however that the energy loss time-scale actually depends on the species and energy of the particle. Accordingly, what is observed on the spallation rates is in fact a mean coulombian timescale, averaged over the EP energy spectrum, and more precisely the part of this spectrum which stands above the energy threshold of the cross-sections. This explains the slight variation observed for the different spallation channels. It is worth emphasizing that the time-scales that we obtain are much shorter than the confinement time-scales inferred from cosmic-ray propagation theories. This indicates that the leakage of the EPs out of the Galaxy has negligible influence on the spallation yields, and justifies our choice of neglecting it. Even for an ambient density of 1 cm −3 , the bulk of the light element production is contributed by nuclear reactions occuring within a few million years after the SN explosion, which is to be compared with Galactic confinement times of order a few 10 7 years. Comparing Fig. 6a with Fig. 6b (or Fig. 6c with Fig. 6d), we can see the influence of the adiabatic losses on the nuclear rates. For inverse spallation reactions, we observe an almost perfect power law decrease, with logarithmic slope ∼ 0.4, in very good agreement with the value derived in Paper I. Indeed, the analytic treatment led us to expect spallation rates proportional to R −3/4 , or equivalently t −3/10 . The slightly quicker decrease found in the numerical results is due to the contribution of the coulombian losses (whose effect is also visible on Figs. 6b and 6d), and to the shape of the spallation cross-sections close to their threshold. Likewise, the time evolution of di- Fig. 6. Process 2 (and 3) 6 Li and 9 Be production rates in numbers of nuclei per second through different spallation reactions as a function of time after the SN explosion. The SN model used is U15A and the ambient density is n 0 = 10 cm −3 . On the left, the calculations include adiabatic losses (U15A+); on the right, they do not (U15A-). rect spallation reactions is also very close to a power law, with logarithmic slope of ∼ 1.6 ∼ (1.2 + 0.3), as expected. As noted above, however instructive the examination of the spallation rates evolution may be, they cannot be directly compared to any observational data. We therefore calculated the (more relevant) integrated yields for different models corresponding to initial metallicities Z = 0 (models Z), Z = 10 −4 Z ⊙ (models U), and Z = 10 −2 Z ⊙ (models T), and normalized them to the expected value, i.e. to the value required to explain the abundances observed in the metal-poor stars. Consequently, normalized yields respectively lower and higher than 1 are equivalent to under-and over-production of Be. A few words of explaination are however required as how the normalization is actually performed. The only assumption here is that the Galactic Be evolution is primary relative to both Fe and O. This means that the Be/Fe and Be/O ratios are approximately constant in metal-poor stars (as is consequently the O/Fe ratio). Then each supernova must lead, on average (over the IMF), to the same Be/Fe and Be/O ratios as those observed. These are thus the values we use to normalize our results. Now, as the Fe and O yields calculated by Woosley and Weaver (1995) are different for each of their SN models, we applied our normalization model by model and obtained the results shown in Fig.7, as a function of the mass of the SN progenitor, for different initial metallicities. As discussed earlier, the approximate constancy of the Be/Fe ratio is well established observationally, over two orders of magnitude in metallicity, from Fe/H < ∼ 10 −3 to 10 −1 times the solar value. On the other hand, we still lack similar measurements of the Be/O ratio in stars with O/H < ∼ 10 −2 times the solar value, while the trend at higher metallicity seems to favour a slightly increasing Be/O, if one is to believe the recent observations by Israelian et al. (1998) and Boesgaard et al. (1998) (see also Fields and Olive, 1999). To this respect, it might seem that our normalization based on the primary behavior of Be is better justified for comparison to Fe than to O. In fact, it is just the opposite. Indeed, the models we are investigating (processes 1 and 2) predict a linear increase of Be as compared to O, whatever the Fe evolution may be. As already noted in Paper I, Be and Fe actually have no direct physical link, as the spallation reactions involve only C and O (and in fact mainly Oxygen, as we have shown; see Figs. 1 and 6). Both processes 1 and 2 could therefore account, in principle, for any value of the Be/Fe ratio, provided we can choose the Iron yield of the SNe Woosley and Weaver, 1995). The yield ratios are normalized to the value required by the observations as explained in the text. (this is however not the case, and even if the SN explosion models entail possibly large uncertainties, the claim for and use of a constant Be/Fe ratio is in fact justified by the observations themselves). On the contrary, the Be/O ratio is entirely determined, at a fundamental level, by the processes we investigate here. A higher mass of Oxygen ejected by the supernova would indeed imply a larger Be yield as well, and conversely. Except for a few 'irregular models' which we shall discuss shortly, Fig. 7 shows that the Be yields obtained by process 2 are significantly smaller than the required values, by about two orders of magnitude when comparison is made with Fe, and roughly one order of magnitude when comparison is made with O. This is again in good quantitative agreement with the results of Paper I, so that we confirm that the processes considered here cannot be responsible for the majority of the Be production in our Galaxy. This conclusion has important implications which have been analysed in Paper I and will be summarized below. Let us now comment the figures in greater detail. For each series of explosion models (Z, U and T), corresponding to different initial metallicities, Woosley and Weaver (1995) have calculated the yields of a number of elements for progenitors of different masses ranging from 12 to 40 M ⊙ . For the more massive progenitors, they found that the yields of Fe, notably, greatly depended on the mass-cut, which in their models is directly linked to the explosion energy. For example, a 30 M ⊙ model with a 'standard' explosion energy of 1.2 10 51 erg ejects virtually no Iron at all. Explosion energies greater than the standard value have therefore been explored, leading to higher Fe yields for the most massive stars. We use the same notations as in Woosley and Weaver (1995), i.e. models A, B and C correspond to increasing explosion energies of order 1.2, 2 and 2.5 10 51 ergs, respectively. In fact, the explosion energy has been adjusted for higher mass progenitors in an ad hoc way in order to obtain approximately the 'standard' Fe yield of ∼ 0.1 M ⊙ . Therefore, passing from model A to model B, and finally to model C as the progenitor's mass increases, amounts to ensure that the SN yields of both O and Fe do not vary in dramatic proportions. This is the reason why the curves for models A, B and C connect so smoothly on Figs. 7a-f. In particular, it is worth emphasizing that the results which we obtain for this 'mixed model' (A, then B, then C), are remarkably similar whatever the initial metallicity and mass of the progenitor may be. We find in this way [Be/Fe] ∼ 0.01 and [Be/O] ∼ 0.1, where the brackets mean that the yield ratios have been normalized to the required value as described above. It should be clear, however, that there is no special reason why we should increase the explosion energy for the most massive SN progenitors. In fact, the great sensitivity of the Fe yield to the explosion energy for these stars mostly means, to our opinion, that the SN explosion models are still unable to predict reliable yields (especially at the lowest metallicities; see the huge differences between the models in Fig. 7a). For instance, if we adopt the standard explosion energy (models A), then it is clear from Figs. 7a,c,e that the observed Be/Fe ratio is very easy to reproduce if one assumes that only the most massive stars formed in the early Galaxy. The reason for this success, however, is not that the massive stars (indirectly) produce a lot of Be, but rather that they produce extremely little Fe. In this case, then, a serious Fe underproduction problem will be encountered by the chemical evolution models, so that the high value of the [Be/Fe] should be regarded as somewhat artificial, and rather irrelevant to the question of Be production in the Galaxy. Moreover, such a behaviour is not expected to be found in the curves showing the Be production as compared to the Oxygen. Indeed, as already alluded to, if a particular SN model happens to not eject any substantial amount of O, then In these cases, indeed, the O yield becomes much lower than the C yield, so that the Be production is actually dominated by spallation reactions involving C. Consequently the Be yield is still quite substantial, while the O yield is very low, which brings about a situation very similar to that encountered with Fe. However that may be, even if we trust the low (or even extremely low) Fe and O yields obtained from models A for high mass progenitors, the contribution of these high mass SNe still has to be weighted by their frequency among the type II SNe. In Figs. 8 and 9 we show the normalized [Be/Fe] and [Be/O] ratios, after averaging over a power-law IMF with logarithmic slope x ranging from 0.5 to 3. This allows us to explore the influence of varying the weight of the more efficient high mass stars relatively to the lower mass SN progenitors. A low IMF slope (towards 0.5) strongly favours high mass star formation, and is therefore expected to lead to a higher [Be/Fe] ratio than a high IMF slope (towards 3). This qualitative behavior is indeed observed on Figs. 8 and 9, but it can be seen that the effect is actually quite weak, even for such a large range of IMFs. Note that we used 'IMFs by number' (of stars), and not 'IMFs by mass', so that the Salpeter IMF corresponds to x = 2.35 in our notations. This means that a slope as low as x = 0.5 corresponds to an IMF in which more mass is locked in high mass than in low mass stars. Even for such an IMF, the Be/Fe ratio obtained is still less than a few percent of the observed value. Comparing Be to O, it is shown in Fig. 9 that the IMF slope has almost no influence on the normalized [Be/O] ratio, which is a consequence of the strong physical link between the ejected Oxygen and the Be production, as discussed above. We have also shown, in Figs. 8 and 9, the results obtained without including adiabatic losses (dashed lines). Both [Be/Fe] and [Be/O] ratios are then found to be higher by a factor of about 3 to 4, which is in good quantitative agreement with the analytical calculations of Paper I (see Fig. 5 there). This result has two simple, but important implications. First, it points out the necessity of including the adiabatic losses in the calculations (unless explicitely shown that they do not apply), and therefore of using time-dependent models. Second, it indicates that a model in which the EPs do not suffer adiabatic losses has more chance to succeed in accounting for the observed amount of Be in the halo stars. Conclusion In conclusion, we have calculated the Be production associated with the explosion of a supernova in the ISM, using a time-dependent model, and confirmed the results of Parizot and Drury (1999) stating that isolated SNe cannot be responsible for the Be observed in the metal-poor stars of the Galactic halo. All the qualitative and quantitative features of the two processes investigated (i.e. acceleration of particles at the forward and the reverse shocks of an isolated supernova) have been found to conform to the analytical expectations. This includes the dependence of the Be yields on the ambient density, the evolution of the spallation rates during and after the Sedov-like phase of the SNR expansion, and the influence of the adiabatic energy losses. The implications of these results for the Galactic chemical evolution of the light elements have been discussed in detail in Paper I. We shall only stress here that it proves very hard for theoretical models to produce the required amount of Be (and similarly 6 Li and B) by isolated SNe, according to conventional shock acceleration theory. Indeed, the processes that we investigated tend to optimize the spallation efficiency, in that they either accelerate the freshly synthesized C and O or confine the EPs in an environement much richer in C and O than the surrounding ISM at this stage of chemical evolution. Shock acceleration efficiencies of order 10 percent are also about the maximum that can be expected of any acceleration process. Thinking of a process involving more energy than that released by a SN and/or a higher concentration of C and O than within a SNR is rather challenging. One promising alternative, however, seems to be a model in which the SNe act collectively, rather than individually, as in the processes investigated in this paper. The idea is that most of the massive stars in the Galaxy are formed in associations (Melnik and Efremov, 1995) and generate superbubbles which expand owing to the cumulated energy released by several consecutive supernovae. This energy leads to strong magnetic turbulence within the superbubble, which is thought to accelerate particles in a very efficient way, according to a specific model developed by Bykov and Fleishman (1992). The interesting feature is that the interior of the superbubble is enriched by significant amounts of C and O previously ejected by stellar winds and SN explosions, so that the accelerated particle should have a primary composition Higdon et al., 1998;Parizot and Knoedlseder, 1998) and therefore be very efficient in producing Be. Moreover, the average energy imparted to the EPs by each supernova is directly related to the explosion energy, instead of only the energy in the reverse shock, as in the process 2 investigated here. Indeed, either that the particles are accelerated directly by the forward shock or that the explosion energy first turns into turbulence and a distrubution of weak secondary shocks (this will be investigated in a forthcoming paper), the total energy imparted to the EPs is expected to be about ten times larger than that assumed for process 2 above (say 10 % of the explosion energy, instead of the ∼ 1 % implied by the use of the reverse shock energy). Further considering that the adiabatic losses would not apply in such a case, we predict an overall factor of about 10 to 30 on the Be yields, depending on the mixing of the ejecta with non enriched ISM within the superbubble. According to the results presented in this paper, this would be enough to account for the [Be/O] ratio observed in the metal-poor halo stars. Apart from the problem of light element production in the early Galaxy, our calculations have shown that the situation is somewhat different whether we compare Be to Fe or O. This obviously indicates that the Galactic evolution of Fe and O are mutually inconsistent, if one uses the yields of Woosley and Weaver (1995), so that a revision of the SN models should be considered. A similar conclusion has been pointed out by Fields and Olive (1999), who ob-served that these theoretical yields cannot reproduce the O/Fe slope measured in the abundance diagram. Since the Be problem is found to be less serious when comparison is made with O rather than Fe, we suggest that the Fe rather than the O yields may be responsible for the Fe-O problem. Further observational and theoretical work are however needed to reach a convincing conclusion. Fig. 2 . 2Process 1 total 9 Be production rates as a function of time after the explosion of SN model U15A, for different ambient densities. Each curve starts shortly after the sweep-up time and ends at the adiabatic time, marking the end of the Sedov-like phase. Fig. 3 . 3Process 1 production rates of the five light element isotopes as a function of time after the explosion of SN model U15A, in a medium of density n 0 = 10 cm −3 . Results are shown for calculations taking adiabatic losses into account (full lines) as well as ignoring them (dashed lines). Fig. 4 . 4Process 1 6 Li production rate in numbers of nuclei per second through different spallation reactions as a function of time after the SN explosion. The SN model used is U15A and the ambient density is n 0 = 10 cm −3 . Fig. 5 . 5Integrated process 1 Be yields for different SN models as a function of ambient density. For the models with progenitor masses of 30 M ⊙ and a density higher than a few 10 3 cm −3 , the sweep-up time t SW is greater than the adiabatic time t end , so that the Sedov-like phase does not exists. Fig. 7 . 7Normalized process 2 [Be/Fe] and [Be/O] yield ratios, as a function of the mass of the progenitor. Models Z, U and T correspond to the indicated initial metallicity of the stars. Models A, B and C correspond to different explosion energies (see text and Fig. 8. Normalized Be/Fe ratio calculated from SN models ZA, UA, and TA, averaged on the IMF, as a function of the IMF logarithmic slope. (Salpeter slope is 2.35). Models labeled with a '+' include adiabatic losses; those labeled with a '-' (dashed lines) do not.Fig. 9. Normalized Be/O ratio calculated from SN models ZA, UA, and TA, averaged on the IMF, as a function of the IMF logarithmic slope. (Salpeter slope is 2.35). Models labeled with a '+' include adiabatic losses; those labeled with a '-' (dashed lines) do not.it will not lead to any significant Be production either, leaving the [Be/O] ratio virtually unchanged. This can be checked on Figs. 7b,d,f, where all the models are shown to give approximately the same results. The only exceptions arise at low metallicity for models A and can be easily understood.10 -2 10 -1 0,5 1 1,5 2 2,5 3 [ 9 Be/ 56 Fe] IMF slope ZA+ ZA- UA+ UA- TA+ TA- 10 -1 10 0 0,5 1 1,5 2 2,5 3 [ 9 Be/ 16 O] IMF slope ZA+ ZA- UA+ UA- TA+ TA- Etienne Parizot & Luke Drury: Spallative Nucleosynthesis in Supernova Remnants Acknowledgements. This work was supported by the TMR programme of the European Union under contract FMRX-CT98-0168. A M Boesgaard, J R King, C P Deliyannis, S S Vogt, G D Fleishman, press Bykov A. M. 255269Boesgaard A. M., King J. R., Deliyannis C. P., Vogt S. S., 1998, in press Bykov A. M., Fleishman G. D., 1992, MNRAS, 255, 269 . D K Duncan, D L Lambert, M Lemke, ApJ. 401584Duncan D. K., Lambert D. L., Lemke M., 1992, ApJ, 401, 584 . D K Duncan, F Primas, L M Rebull, A M Boesgaard, C P Deliyannis, L M Hobbs, J R King, S G Ryan, ApJ. 488338Duncan D. K., Primas F., Rebull L. M., Boesgaard A. M., Deliyannis C. P., Hobbs L. M., King J. R., Ryan S. 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[ "ON TOPOLOGICALLY CONTROLLED MODEL REDUCTION FOR DISCRETE-TIME SYSTEMS", "ON TOPOLOGICALLY CONTROLLED MODEL REDUCTION FOR DISCRETE-TIME SYSTEMS" ]	[ "Fredy Vides " ]	[]	[]	In this document the author proves that several problems in datadriven numerical approximation of dynamical systems in C n , can be reduced to the computation of a family of constrained matrix representations of elements of the group algebra C[Z/m] in C n×n , factoring through the commutative algebra Circ(m) of circulant matrices in C m×m , for some integers m ≤ n.The solvability of the previously described matrix representation problems is studied. Some connections of the aforementioned results, with numerical analysis of dynamical systems, are outlined, a prorotypical algorithm for the computation of the matrix representations, and some numerical implementations of the algorithm, will be presented.Date: June 13, 2019. 2010 Mathematics Subject Classification. 93B28, 47N70 (primary) and 93C57, 93B25 (secondary).	null	[ "https://arxiv.org/pdf/1905.03910v3.pdf" ]	150,374,045	1905.03910	4c922eb842690f206b9463238d11bf2ca53c67bb	ON TOPOLOGICALLY CONTROLLED MODEL REDUCTION FOR DISCRETE-TIME SYSTEMS Fredy Vides ON TOPOLOGICALLY CONTROLLED MODEL REDUCTION FOR DISCRETE-TIME SYSTEMS In this document the author proves that several problems in datadriven numerical approximation of dynamical systems in C n , can be reduced to the computation of a family of constrained matrix representations of elements of the group algebra C[Z/m] in C n×n , factoring through the commutative algebra Circ(m) of circulant matrices in C m×m , for some integers m ≤ n.The solvability of the previously described matrix representation problems is studied. Some connections of the aforementioned results, with numerical analysis of dynamical systems, are outlined, a prorotypical algorithm for the computation of the matrix representations, and some numerical implementations of the algorithm, will be presented.Date: June 13, 2019. 2010 Mathematics Subject Classification. 93B28, 47N70 (primary) and 93C57, 93B25 (secondary). Introduction In this document we will study discrete-time dynamical systems determined by the pair (Σ, {Θ t }), with Σ ⊆ C n , and where {Θ t \|t ∈ Z} is a family of continuous functions from Σ to Σ, such that Θ t • Θ s = Θ t+s and Θ 0 = id Σ , for every pair of integers t, s ≥ 0. We will prove that several important problems in numerical analysis and datadriven discovery of discrete-time dynamical systems of the form (Σ, {Θ t }) in C n , can be reduced to the computation of a family of discrete-time transition matrices {F t } m−1 t=1 ⊆ ρ m (C[Z/m]) ⊂ C n×n of rank at most m with m ≤ n, for some matrix representation of the group algebra C[Z/m], together with two matriceŝ K,T ∈ C n×n of rank at most m, that are related to some evolution history data {v 1 , . . . , v m } ⊆ Σ ⊆ C n , (approximately) generated by the dynamical (Σ, {Θ t }), by the equations v t+1 =KF tT v t for 1 ≤ t ≤ m − 1. We will also show that each variation of the problem corresponding to the computation of the aforementioned transition matrices, can be reduced to solving a constrained matrix representation problem of the group algebra C[Z/m] in C n×n , factoring through the commutative algebra Circ(m) of circulant matrices in C m×m , for some integers m ≤ n. The motivation for the theoretical and computational machinery presented in this document came from some questions raised by M. H. Freedman along the lines of [2], concerning to the implications in linear algebra and matrix computations of the so called Kirby Torus Trick, presented by R. Kirby in [3]. We study the solvability of the previously described matrix representation problems. Some connections of the aforementioned results, with numerical analysis of dynamical systems, are outlined, a prorotypical algorithm for the matrix representation computations, and some numerical implementations of the algorithm will be presented. Preliminaries and Notation Given two positive integers p, q such that p ≥ q, we will write p mod q to denote the integer r, such that p = mq + r for some integer m. We will write Z + 0 to denote the set {z ∈ Z\|z ≥ 0}. Given k ∈ Z + 0 , we will write Z/k to denote the (additive) cyclic group Z/kZ = {0,1,2, . . . , k − 1}. Given any matrix X ∈ C m×n , we will write X ij to denote the ij entry of X, and we will write X * to denote its conjugate transpose X = (X ji ) ∈ C n×m . We will identify elements in C n with elements in C n×1 . As a consequence of this identification, given x, y ∈ C n , y * x ∈ C will determine the Ecuclidean inner product x, y in C n , while xy * ∈ C n×n will determine a rank-one matrix in C n×n . In this document we write 1 n and 0 m×n to denote the identity and zero matrices in C n×n and C m×n , respectively. We will write 0 n to denote the zero matrix in C n×n . A set of m elements v 1 , . . . , v m ∈ C\{0} is said to be an orthogonal m-system if (2.1) v * j v k = δ j,k v * j v k , for 1 ≤ j, k ≤ m. From here on, we will write δ k,j to denote the Kronecker delta defined by (2.2) δ k,j = 1, k = j, 0, k = j. We say that the set of vectors v 1 , . . . , v m ∈ C\{0} is an orthonormal m-system if the vectors v j satisfy (2.1) and in addition (2.3) v j 2 = v * j v j = 1, for 1 ≤ j ≤ m. We will writeê j,n to denote the element in C n×1 represented by the expression. (2.4)ê j,n =           δ 1,j δ 2,j . . . δ k,j . . . δ n,j           Eachê j,n can be interpreted as the j-column of 1 n = [ê 1,nê2,n · · ·ê n,n ]. A matrix B ∈ C n×n is said to be normal if BB * = B * B, a matrix A ∈ C n×n is said to be Hermitian if X * = X, and a hermitian matrix P is said to be an orthogonal projection or just a projection, if P 2 = P = P * . A matrix X ∈ C n×n is said to be unitary if X * X = XX * = 1 n . A matrix A ∈ C n×m is said to be positive if there is a matrix B ∈ C n×n such that A = B * B, we also write A ≥ 0 to indicate that A is positive. We will denote by U(n) and P(n), the sets of unitaries and positive matrices in C n×n , respectively. Given a matrix W ∈ C m×n , we write Ad[W ] to denote the linear map from C n×n to C m×m , defined by the operation Ad[W ](X) = W XW * for any X ∈ C n×n . We say that a matrix A ∈ C n×n is invertible if there is one matrix B ∈ C n×n such that AB = BA = 1 n . We will write GL(n) to denote the set of invertible matrices in C n×n . Given a matrix X ∈ C n×n , we will write σ(X) to denote the spectrum of X, that is the set {z ∈ C\|X − z1 n / ∈ GL(n)}. From here on we write · 2 to denote the Euclidean norm in C n defined by the operation x 2 = √ x * x for any x ∈ C n . In this document we will write · to denote the spectral norm in C n×n defined by the opration X = sup x 2=1 Ax 2 , for any A ∈ C n×n . Definition 2.1. A linear map ϕ : C n×n → C m×m is said to be a completely positive (CP) linear map if ϕ(A) ≥ 0 for every positive A ∈ C n×n , and if it has a Choi's representation of the form ϕ = k j=1 Ad[W j ] for some matrices W j ∈ C m×n . We will write CP (n, m) to denote the set of completely positve linear maps from C n×n to C m×m . Given any X ∈ C n×n and any p ∈ C[z] determined by the formula p(z) = a 0 + a 1 z + a 2 z 2 + · · · + a n z n , we will write p(X) to denote the matrix defined by the expression p(X) = a 0 1 n + a 1 X + a 2 X 2 + · · · + a n X n . Given X ∈ C n×n , we will write C[X] to denote the commutative algebra determined by the set {p(X)\|p ∈ C[z]} = span C {1 n , X, X 2 , . . . , X n−1 }, with respect to the usual addition and multiplication operations in C n×n . We have that in fact C[X] is an algebra since, X m ∈ span C {1 n , X, X 2 , . . . , X n−1 } for every integer m ≥ n, as a consequence of the Cayley-Hamilton Theorem, and that C[X] is commutative as a consequence of the identity (aX k )(bX j ) = abX k+j = baX j+k = bX j aX k , that holds for each a, b ∈ C and for each pair of integers k, j ≥ 1. We will write Circ(k) to denote the commutative algebra of k × k Circulant matrices that is defined by the expression: (2.5) Circ(k) = C[C k ] = {p(C k )\|p ∈ C[z]} where C k is the cyclic permutation matrix defined as follows. (2.6) C k = 0 1×(k−1) 1 1 k−1 0 (k−1)×1 Given a matrix X ∈ C n×n , we write Z(X) to denote the commutant set of X defined by the expression Z(X) = {Y ∈ C n×n \|XY = Y X}. Given a finite group G, we will write C[G] to denote the group algebra over C. For a finite group G In this document we will focus on group algebra representations of C[G] determined by algebra homomorphisms of the form ρ : C[G] → C n×n , g∈G c g g → g∈G c g ρ(g), such that ρ\| G is a group representation of G in GL(n). We will say that an algebra representation ρ : C[G] → C n×n is unitary if ρ\| G (G) ⊂ U(n). Given a discrete-time dynamical system (Σ, {Θ t }), if there is an integer T > 0 such that Θ t+T = Θ t for each every t ∈ Z + 0 , we say that (Σ, {Θ t }) is a discrete-time T -periodic dynamical system. Given a discrete-time dynamical system (Σ, {Θ t }), a set of vectors H[Σ, m] = {v 1 , . . . , v m } ⊆ Σ will be called a m-system of history vectors for (Σ, {Θ t }), if they satisfy the relations v k+1 = Θ 1 (v k ) = Θ k (v 1 ) for 0 ≤ k ≤ m − 1. Given δ, ε > 0, we will say that a discrete- time dynamical system (Σ, {Θ t }) is (T, δ, ε)-almost-periodic dynamical system, if there is a T -periodic discrete-time dynamical system (Σ, {Θ t }) such that for each x ∈ Σ there isx ∈Σ such that x −x ≤ δ, and Θ t (x) −Θ t (x) 2 ≤ ε for each t ∈ Z + 0 and every (x,x) ∈ Σ ×Σ such that x −x ≤ δ. Topological Control Method (TCM) 3.1. Switched Closed Loop Reduced Order Models SCL-ROM. In this section we will stablish the notion of S 1 topological control considered for this study. Given a discrete-time dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , and a m- system of history vectors H[Σ, m] = {v 1 , . . . , v m } for (Σ, {Θ t }), we say that (Σ , {Θ t } , H[Σ, m]) is topologically controlled by a topological manifold M ⊆ C or M-controlled, if there is a matrix Z ∈ C n×n with σ(Z) ⊆ M, an algebra homomorphism ϕ : C[Z] → C n×n , a family of polynomials {f 0 , . . . , f m−1 }, and two projections K, T ∈ C n×n such that Θ k (v 1 ) = Kϕ(f k (Z))T v 1 , for each k ≥ 0. We will call the 6-tuple (M, Z, K, T, ϕ, {f t }) a topological control for (Σ, {Θ t }, H[Σ, m]). Given ε > 0 and manifold M ⊆ C, and a M-controlled discrete-time dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , and a topological control (M, Z, K, T , ϕ , {f t }) for (Σ, {Θ t }, H[Σ, m]), we say that (M, Z, K, T, ϕ, {f t }) is a control of order k, if there is an integer k > 0, together with maps Π k ∈ Cp(n, k), Φ ∈ CP (k, n), such that Θ k (v 1 ) − Kϕ(f k (Z))Tṽ ≤ ε and ϕ(X) − Φ • Π k (X) ≤ ε, for each f k , each X ∈ C[Z], and someṽ ∈ C n . In this case we say that (Σ, {Θ t }, H[Σ, m]) is (M, k, ε)-controlled. Given ε > 0, a discrete-time dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , and a m-system of history vectors H[Σ, m] = {v 1 , . . . , v m } for (Σ, {Θ t }), we say that (Σ, {Θ t }, H[Σ, m]) is ε-approximately topologically controlled by Z/m or (Z/m, ε)-controlled, if there is a unitary representation ρ m : C[Z/m] → C m×m , an algebra homomorphism ϕ : C m×m → C n×n , a family of functions {f 0 , . . . , f m−1 }, and two projections K, T ∈ C n×n such that Θ k (v 1 ) − Kϕ • ρ m (f k (1))Tṽ ≤ ε, for each k ≥ 0 and someṽ ∈ C n , with1 ∈ Z/m.   v 1 = αT v 1 v t+1 =F tvt v t+1 = βKv t+1 , t ≥ 0 for some (α, β,K, {F t },T ) ∈ C × C × C n×n × C n×n × C n×n to be determined. Given ε > 0, the discrete-time system (3.1) is called a ε-approximate swithed closed-loop reduced order model (SCL-ROM) of (Σ, {Θ t }), if v t − Θ t (v 1 ) 2 ≤ ε for each t ∈ Z + 0 . 3.2. Some Connections with Dynamic Mode Decomposition. Given ε > 0, an integer T > 0, and a discrete-time system (Σ, {Θ t }) in C n . Let us consider the evolution history determined by the difference equations. (3.2) Σ : x t+1 = Θ t (x t ), x 0 = x , t ∈ Z + 0 for some x ∈ Σ. Given ε > 0. The computation of a SCL-ROM local ε-approximantΣ of Σ, with respect to some sampled-data history {x t } of Σ, is related to the computations of closed-loop matrix realizations H t ∈ C n×n and triples (P H (t), Q H (t), F H (t)) ∈ C n×n × C n×n × C n×n such that. (3.3)                (P H (t)F H (t) − H t P H (t))Q H (t) ≤ ε, P H (t)H t − H t P H (t) = 0, P H (t)x t − x t = 0, P H (t) 2 = P H (t) = P H (t) * , Q H (t) 2 = Q H (t) = Q H (t) * , F H (t) * F H (t) = F H (t)F H (t) * , 0 ≤ t ≤ T − 1 The matrices {H t } in (3.3) are determined by the connecting operator K for the sampled-data history {x t }, in the sense of [6, §2] and [4, §2.2], that satisfies the equations x t+1 = Kx t , 0 ≤ t ≤ T −1, In particular we will consider H t = K, t ∈ Z + 0 . The objective of topological control methods is to compute matrix realizations (F H (t), K,T (t)) ∈ C n×n × C n×n × C n×n such that: KF H (t)T (t)x t − y t ≤ ε for 0 ≤ t ≤ T − 1. Given ε > 0, an integer T > 0, a family of vectors {x t } T −1 t=0 in C n and matrices H t ∈ C n×n determined by the connecting operator K for {x t }. For any triples (P H (t), Q H (t), F H (t)) ∈ C n×n × C n×n × C n×n that satisfy (3.3). Let us consider the structured matrix equations in C 2n×2n determined by: (3.4)    (Q(t)X(t) − X(t)Q(t))P (t) = 0 2n Q(t) 4 = Q(t) 2 Q(t) 2 = ZQ(t) = (Q(t) 2 ) * , 0 ≤ t ≤ T − 1 where P (t) and Z have the form. P (t) = 1 n − P H (t) 0 n 0 n Q H (t) , Z = 0 n 1 n 1 n 0 n If we set.Q (t) = 0 n P H (t) P H (t) 0 n ,X(t) = H t 0 n 0 n F H (t) It can be seen that (Q(t),X(t)) ∈ (C 2n×2n )\{0 2n }) 2 are ε-approximate solvent pairs for (3.4) in the sense that, (Q(t)X(t) −X(t)Q(t))P (t) ≤ ε, Q(t) 4 = Q(t) 2 and Q(t) 2 = ZQ(t) = (Q(t) 2 ) * for 0 ≤ t ≤ T − 1. Given a positive integer T , together with ε, δ > 0. In this first paper on the subject of topological control of discrete-time systems, we focus on the computation of transition mappings F H (t), avoiding an explicit computation of the connecting operator K, instead we compute the family {F H (t)} by solving some constrained representation problems for C[Z/T ] for a given integer T > 0, restricting our attention to almost (T, ε, δ)-periodic discrete-time systems. Main Objectives. We prove the solvability of the problem of finding a εapproximate SCL-ROMΣ described by (3.1) for a discrete-time system (Σ, {Θ t }) with a m-system of history vectors H[Σ, m] = {v 1 , . . . , v m } ⊆ Σ, by computing matrix representations of C[Z/T ] such that the switching law of the family {F t } is controlled by some family {f t } ⊂ C[Z/T ] subject to almost time-perdiodic constraints on (Σ, {Θ t }). We will then design and implement a prototypical numerical algorithm that numerically solves the aforementioned problems. 3.4. Topologically Controlled Model Order Reduction. Let us consider an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. From here on, we will write C[v 1 \|v m ] to denote the matrix in C n×n defined by the equation. (3.5) C[v 1 \|v m ] = 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 1 n − P [v 1 \|v m ] where P [v 1 \|v m ] is the matrix in C n×n defined by the equation. (3.6) P [v 1 \|v m ] = m j=1 1 v * j v j v j v * j Lemma 3.1. Given an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. The matrix P [v 1 \|v m ] defined by (3.6) is an orthogonal projection such that P [v 1 \|v m ]v j = v j , 1 ≤ j ≤ m. Moreover, 1 n − P [v 1 \|v m ] is an orthogonal projection such that P [v 1 \|v m ](1 n − P [v 1 \|v m ]) = (1 n − P [v 1 \|v m ])P [v 1 \|v m ] = 0 n Proof. Given an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. For each 1 ≤ j ≤ m, let us set (3.7) V j = 1 v * j v j v j v * j , it is clear that each V j satisfies the relation, (3.8) V * j = V j , and we will also have that, (3.9) V 2 j = 1 v * j v j v j v * j 1 v * j v j v j v * j = v * j v j (v * j v j ) 2 v j v * j = 1 v * j v j v j v * j = V j . By (3.8) and (3.9), we will have that V * j = V j = V 2 j , and this implies that each V j is projection, and by orthogonality ot the system v 1 , . . . , v m , we have that (3.10) V j V k = 1 v * j v j v j v * j 1 v * k v k v k v * k = v * j v k v * j v j v * k v k v j v * k = δ j,k v * j v k v * j v j v * k v k v j v * k = δ j,k V j V k . This implies that the projections V 1 , . . . , V m are mutually orthogonal projections, and it can be seen that ( j V j ) * = j V * j = j V j . Moreover, (3.11) P [v 1 \|v m ] 2 = ( m j=1 V j ) 2 = m j,k=1 V j V k = m j=1 V 2 j = m j=1 V j = P [v 1 \|v m ]. We also have that. (3.12) (P [v 1 \|v m ]) * =   m j=1 V j   * = m j=1 V * j = m j=1 V j = P [v 1 \|v m ]. By (3.6) and by orthogonality of the system v 1 , . . . , v m , we will have that for 1 ≤ j ≤ m, P [v 1 \|v m ]v j = m k=1 1 v * k v k v k (v * k v j ) = m k=1 1 v * k v k v k (δ k,j v * k v j ) = 1 v * j v j v j (v * j v j ) = v j . (3.13) By (3.11) we will have that. (1 n − P [v 1 \|v m ]) 2 = 1 n − 2P [v 1 \|v m ] + P [v 1 \|v m ] 2 = 1 n − 2P [v 1 \|v m ] + P [v 1 \|v m ] = 1 n − P [v 1 \|v m ] (3.14) As a consequence of (3.12) we will also have that. (3.15) (1 n − P [v 1 \|v m ]) * = 1 n − P [v 1 \|v m ] * = 1 n − P [v 1 \|v m ] By (3.11) we will have that, (3.16) (1 n − P [v 1 \|v m ])P [v 1 \|v m ] = P [v 1 \|v m ] − P [v 1 \|v m ] 2 = P [v 1 \|v m ] − P [v 1 \|v m ] = 0 n and also that. (3.17) P [v 1 \|v m ](1 n − P [v 1 \|v m ]) = P [v 1 \|v m ] − P [v 1 \|v m ] 2 = P [v 1 \|v m ] − P [v 1 \|v m ] = 0 n This completes the proof. Lemma 3.2. Given an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. The matrices C[v 1 \|v m ] and P [v 1 \|v m ] defined by (3.5) and (3.6), respectively, satisfy the following conditions: • C[v 1 \|v m ]v j = v j+1 , 1 ≤ j ≤ m − 1, • C[v 1 \|v m ]v m = v 1 , • C[v 1 \|v m ]P [v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ], • P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ] = 1 v * m vm v 1 v * m + m−1 j=1 1 v * j vj v j+1 v * j , • C[v 1 \|v m ](1 n − P [v 1 \|v m ]) = (1 n − P [v 1 \|v m ])C[v 1 \|v m ] = 1 n − P [v 1 \|v m ]. Proof. Since C[v 1 \|v m ] is defined by (3.5), by lemma 3.1, we will have that for 1 ≤ j ≤ m: C[v 1 \|v m ]v j = 1 v * m v m v 1 (v * m v j ) + m−1 j=1 1 v * k v k v k+1 (v * k v j ) + v j − P [v 1 \|v m ]v j = δ m,j v * m v j v * m v m v 1 + m−1 j=1 δ k,j v * k v j v * k v k v k+1 + 0 = v j+1 , 1 ≤ j ≤ m − 1, v 1 , j = m. (3.18) By lemma 3.1, and by (3.5) and (3.6), on one hand we will have that, P [v 1 \|v m ]C[v 1 \|v m ] = P [v 1 \|v m ]( 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 1 n − P [v 1 \|v m ]) = 1 v * m v m (P [v 1 \|v m ]v 1 )v * m + m−1 j=1 1 v * j v j (P [v 1 \|v m ]v j+1 )v * j +P [v 1 \|v m ](1 n − P [v 1 \|v m ]) = 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 0 n = 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j (3.19) on the other hand we will have that. C[v 1 \|v m ]P [v 1 \|v m ] = ( 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 1 n − P [v 1 \|v m ])P [v 1 \|v m ] * = 1 v * m v m v 1 (v * m P [v 1 \|v m ] * ) + m−1 j=1 1 v * j v j v j+1 (v * j P [v 1 \|v m ] * ) +(1 n − P [v 1 \|v m ])P [v 1 \|v m ] * = 1 v * m v m v 1 (P [v 1 \|v m ]v m ) * + m−1 j=1 1 v * j v j v j+1 (P [v 1 \|v m ]v j ) * + +(1 n − P [v 1 \|v m ])P [v 1 \|v m ] = 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 0 n = 1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j (3.20) By combining (3.19) and (3.20) we have that (3.21) P [v 1 \|v m ]C[v 1 \|v m ] = C[v 1 \|v m ]P [v 1 \|v m ] Since P [v 1 \|v m ] is an orthogonal projection, we will also have that. (3.22) P [v 1 \|v m ]C[v 1 \|v m ] = P [v 1 \|v m ]P [v 1 \|v m ]C[v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ] By (3.22) and (3.19), it can be seen that C[v 1 \|v m ] can be represented in the form. (3.23) C[v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ] + 1 n − P [v 1 \|v m ] By lemma 3.1, we will have that, (1 n − P [v 1 \|v m ])C[v 1 \|v m ] = (1 n − P [v 1 \|v m ])P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ] +(1 n − P [v 1 \|v m ]) 2 = 0 n C[v 1 \|v m ]P [v 1 \|v m ] + 1 n − P [v 1 \|v m ] = 1 n − P [v 1 \|v m ] (3.24) and also that. C[v 1 \|v m ](1 n − P [v 1 \|v m ]) = P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ](1 n − P [v 1 \|v m ]) +(1 n − P [v 1 \|v m ]) 2 = P [v 1 \|v m ]C[v 1 \|v m ]0 n + 1 n − P [v 1 \|v m ] = 1 n − P [v 1 \|v m ] (3.25) This completes the proof. Lemma 3.3. Given an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. The matrix C[v 1 \|v m ] defined by (3.5) satisfies the equation (3.26) C[v 1 \|v m ] m = 1 n . Moreover, p(z) = z m − 1 is the minimal polynomial of C[v 1 \|v m ]. Proof. Given an orthogonal m-system v 1 , · · · , v m ∈ C n×1 \{0} with m ≤ n. By (3.5), (3.6), and by iterating on lemma 3.2, we will have that. C[v 1 \|v m ] m = C[v 1 \|v m ] m−1   1 v * m v m v 1 v * m + m−1 j=1 1 v * j v j v j+1 v * j + 1 n − P [v 1 \|v m ]   = 1 v * m v m (C[v 1 \|v m ] m−1 v 1 )v * m + m−1 j=1 1 v * k v j (C[v 1 \|v m ] m−1 v j+1 )v * j + C[v 1 \|v m ] m−1 (1 n − P [v 1 \|v m ]) = 1 v * m v m v m−1+1 v * m + m−1 j=1 1 v * j v j v (j+1+(m−1)) mod m v * j + 1 n − P [v 1 \|v m ] = 1 v * m v m v m v * m + m−1 j=1 1 v * j v j v j v * j + 1 n − P [v 1 \|v m ] = m j=1 1 v * j v j v j v * j + 1 n − P [v 1 \|v m ] = P [v 1 \|v m ] + 1 n − P [v 1 \|v m ] = 1 n . (3.27) By orthogonality properties we have that the system v 1 , . . . , v m is linearly independent. This fact combined with (3.27) implies that p(z) = z m − 1 is the minimal polynomial of C[v 1 \|v m ], since if C[v 1 \|v m ] k = 1 n for some k < m, we would have that v 1 = v k , which contradicts the linear independence of the system. Proof. Given an orthonormal m-system {v 1 , . . . , v m } ∈ C n \{0}, since v * j v j = 1 for each 1 ≤ j ≤ m, by (3.5) we will have that the matrix C[v 1 \|v m ] satisfies the equation, (3.28) C[v 1 \|v m ] = v 1 v * m + m−1 j=1 v j+1 v * j + 1 n − P [v 1 \|v m ] and also that the matrix P [v 1 , v m ] satisfies the equation. (3.29) P [v 1 \|v m ] = m j=1 v j v * j By lemma 3.2 we will have that, C[v 1 \|v m ] * (1 n − P [v 1 \|v m ]) = ((1 n − P [v 1 \|v m ])C[v 1 \|v m ]) * = 1 n − P [v 1 \|v m ] (3.30) and also that. C[v 1 \|v m ] * P [v 1 \|v m ] = (P [v 1 \|v m ]C[v 1 \|v m ]) * = (P [v 1 \|v m ]C[v 1 \|v m ]P [v 1 \|v m ]) * = v m v * 1 + m−1 j=1 v j v * j+1 (3.31) Since C[v 1 \|v m ] * = C[v 1 \|v m ] * (P [v 1 \|v m ] + 1 n − P [v 1 \|v m ]) = C[v 1 \|v m ] * P [v 1 \|v m ] + 1 n − P [v 1 \|v m ] and C[v 1 \|v m ] = (P [v 1 \|v m ] + 1 n − P [v 1 \|v m ])C[v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ] + 1 n − P [v 1 \|v m ], by (3.28) and (3.29) we will have that. C[v 1 \|v m ] * C[v 1 \|v m ] = C[v 1 \|v m ] * P [v 1 \|v m ]P [v 1 \|v m ]C[v 1 \|v m ] + (1 n − P [v 1 \|v m ]) 2 =   v m v * 1 + m−1 j=1 v j v * j+1     v 1 v * m + m−1 j=1 v j+1 v * j   +1 n − P [v 1 \|v m ] = P [v 1 \|v m ] + 1 n − P [v 1 \|v m ] = 1 n (3.32) This implies that. C[v 1 \|v m ]C[v 1 \|v m ] * = (C[v 1 \|v m ] * C[v 1 \|v m ]) * = (1 n ) * = 1 n (3.33) This completes the proof. Lemma 3.5. Given m vectors v 1 , . . . , v m ∈ C n \{0} such that 2m ≤ n, there is a orthonormal m-systemv 1 , . . . ,v m ∈ C n , two scalars ρ, κ = C, two projections K , T and a unitary U [v 1 \|v m ] in C n×n such that: (3.34)        T v 1 = ρv 1 , U [v 1 \|v m ]v j =v j+1 , U [v 1 \|v m ]v m =v 1 Kκv j = v j for each 1 ≤ j ≤ m − 1. Proof. Let us consider the matrix H[v 1 \|v m ] ∈ C n×m defined by the expression. (3.35) H[v 1 \|v m ] =   \| \| \| v 1 v 2 · · · v m \| \| \|  H[v 1 \|v m ] =   \| \| \| V 1 V 2 · · · V m \| \| \|         s 1 0 · · · 0 0 s 2 . . . . . . . . . 0 0 · · · 0 s m         \| \| \| W 1 W 2 · · · W m \| \| \|   Where V 1 , . . . , V m and W 1 , . . . , W m are orthonormal m-systems in C n and C m respectively, and with s 1 ≥ s 2 ≥ · · · s m ≥ 0. Since 2m ≤ n, by the Gram-Schmidt orthonormalization theorem, we will have that there is an orthonormal m-system U 1 , . . . , U m ∈ (span C {V 1 , . . . , V m }) ⊥ . Since {v 1 , . . . , v m } ∈ C n \{0} we will have that for each 1 ≤ j ≤ m, s 1 ≥ s j > 0. Let us set t j = 1 − (s j /s 1 ) 2 , for 1 ≤ j ≤ m. We will have that 0 ≤ t j ≤ 1 and (s j /s 1 ) 2 + t 2 j = 1 for each 1 ≤ j ≤ m, since s j /s 1 ≤ 1 for every 1 ≤ j ≤ m. Let us define the matrix CH[v 1 \|v m ] by the expression. (3.37) CH[v 1 \|v m ] =   \| \| \| V 1 V 2 · · · V m \| \| \|         1 0 · · · 0 0 s 2 /s 1 . . . . . . . . . 0 0 · · · 0 s m /s 1         \| \| \| W 1 W 2 · · · W m \| \| \|   and (3.38) SH[v 1 \|v m ] =   \| \| \| U 1 U 2 · · · U m \| \| \|         t 1 0 · · · 0 0 t 2 . . . . . . . . . 0 0 · · · 0 t m         \| \| \| W 1 W 2 · · · W m \| \| \|   , Let us define the matrixV = [v 1 · · ·v m ] ∈ C n×n by the expression. Since (s j /s 1 ) 2 +t 2 j = 1 for each 1 ≤ j ≤ m, by (3.37) and (3.38) and by orthogonality of the 2m-system V 1 , . . . , V m , U 1 , . . . , U m , we will have thatV * V = 1 m . This implies thatv 1 , . . . ,v m is an orthonormal m-system. By lemma 3.4 we have that C[v 1 \|v m ] is a unitary matrix that satisfies the constraints C[v 1 \|v m ]v j =v j+1 , 1 ≤ j ≤ m − 1, and C[v 1 \|v m ]v m =v 1 . Let us set. κ = s 1 ρ = 1 s 1 (v * 1 v 1 ) K =   \| \| \| V 1 V 2 · · · V m \| \| \|     \| \| \| V 1 V 2 · · · V m \| \| \|   * T =v 1v * 1 U [v 1 \|v m ] = C[v 1 \|v m ] (3.40) SinceV * V = 1 n we will have that K 2 =V * VV * V =V * V and K * = (V * V ) * = V * V = K. Since U 1 , . . . , U m ∈ (span C {V 1 , . . . , V m }) ⊥ , we will have that KV = CH[v 1 \|v m ], by (3.37) this implies that. Kκv j = κKVê j,m = s 1 CH[v 1 \|v m ]ê j,m = H[v 1 \|v m ]ê j,m = v j We will first show thatv * 1 v 1 = 0, in fact, since H[v 1 \|v m ] = s 1 CH[v 1 \|v m ], and SH[v 1 \|v m ] * CH[v 1 \|v m ] = 0 n by orthogonality of V 1 , . . . , V m , U 1 , . . . , U m , we will have that.v * 1 v 1 = (Vê 1,m ) * H[v 1 \|v m ]ê 1,m =ê * 1,mV * H[v 1 \|v m ]ê 1,m =ê * 1,m 1 s 1 H[v 1 \|v m ] * H[v 1 \|v m ]ê 1,m = 1 s 1ê * 1,m H[v 1 \|v m ] * H[v 1 \|v m ]ê 1,m = 1 s 1 (H[v 1 \|v m ]ê 1,m ) * H[v 1 \|v m ]ê 1,m = 1 s 1 v * 1 v 1 > 0 (3.41) We will also have that T 2 =v 1v * 1v1v * 1 =v 1v * 1 = T and T * = (v 1v * 1 ) * =v 1v * 1 = T . Sincev * 1 v 1 = 0, by (3.41) we have that T v 1 =v 1 (v * 1 v 1 ) = 1 s1 (v * 1 v 1 )v 1 = ρv 1 . This completes the proof. . . ,v m ∈ C n is an orthonormal m-system, we will have thatV * V = 1 m , by (3.6) we will have that. Π m (XY ) = Π m (P [v 1 \|v m ]XY ) = Π m (XP [v 1 \|v m ]Y ) =V * XP [v 1 \|v m ]YV =V * XVV * YV = Π m (X)Π m (Y ) (3.46) By (3.42) and by orthonormality ofv 1 , . . . ,v m we will also have that for each 1 ≤ i, k ≤ m. Π m (v iv * k ) =V * v jv * kV = (V * v i )(V * v k ) * =   m j=1ê j,mv * jvi     m j=1ê j,mv * jvk   * =   m j=1ê j,m δ j,i     m j=1ê j,m δ j,k   * =ê i,mê * k,m (3.47) By (3.47) we have that {ê i,mê * k,m } m i,k=1 ⊂ Π m (Z(P [v 1 \|v m ])), since Π m = Ad[V * ] ∈ CP (m, n) and C m×m = span C {ê i,mê * k,m } m i,k=1 , we have that Π m is surjective. This completes proof. Given v 1 , . . . , v m ∈ C n \{0} with n ≥ 2m, the matrix U [v 1 , v m ] whose existence is proved in lemma 3.5 will be called a circular shift factor (CSF) for v 1 , . . . , v m . By lemma 3.6 there isV ∈ C n×m such that Π m = Ad[V * ] is CP map from C n×n onto C m×m that preserves products in Z(P [v 1 \|v m ]). Let us set π m = Π m \| C[U [v1\|vm]] . It is clear that π m ∈ CP (n, m). By (3.45) and (3.47) we will have that. π m (U [v 1 \|v m ]) = Π m (P [v 1 \|v m ]C[v 1 \|v m ]) = Π m  v 1v * m + m−1 j=1v j+1v * j   (3.48) = Π m (v 1v * m ) + m−1 j=1 Π m (v j+1v * j ) =ê 1,mê * m,m + m−1 j=1ê j+1,mê * j,m = 0 1×(m−1) 1 1 m−1 0 (m−1)×1 = C m (3.49) The identity (3.45) also implies that. (3.50) π m (1 n ) = Π m (1 n ) = Π m (P [v 1 \|v m ]) = 1 m Since Π m preserves products in Z(P [v 1 \|v m ]), for any two integers j, k ≥ 1 we will have that, (3.51) and also that. π m (U [v 1 \|v m ] j U [v 1 \|v m ] k ) = Π m (U [v 1 \|v m ] j U [v 1 \|v m ] k ) = Π m (U [v 1 \|v m ] j )Π m (U [v 1 \|v m ] k ) = π m (U [v 1 \|v m ] j )π m (U [v 1 \|v m ] k )(3.52) π m (U [v 1 \|v m ] j ) = Π m (U [v 1 \|v m ] j ) = Π m (U [v 1 \|v m ]) j = π m (U [v 1 \|v m ]) j = C j m By (3.50), (3.51) and (3.52), we will have that the map π m ∈ CP (n, m) determines an algebra homomorphism from C[U [v 1 \|v m ]] onto Circ(k). Definition 3.2. Given v 1 , . . . , v m ∈ C n \{0} with n ≥ 2m, with corresponding CSF U [v 1 , v m ] ∈ U(n). The algebra homomorphism π m whose existence is warranteed by lemma 3.7, will be called a Circulant representation (CR) for C[U [v 1 \|v m ]] Theorem 3.1. Given v 1 , . . . , v m ∈ C n \{0} with n ≥ 2m, there is a projection P ∈ C m×m together two maps ϕ ∈ CP (n, n) and Φ ∈ CP (m, n), such that the following diagram commutes, (3.53) Circ(m) Φ C[U [v 1 \|v m ]] πm 8 8 ϕ / / Z(P ) where π m is a CR of C[U [v 1 \|v m ]]. Moreover, Φ preserves products on C m×m and P U [v 1 \|v m ] = U [v 1 \|v m ]P . Proof. By 3.5 there is an orthonormal m-systemv 1 , . . . ,v m ∈ C n together with projection P [v 1 \|v m ] such that P [v 1 \|v m ]v j =v j for each 1 ≤ j ≤ m. As a consequence of the argument implemented in the proof of lemma 3.7, we have that by lemma 3.6, there is a matrixV ∈ C n×m such that π m = Ad[ V * ]\| C[U [v1\|vm]] and P [v 1 \|v m ] =VV * . Let us set. (3.54)    P = P [v 1 \|v m ] ϕ = Ad[P ] Φ = Ad[V ] It is clear that ϕ ∈ CP (n, n) and Φ ∈ CP (m, n). Given X ∈ C n×n , we will have that. (3.55) ϕ(X) = P [v 1 \|v m ]XP [v 1 \|v m ] =VV * XVV * =V Π m (X)V * = Φ(Π m (X)) SinceV * V = 1 m we will have that for any two X, Y ∈ C m×m . (3.56) Φ(XY ) =V XYV * =V XV * V YV * = Φ(X)Φ(Y ) Since P = P [v 1 \|v m ] is a projection, for any X ∈ C n×n , we will have that P ϕ(X) = P 2 XP = P XP 2 = ϕ(X)P . This imples that ϕ(C n×n ) ⊆ Z(P ). By (3.2) we have that P U [v 1 \|v m ] = P [v 1 \|v m ]C[v 1 \|v m ] = C[v 1 \|v m ]P [v 1 \|v m ] = U [v 1 \|v m ]P . By (3.55) we will have that ϕ has a representation of the form. Proof. Given ε > 0, and any vector history H[Σ, m] = {v 1 , . . . , v m } ⊂ C n for (Σ, {Θ t }), since 2T ≤ 2m ≤ n, we can apply lemma 3.5 to compute an orthonormal T -system {v 1 , . . . ,v T } ∈ C n , scalars κ, ρ, a CSF U [v 1 \|v T ] ∈ C n×n and two projectionsK,T ∈ C n×n , that satisfy (3.34). By theorem 3.1 we will have that there are a projection P ∈ C n×n , ϕ ∈ CP (n, n) and a product preserver Φ ∈ CP (T, n) such that ϕ = Φ • π T and ϕ(C[U [v 1 \|v T ]]) ⊆ Z(P ). Since π T and Φ are linear product preservers in C[U [v 1 \|v T ]] and C T ×T , respectively, we will have that for any p ∈ C[z]. (3.59) ϕ(p(U [v 1 \|v T ])) = Φ(π m (p(U [v 1 \|v T ]))) = p(Φ(π m (U [v 1 \|v T ]))) = p(ϕ(U [v 1 \|v T ])) By theorem 3.1 we will have that, (3.60) ϕ(X) − Φ • π T (p(U [v 1 \|v T ]))) = 0 < ε for each X ∈ C[U [v 1 \|v T ]]. By (3.54) we have that P = P [v 1 \|v T ], with P [v 1 \|v T ] defined by equation (3.6), by lemma 3.1 we will have that. (3.61) Pv 1 = P [v 1 \|v T ]v 1 =v 1 Let p k (z) = κ ρ z k , 0 ≤ k ≤ m − 1, this impies that p k ∈ C[z] . By lemma 3.5 and by (3.61), for each 0 ≤ k ≤ m − 1 we will have that. Kϕ(p k (U [v 1 \|v T ]))T v 1 =K κ ρ U [v 1 \|v T ] k P ρv 1 =KκU [v 1 \|v T ] k Pv 1 =KκU [v 1 \|v T ] kv 1 =Kκv 1+k = v k+1 = Θ k (v 1 ) (3.62) By (3.60) and (3.62), we will have that (Σ, We have that theorem 3.2 also implies that there is an algebra homomorphism Φ : {Θ t }, H[Σ, m]) is (S 1 , T, ε)-controlled by (S 1 , U [v 1 \|v T ],K,T , ϕ, {p t }).C m×m → C n×n such that ϕ(X) − Φ • π T (X) ≤ ε, for each X ∈ C[U [v 1 \|v T ]]. Since U [v 1 \|v T ] ∈ U(n), we will have that C T = π m (U [v 1 \|v T ]) ∈ U(m) and also that. C T T = π T (U [v 1 \|v m ]) T = π T (U [v 1 \|v m ] T ) = π T (1 n ) = 1 m By universality of C[Z/T ] there is a representation ρ T : C[Z/T ] → U(T ), deter- mined by the assignment1 → C T for1 ∈ Z/T . Applying universality of C[Z/T ] to π T (C[U [v 1 \|v T ]]) we have that, π T (p t (U [v 1 \|v T ])) = p t (π T (U [v 1 \|v T ])) = p t (C T ) = p t (ρ T (1)) = ρ T (p t (1)) for each 0 ≤ t ≤ m − 1. This completes the proof. Theorem 3.4. Given δ, ε > 0, every (T, δ, ε)-almost-periodic dynamical system (Σ, {Θ t }) with Σ ⊆ C n is (S, T, ε)-controlled, whenever 2T ≤ 2m ≤ n. Proof. Let x ∈ Σ. Since (Σ, {Θ t }) is (T, δ, ε)-almost-periodic, we will have that there is a discrete-time T -periodic dynamical system (Σ,Θ t ) such that there is x ∈Σ such that x −x ≤ δ, and Θ t (x) −Θ t (x) 2 ≤ ε for each t ∈ Z + 0 and every (x,x) ∈ Σ ×Σ. ≤ t ≤ T . Θ t (x) −Kϕ(p t (Z))Tx 2 ≤ Θ t (x) −Θ t (x) 2 + Θ t (x) −Kϕ(p t (Z))Tx 2 ≤ Θ t (x) −Θ t (x) 2 ≤ ε (3.63) By (3.63) we have that (Σ, {Θ t }) is (S 1 , T, ε)-controlled by (S 1 , U [v 1 \|v T ],K ,T , ϕ, {p t }) . This completes the proof. Theorem 3.5. Given δ, ε > 0, every (T, δ, ε)-almost-periodic dynamical system (Σ, {Θ t }) with Σ ⊆ C n is (Z/T, ε)-controlled, whenever 2T ≤ n. Proof. Let x ∈ Σ. Since (Σ, {Θ t }) is (T, δ, ε)-almost-periodic, we will have that there is a discrete-time T -periodic dynamical system (Σ,Θ t ) such that there is . . , f T −1 }, and two projectionsK,T ∈ C n×n such that for1 ∈ Z/T and for each t ≥ 0. x ∈Σ such that x −x ≤ δ, and Θ t (x) −Θ t (x) 2 ≤ ε for each t ∈ Z + 0 and every (x,x) ∈ Σ ×Σ.Θ t (x) −Kϕ • ρ T (f k (1))Tx 2 ≤ Θ t (x) −Θ t (x) 2 + Θ t (x) −Kϕ • ρ T (f k (1))Tx 2 ≤ Θ t (x) −Θ t (x) 2 ≤ ε (3.64) By (3.63) we have that (Σ, {Θ t }) is (Z/T, ε)-controlled. This completes the proof. Given ε > 0, from the definition in (3.1) of SCL-ROM approximants of a discretetime system (Σ, {Θ t }) with Σ ⊆ C n , we have that by lemma 3.5 and theorem 3.4, every discrete-time system (Σ, {Θ t }) that is (S 1 , ε)-controlled by some control (S 1 , Z , K , T , ϕ, {f t }), has a SCL-ROM approximant of the form (3.1) with F t = ϕ(f t (Z)) for 0 ≤ t ≤ T − 1, we say that such approximant is SCL-ROM is determined by the control (S 1 , Z, K , T , ϕ , {f t }). We can think of the evolution lawsv t+1 =F tvt for 0 ≤ t ≤ 1 in SCL-ROM the decomposition (3.1), as an embedding of the original system in a manifold, where the evolution of the embedded system is controlled by a unitary matrix determined the CSF of some vector history for the original system. Algorithm The techniques and compuations used to prove the previous results, can be implemented to derive a prototypical algorithm described by the following block diagram. (4.1) Z −1K x t F t C[Z/m] x t+1 x t The proofs of lemma 3.5 and theorem 3.2 provide a computational procedure that is sketched in algorithm 1, and can be used to compute the elements F t T −1 t=0 in diagram (4.1), where each matrix F t has a reresentation F t =KĤ tT , for some matricesK,Ĥ t ,T to be determined by algorithm 1. Algorithm 1 Data-driven matrix approximation algorithm Data: Real number ε > 0, State History H[Σ, m]: {x t } 0≤t≤m≤T , T ∈ Z + Result: Approximate matrix realizations: (K,Ĥ t ,T ) ∈ C n×n × C n×n × C n×n of (Σ, {Θ t }) (1) Compute state/output sampled-data history {v t } 0≤t≤T of (Σ, {Θ t }) (2) Compute the SVD VSW = [v 1 · · · v m ] of {v t } 0≤t≤m≤T (3) Compute the OHFV = [v 1 · · ·v m ] for {v t } 0≤t≤m≤T (4) SetK = VV * (5) SetT =v 1v * 1 (6) For 0 ≤ t ≤ T − 1: (a) Compute p t ∈ C[z] such that: (i) KV p t (C t m )V * T v 1 − Θ t [v 1 ] ≤ ε (b) SetĤ t =Vp t (C t m )V * return {K,Ĥ t ,T (t)} 0≤t≤T Experimental results Materials and Methods. In order to solve the diagram (4.1), a prototypical GNU Octave code that implements some of the core computations on which the proofs of lemma 3.5 and theorem 3.2 are based, has been developed as part of this project, using GNU Octave 4.4.1 on a five node Ubuntu Linux Beowulf Cluster, at the Scientific Computing Innovation Center of UNAH-CU. Numerical Experiments. We will consider three numerical experiments in this section. In §5.2.1 we will simulate 1D waves, in §5.2.2 we will simulate damped waves on planar material sections, and in §5.2.3 we will simulate vorticity transport. In §5.2.1 the periodicity appears naturally, while in §5.2.2 and §5.2.3, we will think of the model as a movie, that is being streamed more than once. Each example will be used to illustrate the potential applications of topological control to system identification, and for extraction of (almost) periodic patterns from sampled-data discrete-time industrial systems and plants. 5.2.1. 1D Waves. As a first application, let us consider a wave equation under Dirichlet boundary conditions of the form: (5.1)        ∂ 2 t w − c 2 ∂ t x 2 w = 0, w(0, t) = w(L, t) = 0, w(x, 0) = w 0 (x), ∂ t w(x, 0) = 0 for some suitable data of L, c, w 0 . The simulation is computed using second order finite difference method combined with second order Crank-Nicolson method. The computation is performed using the following commands. The graphical output is shown in fig. 1. Cantilever Elastic Plate. As a second application of algorithm 1, let us consider a computational mechanics problem consisting on the description of the deformation a damped aluminium Cantilever Lamé beam model under planar displacement hypotheses, whose deformation displacement vector v is described by a Navier dynamical system of the form: (5.2) ρ∂ 2 t v − N (λ, µ)v = u(t), BIC(v) = 0, where N (λ, µ) is the Navier operator defined by the expression, N (λ, µ) = (λ + µ)∇∇ · +µ∇ 2 where λ, µ are the Lamé's coefficients for generic aluminium, and where BIC(v) = 0 is some system of equations that determines suitable boundary and initial conditions for Cantilever Lamé beam deflection. We will have that the input u(t) is determined by the expression u(t) = c(t)ρ∂ t v for some smooth time dependent coefficient c(t). It is important to consider that the dimensions of the beam model have been normalized, and that relative deformation displacement scale is exaggerated for visualization purposes of the corresponding simulation. In order to create the data corresponding to the beam deformation, we use an Octave m-file function that computes a second order finite difference approximation of the Navier dynamical system (5.2) for sample sizes 50, 30, 10 and 5, as follows. 3. Planar vorticity transport. As a third application of local S 1 -control, let us consider the vorticity transport PDE system of the form.            ∂ t ω = −u∂ x ω − v∂ y ω + 1 Re ∆ω u = ∂ y ψ, v = −∂ x ψ ∆ψ = −ω BIC(ω, ψ, u, v) = 0 For Reynolds number Re = 400 and suitable boundary and initial conditions represented by the system of algebraic differential equaitons BIC. The simulation is computed using second order finite difference method combined with fourth order Runge-Kuta method. The computation is performed using the following commands. The graphical output is shown in fig. 4. From a topological perspective, the notion of topological control that we propose in this document can be seen as an extension of the Torus Trick presented by R. Kirby in [3] to algebraic matrix sets in the sense of [7]. This extension and the corresponding matrix computations, were partially inspired by some questions raised by M. H. Freedman along the lines of [2]. In order to perform the previously mentioned computations, we start embedding the vector history of a given discrete-time system under study into a manifold, where we can then we use elementary tools from matrix analysis and representation theory, to compute matrix analogies of the surgical cuts corresponding to Kirby's torus trick, these matrix surgical cuts have a direct effect on the spectrum of the CSF corresponding to the vector history of the corresponding embedded system. Once we perform the previously mentioned surgical cuts on the spectrum of the unitary matrices that model the dynamical behaviour of the embedded system under study, the computation of the transition matrices of the embedded system can be easily and efficiently computed in terms of the topologically pre-processed matrices. Another interesting effect of the aformentioned matrix surgical procedures, consists on the reduction of the group action that determines the global behaviour of the system, to a finite group action that can be efficiently computed using algorithm 1, without aditional computational cost due to the additional liftings in the original state (matrix realization) space, whose large dimension can make the standard lifting impossible to compute. This approach was inspired by the work of M. Rieffel in [5], and will be the subject of further study. 6.1. Forthcoming Research. We will further explore the numerical solvability of (3.4) together with some additional contraints. We will improve the computational implementation of the prototypical algorithm 1, extending the topological control techniques presnted in this document to higher dimensional compact manifolds like S 1 × S 1 , S n , and so forth. We will implement AI tools like TensorFlow to take advantage of the model training capabilities predicted by the proofs of theorem 3.4 and theorem 3.5. Besides setting the bases for algorithm 1, the family {f 0 , f 2 , . . . , f T } ⊂ C[z] whose existence and computability is guaranteed by theorem 3.5, provides a natural way to compress the mean dynamical behaviour of a given discrete-time system (Σ, {Θ t }). The information compression property of {f 0 , f 2 , . . . , f T } provides a natural connection to video streaming, this connection will be the subject of further study and experimentation. We will also explore further connections to classical and quantum finite automata. Conclusions Given ε, δ > 0, and a state X t of of discrete time almost periodic system (Σ, {Θ t }) that is ε-approximated by a SCL-ROMΣ determined by a topological control (M, Z, K, T, ϕ, {f t }) of (Σ, {Θ t }), the learning cost of a model update does not exceed the solving cost of the problem: arg min p∈C[z] { X t − Kp(Z)T X 0 \| deg(p) ≤ k} for some X 0 ∈ Σ, where k is the control order. The application of this training technique to the extraction of (almost) periodic patterns from sampled-data discrete-time industrial systems and plants, will be further explored. The family {f 1 , f 2 , . . . , f T } ⊂ C[z] derived from the implementation of algorithm 1 provides an effective way to compress the mean dynamical behaviour of a given discrete-time sampled-data system Σ. Acknowledgments The numerical experiments that provided insight and motivation for the results presented in this report, together with its applications to structure preserving matrix approximation, were performed in the Scientific Computing Innovation Center (CICC-UNAH), of the National Autonomous University of Honduras. I am grateful with Terry Loring, Stanly Steinberg, Marc Rieffel, Concepción Ferrufino, Leonel Obando, Mario Molina and William Fúnez for several interesting questions and comments, that have been very helpful for the preparation of this document. For a given a discrete-time dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , and a m-system of history vectors H[Σ, m] = {v 1 , . . . , v m } for (Σ, {Θ t }), we approach the local controllability of (Σ, {Θ t }) by computing a switched closed loop control system (Σ, {Θ t }) in the sense of [1, §4.2,Example 4.2], determined by the decomposition. Lemma 3. 4 . 4Given an orthonormal m-system {v 1 , . . . , v m } ∈ C n \{0} with m ≤ n, we will have that the corresponding matrix C[v 1 \|v m ] is unitary. ( 3 . 339)V = CH[v 1 \|v m ] + SH[v 1 \|v m ] Lemma 3. 6 . 6Given an orthonormal m-systemv 1 , . . . ,v m ∈ C n \{0} with m ≤ n.There isV ∈ C n×m determined byv 1 , . . . ,v m , such that the map Π m = Ad[V * ] ∈ CP (n, m) from Z(P [v 1 \|v m ]) onto C m×m preserves products in Z(P [v 1 \|v m ]), with P [v 1 \|v m ] determinedby (3.6). Moreover, we will have that Π m (P [v 1 \|v m ]) = 1 m , and Π m (X) = Π m (P [v 1 \|v m ]X) for any X ∈ C n×n .Proof. Let us set. SinceV * V = 1 m , by (3.43) we will have that,(3.44) Π m (P [v 1 \|v m ]) =V * VV * V = 1 2 n = 1 n we will also have that for any X ∈ C n×n .Π m (XY ) =V * V Π m (X) =V * VV * XV =V * P [v 1 \|v m ]XV Π m (P [v 1 \|v m ]X) (3.45)By (3.43) and (3.45) we will have that for any two X, Y ∈ Z(P [v 1 \|v m ]). Lemma 3 . 7 . 37Given v 1 , . . . , v m ∈ C n \{0} with n ≥ 2m, there is an algebra homomorphism π m from C[U [v 1 \|v m ]] onto Circ(m). Proof. By lemma 3.5 we have that there is an orthornormal m-systemv 1 , . . . ,v m ∈ C n together with a CSF U [v 1 \|v m ] = C[v 1 \|v m ]. By lemma 3.5 we have that C[v 1 \|v m ] ∈ Z(P [v 1 \|v m ]) with P [v 1 \|v m ] determined by (3.6), this in turn implies that C[U [v 1 \|v m ]] = C[C[v 1 \|v m ]] ⊂ Z(P [v 1 \|v m ]). ( 3 . 357) ϕ = Φ • Π mBy (3.57) we will have that.(3.58) ϕ\| C[U [v1\|vm]] = Φ • Π m \| C[U [v1\|vm]] = Φ • π mThis completes the proof.Given a discrete-time dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , and a msystem of (history) vectors H[Σ, m] = {v 1 , . . . , v m } for (Σ, {Θ t }), the matrixV ∈ C n×m defined by the formula (3.39) for {v 1 , . . . , v m }, will be called the orthonormal history factor (OHF) of H[Σ, m]. Theorem 3. 2 . 2Given ε > 0, an integer T > 0, and a discrete-time T -periodicdynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , then (Σ, {Θ t }, H[Σ, m]) is (S 1 ,T, ε)controlled, for every ε > 0 and each m ∈ Z such that 2T ≤ 2m ≤ n. Theorem 3 . 3 . 33Given ε > 0, an integer T > 0, and a discrete-time T -periodic dynamical system (Σ, {Θ t }) with Σ ⊆ C n×n , then (Σ, {Θ t }, H[Σ, m]) is (Z/T, ε)controlled, for every ε > 0 and each m ∈ Z such that 2T ≤ 2m ≤ n. Proof. Given ε > 0, by theorem 3.2, we will have that (Σ, {Θ t }, H[Σ, m]) is (S 1 , T , ε)-controlled by some control (S 1 , U [v 1 \|v T ],K,T , ϕ, {p t }). will have that (Σ, {Θ t }, H[Σ, T ]) is (S 1 , T, 0)-controlled by some control (S 1 , U [v 1 \|v T ],K,T , ϕ, {p t }), for some T -system of history vectors H[Σ, T ] = {v 1 , . . . , v T } with v 1 =x, this implies that for each 0 By theorem 3.3 we will have that (Σ, {Θ t }, H[Σ, T ]) is (Z/T, 0)-controlled, this implies that for some T -system of history vectors H[Σ, T ] = {v 1 , . . . , v T } with v 1 =x, there is a unitary representation ρ T : C[Z/T ] → C T ×T , an algebra homomorphism ϕ : C T ×T → C n×n , a family of functions {f 0 , . >> [x,t,data_wave,Cx,Sx]=CL_ROM_WaveDS([0 1],10);-------------------------------------------------------------Running simulation: --------------------------------------------------------------------------------------------------------------------------Computing circular matrix representations in C[U[v1\|vm]]: ------------------------------------------------------------- Figure 1 . 1Σ andΣ output histories >> [Bx,By,data_x,data_y,Yx,Yy]=CL_ROM_BeamDS(1,50,40,-5e9,-10,1); >> [Bx,By,data_x,data_y,Yx,Yy]=CL_ROM_BeamDS(1,30,40,-5e9,-10,2); >> [Bx,By,data_x,data_y,Yx,Yy]=CL_ROM_BeamDS(1,10,40,-5e9,-10,3); >> [Bx,By,data_x,data_y,Yx,Yy]=CL_ROM_BeamDS(1,5,40,-5e9,-10,4); -------------------------------------------------------------Running simulation:----------------------------------------------------------is 298.816 seconds. -------------------------------------------------------------Computing circular matrix representations in C[U[v1\|vm]]: -------------------------------------------------------------Elapsed time is 3.51714 seconds. -------------------------------------------------------------Verifying circular mimetic constraints for C[U[v1\|vm]]: -------------------------------------------------------------Verification passed... max{\|\|Kx Rx Usx^k Tx Ux0-Oxk\|\| \| 1<=k<=m} = 2.7082e-27 <= eps max{\|\|Ky Ry Usy^k Ty Uy0-Oyk\|\| \| 1<=k<=m} = 7.4542e-13 <= eps -------------------------------------------------------------For m = 121 For n = 13357 For eps = 7.4542e-13 Elapsed time is 20.7964 seconds. -------------------------------------------------------------This produces the original output history shown in Figure fig. 2. Figure 2 . 2Original model Σ output historyThe output histories of the corresponding SCL-ROM approximants are shown inFigure 3. Figure 3 . 3Output histories for several SCL-ROM approximants Σ of Σ 5.2. >> [x,y,data,Cx,Sx]=CL_ROM_VortexDS(1,10,[0,0.15],[100,750],[1,3]); -------------------------------------------------------------Running simulation: --------------------------------------------------------------------------------------------------------------------------Computing circular matrix representations in C[U[v1\|vm]]: -------------------------------------------------------------Elapsed time is 2.08992 seconds. -------------------------------------------------------------Verifying circular mimetic constraints for C[U[v1\|vm]]: -------------------------------------------------------------max{\|\|Kx Rx Ux^k Tx X0-Yk\|\| \| 1<=k<=m} = 8.2096e-13 <= eps -------------------------------------------------------------time is 111.598 seconds. ------------------------------------------------------------- Figure 4 . 4Σ Bilinear Control Systems Matrices in Action. D L Elliott, Applied Mathematical Sciences. 169SpringerElliott D. L. (2009). Bilinear Control Systems Matrices in Action. Applied Mathematical Sciences Vol. 169. Springer. Truncation of Wavelet Matrices: Edge Effects and the Reduction of Topological Control. M H Freedman, W H Press, Linear Algebra Appl. 2Freedman M. H. and Press W. H. (1996). Truncation of Wavelet Matrices: Edge Effects and the Reduction of Topological Control Linear Algebra Appl. 2:34:1-19 (1996) Stable homeomorphisms and the annulus conjecture. R C Kirby, Ann. of Math., Second Series. 893Kirby R. C. (1969). Stable homeomorphisms and the annulus conjecture. Ann. of Math., Second Series, Vol. 89, No. 3 (May, 1969), pp. 575-582 Dynamic Mode Decomposition with Control. J L Proctor, S L Brunton, J Kutz, SIAM J Appl. Dyn. Syst. 151142161Proctor J. L., Brunton S. L., Kutz J. N. Dynamic Mode Decomposition with Control. SIAM J Appl. Dyn. Syst. 2016, Vol. 15, No. 1, pp. 142161 Actions of finite groups on C-algebras. M A Rieffel, Math. Scand. 47Rieffel M. A. Actions of finite groups on C-algebras. Math. Scand. 47 (1980), 157-176 Dynamic mode decomposition of numerical and experimental data. P J Schmid, J. Fluid Mech. 656Schmid P. J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. (2010), vol. 656, pp. 5-28. On Uniform Connectivity of Algebraic Matrix Sets. F Vides, Banach. J. Math. Anal. To appear inVides F. On Uniform Connectivity of Algebraic Matrix Sets. To appear in Banach. J. Math. Anal. E-mail address: fredy. Tegucigalpa, HondurasScientific Computing Innovation Center, Universidad Nacional Autónoma de [email protected] Computing Innovation Center, Universidad Nacional Autónoma de Hon- duras, Tegucigalpa, Honduras E-mail address: [email protected]	[]
[ "Supplementary Information for Evidence of nematic order and nodal superconducting gap along [110] direction in RbFe 2 As 2", "Supplementary Information for Evidence of nematic order and nodal superconducting gap along [110] direction in RbFe 2 As 2" ]	[ "X Liu ", "R Tao ", "M Ren " ]	[]	[]	Supplementary Figure 1 \| Crystal structure and transport measurement of RbFe2As2 (a) Bulk crystal structure of RbFe2As2. (b) Temperature dependence of the resistance (0 ~ 300K), the red curve below T = 50 K is a fit to T 2 . (c) Temperature dependence of the resistance across the superconducting transition at Tc ~2.5 K. Supplementary Figure 2 \| Calibration of the Teff (a) Topographic image of an Al/Si(111) film of thickness ~20 ML (scale bar: 50 nm). (b) The superconducting gap of the Al/Si(111) film taken at T = 20 mK (Vb = 1 mV, I = 100 pA, ΔV = 30 μV). Red curve is the BCS fit with Δ = 0.19 meV, Teff = 310 mK and Γ = 0.	10.1038/s41467-019-08962-z	null	70,350,140	1803.07304	06c4039d805208caacc42839706511f00d08bbd3	Supplementary Information for Evidence of nematic order and nodal superconducting gap along [110] direction in RbFe 2 As 2 X Liu R Tao M Ren Supplementary Information for Evidence of nematic order and nodal superconducting gap along [110] direction in RbFe 2 As 2 Supplementary Figure 1 \| Crystal structure and transport measurement of RbFe2As2 (a) Bulk crystal structure of RbFe2As2. (b) Temperature dependence of the resistance (0 ~ 300K), the red curve below T = 50 K is a fit to T 2 . (c) Temperature dependence of the resistance across the superconducting transition at Tc ~2.5 K. Supplementary Figure 2 \| Calibration of the Teff (a) Topographic image of an Al/Si(111) film of thickness ~20 ML (scale bar: 50 nm). (b) The superconducting gap of the Al/Si(111) film taken at T = 20 mK (Vb = 1 mV, I = 100 pA, ΔV = 30 μV). Red curve is the BCS fit with Δ = 0.19 meV, Teff = 310 mK and Γ = 0. pre-aligned to X scan and Y scan directions (Vb = 8 mV, I = 1 nA, scale bar: 2 nm)). The lattice constant is measured to be 5.4 Å, and the lattice is rotated 45° with respect to a0, b0. (d) Laue diffraction pattern of a FeSe single crystal. (e) Simulated Laue pattern of FeSe with a0, b0 along X and Y directions. The Miller indices (hkl) of the plane corresponding to the diffraction points near center are also marked. (f) STM image of cleaved FeSe crystal with a0, b0 pre-aligned to X scan and Y scan directions (Vb = 100 mV, I = 100 pA, scale bar: 2 nm). The lattice constant is measured to be 3.75 Å, and the lattice has the same orientation with a0, b0. Note: The spots marked by red and blue circles in panels a, d are corresponding to the spots marked by the same colored indices in the simulated pattern (in panels b, e). They originated from {10X} and {11X} plane families, respectively. Fig. 1b of the main text. Each dI/dV maps are taken at a Vb equal to the mapping energy (labeled on the map) and I = 100 pA; the lock-in modulation (ΔV) for each map has an amplitude of 5% Vb. Details about the FFT symmetrization is described in Supplementary Note 3. Every dI/dV map has 256 × 256 pixels. Scale bar in the dI/dV image is 20 nm, and that in FFTs are 0.5 Å -1 . Supplementary Supplementary Figure 8 \| A full set of dI/dV maps taken on type B surface and their FFTs. The topography of the mapping area is shown in Fig. 1c of the main text. Each dI/dV maps are taken at a Vb equal to the mapping energy (labeled on the map) and I = 100 pA; the lock-in modulation (ΔV) for each map has an amplitude of 5% Vb. Every map has 256 × 256 pixels. Scale bar in the dI/dV image is 20 nm, and that in FFTs are 0.5 Å -1 . Supplementary Supplementary Note 1: Calibration of the effective electron temperature of the STM system Due to the electrical noise and RF radiations, the effective electron temperature (Teff) of a low-T STM is usually higher than the thermometer reading. The Teff of the dilution refrigerator STM used in this work is calibrated by measuring the superconducting gap of an Al film grown Si(111). Supplementary Fig. 2a shows a typical STM image of the Al/Si(111) film, with a thickness of ~ 20 monolayer (ML), and Supplementary Fig. 2b shows its superconducting gap spectrum. A standard BCS fit (red curve) yields Δ = 0.19 meV and Teff = 310 mK. Here we note that in order to make a conservative estimation of Teff, Dynes broadening term is not used in the fitting (Γ = 0). In the presence of finite Γ or other broadening factors, the actual Teff could be slightly lower than the fitted value. Supplementary Note 2: Determining the surface atomic structure of cleaved RbFe2As2 The surface structure of cleaved RbFe2As2 is revealed by resolving the atomic lattice of the defect-free area and that inside of the Rb vacancies. Supplementary Fig. 3a is a typical image of type B surface with multiple Rb vacancies. Supplementary Fig. 3b shows the region marked in Supplementary Fig. 3a in greater detail (shown with a linear color scale). A square lattice inside of the Rb vacancies can be seen and has a lattice constant of ~3.8 Å, matching a0, which is mostly likely from the underlying FeAs layer. The lattice of the defect-free area is hard to see in Supplementary Fig. 3b due to its much smaller corrugation. To enhance the contrast, a nonlinear color mapping is used in Supplementary Fig. 3c, in which both the surface Rb lattice and lattice inside the vacancy can be seen. By comparing the atomic sites of these two lattices (as marked in Supplementary Fig. 3d), the surface lattice model is derived and shown in Supplementary Fig. 3e. The surface Rb forms a √2×√2 (R45°) lattice with respect to the As lattice. For the lattice model in Supplementary Fig. 3e, the surface Rb atoms will have another set of occupation sites, as illustrated by the dashed circles. These two equivalent occupation sites are shifted by 1/2 unit cell with respect to each other (along both a and b directions), which should result in domain structures when both are present. We indeed observed such domain structures on samples cleaved at a lower temperature (~30K), as shown in Supplementary Fig. 4a. There are domain boundaries running through the surface (marked by red arrows). Supplementary Fig. 4b is an atomically resolved image near a domain boundary. One can see aside of the boundary, the surface still displays a √2×√2 lattice. However, the lattice of the upper domain is shifted by 1/2 unit cell along a and b with respect to the lower domain. To illustrate this, we draw a lattice of white spots which matches the atomic lattice of the lower domain, however it mismatches the upper domain by the above offset. The existence of different domains gives further support to the surface lattice model in Supplementary Fig. 3e. The assignment of surface atomic structure above is based on STM imaging. To further confirm the orientation of surface √2×√2 lattice with respect to the bulk FeAs lattice, we performed Laue diffraction measurement to accompany STM imaging. The results are summarized in Supplementary Fig. 5. We first determined the orientation of a0 and b0 (the inplane basic vectors of 2Fe unit cell) of RbFe2As2 single crystal by comparing its measured Laue pattern with a simulated pattern. As shown in the simulation in Supplementary Fig. 5b, the four diffraction spots that closest to the center, labelled by red colored Miller indices, are originated from {10X} plane family (defined by 2Fe unit cell, so they are along the a0 or b0 directions); while the spots with blue colored Miller indices are from {11X} plane family. In the measured Laue pattern, only the spots close to the center show up with significant weight ( Supplementary Fig. 5a), so the crystal orientation can be determined by comparing it to Supplementary Fig. 5b. Then the crystal was glued on STM sample holder with a0 and b0 aligned to X and Y scan directions, respectively. The STM image of cleaved surface ( Supplementary Fig. 5c) then directly shows the surface √2×√2 lattice is rotated 45° with respect to a0 and b0. We also repeated the same procedure on a pure FeSe single crystal, the results are shown in Supplementary Figs. 5d-f. The FeSe has a Laue pattern in analogous to RbFe2As2, and its a0, b0 directions are determined in a similar way. The STM image in Supplementary Fig. 5f show that its surface lattice has a constant of 3.75 Å, with the direction the same as a0 and b0. This is well expected for a Se terminated surface of FeSe. Therefore, combined Laue and STM measurement directly indicates the surface √2×√2 lattice of RbFe2As2 is rotated 45° with respect to a0, b0. Supplementary Figure 3 \| 3Surface atomic lattice of RbFe2As2 (a) An STM image of type B surface (Vb = 0.8V, I = 100pA, scale bar: 10 nm) B. (b) Higher-resolution image (Vb = 6mV, I = 3nA) of the region marked in panel a, shown with a linear color scale (scale bar: 10 nm). (c) The same image as panel b but mapped with a non-linear color scale to reveal the Rb lattice. (d) The same image as panel c, with the atomic sites marked. (e) The surface lattice model derived from panel d. The orientation of 2Fe unit cell is denoted by a0, b0. Supplementary Figure 4 \| Domain structures on the surface of low temperature cleaved RbFe2As2 (a) STM image of RbFe2As2 surface cleaved at T ≈ 30 K (Vb = -4 meV, I = 100 pA, scale bar: 10 nm). Domain boundaries are indicated by red arrows. (b) Atomically resolved STM image (Vb = -4 meV, I = 500 pA, scale bar: 1 nm) taken in the region marked in panel a. A lattice of white spots is intentionally drawn to match the atomic lattice of the lower domain. However it is misaligned from the upper domain lattice by 1/2 unit cell along a and b direction. (Images are taken at T = 4.5 K) Supplementary Figure 5 \| Determine the surface lattice orientation of RbFe2As2 by Laue diffraction and STM imaging. (a) Laue diffraction pattern of RbFe2As2 single crystal. (b) Simulated Laue pattern using SingleCrystal TM software for RbFe2As2 with a0, b0 along X and Y directions. The Miller indices (hkl) of the planes corresponding to the diffraction points near center are marked, which are belong to {01X} (red) and {11X} (blue) plane families. (c) STM image of cleaved RbFe2As2 with a0, b0 Figure 6 \| 6Spatial dependence of the superconducting gap on type A and B surfaces. (a) STM image of a type A surface (Vb = 0.5 V, I = 10 pA, scale bar: 30 nm) (b) superconducting gap spectra taken along the arrows marked in panel a (Vb = 2 mV, I = 100 pA, ΔV = 50 μV). (c) STM image of type B surface (Vb = 1 V, I = 10 pA, scale bar: 15 nm). (d) Superconducting gap spectra taken at the spots marked in panel c (Vb = 2 mV, I = 100 pA, ΔV = 50 μV). The spots are randomly chosen. All the spectra shown in this figure are taken at T = 20 mK and Teff = 310 mK. As seen from panel a-d, the superconducting gaps on both type A and B surfaces are spatially homogenous. Supplementary Figure 7 \| A full set of dI/dV maps taken on type A surface and their FFTs. The topography of the mapping area is shown in Figure 9 \| 9Additional QPI data of surface K-dosed RbFe2As2 (a) Additional dI/dV maps (scale bar: 10 nm), raw FFTs and symmetrized FFTs (scale bars: 0.3 Å -1 ) taken on RbFe2As2 with Kc =0 at T = 4.5 K. All dI/dV maps are taken at the setpoint of Vb = 10 mV, I = 200 pA, ΔV = 1 mV. (b) Additional dI/dV maps (scale bar: 10 nm), raw FFTs and symmetrized FFTs (scale bars: 0.3 Å -1 ) taken on RbFe2As2 with Kc = 0.17 ML at T = 4.5 K. All dI/dV maps are taken at the setpoint of Vb = 10 mV, I = 200 pA, ΔV = 1 mV. Each map has 250 × 250 pixels. Supplementary Figure 10 \| Temperature dependence of the low energy tunneling gap observed at Kc = 0.17 ML (Vb = 15 mV, I = 300 pA, ΔV = 1 mV for all the spectra). The gap closed at about 12 K. Supplementary Note 3: Additional QPI data of as-cleaved and surface K-dosed RbFe2As2, and details of the FFT symmetrization process.Additional dI/dV maps, raw FFTs and symmetrized FFTs taken on type A and B surfaces are shown inSupplementary Fig. 7andSupplementary Fig. 8, respectively. Symmetrized FFTs are all obtained by mirror symmetrizing the raw FFTs along (π, π) and (π, -π) directions. Detailed process is 1): Identify the (π, π) and (π, -π) directions of the raw FFT by using atomically resolved images; 2): Mirror flip the raw FFT along (π, π) and add it to the raw FFT; 3) Flip the results of 2) along (π, -π) and add it to the results of 2).Additional dI/dV maps, raw FFTs and symmetrized FFTs taken on RbFe2As2 with Kc =0 and Kc = 0.17 ML at T = 4.5 K are shown inSupplementary Fig. 9a-b. The symmetrized FFTs are also obtained by mirror symmetrizing the raw FFTs along (π, π) and (π, -π) directions, as described above.	[]
[ "Two-channel Kondo physics due to As vacancies in the layered compound ZrAs 1.58 Se 0.39", "Two-channel Kondo physics due to As vacancies in the layered compound ZrAs 1.58 Se 0.39" ]	[ "T Cichorek ", "L Bochenek ", "M Schmidt ", "A Czulucki ", "G Auffermann ", "R Kniep ", "R Niewa ", "F Steglich ", "S Kirchner ", "\nInstitute of Low Temperature and Structure Research\nMax-Planck-Institute for Chemical Physics of Solids\nInstitute of Inorganic Chemistry\nPolish Academy of Sciences\n50-950, 01187Wrocław, DresdenPoland, Germany\n", "\nMax-Planck-Institute for Chemical Physics of Solids\nCenter for Correlated Matter\nUniversity of Stuttgart\n70569, 01187Stuttgart, DresdenGermany, Germany\n", "\nInstitute of Physics\nZhejiang University\n310058HangzhouZhejiangChina\n", "\nCenter for Correlated Matter\nChinese Academy of Science\n100190BeijingChina\n", "\nZhejiang University\n310058HangzhouZhejiangChina\n" ]	[ "Institute of Low Temperature and Structure Research\nMax-Planck-Institute for Chemical Physics of Solids\nInstitute of Inorganic Chemistry\nPolish Academy of Sciences\n50-950, 01187Wrocław, DresdenPoland, Germany", "Max-Planck-Institute for Chemical Physics of Solids\nCenter for Correlated Matter\nUniversity of Stuttgart\n70569, 01187Stuttgart, DresdenGermany, Germany", "Institute of Physics\nZhejiang University\n310058HangzhouZhejiangChina", "Center for Correlated Matter\nChinese Academy of Science\n100190BeijingChina", "Zhejiang University\n310058HangzhouZhejiangChina" ]	[]	We address the origin of the magnetic-field independent −\|A\| T 1/2 term observed in the lowtemperature resistivity of several As-based metallic systems of the PbFCl structure type. For the layered compound ZrAs1.58Se0.39, we show that vacancies in the square nets of As give rise to the low-temperature transport anomaly over a wide temperature regime of almost two decades in temperature. This low-temperature behavior is in line with the non-magnetic version of the twochannel Kondo effect, whose origin we ascribe to a dynamic Jahn-Teller effect operating at the vacancy-carrying As layer with a C4 symmetry. The pair-breaking nature of the dynamical defects in the square nets of As explains the low superconducting transition temperature Tc ≈ 0.14 K of ZrAs1.58Se0.39, as compared to the free-of-vacancies homologue ZrP1.54S0.46 (Tc ≈ 3.7 K). Our findings should be relevant to a wide class of metals with disordered pnictogen layers.	10.1103/physrevlett.117.106601	[ "https://arxiv.org/pdf/1608.04263v1.pdf" ]	2,012,767	1608.04263	04f5934e200432fe23245adde4adf4572516ae5a	Two-channel Kondo physics due to As vacancies in the layered compound ZrAs 1.58 Se 0.39 15 Aug 2016 T Cichorek L Bochenek M Schmidt A Czulucki G Auffermann R Kniep R Niewa F Steglich S Kirchner Institute of Low Temperature and Structure Research Max-Planck-Institute for Chemical Physics of Solids Institute of Inorganic Chemistry Polish Academy of Sciences 50-950, 01187Wrocław, DresdenPoland, Germany Max-Planck-Institute for Chemical Physics of Solids Center for Correlated Matter University of Stuttgart 70569, 01187Stuttgart, DresdenGermany, Germany Institute of Physics Zhejiang University 310058HangzhouZhejiangChina Center for Correlated Matter Chinese Academy of Science 100190BeijingChina Zhejiang University 310058HangzhouZhejiangChina Two-channel Kondo physics due to As vacancies in the layered compound ZrAs 1.58 Se 0.39 15 Aug 2016(Dated: March 7, 2018) We address the origin of the magnetic-field independent −\|A\| T 1/2 term observed in the lowtemperature resistivity of several As-based metallic systems of the PbFCl structure type. For the layered compound ZrAs1.58Se0.39, we show that vacancies in the square nets of As give rise to the low-temperature transport anomaly over a wide temperature regime of almost two decades in temperature. This low-temperature behavior is in line with the non-magnetic version of the twochannel Kondo effect, whose origin we ascribe to a dynamic Jahn-Teller effect operating at the vacancy-carrying As layer with a C4 symmetry. The pair-breaking nature of the dynamical defects in the square nets of As explains the low superconducting transition temperature Tc ≈ 0.14 K of ZrAs1.58Se0.39, as compared to the free-of-vacancies homologue ZrP1.54S0.46 (Tc ≈ 3.7 K). Our findings should be relevant to a wide class of metals with disordered pnictogen layers. In the last years, several exciting phenomena have been discovered in pnictogen-containing materials. This holds particularly true for high-temperature superconductivity in a class of materials based on iron [1]. Another example is the family of filled skutterudites with pnictogen atoms in the cage [2,3]. Topologically nontrivial phases of certain 3D insulators as well as Dirac and Weyl semimetals have been first observed in pnictogen-based systems, such as Bi 1−x Sb x [4], Cd 3 As 2 [5], and TaAs [6]. Here, we address the origin of the magnetic-field independent −\|A\| T 1/2 term observed in the low-temperature resistivity of several As-based metallic systems. We show that As vacancies in the layered compound ZrAs 1.58 Se 0.39 give rise to an orbital two-channel Kondo effect (2CK) that is symmetry-protected against level splitting [8,22] The standard theory of metals states that at sufficiently low temperatures (T ), the electrical resistivity ρ(T ) is expected to vary quadratically with T due to the electron-electron interaction (EEI) [9]. Deviations from this behavior can occur, but generally ρ(T ) continues to decrease upon cooling. Only a few mechanisms are known to produce a low-T resistivity minimum. The paradigmatic example is the Kondo effect where spin-flip scattering of conduction electrons off dynamic centers as-sociated with the local magnetic moments gives rise to a logarithmic increase of the resistivity upon cooling [9]. Theoretical studies have revealed that an even more exotic 2CK effect can occur if two degenerate channels of the conduction electrons exist, which independently scatter off centers with a local quantum degrees of freedom [10]. The low-energy physics in this case is governed by a non-Fermi liquid fixed point and results in a resistivity minimum followed at lower temperature by a √ T increase of ρ(T ). Realizing the nonmagnetic version of the 2CK effect requires a degeneracy between orbitals that is difficult to maintain in real systems. Here, the orbital degree of freedom plays the role of the (pseudo-)spin and the electron spin represents the degenerate channel index. Such a scenario was proposed for metals with dynamic structural defects modeled by double-well potentials [18][19][20]. However, in spite of considerable interest [14][15][16][17][18], this type of 2CK physics has never been conclusively demonstrated in any bulk metallic system. This appears to be in line with further theoretical efforts that established two-level systems as an unlikely source of the non-Fermi behavior [8,19,20,22]. Additional complications arise from the fact that the disorder-enhanced EEI in three-dimensional metals can also give rise to a √ T term in the resistivity [10,11]. Thus, the clearest evidence to date for the 2CK effect has come from an artificial nanostructure that can be judiciously tuned to show a T −1/2 term in the differential conductance for about a decade in T [23]. On the other hand, there are a number of experimental results which point to unconventional scattering mechanisms whose precise nature has remained enigmatic [12]. In particular, several As-based metallic systems with the layered PbFCl structure display a low-T resistivity minimum. Investigations on e.g. diamagnetic ThAsSe (nominal composition) single crystals revealed a magneticfield-(B)-independent −\|A\|T 1/2 term in the low-T resistivity [25]. Furthermore, the thermal conductivity and specific heat of ThAsSe showed glass-type temperature dependences which support the presence of structural defects with internal degrees of freedom [25]. Investigations of the closely related Zr-and Hf-based arsenide selenides have pointed to a B-independent −\|A\|T 1/2 contribution to ρ(T ) as a generic feature of metallic arsenide selenides crystallizing in the PbFCl structure [26]. To address the physical mechanism leading to the B-independent −\|A\|T 1/2 term in the low-T ρ(T ) we have identified two homologues, ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 , that allow us to identify the microscopic origin of the anomalous behavior. By combining precise physical property measurements with chemical and structural investigations performed on the same single crystals, we show that the only viable explanation for the observed transport anomalies of ZrAs 1.58 Se 0.39 is in terms of a 2CK. We argue that 2CK physics is possible due to vacancies in the square nets of As atoms. The ternary pnictide-chalcogenides (P n-Ch) contain intermediate phases which crystallize in the tetragonal (P 4/nmm) PbFCl structure, a substitution variant of the Fe 2 As type. This crystal structure consists of square-planar 4 4 nets stacked along the [001] direction. For compounds with exact chemical composition M :P n:Ch=1:1:1, each of the 4 4 nets is exclusively occupied by one element resulting in a layer-sequence ...P n-M -Ch-Ch-M -P n, and forming a puckered double-layer M 2 Ch 2 with an ordered distribution of M and Ch. Phases with M , being a metal with the fixed +4 oxidation state such as Zr, Hf, and Th, are characterized by significant excess of pnictogen atoms (see Fig. 1). In case that the homogeneity range is restricted to the tie-line between M P n 2 and M Ch 2 the chemical formula corresponds to M P n x Ch y with x + y = 2. For M =Zr, it has been shown that only the 2c site is randomly occupied by P n and Ch. Exceptionally for the arsenide selenide, however, the homogeneity range is located right next to the tie-line ZrAs 2 -ZrSe 2 on the As depleted side and hence 1.90 ≤ x + y ≤ 1.99 [2] and Fig. S4 does not distinctly alter basic physical properties, such as electrical resistivity and specific heat of the tetragonal Zr-P n-Ch phases. In fact, differences in the metallic behavior of ρ(T ) are of minor significance only, as depicted in Fig. 2(a). For both homologues ZrP n x Ch y , a drop of ρ(T ) signals the onset of a superconducting phase transition at T c ≈ 0.14 K (ZrAs 1.58 Se 0.39 ) and 3.9 K (ZrP 1.54 S 0.46 ), respectively. However, ρ = 0 is found only substantially below the onset temperature [see Fig. 2(b)]. The complex behavior of ρ(T ) in the vicinity of T c is ascribed to a likely delicate variation of chemical composition, i.e., below 0.5 wt. % which is the resolution limit of our analysis. The normal-state specific heats C(T ) for both ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 are very similar, leading to virtually the same value of the Sommerfeld coefficient of the electronic specific heat γ = 1.7(±0.1) mJK −2 mol −1 [see Fig. 2(c)]. Summing up, the main physical properties of ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 demonstrate a farreaching similarity between both homologues. Therefore, the observed difference in their T c 's by a factor ≈ 30 is an unexpected result, especially in view of the tiny concentration of magnetic impurities in the arsenide selenide crystals, i.e. less than 0.10 wt. % [26]). Note that the As vacancies themselves, as (static) potential scatterers, cannot be the source of strong pair-breaking. A hallmark of the 2CK effect is the √ T dependence of the low-T resistivity. In the case of dynamic structural defects, the resulting non-Fermi liquid properties are not expected to depend on an applied magnetic field as long . For clarity, the temperature scale of the ρ(T ) data for the P-based system was multiplied by a factor of 5. (c) Low-temperature specific heat for ZrAs1.58Se0.39 and ZrP1.54S0.46. Inset: Electronic specific heat of ZrP1.54S0.46 in the vicinity of the superconducting phase transition, as Ce/T vs temperature. Ce = C − C ph , where C ph is the phonon contribution, estimated from the normal-state B = 0.5 T data. (d) Zero-field-cooled (ZFC) and field-cooled (FC) dc magnetic susceptibility as a function of temperature for ZrP1.54S0.46. Inset: Evidence for large shielding in ZrAs1.58Se0.39 is provided by the sizable diamagnetic signal of the ac magnetic susceptibility below about T = 0.125 K. as the Zeeman splitting does not cause a difference in the conduction-electron DOS at the Fermi level between up and down subbands. The only other mechanism that can in principle result in a B-independent −\|A\|T 1/2 correction to the resistivity is the electron-electron interaction (EEI) in a three-dimensional, disordered metal [10,11]. This could happen if electron screening is strongly reduced, yielding the unique case of a screening factor, F σ , very close or even equal to zero [10,11]. To analyze possible corrections due to the EEI in ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 , we thus have plotted in Fig. 3(a) the relative change of the resistivity normalized to the minimum value, (ρ − ρ min )/ρ min , as a function of T 1/2 . Below T min ≈ 15.0 K, the zero-field ρ(T ) data for ZrAs 1.58 Se 0.39 depend strictly linearly on T 1/2 over almost two decades Figure 3. Evidence for the 2CK effect due to As vacancies. (a) Despite very similar basic low-temperature properties of ZrAs1.58Se0.39 and ZrP1.54S0.46, these sister compounds display qualitatively different responses of (ρ − ρmin)/ρmin vs. T 1/2 . Note the magnetic-field-independent −\|A\|T 1/2 contribution to ρ(T ) for ZrAs1.58Se0.39, i.e., for a system with vacancies in the pnictogen layer. Inset: Low-T resistivity of ZrP1.54S0.46 measured in zero and varying magnetic fields up to 14 T. For B = 0, ρmin = ρ(9 K) was taken. (b) Remarkable differences of the B-independent low-T ρ(T ) as (ρ−ρmin)/ρmin vs. T 1/2 for two single-crystalline ZrAs1.58Se0.39 specimens with similar residual resistivities. Dashed line shows the magnitude of a hypothetical −\|A\|T 1/2 correction to ρ(T ) for sample #2 due to 3D EEI assumingFσ = 0. Inset: Magnetic field dependence of ρ(T ) of ZrAs1.58Se0.39 and ZrP1.54S0.46 single crystals. Note that the data for ZrAs1.58Se0.39 are divided by a factor of 10. (c)-(f) Dynamic structural scattering centers in metallic arsenide selenides. Shown are different possible arrangements of dimers or oligomers within the As layer triggered by the vacancy (red square). Arsenic atoms may for example arrange as dimers, trimers, oligomers of several As atoms, or infinite chains with zigzag or sawtooth configuration. (f) is obtained from (e) by a π/2 rotation of the arrangement of (e) indicating how the C4 symmetry is dynamically restored. in temperature. A magnetic field of B = 14 T, being the largest field accessible in our experiment, does not alter the −\|A\|T 1/2 dependence for the system containing vacancies in the pnictogen layers. For the free-of-vacancies system, the low-T upturn displays a completely different response to an applied field. (In the absence of a magnetic field, superconducting fluctuations dominate the transport properties of ZrP 1.54 S 0.46 leading to a strong decrease of ρ(T ) in a temperature region which significantly exceeds T c ≈ 3.7 K.) Although at B = 1 T a small fraction of the ZrP 1.54 S 0.46 sample still displays surface superconductivity below 1 K, at elevated temperatures, we observe an upturn in ρ(T ) with a rather complex T behavior. More remarkably, however, the (ρ − ρ min )/ρ min correction is distinctly smaller at intermediate fields of about 2 T [cf. the inset of Fig. 3(a)]. For 14 T, the low-T upturn is strongly reduced, and its magnitude is about 7 times smaller than for B = 1 T. Furthermore, the ρ(T ) rise is restricted to a rather narrow temperature window, i.e. from above 4.2 K to around 1 K. At T < 1 K, the resistivity tends to saturate. These findings point to a negligible EEI in single crystals of ZrP n x Ch y and yield striking evidence for an entirely different origin of the −\|A\|T 1/2 term in the low-T ρ(T ) occurring in the material without (ZrAs 1.58 Se 0.39 ) and with (ZrP 1.54 S 0.46 ) a full occupancy of the square-planar pnictogen layers. For an in-depth analysis, see [35]. In the latter case, structural disorder (driven from the mixed occupied 2c sites only) leads to weak localization. Its negative contribution to the magnetoresistance MR= [(ρ(B) − ρ(0)]/ρ(0) increases upon cooling and amounts to about 0.02 % at T = 2 K. The weak-localization contribution is nearly two orders of magnitude smaller than the classical MR [cf. the inset of Fig. 3(b)]. Therefore, the total MR of ZrP 1.54 S 0.46 is essentially temperature independent at T ≤ 10 K and approaches 1.4 % at B = 14 T. Figure 3(b) shows the temperature dependence of (ρ − ρ min )/ρ min for two ZrAs 1.58 Se 0.39 specimens with similar elastic relaxation times. In spite of this moderate variation, the size of the magnetic-field-insensitive −\|A\|T 1/2 term between these samples differs by more than a factor of 3 and hence the A coefficient amounts to 0.038 and 0.167 Ωcm/K 1/2 for samples #1 and #2, respectively. Similarly to the afore mentioned results, the experimental observations shown in Fig. 3(b) are at strong variance to the expectation based on enhanced EEI in 3D specimens of the same disordered metal [10,11]. Indeed, a nearly identical screening factorF σ , see Ref. [35], implies that the magnitude of a hypothetical (ρ − ρ min )/ρ min anomaly would only vary with ρD −1/2 ∝ ρ 3/2 . This, however, is not observed in ZrAs 1.58 Se 0.39 . In fact, a supposed −\|A\|T 1/2 correction, calculated in respect to the \|A\|-coefficient value of sample #1 and schematically sketched by a dashed line in Fig. 3(b), would be substantially smaller than what is experimentally found for sample #2. (Since A is field independent,F σ = 0 was assumed in our calculations [35]). We thus conclude that the B-field independent −\|A\|T 1/2 correction to ρ(T ) in ZrAs 1.58 Se 0.39 cannot be caused by EEI and can only be explained by the existence of non-magnetic defects with degenerated ground state which, at low temperatures, place the system near the 2CK fixed point. Note that in the dilute limit, where the dynamic scattering centers are independent of each other, the amplitude \|A\| is proportional to the concentration of dynamic scattering centers in the 2CK regime which is in general smaller than, and not expected to scale in a simple fashion with, the concentration of As vacancies. The chemical composition of ZrAs 1.58 Se 0.39 implies that the vacancies in the As layer are in the dilute limit. As discussed above, our analysis shows that interstitial As does not occur in the pnictogen layer. Each vacancy thus preserves the C 4 symmetry of the pnictogen layer. As a result, a Jahn-Teller distortion forms in the pnictogen layer concomitant with a formation of As dimers or oligomers. This phenomenon is well known to occur in such square nets of the PbFCl structure-type compounds [5]. The flattened displacement ellipsoids shown in Fig. 1(b) are indicative of the occurrence of the dynamic Jahn-Teller effect in the As (2a) layer. As the Jahn-Teller distortion develops, the doublet states split and one of them becomes the new ground state. A finite tunneling rate between the different impurity positions compatible with the overall square symmetry restores the square symmetry through a dynamic Jahn-Teller effect, see Figs. 3(c)-(f). The group C 4 possesses only one-dimensional irreducible representations and one twodimensional irreducible representation (IRREP). Thus, a non-Kramers doublet associated with the dynamic Jahn-Teller distortion transforms as the two-dimensional irreducible representation of the group C 4 , and allows for a two-channel Kondo fixed point to occur at sufficiently low energies [16,17,23]. The pseudo-spin index ± of the effective 2CK Hamiltonian labels the basis states of this two-dimensional subspace [35]. As a result of the coupling to the conduction electrons the doublet can become the ground state [24]. In fact, the doublet is renormalized below the singlet in a wide parameter regime [33]. Our conclusion that 2CK centers exist in ZrAs 1.58 Se 0.39 is further corroborated by the low superconducting transition temperature, as compared to ZrP 1.54 S 0.46 , which points to the presence of efficient Cooper pair breakers [26]. The 2CK effect in ZrAs 1.58 Se 0.39 has been argued to arise out of the non-Kramers doublet transforming as the two-dimensional IRREP of C 4 . The associated basis states of the two-dimensional IRREP, labeled by + and −, are time-reversed partners. Expanding the BCS order parameter around the quantum defect will thus have to involve singlets of + and −. The full Hamiltonian including the quantum defect involves scattering from one of the basis states of the two-dimensional IRREP to the other. These scattering processes therefore have to break up Cooper pairs and thus reduces T c . For details, see [35]. Thus, our model is in line with all observed properties of ZrAs 1.58 Se 0.39 and is not susceptible to the difficulties that exist for the 2CK scenario based on two-level systems. The observed suppression of superconductivity in ZrAs 1.58 Se 0.39 as compared to ZrP 1.54 S 0.46 is naturally explained. A logT -behavior of ρ(T ) is expected for the T -behavior right above the √ T regime for a 2CK system, which is indeed observed [35]. Direct evidence of the dynamic scattering centers could come from scanning tunneling spectroscopy apt to zoom into one of these dynamic scattering centers. In conclusion, we have shown that the B-independent −\|A\|T 1/2 -term in ρ(T ) of ZrAs 1.58 Se 0.39 is triggered by non-magnetic centers with a local quantum degree of freedom. Our analysis indicates that a dynamic Jahn-Teller effect concomitant with the formation of As oligomers places the scattering center in the vicinity of the orbital two-channel Kondo fixed point. These quantum impurities act as efficient Cooper pair breakers which explains the suppression of T c of ZrAs 1.58 Se 0.39 as compared to the homologue ZrP 1.54 S 0. 46 We expect that similar dynamic scattering centers occur in other materials PbFCl structure type with square nets of pnictogen. Single crystals of the ternary phases with PbFCl structure type were grown by Chemical Transport Reaction (CTR) using iodine as transport agent [S1] starting from microcrystalline powders of pre-reacted materials, which were obtained by gradual temperature treatment of mixtures of the elements under inert conditions (vacuum). The amount and chemical composition of deposited crystals depends on the quantity and composition of the source material. A large quantity of source material leads to almost constant transport conditions by keeping temperature of the source and sink as well as the total pressure in the transport ampoule constant. Various positions of the growing crystals within the sink area of the ampoule may cause differences in chemical composition due to different temperature gradients. For this reason and because of the homogeneity regions of the ternary phases under consideration, it is of crucial importance not only to check the total chemical composition of a crystal but to also investigate the constancy of the chemical composition over the crystal individual by WDXS-analyzes as a function of position. Microcrystalline samples of chemical composition ZrP 1.54 S 0.46 and ZrAs 1.58 Se 0.39 were prepared by reaction of the elements using glassy carbon crucibles as containers which were sealed in fused-silica ampoules. Starting from these powders, tetragonal single crystals (up to 2 mm in length) of the ternary phases were grown by exothermal CTR in temperature gradients from 875 • C (source) to 975 • C (sink) by using iodine as the transport agent. Figure S1. Isothermal section of the ternary system Zr-As-Se at 1223 K. The homogenity range of the PbFCl type ternary phase is enlarged, with the dashed line indicating the tie-line between the binary compounds ZrAs2 and ZrSe2 [S2]. Single crystals grown by the CTR in the system Zr-P-S reveal a rather close chemical composition with variations for Zr, P and S below 0.5 wt %: WDXS analyses (Cameca SX 100) on a single crystal resulted in Zr 1.000(4) P 1.543(5) S 0.460(4) ; IC P-OES analyses (Vista RL, Varian) on crystals from the same batch gave Zr 1.00(2) P 1.56(1) S 0.461 (4) , confirming an only narrow range in homogeneity. Crystallographic data: Tetragonal, P 4/nmm, a = 3.5851(1) Å, c = 7.8036(2) Å, Z = 2. A single crystal grown by the CTR in the system Zr-As-Se was investigated by WDXS. The chemical composition was determined as Zr 1.000(3) As 1.595(3) Se 0.393 (1) which is consistent with the results of crystal structure determination and refinements (ZrAs 1.59(1) Se 0.39 (1) , tetragonal, P 4/nmm, a = 3.7576(2) Å, c = 8.0780(5) Å, Z = 2). A linescan of 50 measuring points over a distance of 500 µm along the crystal revealed a homogeneous distribution of Zr and Se, with only small fluctuations in the As content. The ternary systems Zr-As-Se and Zr-P-S contain intermediate phases which crystallize in the tetragonal (P 4/nmm) PbFCl [S3] structure, a substitution variant of the Fe 2 As type. The crystal structure consists of squareplanar 4 4 nets stacked along the [001] direction. For ternary pnictide-chalcogenides (P n-Ch) with precise chemical composition M :P n:Ch=1:1:1 each of the 4 4 nets is exclusively occupied by one element resulting in a layer-sequence ...P n-M -Ch-Ch-M -P n, and forming a puckered double-layer M 2 Ch 2 with an ordered distribution of M and Ch. A representative for ternary pnictide-chalcogenides with precise 1:1:1 composition is given by CeAsSe [S4]. The ternary phases adopting the PbFCl structure type (or more specifically the ZrSiS type) in the systems Zr-As-Se and Zr-P-S are characterized by homogeneity ranges and significant excess of P n (As and P, respectively). In case that the homogeneity range is restricted to the tie-line between ZrP n 2 and ZrCh 2 (Ch = S and Se, respectively), the chemical formula of the ternary phase corresponds to ZrP n x Ch y with x + y = 2, a situation which holds for the zirconium phosphide sulfide. The homogeneity range of the arsenide selenide, however, is shifted from the tieline ZrAs 2 -ZrSe 2 (see Fig. 1) and forms a triangle between the chemical compositions ZrAs 1.65 Se 0.32 , ZrAs 1.38 Se 0.61 and ZrAs 1.40 Se 0.50 [S2]. Detailed structural investigations on single crystals belonging to this homogeneity triangle revealed that the Q(2c) site (see Fig. 1 of the main text) is randomly (but fully) occupied by As and Se. At the same time, vacancies are present within the As layers (2a site) [S2]. In case of the ZrP x S y (x + y = 2) series, vacancies within the P layers are missing (full occupation of the 2a site by P). Quadratic nets in metallic PbFCl-type compounds are expected to distort due to a second-order Jahn-Teller effect, i.e., chemical bond formation [S5] as was observed frequently in rare-earth metal chalcogenides [S6] as well as CeAsSe [S4]. However, for the compounds under consideration no crystallographic symmetry reduction, but rather local displacements of As atoms were observed in X-ray and neutron diffraction. These displacements of As within the quadratic nets in the direction of neighboring As atoms indicate the presence of dimers or oligomers As n , as illustrated in Figs. 3(d)-3(g) of the main text. The displacement factors of As within the plane of the 2a sites clearly increase with increasing vacancy concentration, whereas the phase without a significant amount of vacancies, namely Zr 1.000(4) P 1.543(5) S 0.460(4) , does not show any indication of P displacements. II PHYSICAL MEASUREMENTS After the (destruction-free) chemical analysis and crystallographic investigations, measurements of physical properties were performed on one and the same crystal or by use of parts of the well-characterized specimen. Specific heat measurements in the range 0.4 K ≤ T ≤ 5 K were performed on a ZrAs 1.58 Se 0.39 single crystal with a mass of 19.87(1) mg and a cubic-like shape of approximate edge length 1.4 mm, and a single crystal of ZrP 1.54 S 0.46 with a mass of 5.2(1) mg and a dimension of about 1.4 mm along the c axis via the thermal-relaxation method using a commercial 3 He microcalorimeter (PPMS). For ZrAs 1.58 Se 0.39 , the low-temperature ac susceptibility χ ac (T ) was investigated with the driving field B ac = 0.01 mT utilizing a 3 He-4 He dilution refrigerator. For ZrP 1.54 S 0.46 , dc magnetization measurements were performed for 1.8 K≤ T ≤ 6.5 K and B = 1 mT using a superconducting quantum interference device magnetometer (MPMS). The electrical resistivity was studied by a standard four-point ac technique in zero and applied magnetic fields up to 14 T. For low-temperature measurements, a Linear Research ac resistance bridge (model 700) was utilized applying electrical currents as low as 150 µA in the mK temperature range. In all samples, the resistivity was measured parallel to the crystallographic c axis. From the single crystal of ZrAs 1.58 Se 0.39 two specimens have been cut off, with a length of 1.12 mm (#1) and 0.94 mm (#2) and a cross section of 0.023 mm 2 and 0.032 mm 2 , respectively. For ZrP 1.54 S 0.46 , the specimen had a length of 1.18 mm and a cross section of 0.15 mm 2 . Because of the high fragility of the crystals, electrical contacts for temperature-dependent experiments were made by electrochemical deposition of copper, and a two-needle voltage probe with a fixed distance of 0.548 mm was used to determine absolute values of the resistivity. For the As-based system, the c-axis residual resistivity ρ 0 amounts to about 500 µΩ cm. For the P-based system, one finds a lower value of ρ 0 = 270(±40) µΩ cm and a residual resistivity ratio RRR = 1.65, compared to RRR ≈ 1.1 for ZrAs 1.58 Se 0.39 . This points to a similar, but somewhat less pronounced structural disorder in the free-of-vacancies phase. The magnetoresistance, which is a dimensionless relaxation-time-dependent quantity, provides further evidence for a far-reaching similarity between both ZrP n x Ch y homologues. In fact, also for two ZrAs 1.58 Se 0.39 specimens, neither any temperature dependence nor any deviation from the B 2 behavior was observed up to 20 K, as shown in the inset of Fig. 3(b). From the values of MR ∝ (1/ρ) 2 one can anticipate that ρ 0 of sample #2 excesses that of sample #1 by about 18 %. This underlines that our estimation of a minor (12 %) difference between the measured ρ 0 values is accurate. III SCREENING FACTOR, ELECTRON DIFFUSION CONSTANTS AND OTHER PHYSICAL PARAMETERS OF Zr-BASED PNICTIDE CHALCOGENIDES In order to estimate the screening factor F for single crystalline ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 , we turn to the Hartree-Fock approximation, following the formalism presented in [S7]. A measure of the strength of the electronic correlations is provided by the dimensionless parameter a, which is the ratio between Coulomb potential energy and kinetic energy. The parameter a is thus proportional to the average distance between electrons r s , measured in multiples of the Bohr radius a 0 . We estimate a from a = r s a 0 = 3 4πn where n is the density of the electron gas in three dimensions. Taking the experimental value of n ≈ 0.45 × 10 22 cm −3 , for the ferromagnetic isostructural system UAsSe [S8], we obtain a = 7.1. [It is worth to mentioning that UAsSe (nominal composition) with the Curie temperature of about 110 K displays sample-dependent anomalies in the resistivity far below the ferromagnetic transition [S9].] Consequently, the ratio between the screening vector and the Fermi wavevector k/k F is given by: k k F = 16 3π 2 1 3 m * m 0 1 2 √ a. (S.2) The value of the effective mass m * in ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 can be estimated from the electronic specificheat coefficient γ, which results in k/k F = 2.06 for both Zr-based pnictide chalcogenides. For a 3D electron gas the screening factor F is defined by Finally, it has been pointed out [S10, S11] that in all formulas F should be replaced byF σ given bỹ F σ (F ) = − 32 −(0.5F + 1) 3 In disordered metals with random scattering potential the electron motion is diffusive due to repeated scatterings with the random potential. The diffusive character of electron conduction is characterized by the diffusion constant D, which can be estimated through the Einstein relation, F = k 2 4k 2 F ln 1 + 4k 2 F k 2 .1/ρ 0 = N (E F )e 2 D, (S.5) where N (E F ) is the conduction electron density of states at the Fermi level and ρ 0 is the residual resistivity. N (E F ) can be estimated from the free-electron formula: N (E F ) = 3γ (k B π) 2 , (S.6) where γ is the Sommerfeld coefficient of the molar electronic specific heat and k B is the Boltzmann constant. For both pnictide chalcogenides, we have obtained similar values of N (E F ), i.e., approximately 7.9 × 10 40 J −1 cm −3 and 9.0 × 10 40 J −1 cm −3 for ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 , respectively. Consequently, the electron diffusion constant amounts to D ≈ 1.0 and 1.6 cm 2 s −1 for the phase with and without vacant lattice sites, respectively. The disorder parameter k F l (with l being the mean free path and k F the Fermi wavenumber) turns out to be k F l ≈ 4.2 for ZrAs 1.58 Se 0.39 and k F l ≈ 7.1 for ZrP 1.54 S 0.46 . This estimate has been obtained using the free-electron relation k F l = 3m ⋆ D h , (S.7) where m ⋆ ≈ 1.7m is the effective electron mass obtained from the specific heat (m being the bare electron mass). Because of the small diffusion constants, a strong rather than weak disorder effect is anticipated for both Zr-based pnictide chalcogenides. Table I lists several physical properties for both materials. Finally, we emphasize that our estimate of the residual resistivity (cf. Table I) is accurate for the rather small samples of ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 . This is inferred from the isothermal response of the resistivity to a magnetic field that was studied by taking advantage of the fact that the magnetoresistivity MR = [(ρ(B) − ρ(0)]/ρ(0) is a dimensionless quantity. The MR data are depicted in the inset of Fig. 3(c) of the main text. Neither any temperature dependence nor any deviation from the B 2 behavior was observed in the temperature window 0.1 K -20 K and for B > 1 T. In this field range, the Lorentz force leads to the curvature of the electronic trajectories and hence, a change of the classical transverse MR ∝ (B/ρ) 2 . At B = 14 T, we have observed an MR of, respectively, 0.27 % and 0.18 %, i.e., a difference of about 35 % between the two specimens of ZrAs 1.58 Se 0.39 . Thus, one can anticipate that ρ 0 of sample #2 exceeds that of sample #1 by about 18 %. Note that this estimate, being in satisfactory agreement with the 12 % difference between the measured ρ 0 values, does not alter the discussion presented in Sec. IV. For ZrP 1.54 S 0.46 we have found that the MR along the c axis is 1.4 % at B = 14 T. This is in accord with the lower residual resistivity compared to that in ZrAs 1.58 Se 0.39 , which lends additional support to the similarity in basic physical properties of both Zr-based pnictide chalcogenides. IV ANALYSIS OF THE DEPHASING RATES, NEGLIGIBLE ELECTRON-ELECTRON INTERACTION, AND DYNAMIC SCATTERING CENTERS This section contains our in-depth analysis of the dephasing rates of ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 . It will lead us to the conclusion that dynamic scattering centers in the vacancy-containing pnictogen layer of ZrAs 1.58 Se 0.39 are responsible for the observed low-temperature transport anomalies in ZrAs 1.58 Se 0.39 . Finally, we show that the dynamic scattering centers act as superconducting pair breakers. Thus, the difference in T c of the two systems is a consequence of this particular type of 2CK "impurities". Quantum interference effects in ZrP1.54S0.46 and dephasing in ZrAs1.58Se0.39 In the absence of a magnetic field, superconducting fluctuations dominate the transport properties of ZrP 1.54 S 0.46 leading to a strong decrease of ρ(T ) in a temperature region which significantly exceeds T c ≈ 3.7 K. Under these conditions it is extremely difficult to convincingly separate corrections to the resistivity due to weak localization and EEI [S10, S11]. Thus, we have performed experiments at finite magnetic field to suppress superconductivity and to explore the field dependence of the normal-state resistivity. In Fig. S2, the relative change of the resistivity normalized to the minimum value, (ρ − ρ min )/ρ min , is shown as a function of T 1/2 for fields up to 14 T (with field direction parallel to the current along the c axis). At B = 1 T a small fraction of the ZrP 1.54 S 0.46 sample still displays surface superconductivity below ≈1 K, while at elevated temperatures, we observe an upturn in ρ(T ) which is more complex than the −\|A\|T 1/2 behavior found in ZrAs 1.58 Se 0.39 and is ascribed to weak localization. Note that in three dimensions, the corrections to the resistivity coming from EEI and the weak-localization correction have different temperature dependencies. In the case of the weak-localization correction, the temperature enters only through the relaxation rates for inelastic scattering processes. The standard way to experimentally extract the correction due to EEI is to apply a magnetic field in order to suppress the weak localization [S10, S11]. The effect of an external magnetic field is to destroy the phase coherence of the partial electron waves. Since weak localization leads to an increase of the resistivity upon cooling, its suppression by a magnetic field results in a negative magnetoresistivity. A negative MR in the ZrP 1.54 S 0.46 compound is inferred from the B ≥ 1 T data (cf. Fig. S2). Already at intermediate fields of about 2 T, the (ρ − ρ min )/ρ min correction is smaller than for B = 1 T. For the largest available field of 14 T, the low-T upturn is strongly reduced, and its magnitude is about 7 times smaller than for B = 1 T. Several additional observations can be made: The application of a magnetic field shifts the position of ρ min towards lower temperatures. As a result, at B = 14 T the resistivity of ZrP 1.54 S 0.46 is essentially temperature-independent down to liquid-helium temperature, and its rise is restricted to a rather narrow temperature window, i.e., from above 4.2 K to around 1 K. At T < 1 K, the resistivity is found to saturate. (Similar complex behavior in the same temperature regime is already observed for B = 2 T). Taken together, these experimental findings imply that the absolute value of the weak-localization MR increases upon cooling and amounts to about 0.02 % at T = 2 K. Thus, it is nearly two orders of magnitude smaller than the classical MR (cf. the inset of Fig. 3(c) of the main text). Therefore, the total MR of ZrP 1.54 S 0.46 is essentially temperature independent at T ≤ 10 K and approaches 1.4 % at B = 14 T. The weak-localization correction to ρ(T ) in ZrP 1.54 S 0.46 is in striking contrast to the field-independent −\|A\|T 1/2 term in ρ(T ) of ZrAs 1.58 Se 0.39 . As we shall conclude below, this term must be caused by the presence of dynamic "impurities" that lead to two-channel Kondo physics. Since two-channel Kondo "impurities" act as dephasing centers for conduction electrons due to their dynamic nature, no weak-localization correction in ρ(T ) is expected. Note that the strong dephasing in ZrAs 1.58 Se 0.39 cannot be caused simply by the vacancies. Because of their static nature, vacancies act as elastic scatterer and do not destroy phase coherence. However, this situation changes dramatically when the vacant sites trigger the formation of dynamic scattering centers. Quite generally, internal quantum degrees of freedom of scatterers lead to dephasing in metals [S12]. A well known example are magnetic impurities giving rise to spin-flip scattering [S13]. If, for the specific case of a Kondo impurity, the temperature is reduced to below the Kondo temperature, the dephasing time for the standard (single-channel) Kondo effect diverges for T → 0. In contrast to (standard) magnetic Kondo impurities where the ground-state singlet has zero entropy, a doubly degenerate twochannel Kondo impurity in the non-Fermi liquid regime has a nonzero ground state entropy of 1 2 ln2. This implies that the dephasing time for the two-channel Kondo effect is finite even at T = 0 [S14]. The resistivity results for ZrP 1.54 S 0.46 displayed in Fig. S2 lead to the conclusion that EEI very likely does not contribute to the field-independent −\|A\|T 1/2 term in ρ(T ) of ZrAs 1.58 Se 0.39 : For ZrP 1.54 S 0.46 , the ρ(T ) upturn at B=14 T is about 7 times smaller than the one at B = 1 T. Within the EEI theory for three-dimensional disordered metals [see Eqs. (S.8)-(S.11) and discussion below], a field-independent correction to the resistivity requires very weak screening (F σ ≈ 0) and consequently, a very large value of the diffusion constant, i.e., D = 647 cm 2 s −1 (cf. dashed line in Fig. S2). Estimates of the required diffusion constants for such a field-independent contribution, based on Eq. (S.11), are more than three orders of magnitude larger than the afore-mentioned values obtained from the Einstein relation. Furthermore, for a system with ρ 0 = 270 µΩ cm and D = 1.6 cm 2 s −1 , like e.g. for ZrP 1.54 S 0.46 , one would expect the B-independent (ρ − ρ min )/ρ min term to be as large as about 0.12 % at the lowest temperatures. Thus, if dominant, EEI would cause an easy-to-detect correction to the low-temperature resistivity, as indicated by the dotted line in Fig. S2. Thus, we conclude that the observed −\|A\|T 1/2 term in ρ(T ) of ZrAs 1.58 Se 0.39 cannot be caused by EEI. For more quantitative arguments, see the following subsection. The only mechanism, other than the proximity to a two-channel Kondo fixed point, that can result in a Bindependent −\|A\|T 1/2 correction to the resistivity is the three-dimensional (3D) EEI in the so-called diffusion channel of disordered metals. This may occur in systems with an exceptionally weak electron screening, i.e.F σ ≈ 0. In the presence of a magnetic field, the diffusion-channel correction to the resistivity can be written as a sum of two terms [S10, S11]: ∆ρ ρ EEI (B, T ) = ∆ρ ρ ′ EEI (T ) + ∆ρ ρ ′′ EEI (B, T ), (S.8) where we assume that, in a metal at low temperatures, the corrections to the resistivity, δρ = ρ − ρ 0 , are small (δρ ≪ ρ 0 ). The first term represents the field-independent exchange, S z = 0 Hartree contribution, given by: ∆ρ ρ ′ EEI (T ) = −ρ 0.919e 2 4π 2h 4 3 − 1 2F σ k B T hD , (S.9) where D is the electron diffusion constant and the screening factorF σ is defined in Eq. (S.4). The second term in Eq. (S.8) is the remaining \|S z \| = 1 triplet contribution. Its field dependence is a result of Zeeman spin splitting and can be written as: ∆ρ ρ ′′ EEI (B, T ) = ρ e 2 4 √ 2π 2h g 3 (h)F σ k B T hD , (S.10) with h = gµBB kBT , where g is the Landé factor. Assuming g = 2, the possible values of g 3 (h) at B = 14 T are strongly T dependent and hence vary between 12 to 0.2 when increasing temperature from 0.1 to 10 K (see Fig. S3). In the absence of a magnetic field the EEI correction to the resistivity reduces to the following expression [S11]: with contributions coming both from the singlet as well as the triplet term. ∆ρ ρ EEI (0, T ) = −ρ 0.919e 2 4π 2h 4 3 − 3 2F σ k B T hD , (S.11) It follows from Eq. (S.10) that a B-independent −\|A\|T 1/2 correction to ρ(T ) is in principle possible ifF σ ≈ 0. According to Section III, theoretical values of the corrected screening factor for single crystalline ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 areF σ ≈ 0.66. Therefore, it is inconsistent to attribute the B-field independent −\|A\|T 1/2 term in the resistivity of ZrAs 1.58 Se 0.39 to EEI. To further expand on this, we will now assumeF assumed σ ≈ 0 for ZrAs 1.58 Se 0.39 and demonstrate that this is indeed inconsistent with our experimental results. With this assumption, we have fitted the zero-field ρ(T ) data for samples #1 (ρ #1 = 490 µΩ cm) and #2 (ρ #2 = 550 µΩ cm) leaving D as a free parameter. Satisfactory fits, which are displayed as continuous, straight lines in Fig. S4, require electron diffusion constants D #1 = 124 cm 2 s −1 and D #2 = 26 cm 2 s −1 , respectively. However, such values are unrealistic for several reasons: First of all, a difference in electron diffusion constants by a factor of 6 is not expected between specimens with very similar residual resistivities and densities of states at the Fermi level. Note that the same Sommerfeld coefficient γ = 1.7 mJ/K 2 mol as in ZrAs 1.58 Se 0.39 was reported for ZrAs 1.40 Se 0.50 [S15]. This implies a negligible variation of N (E F ) inside the narrow homogeneity range of the tetragonal Zr-As-Se phase. Furthermore, an essentially sample-independent N (E F ) is expected from the very similar superconducting transition temperatures These values are comparable to typical values of the electron diffusion constant for sp-band amorphous metals or d-band amorphous alloys, which amount to a few of tenths of cm 2 s −1 [S12]. T c ∝ exp[−1/N (E F )] of Hence, we reach the important conclusion that the assumption ofF σ ≈ 0 for ZrAs 1.58 Se 0.39 leads to a non-physical variation of the diffusive motion in (single crystalline) ZrAs 1.58 Se 0.39 . In turn, it is also possible to fit the zero-field ρ(T ) data for samples #1 (ρ #1 = 490 µΩ cm and D #1 = 1.01 cm 2 s −1 ) and #2 (ρ #2 = 550 µΩ cm and D #2 = 0.91 cm 2 s −1 ) to Eq. (S.11) leavingF σ as a free parameter. Least-squares fits yieldF hypo σ = 0.81 and 0.72 for samples #1 and #2, respectively. In such a case of very efficient screening, the magnetic-field-dependent part of the S z = 1 Hartree term is expected to dephase electrons giving rise to a positive magnetoresistivity, whose magnitude significantly increases with decreasing temperature, as shown in the inset of Fig. S3. Consequently, this would lead to large changes of the −\|A\|T 1/2 term in ρ(T ) (cf. the dotted lines in Fig. S4). The results on ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 reported here together with our analysis of possible electron-electron interaction and weak localization corrections imply the existence of a dephasing mechanism associated with vacancies in the pnictogen layers that is compatible only with 2CK scattering centers. Here, we argue that such scattering centers are possible due to the square lattice symmetry of the vacancy carrying As layer and the tendency of such square nets with the PbFCl structure-type compounds to develop a dynamic Jahn-Teller distortion [S5]. The (single-channel) Kondo effect occurs when magnetic impurities interact antiferromagnetically with delocalized electronic states. The interplay of Kondo screening and the Jahn-Teller distortion has been discussed by Gogolin [S16] and the effect of a dynamical Jahn-Teller distortion in cage compounds like skutterudites has been addressed by Hotta [S17]. In contrast, 2CK physics arises when the local moment is coupled to two identical fermionic baths. In this case, overscreening of the impurity degree of freedom occurs and even the strong-coupling fixed point becomes unstable. The system flows to some intermediate effective coupling and displays non-Fermi liquid behavior with a non-vanishing zero-temperature entropy, a logarithmically increasing spin susceptibility, and square-root-in-temperature behavior of the scattering rate. The 2CK model, has turned out to be extremely difficult to realize, as e.g. any channel-symmetry breaking terms will drive the model away from the non-Fermi liquid fixed point. A very interesting proposal to realize two-channel Kondo physics has been made by Zawadowski and Vladár [S18-S20] to explain transport anomalies often seen in metallic glasses. The model by Vladár and Zawadowski is based on atoms in double-well potentials that may tunnel from one minimum to the other. These two minima of the atomic energy give raise to a pseudospin variable. Direct tunneling of the atom between the two minima necessarily splits the degeneracy between the two states and thus corresponds to an effective magnetic field. The conduction electron assisted tunneling of the atom assumes the role of a hybridization term. The conduction electron spin does not participate in assisted tunneling processes and thus gives rise to two degenerate scattering channels protected by time-reversal symmetry. This opens up the possibility for the two-level system to flow to the 2CK fixed point. For energies well below the associated Kondo temperature, the system would therefore display a √ T behavior insensitive to an applied magnetic fields as the Kondo scattering processes are non-magnetic in character [S18-S21]. However, as pointed out by Aleiner et al., this proposal also implies the existence of an effective magnetic field in excesses the associated Kondo scale so that a two-level system may never reach the regime where 2CK physics ensues [S22]. It thus would seem that the existence of non-magnetic 2CK scattering centers in ZrAs 1.58 Se 0.39 is unlikely although our analysis strongly supports their existence. A way out of this conundrum is to consider dynamical defects with higher symmetry [S23]. If the dynamic defect is compatible with a triangular or C 3 symmetry, there exists a doublet whose degeneracy is ensured by symmetry. In this case, the low-energy properties are governed by a fixed point that is identical to the two-channel Kondo fixed point up to irrelevant operators. A difficulty with this proposal is that the doublet is usually higher in energy than the singlet. Under certain conditions, the doublet can become the ground state. For the case of a defect with triangular symmetry, this has been investigated in Ref. [S23]. More recently, the case of an SU(3)-symmetric defect in a metal was investigated by perturbative and NRG methods and it was found that the level spacing between doublet and singlet always renormalizes down as a result of the coupling to the electronic bath [S24]. As a result, the doublet is renormalized below the singlet in a wide parameter range [S24]. Here, we carry over the analysis of Refs. [S23, S24] to vacancies in square nets of the PbFCl structure-type compounds and show that the vacancies in the pnictogen layer give rise to dynamic defects with a C 4 symmetry. The square symmetry implies the existence of doubly generate eigenstates that form an irreducible representation of the group C 4 . The comparably small composition range 1.90 ≤ x + y ≤ 1.99 for which two-channel Kondo physics has been observed in ZrAs x Se y implies that the vacancies in the pnictogen layer are in the dilute limit. Furthermore, as discussed above (see Sec. I), the pnictogen layer P n(2a)-site is only occupied up to 97 % with As. Interstitial As does not occur in this layer. Thus, each vacancy preserves the C 4 symmetry of the pnictogen layer. As a result, a Jahn-Teller distortion forms in the pnictogen layer concomitant with a formation of As dimers or oligomers. This phenomenon is well known to occur in such square nets of the PbFCl structure-type compounds [S5]. The flattened displacement ellipsoids shown in Fig. 2(b) are indicative of the occurrence of the dynamic Jahn-Teller effect in the As (2a) layer. As the Jahn-Teller distortion develops, the doublet states split and one of them becomes the new ground state. A finite tunneling rate between the different impurity positions compatible with the overall square symmetry restores the square symmetry through a dynamic Jahn-Teller effect. The group C 4 possesses only one-dimensional irreducible representations and one two-dimensional irreducible representation (IRREP). Thus, a non-Kramers doublet associated with the dynamic Jahn-Teller distortion transforms as the two-dimensional irreducible representation of the group C 4 , and allows for a two-channel Kondo fixed point to occur at sufficiently low energies [S23]. In this twodimensional subspace, the basis states (labelled ±) allow to introduce a pseudospin label. Expanding the conduction electron states around the quantum impurity yields H = H loc (S.12) + Q 1 σ (d † + d + − d † − d − )(c † +,σ c +,σ − c † −,σ c −,σ ) + ∆ 1 σ (d † + d − c † −,σ c +,σ + d † − d + c † +,σ c −,σ ) + H additional , where H loc contains the local part of the dynamic defect, d † ± creates an electron in the basis state ±, c † ±,σ is the local (energy integrated) conduction electron creation operator projected onto the basis states of the set of irreducible representations of the local symmetry using the great orthogonality theorem to obtain the invariant coupling between the local doublet subspace and the conduction electrons. The term proportional to Q 1 describes the coupling of the z-component of the pseudospins wheres ∆ 1 is the (pseudo-) spin-flip component responsible for Kondo-scattering processes and H additional contains all additional terms in the Hamiltonian (see Ref. [S23] for a discussion of their relevance/irrelevance near the two-channel Kondo fixed point). As the conduction electron spin degree of freedom σ only enters as an overall summation index and the degeneracy in σ is protected by time-reversal symmetry, the model is equivalent to the two-channel Kondo model. A magnetic field breaks the spin degeneracy of the conduction electrons but this only leads to observable effects for a field strength comparable to the bandwidth. That a magnetic field up to 14 Tesla does not affect the anomalous low-temperature behavior of ZrAs 1.58 Se 0.39 is a strong indication for two-channel physics. The Kondo temperature scale below which one observes 2CK physics in ZrAs 1.58 Se 0.39 is comparatively large. This points to a correspondingly large tunneling rate associated with the dynamic defect. This is possible if the associated effective mass is small or the coupling between the As electrons and the conduction electrons is strong enough. The formation of As dimers or oligomers associated with the distortion does imply that the coupling between the As and conduction electrons can be large and can result in a tunneling potential of small effective mass. and transforming back to (radial) momentum c l,m,σ (k) = (−i) l drkrj l (kr)a † l,m,σ (r) (S. 19) where j l (kr) is the lth Bessel function of the first kind. Inserting this expansion into the equation for the superconducting order parameter yields As Cooper pairs are singlets in spin and pseudospin space, the quantum impurity will break Cooper pairs via electron assisted tunneling, i.e. through flipping the pseudospin of an electron in much the same way as magnetic quantum impurities. As a consequence, Anderson's theorem does not apply. ∆ = −V l, For ZrAs 1.58 Se 0.39 , with a small number of such structural dynamic scattering centers, bulk superconductivity still sets in but at much lower temperature than it would without such quantum impurities. It is worth noting that here, the pseudospin is not screened thus giving rise to non-Fermi liquid behavior and the dynamic defects remain pair breaking at lowest temperatures. This is in contrast to Kondo impurities, which obey Anderson's theorem in the limit where the Kondo temperature is much larger than the superconducting transition temperature. VII TEMPERATURE DEPENDENCE OF ρ(T ) ABOVE THE −\|A\| √ T REGIME Figure S5. Low-temperature resistivity (ρ(T ) − ρmin)/ρmin vs. T A logarithmic temperature dependence of the resistivity is expected for the 2CK at temperatures above the T 1/2 behavior. This will be overshadowed by the increase in resistivity due to additional scattering processes, i.e. scattering due to phonons. As a result, the log(T) behavior is observed only in a limited temperature range right below the temperature minimum. We greatly acknowledge helpful discussions with Z. Henkie, J. Kroha, J. J. Lin, P. Ribeiro, F. Zamani and A. Zawadowski. Experimental work on non-magnetic Kondo effect at the Institute of Low Temperature and Structure Research, Polish Academy of Sciences in Wroclaw was supported by the Max Planck Society through the Partner Group Program. S. Kirchner acknowledges partial support by the National Natural Science Foundation of China, grant No.11474250 and the National Science Foundation under Grant No. PHY11-25915. I CRYSTAL GROWTH, CHEMICAL CHARACTERIZATION, AND STRUCTURAL ANALYSIS -Fermi theory one finds F varying between 1 for strong and ≈ 0 for weak screening. For ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 one obtains F = 0.81. Thus, m * and n are compatible with very efficient screening in ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 . Figure S2 . S2Low-temperature electrical resistivity of ZrP1.54S0.46, as (ρ − ρmin)/ρmin vs. T 1/2 , measured along c axis in varying magnetic fields up to 14 T. The slope of the dashed line is a prediction of the 3D EEI theory withFσ ≈ 0 and an electron diffusion constant as large as 647 cm 2 s −1 . The dotted line shows the magnitude of a hypotheticalFσ ≈ 0 EEI correction to the resistivity for a disordered metal with ρ0 = 270 µΩ cm and D = 1.6 cm 2 s −1 , as experimentally found for ZrP1.54S0.46.Quantitative arguments for negligible electron-electron interactions in ZrAs1.58Se0.39 Figure S3 . S3Temperature dependence of the g3(h) function. Calculations, presented on a T 1/2 scale, were done for the constant field B = 14 T. Inset: The diffusion-channel contribution to the magnetoresisitivity for various temperatures. D = 0.91 cm 2 s −1 , ρ = 550 µΩ cm,Fσ = 0.72, g = 2. the ZrAs 1.58 Se 0.39 samples. Finally, estimates of diffusion constants based on Eq. (S.11) are by up to 3 orders of magnitude larger than the corresponding values obtained from the Einstein relation: as shown in Section III, combining the resistivity and specific-heat results yields D #1 = 1.01 cm 2 s −1 and D #2 = 0.91 cm 2 s −1 . Figure S4 . S4Quantitative arguments for negligible EEI in the Zr-As-Se system. Temperature dependence of the normalized zero-field ρ(T ) for the ZrAs1.58Se0.39 samples as (ρ − ρmin)/ρmin vs.T 1/2 . Continuous, straight lines display the 3D EEI theoretical correction forF assumed σ = 0, which implies the unrealistic values D = 124 cm 2 s −1 and 26 cm 2 s −1 for sample #1 and #2 at B = 0, respectively. The Einstein relation yields D = 1.01 cm 2 s −1 and 0.91 cm 2 s −1 instead. Dotted lines indicate the expected results at B = 14 T for F hypo σ = 0.71 and 0.81 respectively, if the T 1/2 term would be caused by EEI. V TWO-CHANNEL KONDO PHYSICS FROM DYNAMIC STRUCTURAL SCATTERING CENTERS l+m c l,m,σ (k)c l,−m,−σ (k) . (S.20) of [35]. As a result, vacancies are solely present within the As layers (2a site) [35].Replacement of As and Se by P and S, respectively,Figure 1. Structural disorder in zirconium pnictide chalco- genides. (a) The PbFCl structure type with the fully oc- cupied Q(2c) site (green) by Ch together with P n. The Pn(2a) site (blue) arranged within planar layers is only oc- cupied to 97 % by As in ZrAs1.58Se0.39, but fully occupied by P in ZrP1.54S0.46. (b) The vacancies in ZrAs1.58Se0.39 (left) manifest random displacements of As within the layer due to homoatomar covalent bond formation. This is in- dicated by flattened displacement ellipsoids in the struc- ture refinements, while the refined displacement ellipsoids in ZrP1.54S0.46 (right) are nearly spherical. 54S0.46 (the two specimens of the former material were cut off from the same single crystal). (b) The same results, but with focus on the vicinity of the superconducting transition, as ρ/ρ0 vs. T /T onset c0.75 1.00 0.05 0.10 0.15 0.20 3.25 3.75 4.25 2 3 4 ( b) (d) (c) ( cm) (a) #1 #2 { T (K) 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Zr P 1.54 S 0.46 5 T T /T c onset (T ) / 0 ZrAs 1.58 Se 0.39 { Zr P 1.54 S 0.46 { 0 T 0.5 T ZrAs 1.58 Se 0.39 C (mJ/Kmol) T (K) Zr P 1.54 S 0.46 ZrAs 1.58 Se 0.39 Zr P 1.54 S 0.46 FC ZFC B = 10 Gs 4 d V T (K) T (K) ac (a.u.) B ac = 0.1 Gs ZrAs 1.58 Se 0.39 0.90 0.92 0.94 0.96 0.98 1.00 C e /T (mJ/K 2 mol) T (K) Figure 2. Characteristics of the pnictide chalcogenide su- perconductors ZrAs1.58Se0.39 and ZrP1.54S0.46. (a) Normal- state electrical resistivity along the c axis of ZrAs1.58Se0.39 and ZrP1. Table I . INormal state and superconducting properties of the ZrAs1.58Se0.39 and ZrP1.54S0.46 single crystals studied in this paper. Note that the transport property estimates are for transport along the c direction.ZrAs1.58Se0.39 #1 ZrAs1.58Se0.39 #2 ZrP1.54S0.46 Residual resistivity ρ0 (µΩ cm) 490 (±40) 550(±40) 270(±40) Residual resistivity ratio RRR 1.19 1.09 1.65 Sommerfeld coefficient γ (mJK −2 mol −1 ) 1.7 1.7 1.7 Debye temperature ΘD (K) 356 356 477 Density of states N (EF) (×10 40 J −1 cm −3 ) 7.9 7.9 9.0 Superconducting critical temperature Tc (K) 0.13 0.14 3.7 Upper critical field Bc2 (T) <0.05 <0.05 <0.5 Electron diffusion constant D (cm 2 s −1 ) 1.0 0.9 1.6 Disorder parameter kFl 4.5 4.0 7.1 if resistivity corrections due to EEI are calculated. Note that for very weak electron screening, i.e. F ≪ 1, the expression (S.4) reduces toF σ ≈ F . For F = 0.81, the value we obtained for ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 ,/2 + 3F 4 + 1 3F , (S.4) F σ (F = 0.81) ≈ 0.76. VI PAIR BREAKING DUE TO DYNAMIC STRUCTURAL SCATTERING CENTERSBoth homologues ZrP n x Ch y develop superconductivity at sufficiently low temperatures, i.e. below T c ≈ 0.14 K for ZrAs1.58Se 0.39 and T c ≈ 3.7 K for ZrP 1.54 S 0.46 (seeTable I). These compounds are standard (BCS) singlet superconductors with fully gapped Fermi surfaces, as can be inferred e.g. from the exponential behavior of the heat capacity in the superconducting state [cf.Fig. 2(c) of the main text]. Despite their similar physical and chemical properties, the transition temperatures of both compounds differ by roughly a factor of 30. This is unexpected. We show here, that this difference is a consequence of the presence of dynamic scattering centers in ZrAs 1.58 Se 0.39 .For weak-coupling superconductors, T c is related to the superconducting gap at zero temperature, ∆(T = 0)which, within BCS theory, obeyswhere ω D is the Debye frequency, N (E F ) is the density of states and V is the strength of the net attractive coupling. The Sommerfeld coefficients for ZrAs 1.58 Se 0.39 and ZrP 1.54 S 0.46 are virtually identical so that N (E F ) of both compounds must be very similar. The difference in the phonon contribution to the heat capacity mainly reflects the difference in the atomic weights of the constituting elements of the two compounds, e.g. As and Se vs. P and S, and thus should result in only a moderate variation of T c . The strong suppression of T c in ZrAs 1.58 Se 0.39 as compared to ZrP 1.54 S 0.46 and its variation across samples from the same sample growth (seeTable I) is another indication that dynamic, non-magnetic quantum impurities exist and that the low-energy behavior in the ZrAs 1.58 Se 0.39 compound is in line with 2CK physics, as we will now demonstrate. Note that according to Anderson's theorem, neither ∆ nor T c of s-wave superconductors are affected by the presence of weak disorder[S25]. (This theorem does not apply when magnetic impurities are present which act as pair-breakers.) To address the effect of Cooper pair scattering off the dynamic impurity with C 4 symmetry, we model the superconducting host by the BCS Hamiltonian,where (within BCS) ∆ = −V k,σ c −k,−σ c k,σ . The order parameter describes the pairing of time-reversed conduction electron states. 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[ "Elastic Graph Neural Networks", "Elastic Graph Neural Networks" ]	[ "Xiaorui Liu ", "Wei Jin ", "Yao Ma ", "Yaxin Li ", "Hua Liu ", "Yiqi Wang ", "Ming Yan ", "Jiliang Tang " ]	[]	[]	While many existing graph neural networks (GNNs) have been proven to perform 2 -based graph smoothing that enforces smoothness globally, in this work we aim to further enhance the local smoothness adaptivity of GNNs via 1 -based graph smoothing. As a result, we introduce a family of GNNs (Elastic GNNs) based on 1 and 2 -based graph smoothing. In particular, we propose a novel and general message passing scheme into GNNs. This message passing algorithm is not only friendly to back-propagation training but also achieves the desired smoothing properties with a theoretical convergence guarantee. Experiments on semi-supervised learning tasks demonstrate that the proposed Elastic GNNs obtain better adaptivity on benchmark datasets and are significantly robust to graph adversarial attacks. The implementation of Elastic GNNs is available at https: //github.com/lxiaorui/ElasticGNN.	null	[ "https://arxiv.org/pdf/2107.06996v1.pdf" ]	235,826,319	2107.06996	f5e4b58e27229e022b59e71d225efbae2fd9dfe3	Elastic Graph Neural Networks Xiaorui Liu Wei Jin Yao Ma Yaxin Li Hua Liu Yiqi Wang Ming Yan Jiliang Tang Elastic Graph Neural Networks While many existing graph neural networks (GNNs) have been proven to perform 2 -based graph smoothing that enforces smoothness globally, in this work we aim to further enhance the local smoothness adaptivity of GNNs via 1 -based graph smoothing. As a result, we introduce a family of GNNs (Elastic GNNs) based on 1 and 2 -based graph smoothing. In particular, we propose a novel and general message passing scheme into GNNs. This message passing algorithm is not only friendly to back-propagation training but also achieves the desired smoothing properties with a theoretical convergence guarantee. Experiments on semi-supervised learning tasks demonstrate that the proposed Elastic GNNs obtain better adaptivity on benchmark datasets and are significantly robust to graph adversarial attacks. The implementation of Elastic GNNs is available at https: //github.com/lxiaorui/ElasticGNN. Introduction Graph neural networks (GNNs) generalize traditional deep neural networks (DNNs) from regular grids, such as image, video, and text, to irregular data such as social networks, transportation networks, and biological networks, which are typically denoted as graphs (Defferrard et al., 2016;Kipf & Welling, 2016). One popular such generalization is the neural message passing framework (Gilmer et al., 2017): x (k+1) u = UPDATE (k) x (k) u , m (k) N (u)(1) where x (k) u ∈ R d denotes the feature vector of node u in k-th iteration of message passing and m (k) N (u) is the message aggregated from u's neighborhood N (u). The specific architecture design has been motivated from spectral domain (Kipf & Welling, 2016;Defferrard et al., 2016) and spatial domain (Hamilton et al., 2017;Veličković et al., 2017;Scarselli et al., 2008;Gilmer et al., 2017). Recent study has proven that the message passing schemes in numerous popular GNNs, such as GCN, GAT, PPNP, and APPNP, intrinsically perform the 2 -based graph smoothing to the graph signal, and they can be considered as solving the graph signal denoising problem: arg min F L(F) := F − X in 2 F + λ tr(F LF),(2) where X in ∈ R n×d is the input signal and L ∈ R n×n is the graph Laplacian matrix encoding the graph structure. The first term guides F to be close to input signal X in , while the second term enforces global smoothness to the filtered signal F. The resulted message passing schemes can be derived by different optimization solvers, and they typically entail the aggregation of node features from neighboring nodes, which intuitively coincides with the cluster or consistency assumption that neighboring nodes should be similar (Zhu & Ghahramani;Zhou et al., 2004). While existing GNNs are prominently driven by 2 -based graph smoothing, 2based methods enforce smoothness globally and the level of smoothness is usually shared across the whole graph. However, the level of smoothness over different regions of the graph can be different. For instance, node features or labels can change significantly between clusters but smoothly within the cluster (Zhu, 2005). Therefore, it is desired to enhance the local smoothness adaptivity of GNNs. Motivated by the idea of trend filtering (Kim et al., 2009;Tibshirani et al., 2014;Wang et al., 2016), we aim to achieve the goal via 1 -based graph smoothing. Intuitively, compared with 2 -based methods, 1 -based methods penalize large values less and thus preserve discontinuity or nonsmooth signal better. Theoretically, 1 -based methods tend to promote signal sparsity to trade for discontinuity (Rudin et al., 1992;Tibshirani et al., 2005;Sharpnack et al., 2012). Owning to these advantages, trend filtering (Tibshirani et al., 2014) and graph trend filter (Wang et al., 2016;Varma et al., 2019) demonstrate that 1 -based graph smoothing can adapt to inhomogenous level of smoothness of signals and yield estimators with k-th order piecewise polynomial functions, such as piecewise constant, linear and quadratic functions, depending on the order of the graph difference operator. arXiv:2107.06996v1 [cs.LG] 5 Jul 2021 While 1 -based methods exhibit various appealing properties and have been extensively studied in different domains such as signal processing (Elad, 2010), statistics and machine learning (Hastie et al., 2015), it has rarely been investigated in the design of GNNs. In this work, we attempt to bridge this gap and enhance the local smoothnesss adaptivity of GNNs via 1 -based graph smoothing. Incorporating 1 -based graph smoothing in the design of GNNs faces tremendous challenges. First, since the message passing schemes in GNNs can be derived from the optimization iteration of the graph signal denoising problem, a fast, efficient and scalable optimization solver is desired. Unfortunately, to solve the associated optimization problem involving 1 norm is challenging since the objective function is composed by smooth and non-smooth components and the decision variable is further coupled by the discrete graph difference operator. Second, to integrate the derived messaging passing scheme into GNNs, it has to be composed by simple operations that are friendly to the back-propagation training of the whole GNNs. Third, it requires an appropriate normalization step to deal with diverse node degrees, which is often overlooked by existing graph total variation and graph trend filtering methods. Our attempt to address these challenges leads to a family of novel GNNs, i.e., Elastic GNNs. Our key contributions can be summarized as follows: • We introduce 1 -based graph smoothing in the design of GNNs to further enhance the local smoothness adaptivity, for the first time; • We derive a novel and general message passing scheme, i.e., Elastic Message Passing (EMP), and develop a family of GNN architectures, i.e., Elastic GNNs, by integrating the proposed message passing scheme into deep neural nets; • Extensive experiments demonstrate that Elastic GNNs obtain better adaptivity on various real-world datasets, and they are significantly robust to graph adversarial attacks. The study on different variants of Elastic GNNs suggests that 1 and 2 -based graph smoothing are complementary and Elastic GNNs are more versatile. Preliminary We use bold upper-case letters such as X to denote matrices and bold lower-case letters such as x to define vectors. Given a matrix X ∈ R n×d , we use X i to denote its i-th row and X ij to denote its element in i-th row and j-th column. We define the Frobenius norm, 1 norm, and 21 norm of matrix X as X F = ij X 2 ij , X 1 = ij \|X ij \|, and X 21 = i X i 2 = i j X 2 ij , respectively. We define X 2 = σ max (X) where σ max (X) is the largest singular value of X. Given two matrices X, Y ∈ R n×d , we define the inner product as X, Y = tr(X Y). Let G = {V, E} be a graph with the node set V = {v 1 , . . . , v n } and the undirected edge set E = {e 1 , . . . , e m }. We use N (v i ) to denote the neighboring nodes of node v i , including v i itself. Suppose that each node is associated with a d-dimensional feature vector, and the features for all nodes are denoted as X fea ∈ R n×d . The graph structure G can be represented as an adjacent matrix A ∈ R n×n , where A ij = 1 when there exists an edge between nodes v i and v j . The graph Laplacian matrix is defined as L = D−A, where D is the diagonal degree matrix. Let ∆ ∈ {−1, 0, 1} m×n be the oriented incident matrix, which contains one row for each edge. If e = (i, j), then ∆ has -th row as: ∆ = (0, . . . , −1 i , . . . , 1 j , . . . , 0) where the edge orientation can be arbitrary. Note that the incident matrix and unnormalized Laplacian matrix have the equivalence L = ∆ ∆. Next, we briefly introduce some necessary background about the graph signal denoising perspective of GNNs and the graph trend filtering methods. GNNs as Graph Signal Denoising It is evident from recent work ) that many popular GNNs can be uniformly understood as graph signal denoising with Laplacian smoothing regularization. Here we briefly describe several representative examples. GCN. The message passing scheme in Graph Convolutional Networks (GCN) (Kipf & Welling, 2016), X out =ÃX in , is equivalent to one gradient descent step to minimize tr(F (I −Ã)F) with the initial F = X in and stepsize 1/2. HereÃ =D − 1 2ÂD − 1 2 withÂ = A + I being the adjacent matrix with self-loop, whose degree matrix isD. PPNP & APPNP. The message passing scheme in PPNP and APPNP (Klicpera et al., 2018) follow the aggregation rules X out = α I − (1 − α)Ã −1 X in , and X (k+1) = (1 − α)ÃX (k) + αX in . They are shown to be the exact solution and one gradient descent step with stepsize α/2 for the following problem min F F − X in 2 F + (1/α − 1) tr(F (I −Ã)F). (3) For more comprehensive illustration, please refer to . We point out that all these message passing schemes adopt 2 -based graph smoothing as the signal differences between neighboring nodes are penalized by the square of 2 norm, e.g., (vi,vj )∈E Fi √ di+1 − Fj √ dj +1 2 2 with d i being the node degree of node v i . The resulted message passing schemes are usually linear smoothers which smooth the input signal by their linear transformation. Graph Trend Filtering In the univariate case, the k-th order graph trend filtering (GTF) estimator (Wang et al., 2016) is given by arg min f ∈R n = 1 2 f − x 2 2 + λ ∆ (k+1) f 1(4) where x ∈ R n is the 1-dimensional input signal of n nodes and ∆ (k+1) is a k-th order graph difference operator. When k = 0, it penalizes the absolute difference across neighboring nodes in graph G: ∆ (1) f 1 = (vi,vj )∈E \|f i − f j \| where ∆ (1) is equivalent to the incident matrix ∆. Generally, k-th order graph difference operators can be defined recursively: ∆ (k+1) = ∆ ∆ (k) = L k+1 2 ∈ R n×n for odd k ∆∆ (k) = ∆L k 2 ∈ R m×n for even k. It is demonstrated that GTF can adapt to inhomogeneity in the level of smoothness of signal and tends to provide piecewise polynomials over graphs (Wang et al., 2016). For instance, when k = 0, the sparsity induced by the 1 -based penalty ∆ (1) f 1 implies that many of the differences f i −f j are zeros across edges (v i , v j ) ∈ E in G. The piecewise property originates from the discontinuity of signal allowed by less aggressive 1 penalty, with adaptively chosen knot nodes or knot edges. Note that the smoothers induced by GTF are not linear smoothers and cannot be simply represented by linear transformation of the input signal. Elastic Graph Neural Networks In this section, we first propose a new graph signal denoising estimator. Then we develop an efficient optimization algorithm for solving the denoising problem and introduce a novel, general and efficient message passing scheme, i.e., Elastic Message Passing (EMP), for graph signal smoothing. Finally, the integration of the proposed message passing scheme and deep neural networks leads to Elastic GNNs. Elastic Graph Signal Estimator To combine the advantages of 1 and 2 -based graph smoothing, we propose the following elastic graph signal estimator: arg min F∈R n×d λ 1 ∆F 1 g1(∆F) + λ 2 2 tr(F LF) + 1 2 F − X in 2 F f (F)(5) where X in ∈ R n×d is the d-dimensional input signal of n nodes. The first term can be written in an edge-centric way: ∆ (1) F 1 = (vi,vj )∈E F i − F j 1 , which penalizes the absolute difference across connected nodes in graph G. Similarly, the second term penalizes the difference quadratically via tr(F LF) = (vi,vj )∈E F i − F j 2 2 . The last term is the fidelity term which preserves the similarity with the input signal. The regularization coefficients λ 1 and λ 2 control the balance between 1 and 2 -based graph smoothing. Remark 1. It is potential to consider higher-order graph differences in both the 1 -based and 2 -based smoothers. But, in this work, we focus on the 0-th order graph difference operator ∆, since we assume the piecewise constant prior for graph representation learning. Normalization. In existing GNNs, it is beneficial to normalize the Laplacian matrix for better numerical stability, and the normalization trick is also crucial for achieving superior performance. Therefore, for the 2 -based graph smoothing, we follow the common normalization trick in GNNs: L = I −Ã, whereÃ =D − 1 2ÂD − 1 2 ,Â = A + I andD ii = d i = jÂ ij . It leads to a degree normalized penalty tr(F L F) = (vi,vj )∈E F i √ d i + 1 − F j d j + 1 2 2 . In the literature of graph total variation and graph trend filtering, the normalization step is often overlooked and the graph difference operator is directly used as in GTF (Wang et al., 2016;Varma et al., 2019). To achieve better numerical stability and handle diverse node degrees in real-world graphs, we propose to normalize each column of the incident matrix by the square root of node degrees for the 1 -based graph smoothing as follows 1 :∆ = ∆D − 1 2 . It leads to a degree normalized total variation penalty 2 ∆ F 1 = (vi,vj )∈E F i √ d i + 1 − F j d j + 1 1 . Note that this normalized incident matrix maintains the relation with the normalized Laplacian matrix as in the unnormalized caseL =∆ ∆ given that L =D − 1 2 (D −Â)D − 1 2 =D − 1 2 LD − 1 2 =D − 1 2 ∆ ∆D − 1 2 . With the normalization, the estimator defined in (5) becomes: arg min F∈R n×d λ 1 ∆ F 1 g1(∆F) + λ 2 2 tr(F L F) + 1 2 F − X in 2 F f (F) .(7) Capture correlation among dimensions. The node features in real-world graphs are usually multi-dimensional. Although the estimator defined in (7) is able to handle multidimensional data since the signal from different dimensions are separable under 1 and 2 norm, such estimator treats each feature dimension independently and does not exploit the potential relation between feature dimensions. However, the sparsity patterns of node difference across edges could be shared among feature dimensions. To better exploit this potential correlation, we propose to couple the multi-dimensional features by 21 norm, which penalizes the summation of 2 norm of the node difference ∆ F 21 = (vi,vj )∈E F i √ d i + 1 − F j d j + 1 2 . This penalty promotes the row sparsity of∆F and enforces similar sparsity patterns among feature dimensions. In other words, if two nodes are similar, all their feature dimensions should be similar. Therefore, we define the 21 -based estimator as arg min F∈R n×d λ 1 ∆ F 21 g21(∆F) + λ 2 2 tr(F L F) + 1 2 F − X in 2 F f (F) (8) where g 21 (·) = λ 1 · 21 . In the following subsections, we will use g(·) to represent both g 1 (·) and g 21 (·). We use 1 to represent both 1 and 21 if not specified. Elastic Message Passing For the 2 -based graph smoother, message passing schemes can be derived from the gradient descent iterations of the graph signal denoising problem, as in the case of GCN and APPNP . However, computing the estimators defined by (7) and (8) is much more challenging because of the nonsmoothness, and the two components, i.e., f (F) and g(∆F), are non-separable as they are coupled by the graph difference operator∆. In the literature, researchers have developed optimization algorithms for the graph trend filtering problem (4) such as Alternating Direction Method of Multipliers (ADMM) and Newton type algorithms (Wang et al., 2016;Varma et al., 2019). However, these algorithms require to solve the minimization of a nontrivial sub-problem in each single iteration, which incurs high computation complexity. Moreover, it is unclear how to make these iterations compatible with the back-propagation training of deep learning models. This motivates us to design an algorithm which is not only efficient but also friendly to back-propagation training. To this end, we propose to solve an equivalent saddle point problem using a primal-dual algorithm with efficient computations. Saddle point reformulation. For a general convex function g(·), its conjugate function is defined as g * (Z) := sup X Z, X − g(X) . (7) and (8) can be equivalently written as the following saddle point problem: By using g(∆F) = sup Z ∆ F, Z − g * (Z), the problemmin F max Z f (F) + ∆ F, Z − g * (Z).(9) where Z ∈ R m×d . Motivated by Proximal Alternating Predictor-Corrector (PAPC) (Loris & Verhoeven, 2011;Chen et al., 2013), we propose an efficient algorithm with per iteration low computation complexity and convergence guarantee:     F k+1 = F k − γ∇f (F k ) − γ∆ Z k ,(10)Z k+1 = prox βg * (Z k + β∆F k+1 ),(11)F k+1 = F k − γ∇f (F k ) − γ∆ Z k+1 ,(12) where prox βg * (X) = arg min Y 1 2 Y − X 2 F + βg * (Y) . The stepsizes, γ and β, will be specified later. The first step (10) obtains a prediction of F k+1 , i.e.,F k+1 , by a gradient descent step on primal variable F k . The second step (11) is a proximal dual ascent step on the dual variable Z k based on the predictedF k+1 . Finally, another gradient descent step on the primal variable based on (F k , Z k+1 ) gives next iteration F k+1 (12). Algorithm (10)-(12) can be interpreted as a "predict-correct" algorithm for the saddle point problem (9). Next we demonstrate how to compute the proximal operator in Eq. (11). Proximal operators. Using the Moreau's decomposition principle (Bauschke & Combettes, 2011) X = prox βg * (X) + βprox β −1 g (X/β), we can rewrite the step (11) using the proximal operator of g(·), that is, prox βg * (X) = X − βprox 1 β g ( 1 β X).(13) We discuss the two options for the function g(·) corresponding to the objectives (7) and (8). • Option I ( 1 norm): g 1 (X) = λ 1 X 1                        Y k+1 = γX in + (1 − γ)ÃF k F k+1 = Y k − γ∆ Z k Z k+1 = Z k + β∆F k+1    Z k+1 = min(\|Z k+1 \|, λ 1 ) · sign(Z k+1 ) (Option I: 1 norm) Z k+1 i = min( Z k+1 i 2 , λ 1 ) ·Z k+1 i Z k+1 i 2 , ∀i ∈ [m] (Option II: 21 norm) F k+1 = Y k − γ∆ Z k+1 By definition, the proximal operator of 1 β g 1 (X) is prox 1 β g1 (X) = arg min Y 1 2 Y − X 2 F + 1 β λ 1 Y 1 , which is equivalent to the soft-thresholding operator (component-wise): (S 1 β λ1 (X)) ij =sign(X ij ) max(\|X ij \| − 1 β λ 1 , 0) =X ij − sign(X ij ) min(\|X ij \|, 1 β λ 1 ). Therefore, using (13), we have (prox βg * 1 (X)) ij = sign(X ij ) min(\|X ij \|, λ 1 ). (14) which is a component-wise projection onto the ∞ ball of radius λ 1 . • Option II ( 21 norm): g 21 (X) = λ 1 X 21 By definition, the proximal operator of 1 β g 21 (X) is prox 1 β g21 (X) = arg min Y 1 2 Y − X 2 F + 1 β λ 1 Y 21 with the i-th row being prox 1 β g21 (X) i = X i X i 2 max( X i 2 − 1 β λ 1 , 0). Similarly, using (13), we have the i-th row of prox βg * 21 (X) being (prox βg * 21 (X)) i = X i − βprox 1 β g21 (X i /β) = X i − β X i /β X i /β 2 max( X i /β 2 − λ 1 /β, 0) = X i − X i X i 2 max( X i 2 − λ 1 , 0) = X i X i 2 ( X i 2 − max( X i 2 − λ 1 , 0)) = X i X i 2 min( X i 2 , λ 1 ),(15) which is a row-wise projection on the 2 ball of radius λ 1 . Note that the proximal operator in the 1 norm case treats each feature dimension independently, while in the 21 norm case, it couples the multi-dimensional features, which is consistent with the motivation to exploit the correlation among feature dimensions. The Algorithm (10)-(12) and the proximal operators (14) and (15) enable us to derive the final message passing scheme. Note that the computation F k − γ∇f (F k ) in steps (10) and (12) can be shared to save computation. Therefore, we decompose the step (10) into two steps: Y k = F k − γ∇f (F k ) = (1 − γ)I − γλ 2L F k + γX in ,(16)F k+1 = Y k − γ∆ Z k .(17) In this work, we choose γ = 1 1+λ2 and β = 1 2γ . Therefore, withL = I −Ã, Eq. (16) can be simplified as Y k+1 = γX in + (1 − γ)ÃF k .(18) LetZ k+1 := Z k + β∆F k+1 , then steps (11) and (12) become Z k+1 = prox βg * (Z k+1 ),(19)F k+1 = F k − γ∇f (F k ) − γ∆Z k+1 = Y k − γ∆ Z k+1 .(20) Substituting the proximal operators in (19) with (14) and (15), we obtain the complete elastic message passing scheme (EMP) as summarized in Figure 1. Interpretation of EMP. EMP can be interpreted as the standard message passing (MP) (Y in Fig. 1) with extra operations (the following steps). The extra operations computẽ ∆ Z to adjust the standard MP such that sparsity in∆F is promoted and some large node differences can be preserved. EMP is general and covers some existing propagation rules as special cases as demonstrated in Remark 2. Remark 2 (Special cases). If there is only 2 -based regularization, i.e., λ 1 = 0, then according to the projection operator, we have Z k = 0 m×n . Therefore, with γ = 1 1+λ2 , the proposed message passing scheme reduces to F k+1 = 1 1 + λ 2 X in + λ 2 1 + λ 2Ã F k . If λ 2 = 1 α − 1, it recovers the message passing in APPNP: F k+1 = αX in + (1 − α)ÃF k . If λ 2 = ∞, it recovers the simple aggregation operation in many GNNs: F k+1 =ÃF k . Computation Complexity. EMP is efficient and composed by simple operations. The major computation cost comes from four sparse matrix multiplications, includẽ AF k ,∆ Z k ,∆F k+1 and∆ Z k+1 . The computation complexity is in the order O(md) where m is the number of edges in graph G and d is the feature dimension of input signal X in . Other operations are simple matrix additions and projection. The convergence of EMP and the parameter settings are justified by Theorem 1, with a proof deferred to Appendix B. Theorem 1 (Convergence). Under the stepsize setting γ < 2 1+λ2 L 2 and β ≤ 4 3γ ∆∆ 2 , the elastic message passing scheme (EMP) in Figure 1 converges to the optimal solution of the elastic graph signal estimator defined in (7) (Option I) or (8) (Option II). It is sufficient to choose any γ < 2 1+2λ2 and β ≤ 2 3γ since L 2 = ∆ ∆ 2 = ∆∆ 2 ≤ 2. Elastic GNNs Incorporating the elastic message passing scheme from the elastic graph signal estimator (7) and (8) into deep neural networks, we introduce a family of GNNs, namely Elastic GNNs. In this work, we follow the decoupled way as proposed in APPNP (Klicpera et al., 2018), where we first make predictions from node features and aggregate the prediction through the proposed EMP: Y pre = EMP h θ (X fea ), K, λ 1 , λ 2 .(21) X fea ∈ R n×d denotes the node features, h θ (·) is any machine learning model, such as multilayer perceptrons (MLPs), θ is the learnable parameters in the model, and K is the number of message passing steps. The training objective is the cross entropy loss defined by the final prediction Y pre and labels for training data. Elastic GNNs also have the following nice properties: • In addition to the backbone neural network model, Elastic GNNs only require to set up three hyperparameters including two coefficients λ 1 , λ 2 and the propagation step K, but they do not introduce any learnable parameters. Therefore, it reduces the risk of overfitting. • The hyperparameters λ 1 and λ 2 provide better smoothness adaptivity to Elastic GNNs depending on the smoothness properties of the graph data. • The message passing scheme only entails simple and efficient operations, which makes it friendly to the efficient and end-to-end back-propagation training of the whole GNN model. Experiment In this section, we conduct experiments to validate the effectiveness of the proposed Elastic GNNs. We first introduce the experimental settings. Then we assess the performance of Elastic GNNs and investigate the benefits of introducing Experimental Settings Datasets. We conduct experiments on 8 real-world datasets including three citation graphs, i.e., Cora, Citeseer, Pubmed (Sen et al., 2008), two co-authorship graphs, i.e., Coauthor CS and Coauthor Physics (Shchur et al., 2018), two co-purchase graphs, i.e., Amazon Computers and Amazon Photo (Shchur et al., 2018), and one blog graph, i.e., Polblogs (Adamic & Glance, 2005). In Polblogs graph, node features are not available so we set the feature matrix to be a n × n identity matrix. Baselines. We compare the proposed Elastic GNNs with representative GNNs including GCN (Kipf & Welling, 2016), GAT (Veličković et al., 2017), ChebNet (Defferrard et al., 2016), GraphSAGE (Hamilton et al., 2017), APPNP (Klicpera et al., 2018) and SGC (Wu et al., 2019). For all models, we use 2 layer neural networks with 64 hidden units. Parameter settings. For each experiment, we report the average performance and the standard variance of 10 runs. For all methods, hyperparameters are tuned from the following search space: 1) learning rate: {0.05, 0.01, 0.005}; 2) weight decay: {5e-4, 5e-5, 5e-6}; 3) dropout rate: {0.5, 0.8}. For APPNP, the propagation step K is tuned from {5, 10} and the parameter α is tuned from {0, 0.1, 0.2, 0.3, 0.5, 0.8, 1.0}. For Elastic GNNs, the propagation step K is tuned from {5, 10} and parameters λ 1 and λ 2 are tuned from {0, 3, 6, 9}. As suggested by Theorem 1, we set γ = 1 1+λ2 and β = 1 2γ in the proposed elastic message passing scheme. Adam optimizer (Kingma & Ba, 2014) is used in all experiments. Performance on Benchmark Datasets On commonly used datasets including Cora, CiteSeer, PubMed, Coauthor CS, Coauthor Physics, Amazon Computers and Amazon Photo, we compare the performance of the proposed Elastic GNN ( 21 + 2 ) with representative GNN baselines on the semi-supervised learning task. The detail statistics of these datasets and data splits are summarized in Table 5 in Appendix A. The classification accuracy are showed in Table 1. From these results, we can make the following observations: • Elastic GNN outperforms GCN, GAT, ChebNet, Graph-SAGE and SGC by significant margins on all datasets. For instance, Elastic GNN improves over GCN by 3.1%, 2.0% and 1.8% on Cora, CiteSeer and PubMed datasets. The improvement comes from the global and local smoothness adaptivity of Elastic GNN. • Elastic GNN ( 21 + 2 ) consistently achieves higher performance than APPNP on all datasets. Essentially, Elastic GNN covers APPNP as a special case when there is only 2 regularization, i.e., λ 1 = 0. Beyond the 2 -based graph smoothing, the 21 -based graph smoothing further enhances the local smoothness adaptivity. This comparison verifies the benefits of introducing 21 -based graph smoothing in GNNs. Robustness Under Adversarial Attack Locally adaptive smoothness makes Elastic GNNs more robust to adversarial attack on graph structure. This is because the attack tends to connect nodes with different labels, which fuzzes the cluster structure in the graph. But EMP can tolerate large node differences along these wrong edges, and maintain the smoothness along correct edges. To validate this, we evaluate the performance of Elastic GNNs under untargeted adversarial graph attack, which tries to degrade GNN models' overall performance by deliberately modifying the graph structure. We use the MetaAttack (Zügner & Günnemann, 2019) implemented in Deep-Robust (Li et al., 2020) 3 , a PyTorch library for adversarial attacks and defenses, to generate the adversarially attacked graphs based on four datasets including Cora, CiteSeer, Polblogs and PubMed. We randomly split 10%/10%/80% of nodes for training, validation and test. The detailed data statistics are summarized in Table 6 in Appendix A. Note that following the works (Zügner et al., 2018;Zügner & Günnemann, 2019;Entezari et al., 2020;Jin et al., 2020), we only consider the largest connected component (LCC) in the adversarial graphs. Therefore, the results in Table 2 are not directly comparable with the results in Table 1. We Variants of Elastic GNNs. To make a deeper investigation of Elastic GNNs, we consider the following variants: (1) 2 (λ 1 = 0); (2) 1 (λ 2 = 0, Option I); (3) 21 (λ 2 = 0, Option II); (4) 1 + 2 (Option I); (5) 21 + 2 (Option II). To save computation, we fix the learning rate as 0.01, weight decay as 0.0005, dropout rate as 0.5 and K = 10 since this setting works well for the chosen datasets and models. Only λ 1 and λ 2 are tuned. The classification accuracy under different perturbation rates ranging from 0% to 20% is summarized in Table 2. From the results, we can make the following observations: • All variants of Elastic GNNs outperforms GCN and GAT by significant margins under all perturbation rates. For instance, when the pertubation rate is 15%, Elastic GNN ( 21 + 2 ) improves over GCN by 12.1%, 7.4%, 13.7% and 7.7% on the four datasets being considered. This is because Elastic GNN can adapt to the change of smoothness while GCN and GAT can not adapt well when the perturbation rate increases. • 21 outperforms 1 in most cases, and 21 + 2 outperforms 1 + 2 in almost all cases. It demonstrates the benefits of exploiting the correlation between feature channels by coupling multi-dimensional features via 21 norm. • 21 outperforms 2 in most cases, which suggests the benefits of local smoothness adaptivity. When 21 and 2 is combined, the Elastic GNN ( 21 + 2 ) achieves significantly better performance than solely 2 , 21 or 1 variant in almost all cases. It suggests that 1 and 2 -based graph smoothing are complementary to each other, and combining them provides significant better robustness against adversarial graph attacks. Ablation Study We provide ablation study to further investigate the adaptive smoothness, sparsity pattern, and convergence of EMP in Elastic GNN, based on three datasets including Cora, Cite-Seer and PubMed. In this section, we fix λ 1 = 3, λ 2 = 3 for Elastic GNN, and α = 0.1 for APPNP. We fix learning rate as 0.01, weight decay as 0.0005 and dropout rate as 0.5 since this setting works well for both methods. Adaptive smoothness. It is expected that 1 -based smoothing enhances local smoothness adaptivity by increasing the smoothness along correct edges (connecting nodes with same labels) while lowering smoothness along wrong edges (connecting nodes with different labels). To validate this, we compute the average adjacent node differences (based on node features in the last layer) along wrong and correct edges separately, and use the ratio between these two averages to measure the smoothness adaptivity. The results are summarized in Table 3. It is clearly observed that for all datasets, the ratio for ElasticGNN is significantly higher than 2 based method such as APPNP, which validates its better local smoothness adaptivity. Sparsity pattern. To validate the piecewise constant property enforced by EMP, we also investigate the sparsity pattern in the adjacent node differences, i.e.,∆F, based on node features in the last layer. Node difference along edge e i is defined as sparse if (∆F) i 2 < 0.1. The sparsity ratios for 2 -based method such as APPNP and 1 -based method such as Elastic GNN are summarized in Table 4. It can be observed that in Elastic GNN, a significant portion of∆F are sparse for all datasets. While in APPNP, this portion is much smaller. This sparsity pattern validates the piecewise constant prior as designed. Convergence of EMP. We provide two additional experiments to demonstrate the impact of propagation step K on classification performance and the convergence of message passing scheme. Figure 2 shows that the increase of classifi-cation accuracy when the propagation step K increases. It verifies the effectiveness of EMP in improving graph representation learning. It also shows that a small number of propagation step can achieve very good performance, and therefore the computation cost for EMP can be small. Figure 3 shows the decreasing of the objective value defined in Eq. (8) during the forward message passing process, and it verifies the convergence of the proposed EMP as suggested by Theorem 1. Related Work The design of GNN architectures can be majorly motivated in spectral domain (Kipf & Welling, 2016;Defferrard et al., 2016) and spatial domain (Hamilton et al., 2017;Veličković et al., 2017;Scarselli et al., 2008;Gilmer et al., 2017). The message passing scheme (Gilmer et al., 2017; for feature aggregation is one central component of GNNs. Recent works have proven that the message passing in GNNs can be regarded as low-pass graph filters (Nt & Maehara, 2019;Zhao & Akoglu, 2019). Generally, it is recently proved that message passing in many GNNs can be unified in the graph signal denosing framework Pan et al., 2020;Zhu et al., 2021;Chen et al., 2020). We point out that they intrinsically perform 2 -based graph smoothing and typically can be represented as linear smoothers. 1 -based graph signal denoising has been explored in graph trend filtering (Wang et al., 2016;Varma et al., 2019) which tends to provide estimators with k-th order piecewise polynomials over graphs. Graph total variation has also been utilized in semi-supervised learning (Nie et al., 2011;Jung et al., 2016;Jung et al., 2019;Aviles-Rivero et al., 2019), spectral clustering (Bühler & Hein, 2009;Bresson et al., 2013b) and graph cut problems (Szlam & Bresson, 2010;Bresson et al., 2013a). However, it is unclear whether these algorithms can be used to design GNNs. To the best of our knowledge, we make first such investigation in this work. Conclusion In this work, we propose to enhance the smoothness adaptivity of GNNs via 1 and 2 -based graph smoothing. Through the proposed elastic graph signal estimator, we derive a novel, efficient and general message passing scheme, i.e., elastic message passing (EMP). Integrating the proposed message passing scheme and deep neural networks leads to a family of GNNs, i.e., Elastic GNNs. Extensitve experiments on benchmark datasets and adversarially attacked graphs demonstrate the benefits of introducing 1 -based graph smoothing in the design of GNNs. The empirical study suggests that 1 and 2 -based graph smoothing is complementary to each other, and the proposed Elastic GNNs has better smoothnesss adaptivity owning to the integration of 1 and 2 -based graph smoothing. We hope the proposed elastic message passing scheme can inspire more powerful GNN architecture design in the future. B. Convergence Guarantee We provide Theorem 1 to show the convergence guarantee of the proposed elastic messsage passing scheme and the practical guidance for parameter settings in EMP. Theorem 1 (Convergence of EMP). Under the stepsize setting γ < 2 1+λ2 L 2 and β ≤ 4 3γ ∆∆ 2 , the elastic message passing scheme (EMP) in Figure 1 converges to the optimal solution of the elastic graph signal estimator defined in (7) (Option I) or (8) (Option II). It is sufficient to choose any γ < 2 1+2λ2 and β ≤ 2 3γ since L 2 = ∆ ∆ 2 = ∆∆ 2 ≤ 2. Proof. We first consider the general problem min F f (F) + g(BF)(22) where f and g are convex functions and B is a bounded linear operator. It is proved in (Loris & Verhoeven, 2011;Chen et al., 2013) that the iterations in (10)-(12) guarantee the convergence of F k to the optimal solution of the minimization problem (22) if the parameters satisfy γ < 2 L and β ≤ 1 γλmax(BB ) , where L is the Lipschitz constant of ∇f (F). These conditions are further relaxed to γ < 2 L and β ≤ 4 3γλmax(BB ) in (Li & Yan, 2017). For the specific problems defined in (7) and (8), the two function components f and g are both convex, and the linear operator ∆ is bounded. The Lipschitz constant of ∇f (F) can be computed by the largest eigenvalue of the Hessian matrix of f (F): L = λ max (∇ 2 f (F)) = λ max (I + λ 2L ) = 1 + λ 2 L 2 . Therefore, the elastic message passing scheme derived from iterations (10)-(12) is guaranteed to converge to the optimal solution of problem (7) (Option I) or problem (8) (Option II) if the stepsizes satisfy γ < 2 1+λ2 L 2 and β ≤ 4 3γ ∆∆ 2 . Elastic Graph Neural Networks Let∆ = UΣV be the singular value decomposition of∆, and we derive ∆∆ 2 = UΣV VΣU 2 = UΣ 2 U 2 = VΣ 2 V 2 = VΣU UΣV 2 = ∆ ∆ 2 . The equivalenceL =∆ ∆ in (6) further gives L 2 = ∆ ∆ 2 = ∆∆ 2 . Since L 2 ≤ 2 (Chung & Graham, 1997), we have 2 1+2λ2 ≤ 2 1+λ2 L 2 and 2 3γ ≤ 4 3γ ∆∆ 2 . Therefore, γ < 2 1+2λ2 β ≤ 2 3γ are sufficient for the convergence of EMP. Figure 1 . 1Elastic Message Passing (EMP). F 0 = Xin and Z 0 = 0 m×d . Figure 2 . 2Classification accuracy under different propagation steps. Figure 3 . 3Convergence of the objective value for the problem in Eq. (8) during message passing. Table 1 . 1Classification accuracy (%) on benchmark datasets with 10 times random data splits.Table 2. Classification accuracy (%) under different perturbation rates of adversarial graph attack.Model Cora CiteSeer PubMed CS Physics Computers Photo ChebNet 76.3 ± 1.5 67.4 ± 1.5 75.0 ± 2.0 91.8 ± 0.4 OOM 81.0 ± 2.0 90.4 ± 1.0 GCN 79.6 ± 1.1 68.9 ± 1.2 77.6 ± 2.3 91.6 ± 0.6 93.3 ± 0.8 79.8 ± 1.6 90.3 ± 1.2 GAT 80.1 ± 1.2 68.9 ± 1.8 77.6 ± 2.2 91.1 ± 0.5 93.3 ± 0.7 79.3 ± 2.4 89.6 ± 1.6 SGC 80.2 ± 1.5 68.9 ± 1.3 75.5 ± 2.9 90.1 ± 1.3 93.1 ± 0.6 73.0 ± 2.0 83.5 ± 2.9 APPNP 82.2 ± 1.3 70.4 ± 1.2 78.9 ± 2.2 92.5 ± 0.3 93.7 ± 0.7 80.1 ± 2.1 90.8 ± 1.3 GraphSAGE 79.0 ± 1.1 67.5 ± 2.0 77.6 ± 2.0 91.7 ± 0.5 92.5 ± 0.8 80.7 ± 1.7 90.9 ± 1.0 ElasticGNN 82.7 ± 1.0 70.9 ± 1.4 79.4 ± 1.8 92.5 ± 0.3 94.2 ± 0.5 80.7 ± 1.8 91.3 ± 1.3 Dataset Ptb Rate Basic GNN Elastic GNN GCN GAT 2 1 21 1 + 2 21 + 2 Cora 0% 83.5±0.4 84.0±0.7 85.8±0.4 85.1±0.5 85.3±0.4 85.8±0.4 85.8±0.4 5% 76.6±0.8 80.4±0.7 81.0±1.0 82.3±1.1 81.6±1.1 81.9±1.4 82.2±0.9 10% 70.4±1.3 75.6±0.6 76.3±1.5 76.2±1.4 77.9±0.9 78.2±1.6 78.8±1.7 15% 65.1±0.7 69.8±1.3 72.2±0.9 73.3±1.3 75.7±1.2 76.9±0.9 77.2±1.6 20% 60.0±2.7 59.9±0.6 67.7±0.7 63.7±0.9 70.3±1.1 67.2±5.3 70.5±1.3 Citeseer 0% 72.0±0.6 73.3±0.8 73.6±0.9 73.2±0.6 73.2±0.5 73.6±0.6 73.8±0.6 5% 70.9±0.6 72.9±0.8 72.8±0.5 72.8±0.5 72.8±0.5 73.3±0.6 72.9±0.5 10% 67.6±0.9 70.6±0.5 70.2±0.6 70.8±0.6 70.7±1.2 72.4±0.9 72.6±0.4 15% 64.5±1.1 69.0±1.1 70.2±0.6 68.1±1.4 68.2±1.1 71.3±1.5 71.9±0.7 20% 62.0±3.5 61.0±1.5 64.9±1.0 64.7±0.8 64.7±0.8 64.7±0.8 64.7±0.8 Polblogs 0% 95.7±0.4 95.4±0.2 95.4±0.2 95.8±0.3 95.8±0.3 95.8±0.3 95.8±0.3 5% 73.1±0.8 83.7±1.5 82.8±0.3 78.7±0.6 78.7±0.7 82.8±0.4 83.0±0.3 10% 70.7±1.1 76.3±0.9 73.7±0.3 75.2±0.4 75.3±0.7 81.5±0.2 81.6±0.3 15% 65.0±1.9 68.8±1.1 68.9±0.9 72.1±0.9 71.5±1.1 77.8±0.9 78.7±0.5 20% 51.3±1.2 51.5±1.6 65.5±0.7 68.1±0.6 68.7±0.7 77.4±0.2 77.5±0.2 Pubmed 0% 87.2±0.1 83.7±0.4 88.1±0.1 86.7±0.1 87.3±0.1 88.1±0.1 88.1±0.1 5% 83.1±0.1 78.0±0.4 87.1±0.2 86.2±0.1 87.0±0.1 87.1±0.2 87.1±0.2 10% 81.2±0.1 74.9±0.4 86.6±0.1 86.0±0.2 86.9±0.2 86.3±0.1 87.0±0.1 15% 78.7±0.1 71.1±0.5 85.7±0.2 85.4±0.2 86.4±0.2 85.5±0.1 86.4±0.2 20% 77.4±0.2 68.2±1.0 85.8±0.1 85.4±0.1 86.4±0.1 85.4±0.1 86.4±0.1 Table 3 . 3Ratio between average node differences along wrong and correct edges.Table 4. Sparsity ratio (i.e., (∆F)i 2 < 0.1) in node differences ∆F.Model Cora CiteSeer PubMed 2 (APPNP) 1.57 1.35 1.43 21 + 2 (ElasticGNN) 2.03 1.94 1.79 Model Cora CiteSeer PubMed 2 (APPNP) 2% 16% 11% 21 + 2 (ElasticGNN) 37% 74% 42% Table 6 . 6Dataset Statistics for adversarially attacked graph. N LCC E LCC Classes FeaturesCora 2,485 5,069 7 1,433 CiteSeer 2,110 3,668 6 3,703 Polblogs 1,222 16,714 2 / PubMed 19,717 44,338 3 500 It naturally supports read-value edge weights if the edge weights are set in the incident matrix ∆.2 With the normalization, the piecewise constant prior is up to the degree scaling, i.e., sparsity in∆F. -based graph smoothing into GNNs with semi-supervised learning tasks under normal and adversarial settings. 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[ "Twin-Load: Building a Scalable Memory System over the Non-Scalable Interface", "Twin-Load: Building a Scalable Memory System over the Non-Scalable Interface" ]	[ "Zehan Cui \nInsitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Tianyue Lu \nInsitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Haiyang Pan [email protected]@[email protected] \nInsitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Sally A Mckee \nChalmers University of Technology\nSweden\n", "Mingyu Chen \nInsitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin\n" ]	[ "Insitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin", "University of Chinese Academy of Sciences\nBeijingChina", "Insitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin", "University of Chinese Academy of Sciences\nBeijingChina", "Insitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin", "University of Chinese Academy of Sciences\nBeijingChina", "Chalmers University of Technology\nSweden", "Insitute of Computing Technology\nChinese Academy of Sciences\nBeijingChin" ]	[]	Commodity memory interfaces have difficulty in scaling memory capacity to meet the needs of modern multicore and big data systems. DRAM device density and maximum device count are constrained by technology, package, and signal integrity issues that limit total memory capacity. Synchronous DRAM protocols require data to be returned within a fixed latency, and thus memory extension methods over commodity DDRx interfaces fail to support scalable topologies. Current extension approaches either use slow PCIe interfaces, or require expensive changes to the memory interface, which limits commercial adoptability.Here we propose twin-load, a lightweight asynchronous memory access mechanism over the synchronous DDRx interface. Twin-load uses two special loads to accomplish one access request to extended memory -the first serves as a prefetch command to the DRAM system, and the second asynchronously gets the required data. Twin-load requires no hardware changes on the processor side and only slight software modifications. We emulate this system on a prototype to demonstrate the feasibility of our approach. Twin-load has comparable performance to NUMA extended memory and outperforms a page-swapping PCIe-based system by several orders of magnitude. Twin-load thus enables instant capacity increases on commodity platforms, but more importantly, our architecture opens opportunities for the design of novel, efficient, scalable, cost-effective memory subsystems. arXiv:1505.03476v1 [cs.AR] 13 May 2015 Command Bus Address Bus Data Bus col RD col RD PRE ACT row tCCD tRTP tRL tBURST Data1 Data2 ba RD col 1 2 3 tRP tRCD Latency Constraints: Interval Constraints:	null	[ "https://arxiv.org/pdf/1505.03476v1.pdf" ]	9,634,813	1505.03476	95dcc04bb3f85209282a203d22714ee673b195c9	Twin-Load: Building a Scalable Memory System over the Non-Scalable Interface Zehan Cui Insitute of Computing Technology Chinese Academy of Sciences BeijingChin University of Chinese Academy of Sciences BeijingChina Tianyue Lu Insitute of Computing Technology Chinese Academy of Sciences BeijingChin University of Chinese Academy of Sciences BeijingChina Haiyang Pan [email protected]@[email protected] Insitute of Computing Technology Chinese Academy of Sciences BeijingChin University of Chinese Academy of Sciences BeijingChina Sally A Mckee Chalmers University of Technology Sweden Mingyu Chen Insitute of Computing Technology Chinese Academy of Sciences BeijingChin Twin-Load: Building a Scalable Memory System over the Non-Scalable Interface Commodity memory interfaces have difficulty in scaling memory capacity to meet the needs of modern multicore and big data systems. DRAM device density and maximum device count are constrained by technology, package, and signal integrity issues that limit total memory capacity. Synchronous DRAM protocols require data to be returned within a fixed latency, and thus memory extension methods over commodity DDRx interfaces fail to support scalable topologies. Current extension approaches either use slow PCIe interfaces, or require expensive changes to the memory interface, which limits commercial adoptability.Here we propose twin-load, a lightweight asynchronous memory access mechanism over the synchronous DDRx interface. Twin-load uses two special loads to accomplish one access request to extended memory -the first serves as a prefetch command to the DRAM system, and the second asynchronously gets the required data. Twin-load requires no hardware changes on the processor side and only slight software modifications. We emulate this system on a prototype to demonstrate the feasibility of our approach. Twin-load has comparable performance to NUMA extended memory and outperforms a page-swapping PCIe-based system by several orders of magnitude. Twin-load thus enables instant capacity increases on commodity platforms, but more importantly, our architecture opens opportunities for the design of novel, efficient, scalable, cost-effective memory subsystems. arXiv:1505.03476v1 [cs.AR] 13 May 2015 Command Bus Address Bus Data Bus col RD col RD PRE ACT row tCCD tRTP tRL tBURST Data1 Data2 ba RD col 1 2 3 tRP tRCD Latency Constraints: Interval Constraints: Introduction Commodity memory interfaces have difficulty in scaling memory capacity to meet the needs of modern multicore and big data systems. For instance, the number of cores in chip multiprocessors (CMPs) is growing such that memory capacity per core drops by 30% every two years [36]. In virtualized environments, running many consolidated virtual machines per core further increases memory requirements [13]. Sufficient capacity to hold frequently used "hot" data becomes critical to avoid slow disk accesses. For example, the total memory used for data caching in Facebook is about 75% of the size of its non-image data [44]. In-memory databases can perform queries 100 times faster than traditional disk-base approaches [9,22]. And Google, Yahoo, and Bing store their search indices entirely in DRAM [34]. This "capacity wall" has several causes. First, the number of channels is limited by the processor's pin count, estimated to increase only by 6.5% each year [29]. Second, the DRAM channel connects multiple dual in-line memory modules (DIMMs) via a multi-drop bus. Signal integrity (SI) requires that higher-frequency bus supports fewer DIMMs per channel, e.g., the newest Intel Xeon only supports one dualrank DIMM per channel for DDR3-1866 [27]. Third, it is challenging to scale DRAM feature sizes below 20nm [29,43]. Buffer chips can be used to mitigate pin count and SI limitations [19,18,39], but the processor's maximum tolerable access latency still restricts current solutions to one-layer extensions. This constraint can be avoided by accessing memory via packet-based asynchronous protocols [24,49,36,38,23]. For instance, standard PCIe can be used to access DRAM in remote servers [23] and disaggregated memory blades [36,38], but random accesses to large working sets can suffer 100× slowdowns [23]. Asynchronous protocols can be implemented over high-speed serialized links [24] or photonics [49], but the expense has thus far limited their widely adoption. We believe that an industry-standard asynchronous interface is the right way to go, But any solution that requires changes to the processor interface slows commercial adoption. We seek a practical solution to the capacity wall that is much more scalable, requires no changes to commodity processors and memory modules, and delivers acceptable performance for big-memory applications. We use Memory Extending Chips (MECs) to build a multi-layer memory system, the extra propagation delay for which violates DRAM latency constraints. To address this, each access to the extended memory is replaced by two special twin-loads, where the first one prefetches data into the MEC buffer, and the second one brings it into the processor. The twin-load addresses point to the same location, but we manipulate them so that 1) the second load reaches the MEC rather than hitting in cache and 2) the commodity processor is tricked into serializing them on the memory interface to ensure time to complete the prefetch. Our contributions are: • We implement an asynchronous protocol over the synchronous DDRx interface by introducing a twin-load mechanism that coordinates software and hardware. • We propose to use commodity processors and memory modules to create a scalable extended memory system based on twin-load. We study two different mechanisms to guarantee prefetch-to-load order and enough time for the prefetch. One of them enables exploiting the memory concurrency. • We implement a software prototype that reserves memory from the operating system to emulate the extended memory and MECs, and we implement a lightweight extended memory manager. • We use our prototype to evaluate the feasibility of our proposals, finding that our best solution has comparable performance with NUMA extensions, but better scalability and performance per dollar. For applications with large working sets and irregular accesses, our twin-load solutions perform orders of magnitude better than page-swapping with PCIeconnected remote memory. Even when compared to an ideal system with all local memory, twin-load incurs an acceptable slowdown (about 26%). The ability to quickly, easily, and inexpensively increase memory capacity delivers good return-on-investment, but it is not the most compelling benefit of our memory architecture. Rather, our approach opens new opportunities to build innovative, efficient memory systems such as integrating remote memory pools, heterogeneous DRAM/NVM, direct remote memory access, and even accelerators into the MECs. Background and Related Work DRAM memory systems usually consist of multiple channels that drive DIMMs composed of multiple ranks, and all ranks on a channel share a command, address, and data bus. A rank contains multiple storage arrays, or banks. Total memory capacity is thus determined by the number of channels, the ranks per channel, and the rank capacity, which are limited by the pin count, signal integrity, and chip density, respectively. Simple are commonly used to access the data arrays; in such protocols, data are placed on the bus a fixed latency after being requested, and thus no handshake occurs between the producer and consumer. An activate (ACT) command "opens" a given row, loading the target row's data into a bank of sense amplifiers from which read (RD) and write (WR) commands access the data at specified columns. Subsequent accesses to an open row (row hits) require no ACT command to resend the row address. When data from a different row are needed (row misses), the memory controller sends a precharge (PRE) command to "close" the row, which writes the data back to the storage array and precharges the bank's sense amplifiers. It then issues an ACT command to open the new row. Figure 1 illustrates the basic operation, and Table 1 shows the related timing parameters. For example, RD 2 in Figure 1 is a row hit. The ACT is omitted, and the RD can be issued after a short tCCD latency. RD 3 is a row miss, and so a PRE command is sent after tRT P time to close the row. After tRP time to finish the precharge, an ACT command with the new row address is sent, and the RD command with the column address can finally be issued after tRCD time. Memory Extension over Standard DDRx Interfaces Buffer chips can be used to alleviate pin count and SI constraints within the latency requirements of DDRx protocols. For instance, Cisco's unified computing system extended technology uses an ASIC buffer to expand a DDR3 DRAM channel into four distinct channels [18], yielding up to 4× the capacity. Such buffer chips with five DDRx interfaces become expensive due to large die area and packaging costs. LRDIMM uses a memory buffer to re-drive the DRAM bus, alleviating SI issues and enabling more DIMMs/channel [39]. Scalability is still limited at higher frequencies: e.g., at DDR3-1866 the newest Intel Xeon only supports one LRDIMM per channel [26]. Both methods have limited scalability because they can only support one-layer extensions. When extending hardware is interposed between the memory controller and DRAM chips, commands and data experience extra propagation delays. For DRAM writes, commands and data propagate in the same direction, remaining synchronous on the channel. DRAM reads require longer round-trip times because data and commands move in opposite directions. Figure 2 shows the difference in read timing. For the simplest scenario where the extending hardware just forwards commands and data without any processing, the extra delay is 3.4ns in each direction [25]. The DRAM read latency in the memory controller, tRL, can be increased by 6.8ns, which is still within the adjustable range of commodity processors. For a slightly more complex system with two layers of extending hardware and minimal logic processing, the propagation delay will likely approach 20ns, which is difficult for commodity processors to tolerate. Memory Extension over Custom Memory Interfaces There are many proposals for replacing the synchronous memory interface and breaking the tight processor-memory coupling. Typically, an interface buffer/die is introduced to bridge processor and memory, such as FBDIMM [30], BOB [19], and HMC [24]. Other researchers study more exotic organizations; for instance, Chen et al. [17] study a message-based memory subsystem with high-speed serial links; Fang et al. [21] incorporate emerging technologies on DDRx-like buses; and Udipi et al. [49] look at using 3D stacking and photonics to create scalable memory systems. The protocol between the buffer/die and memory is still a synchronous DRAM protocol, but the protocol between the buffer/die and processor is replaced by a packet-based access protocol. Although such methods are effective, they require changes to the memory controller and processor-memory interface. It is uncertain whether processor vendors will accept such solutions. Even if they do, high cost and increased access latency may still limit adoption, e.g., BOB is only supported in high-end product lines. Memory Extension over Inter-Processor Interfaces A coherent network can connect multiple server processors to form a NUMA (Non-Uniform Memory Access) node. Each processor can access memory on other nodes with additional latency, which extends the total capacity of directly addressable memory. Various NUMA systems are available from low-end, dual-socket systems to those with 100s of CPUs [48]. When it comes to memory capacity, though, NUMA is expensive. First, adding more memory modules necessitates adding more processors, which may be wasteful for memory-bound applications. Second, maintaining cache coherence across shared memory incurs significant overheads. Both complexity and cost are added to the processor, e.g., only high-end processors support NUMA with more than two processors. Third, the access latency across a NUMA interconnect is relatively high, e.g., the Intel Quick Path Interconnect (QPI) adds about 58-110ns latency per hop [42]. Memory Extension over Network Interfaces PCIe is an asynchronous, packet-based protocol. The lack of a latency constraint facilitates the design of scalable DRAM organizations over standard PCIe interface, but the latency of accessing memory via PCIe is several microseconds. Lim et rank MEC Figure 3: Four-layer tree topology, assuming that a channel can only drive dual ranks at high frequency [27] al. [36,38] find that page-swapping between local and remote memory via DMA performs reasonably for applications with high locality, but swapping is inefficient for applications with large working sets and irregular access patterns. Besides, data accessed via PCIe are not directly cacheable. Other proposals can use memory on remote servers or memory blades via a network interface. Software approaches like vSMP [12] and MemX [20] access memory on remote servers over commodity InfiniBand or Ethernet. For all I/O interface based schemes, the latencies are difficult to go below one micro-second. Memory Controller Memory Extension with Emerging Technologies Emerging storage class memory (SCM) technologies such as PCM and ReRAM could provide higher storage density and lower power consumption than DRAM at comparable access latencies, making it a potentially good candidate for capacity extension. However SCM has more timing constraints that are quite different with DRAMs. For example, write latencies are about 10× longer than read latencies, and reads are still 2-3× slower than DRAM [35]. So it is not possible to access SCM through the commodity DDRx SDRAM interfaces without modifying the processor-integrated memory controller. Micron recent PCM chip [41] uses a special JEDEC LPDDR2-N [33] interface that adds a PREACTIVE command and an overlay window especially designed for NVM. San-Disk's UlltraDIMM connects NAND flash to CPU through DDR3 Interface but only supports direct access to internal buffer [11]. Since SCM technology is still evolving, it is unlikely that a universal interface will be well defined and adopted by processor community soon. Overview Both the DDRx and PCIe are widely-used, open standards, which makes their interfaces good candidates for memory extension. We choose the memory interface because of two advantages: latency and concurrency. Even with the extra propagation delay, the access latency to extended memory is still within tens of nanoseconds. And since these accesses are cacheable, non-blocking caches help mask delays. Figure 3 illustrates one potential memory extension topology that allows us to populate more than two ranks per channel. The Memory Extending Chips implement one slave and one master DDRx interface. The slave of the top-level MEC (MEC1) connects to the processor's commodity memory interface, and the master connects to the DIMMs or to the slave interfaces of other MECs. The MEC interfaces are similar to those of LRDIMM [18], but the internal logic and associated software break the one-layer constraint. Since DRAM reads to extended memory experience intolerable round-trip times, we must access the data in a way that breaks the tRL constraint. Note that both loads and stores cause DRAM reads. Stores first trigger read-for-ownership (RFO) operations to bring data into cache. They update the data in cache, and on eviction they write back to memory. We thus first discuss how loads work before discussing stores. Load Operations To break the latency constraint, we could first prefetch the target data into the buffer of MEC1 and insert a line of fake data (e.g., repetitive patterns of 0x5a) as a placeholder in the processor's caches. A second demand load could then fetch the real data from the MEC. This scheme presents three challenges. First, we can neither issue two normal loads to the target address nor use a software prefetch instruction, since those would cause the demand load to hit in cache and load the placeholder data. The demand load must reach MEC1 to get correct data. Second, MEC1 must process the prefetch first. Modern processors typically employ out-of-order (OoO) execution in the instruction pipeline or the memory controller queue, which makes the order in which the loads reach the MEC unpredictable. Third, the MEC must issue the prefetch early enough to guarantee that the data will be loaded into its buffer before the demand load fetches the data. To address the first challenge, we manipulate the data addresses so that the processor thinks the prefetching load is to a different location from the demand load, but the MEC knows that they correspond to the same target location. For instance, adding a flag bit suffices to distinguish the addresses inside the processor. The MEC simply ignores this bit. This creates a "shadow" address for each location in the extended memory. Figure 4 shows the relationships among the local, extended, and shadow memory spaces. To address the second and third challenges, we design two different twin-load mechanisms. We also investigated an extreme way to guarantee strong ordering by making the extended memory uncacheable [28], but we omit those results here due to its poor performance. The first mechanism, TL-LF, inserts a load fence instruction between the loads. The fence guarantees that the second load will not execute until the first receives data from memory [28]. The spacing between the loads is the round-trip time of a memory load, which is large enough to tolerate the propagation delay. This mechanism retains the benefit of cache locality, but blocks all loads following the fence. To better exploit memory concurrency, we need to allow more out-of-order execution. Our second solution, TL-OoO, does not designate which is the prefetch or demand load at the software level, but assigns appropriate identities dynamically via both hardware and software. The load that arrives first triggers the prefetch and returns fake data, and the one that follows returns the true data. Software identifies which data is correct on-the-fly. If both the extended and shadow addresses map to the same DRAM bank, the second twin-load artificially triggers a DRAM row miss that forces TL-OoO to delay the load's MEC arrival. Recall that an RD command to the same bank but different row must wait tRT P time before issuing the PRE to close the current row, tRP time to complete the PRE and issue the ACT for the new row, and finally tRCD time to complete the ACT and issue the RD. The minimum total delay is about 35ns at DDR3-1600, which is enough to tolerate propagation delays for up to five MEC layers. Store Operations We need twin-loads to bring data into cache before we perform store operations. Memory consistency requires us to ensure that data always be written to the true cache line. In the unlikely event that an interrupt happens between the twin-load and the store, the correct cache line may have been evicted by the time the store is resumed. The store has to trigger an RFO operation to load a fake line into cache, the modification of which will cause an error. To avoid this, an atomic com-pare_and_swap (CAS) instruction 1 first compares the correct value obtained from the twin-load with the value in the cache line, and swaps (stores) the new value into the cache line only when the comparison succeeds. Then if the RFO after the interrupt brings the fake data into cache, the comparison will fail, the cache line remains unmodified, and the store is retried. Implementation Details The MECs organize the physical DIMMs/ranks/banks into logical DIMMs/ranks/banks that the memory controller sees. MEC1 asserts a fake serial presense detect (SPD) [40] Figure 5: Semantics of twin-loads in TL-OoO tables. MEC1 chooses one address bit to differentiate the extended and shadow memories. For TL-OoO, that bit must be a row address bit, and since memory controllers generally use the most significant bit (MSB) of the physical address in the row address, we choose it. For simplicity, TL-LF also uses the MSB, even though it affords more flexibility in how memory capacity grows. The physical memory space consists of the local memory, extended memory, and shadow memory. Figure 4 shows that only local memory and extended memory physically store data. Software Modifications The programmer must identify which objects to place in extended memory. Large data objects make good candidates, whereas the OS, code, stack, and small objects should be in local memory. The programmer use a special interface to allocate objects in extended memory. For the most complicated application we evaluate, the modifications took less than two days (including time to understand the code). For applications that index large arrays to access data, they took less than half a day. Figure 5 shows how loads and stores to identified objects in the program are replaced by (inlined) functions that implement TL-OoO. Loads to virtual address p are replaced by the function load_type(p), which loads both p and p concurrently, and compares their return values to identify the correct one. Stores to virtual address p are replaced by the function store_type(p,val). Two functions, retry_load_type(p) and retry_store_type(p,val), handle the cases in which both loads return fake values or the atomic CAS fails. Such modifications can be done automaticlly by a compiler with user-annotations. This work will be introduced in our future paper. Extended Memory Management Modifying the OS is a practical means of managing the extended memory, but for this study we choose to implement a lightweight manager outside the OS. Big memory applications usually allocate most memory during initialization, with few changes to the allocations throughout execution [15,45] example, Memcached [5] preallocates a big chunk of memory at startup and self-manages it to allocate items internally. Such applications need no complex managers to minimize fragmentation due to frequent allocations and deallocations. The extended and shadow memory spaces are reserved by the OS at boot time and can be allocated in user space via mmap(). To simplify memory management, we allocate/deallocate extended and shadow memory together in large blocks (e.g., 64MB). When allocating a block, two virtual memory regions at a distance of EXT_MEM_SIZE are allocated, as well. Both the virtual and block physical addresses are passed to two mmap() calls to construct corresponding virtual-to-physical mappings in the page table. If the virtual address of an object in extended memory is p, the corresponding virtual address for its shadow p' is simply p+EXT_MEM_SIZE, as shown in Figure 4. Twin-Load Processing In a multi-layer extended system, MEC1 identifies the two loads and forwards the request on the first load, temporarily buffers the data, and returns it on the second load. The MECs on the other layers are much simpler, either executing the received commands or forwarding them to the next layer. MEC1 maintains two main structures not required in lower MECs: the Bank State Table (BST) and the Load Value Cache (LVC), illustrated in Figure 6. The routing table required to implement tree topologies is not shown. For each logical bank, a BST entry indicates whether the bank is open and stores the address of the last row opened. The LVC entries temporarily store prefetched values for the first loads. The tag is the address, and the valid bit indicates whether the entry is in use. Our LVC uses LRU replacement. Upon receiving an ACT command, MEC1 records the bank's row address in the BST. When the RD arrives with its bank address, the MEC accesses the BST to retrieve the previously issued row address. The load address is reconstructed as <row, column, bank>. For TL-OoO, the address is the tag for the LVC lookup: if the lookup misses, the access should be the first load, otherwise the second load. When MEC1 sees the the first load, it allocates an LVC entry, sets the tag to the load address, sets In general, the LVC size M should try to guarantee that the corresponding entry will not have been evicted when the data return, i.e., M > (2 ×tPD +tRL)/tCCD, where 2 ×tPD +tRL is the round-trip time for returning data, and tCCD defines the minimum interval between consecutive RDs. For TL-OoO, the maximum tolerable propagation delay is 35ns, and thus M > 10 suffices. TL-LF can tolerate much longer delays, and so the LVC must be larger. When MEC1 identifies the RD as the second load, it returns the data to the memory controller after tRL time. The valid bit is cleared to free the entry. If the second load arrives too late, the data may have been evicted from the MEC1 LVC. For TL-LF, the second load then returns the fake value. For TL-OoO, the MEC will identify the intended second load as the first, reallocate a new cache entry, return fake values, and prefetch the data again. Software retry ensures that the load gets correct data, but we want to avoid such cases, which waste prefetches and hurt performance. By monitoring our prototype's DRAM command bus, we find that twinned loads are separated by an average of six other loads, which can guide the design choice of M. The middle MECs forward commands to the target leaf MECs to execute. They use the high bits of the row address as the physical DIMM ID for command forwarding. The routing table determines the forwarding port. For the ACT command, the ID is in the row address. For other commands, MEC1 gets the ID from the BST and passes it with the command. Cache State and Correctness Since both extended and shadow memory are cacheable, one or both of the twin-load accesses might not reach MEC1. Let v be the correct value and v the fake value. Table 2 lists their possible states with respect to the cache. In State 1, the initial state, both loads trigger DRAM reads. The MEC takes the first read as the prefetch and returns the fake value and then returns the correct data on the second read. In State 2, the MECs are not involved. Both loads commit quickly, one with the correct value and one with the fake value. In State 3, one load returns the correct value directly from cache, and the MEC identifies the other load as the prefetch and returns the fake value. The corresponding LVC entry will eventually be evicted. In State 4, one load hits in cache and returns the fake value, and the other causes a DRAM read that also returns the fake value, since no former prefetch has reached the MEC. We fall back to a software retry to handle this case: both load addresses are first invalided to return to state 1, and then we use another twin-load to get the correct data. A memory fence instruction is required to complete the invalidation before the following twin-load. If the retry also gets the fake value, we throw an exception that invokes a safe path to memory. We are investigating other strategies for better performance, but discussing them is beyond the scope of this paper. Exception Handling There are two rare cases in which the retry may fail: the LVC entry gets evicted before the second load arrives, or the correct data is the same as the fake value. Our solution is to implement a slow but safe path by which to load the data. We add three uncacheable memory mapped registers in MEC1: an address register to receive the physical load address, a flag register to indicate load completion, and a data register to hold the loaded data. The exception handler actions are like reading I/O ports. Evaluation Methodology To evaluate our twin-load implementation, we emulate multiple sytems for comparison: • TL-LF and TL-OoO: our twin-load mechanisms with local, extended memory, and shadow memory; • NUMA: a system using QPI to connect more processors so they can attach more memory; • PCIe: a system using PCIe to connect more memory, which is accessed using page swapping [36,38]; and • Ideal: an ideal system with all memory locally attached. Our host system has two processors and eight 8GB DIMMs (64GB in total). Table 3 shows how the host memory is used to emulate the extended memory systems. For the TL, PCIe, and Ideal systems, we attach all DIMMs to a single processor (and execute only on that processor) to avoid performance variations among different runs due to nondeterministic memoryto-processor affinities. Our experiments are independent of any specific topology -as long as the propagation delay in within 35ns, the software behaves the same. For the TL systems, both the extended and shadow memories are emulated using reserved host memory. The shadow memory is initialized to hold fake values to emulate the MEC functionality. The twinned addresses cause DRAM row misses in the host memory controller. We implement the required Note that there are some deviations between a real twin-load system and the emulated system. In the emulated system: 1. loads to the extended and shadow memories always return the correct and fake values, respectively, thus it is possible that the correct value returns earlier and advances program prematurely; and 2. for the fourth case in Table 2, the missed load always returns the correct value from memory, whereas it should return the fake value and trigger a retry. To avoid the first situation, we choose to advance the program only when both the values have been returned and checked. This is a conservative choice, since for the first and third cases in Table 2, it could be that the correct value in cache is compared first, and the program could proceed without waiting for the result of another load. It is difficult to avoid the second situation because the software cannot know the cache state. However, by recording the memory requests on the memory bus using a tool like a DDR3 protocol analyzer [4], we find that over 96% of loads to extended memory are twinned. This can be easily explained, since the two addresses are always accessed synchronously and are very likely to be brought into and evicted from cache together. Taking into account the conservative policy for the first deviation, we believe our emulation reasonably approximates a real extended system. Table 4 lists the workloads we use in our evaluations. From a variety of application domains, we select 10 benchmarks with footprints that scale easily and code sizes that are reasonable for manual modification. For Memcached, a client running memslap [6] is connected to the Memcached server via Gigabit Ethernet; to avoid the network bandwidth becoming bottleneck, we test small objects [37] and only use four threads on the server side. For all benchmarks, we evaluate two footprints -a medium one around 4GB and a large one around 16GB. For the TL and NUMA systems, we modify source code to allocate large objects in extended memory. Table 4 shows the proportion of data in extended memory. For the PCIe system, we let the Linux swap mechanism manage data placement. We use performance counters to gather architectural statistics. Results We compare TL-LF and TL-OoO against the NUMA and Ideal systems described in Section 5. Figure 7 shows experimental results for these mechanisms on our emulated prototypes. We normalize performance relative to Ideal. TL-LF, TL-OoO, and TL-NUMA achieve 45%, 75%, and 73% of Ideal performance for medium footprints, and 49%, 74%, and 76% of Ideal performance for large footprints. This suggests that footprint size does not significantly affect performance, so we restrict our discussion to large-footprint results. TL vs. Ideal We first discuss the potential penalties for twin-load compared to having all local memory. Then we discuss how TL-OoO might alleviate certain penalties. Finally, we discuss the shortcomings of TL-LF and discuss future optimizations. Potential Penalties. Twin-load obviously increases the number of instructions and data accesses, potentially causing more cache misses. Also, since we double the address space, TLB conflicts will be exacerbated. of local data may contribute to a significant portion of the accesses, thus twin-load only modestly increases LLC misses. Figure 10 shows that workloads with significant TLB conflicts can be classified into two categories: graph applications and applications that store most data structures in extended memory. For instance, doubling the extended address space roughly doubles the TLB misses for GUPS and Radix. For the graph applications, our results suggest that the relative small but frequently accessed vertex-associated metadata (rather than the large graph) contribute to most of the TLB misses. This is because such metadata are randomly accessed and large enough to exceed the TLB coverage (2MB for a 512entry TLB with 4KB pages), and the graph traversal thrashes the TLB. Workloads with relatively good locality (e.g., CG and ScalParC) cause few TLB conflicts. Increases in TLB MPKI range from 3% to 179% (39% on average). Potential Benefits for TL-OoO. Although twin-load increases the number of executed instructions by 64% and LLC misses (long latency memory accesses) by 71% compared to Ideal, average performance slowdown for TL-OoO is only 25% and 26% for medium and large footprints, respectively. This is due to twin-load's better processor utilization: Figure 8 shows that even though twin-load increases the number of instructions, it delivers higher IPCs for most workloads. (The remaining gap reflects the performance slowdown.) The pipeline usually stalls on long-latency memory accesses, but our twin-load instructions can exploit such stall slots, masking the increase in non-memory instructions. Achievable memory-level parallelism (MLP) of most applications is limited, which is far from saturating the processor's available memory access concurrency, which is defined by the Figure 11 shows that the average number of outstanding off-core reads increases from 11.8 to 14.3, especially for those workloads with significant increases in LLC misses (except GUPS and CG). GUPS's concurrency is likely limited by the many TLB misses, while CG seems to saturate the hardware support for concurrency. Since we increase memory concurrency, the achievable memory bandwidth also increases, as shown in Figure 12. Shortcomings of TL-LF. The most obvious shortcoming of TL-LF is its limited memory concurrency. Consecutive accesses to extended memory are serialized by the load fence. Figure 11 and Figure 12 show that the number of outstanding off-core reads and the memory read bandwidth are both decreased by 34%. Although it incurs more than 50% slowdowns, TL-LF can potentially tolerate higher latencies than TL-OoO, making it adaptable to more application cases. A possible optimization for TL-LF is not to insert a fence per data access, but to batch the first twin-load instructions for several accesses, insert the fence, and then perform the second twin-loads and software checks. We leave this for future work. Figure 7 shows that TL-OoO exhibits comparable performance to NUMA. On our host system, the access latency to local memory and remote memory (via QPI) is about 100ns and 170ns, respectively. For NUMA, the long latency to extended memory and the limited memory concurrency cause memory throughput (bandwidth) to decrease by an average of 30% (and up to 51%) compared to Ideal, resulting in an average 24% (and up to 51%) performance slowdown. TL-OoO performs much better than NUMA for graph applications, in particular. The irregular access behaviors and corresponding limited intrathread memory concurrency make graph applications latencysensitive. Compared to the 70ns latency increase for NUMA, TL-OoO incurs less relative penalty -recall that a row miss causes only 35ns extra latency. TL vs. NUMA TL vs. PCIe We evaluate the performance slowdown of using PCIe extended memory paging. In the experiment, we change the ratio of data to be placed on extended memory, ranging from 0% to 90%. The original replace procedure in Linux is designed for swapping with much slower hard disk so that it may be quite complicated and slow for swapping with fast PCIe remote memory. It takes about 7.8us to swap a page on our prototype, which is 1.4× of the fastest PCIe replacement policy [38]. Thus, to compensate for such extra software overhead, we double the measured performance of the emulated PCIe system in the comparison result. We choose five representative benchmarks -GUPS, CG, BFS, ScaleParC, and Memcached -from Table 4. Figure 13 shows results normalized to those of a nonswapping (0% data in extended memory) system. Placing only 25% of data into extended memory slows performance of most workloads by quite much. Performance further degrades as we move more data into extended memory. At 90%, swapping pages with PCIe extended memory yields slowdowns of one to four orders of magnitude. When 25% of data reside in extended memory, ScalParC has the best performance (0.53×), mainly due to its low LLC and TLB MPKIs, which suggest good locality and less pressure on the memory system. The extremely random-access GUPS only achieves 0.0003× the performance. Even for Memcached, which shows insensitivity to the memory system (Figure 7), performance is only 0.13×. For CG and BFS, resulting performance is 0.12× and 0.27×, respectively. TL-LF and TL-OoO perform much better than this, even if we put over 90% of data into extended memory. Discussion We recognize that accurate cost projections can be difficult, we nonetheless try to put relative costs in perspective before examining the impact of extending the DRAM tRL time. Cost Analysis We compare three ways to extend memory capacity: TL coordinates software and MECs, NUMA adds more processors, and Cluster adds more servers. The PCIe scheme experiences Cost Model. The baseline system has two processors, which is the most cost-effective configuration, but it only supports one dual-rank RDIMM per channel (a currently common situation at higher frequencies and a likely continuing trend). We choose Intel's mid-end Xeon E5-2650v2 processor with four memory channels and 16GB RDIMM for our comparisons. Our baseline system has 128GB memory in total. We take costs of server components from the Intel and Amazon websites. We derive other costs from Barroso and Hölzle [14] -the server cost of a three-year amortization and cost of server power take 50% and 8% of the TCO (total cost of ownership) for a datacenter with mid-end servers 2 . Other costs include capital outlay and operating expenses. Note that for NUMA systems, we must use more expensive processors that support four cores (here Xeon E5-4650v2). We expect that the MEC costs about the same as the LRDIMM buffer, since both contain two DDRx interfaces. The die area of such a chip is mainly determined by pin count, rather than logic. We only add a table and a cache with tens of entries, and thus we do not consume much logic. We conservatively assume such a MEC costs $100. Performance Model. We assume that by doubling the memory capacity, performance can at most be improved by a factor of x. This factor can be quite large in certain cases: Graefe et al. [22] find x ≈100 when the extended memory capacity can cover an in-memory database's datasets. NUMA and Cluster also double the number of processors, so ideal speedup would be 2×. However, each method brings certain penalties. Our results in Section 6 shows that TL and NUMA achieve 74% and 76% of Ideal performance due to twin-load software and long latency memory accesses. In addition, NUMA and Cluster also face the challenge of efficient parallelization. For applications that are difficult to distribute, e.g., graph applications, the penalty for the Cluster method can be large. Performance per Dollar. Table 5 shows the potential speedups/slowdowns and costs of doubling memory capacity for the three systems. The table shows that relative performance per dollar among the mechanisms has no relation to x but rather to the correction factor c due to twin-load software, cross-processor access latency, or efficiency of parallel implementation. Figure 14 draws the performance per dollar relative to the parallel efficiency. TL can improve performance per dollar by at least 7% compared to NUMA when doubling memory capacity. At the meantime, TL has the better scalability: the standard Intel solution only supports up to eight processors, which limits the system to 4× the memory capacity. Clustering has better scalability with respect to memory capacity, but it is difficult to scale performance. TL outperforms Cluster whenever the distributed application achieves below 60% of Ideal performance, which is a challenge for many applications. 2 Server amortization costs 29.5% (65.9%) of TCO for a datacenter with low-end (high-end) servers, and server power costs represent 14.3% (3.8%). Comparison with Increased tRL To support the larger latency of extended memory, why not just increase the maximum latency constraint of JEDEC standard? Since tRL, which determines data transmission time from memory chip to memory controller can be increased, the JEDEC DDRx standard could also adapt to extended memory with larger latencies. Although this scheme needs a tiny modification to memory controller hardware, it is still acceptable. However, according to the DRAM protocol, a memory bank will also be held for a longer time, preventing other accesses to that bank. This reduces memory bus concurrency, which reduces the benefit of this potential approach. We compare TL extended memory to one using a single load with increased read latency. We use trace-driven DRAMSim2 [46] with dependences between memory instructions [47] to simulate the systems, and we choose more benchmarks than just those in Table 4. In the TL system, tRL remains unchanged, but we insert a second load after each read to cause a row miss. To support latencies greater than 35ns, the second load is delayed for certain time, but does not block following loads with no data dependence 3 . We compare extra latencies to tolerate of 0-135ns. Figure 15 summarizes results. The four mechanisms from Section 6 correspond to special points in the figure and have coincident results expected for TL-LF. TL-LF tolerates latencies greater than 100ns, but results on our emulated prototype are worse than in simulation, mainly because we simulate a TL mechanism that does not fence the following loads. In general, increasing tRL performs better for relatively small latencies, but as tRL grows, performance degrades faster than for TL because high tRL values limit memory concurrency. In contrast, the interval between twinned loads can be used to execute other memory requests. Comparison with LRDIMM Load-Reduced DIMMs are already used to maxmize the server memory capacity. For example the newest Intel twosocket Xeon E5 server can support up to 1.5TB memory with LRDIMMs, if not considering the cost. In fact,every buffer-based approach (including MEC) can be considered as reducing the electrical load. However, each buffer is typically capable of 2∼4X load reduction, related to frequency. The real limitation of LRDIMM is one-layer extension, restricted by CPU's synchronous DRAM interface and access protocol -to be specific, the propagation of more layers violate CPU's timing constraint. To the best of our knowledge, commodity processors don't support cascading of LRDIMM buffers. Our proposal breaks such limitation towards more layers and much larger memory capacity using software supports, while LRDIMMs still can be used as local memory or extended memory after MEC. Since LRDIMM has to put all memory chips within single level , the highest LRDIMM model already uses DDP(Dual-Die-Package) or QDP(Quad-Die-Package), or even 3DS devices. It is not supprised that a single LRDIMM is more expensive than a server CPU. While for multi-level MEC extension, more cost-effective RDIMM modules can be incorporated to build a large memory system. Energy The software overheads of twin-load would increase the energy consumption compared to the ideal system. For example, the retired instructions of TL-OoO are 1.64X that of ideal system, indicating more energy consumption. However, when compared to a real commodity system, the potential performance improvement due to twin-load enabled in-memory processing (up to 100X [22]), can actually greatly reduce the total energy consumption (Energy = Power × Delay). Conclusions We propose twin-load, a mechanism to build a lightweight, asynchronous data-access protocol that requires no hardware changes on the processor side. To achieve this, we coordinate software and the Memory Extending Chips on a standard DDRx interface. Data access is accomplished by two special loads, the first of which prefetches data into the top MEC buffer and the second of which brings it into the processor. Using this mechanism, we can easily attach a multi-layer memory system to commodity processors to instantly address the capacity wall problem. We create an emulation-based software prototype to demonstrate the feasibility of our proposal. Our best mechanism can achieve 74% of the performance of an ideal system with all local memory. Our mechanism performs similarly to NUMA extended memory, but delivers much better scalability and performance per dollar. Twinnload also outperforms PCIe-based systems by several orders of magnitude. In addition to facilitating easy, cost-effective memory extionsions, our mechanism opens opportunities to build innovative memory systems on commodity platform using the lowlatency, high-concurrency standard DDRx interface; examples include remote memory pools, heterogeneous DRAM/NVM systems, direct remote memory accesses, and even MECs with integrated accelerators. Relying on open standards and avoiding changes to processor interfaces enables more system designers -including those in academia -to build productionquality systems. Figure 1 : 1DRAM Access Protocol. RD 1 and RD 2 map to the same bank and row, while RD 3 maps to another row in that bank. The rank and bank addresses associated with each command are omitted for readability. Figure 2 : 2DRAM Access with Extending Hardware. tPD is the propagation delay of commands/data between the memory controller and DRAM chips. Figure 4 : 4Relationship between virtual and physical memory spaces. The shadow space does not map to real DRAM. Figure 8 ,Figure 7 :Figure 8 :Figure 9 :Figure 10 : 878910Figure 9, and Figure 10 show the effects of twin-load on instruction execution, LLC, and TLB behaviors. (Note that the LLC MPKI and TLB MPKI of TL-OoO are relative to the number of retired instructions in the Ideal case, to show the absolute miss increase.) Since twin-load replaces some load/store instructions with inline functions, the increas in number of instructions retired depends on the proportion of memory accesses and their relative proportion targeting extended memory. TL-OoO's retired instruction count increases by 64%, on average. LLC misses increase by 11-156% (71%, on average). If all data are in extended memory, the number of LLC misses can potentially double, as is the case for GUPS, Radix, CG, and BFS. For others (e.g., BC, PageRank, and ScalParC) a small portion Normalized Instruction Count and IPC of TL-LLC TLB MPKI Figure 11 :Figure 12 : 1112Average Number of Off-Average Read Bandwidth Figure 13 : 13Performance of PCIe Swapping mechanism number of MSHRs. Thus TL-OoO can take advantage of the remaining capacity for concurrency to overlap execution of the extra loads. Figure 14 :Figure 15 : 1415Normalized performance per dollar relative to TL-OoO. Simulated Results of TL vs. Increased tRL (normalized to tRL=15ns without TL) Table 1 : 1DDRx Timing ParametersTiming a Description Typical Parameter Value tRL Fixed latency from RD command to first data 13.75ns tBURST Fixed duration of data transfer 4 cycles tCCD Minimum delay between two RD commands 4 cycles tRT P Minimum delay between RD and PRE commands 7.5ns tRP Minimum delay between PRE and ACT commands 13.75ns tRCD Minimum delay between ACT and RD commands 13.75ns a Delays between commands to the same bank synchronous protocols like JEDEC DDRx[31,32] Data Bus Command Bus Data Bus Command Bus to the memory interface, and all MECs maintain simple mapping Original Code: val = p; Replaced by: val = load_type(p);type load_type(type p1) { type v1, v2; typep2=(void)p+EXT_MEM_SIZE; v1=p1; v2=p2; if(v1 != FAKE_VALUE) return v1; else if(v2 != FAKE_VALUE) return v2; else return retry_load_type(p); } void store_type(type p1, type newval) { type v1, v2; typep2=(void)p+EXT_MEM_SIZE; v1=p1; v2=p2; if(v1 != FAKE_VALUE) if(!type_cas(p1, v1, newval)) retry_store_type(p1, newval) else if(v2 != FAKE_VALUE) if(!type_cas(p2, v2, newval)) retry_store_type(p1, newval) else retry_store_type(p1, newval); } Original Code: p = val; Replaced by: store_type(p,val); . For ...<open/close, opened row> row col valid data ... ... ... ... N M Bank State Table Load Value Cache bank Figure 6: The Bank State Table and Load Value Cache in the MEC1 hardware. N and M are the number of entries, respec- tively, where N is equal to the number of logical banks, and M is a design parameter. Table 2 : 2Twin-Load Results with Respect to Cache StateState v v DRAM Reads Result 1 not in cache not in cache two v, v 2 in cache in cache zero v, v 3 in cache not in cache one v, v 4 not in cache in cache one v v v , , , v v v the valid bit, and forwards the RD to the target MEC. After tRL time, it puts the fake values on the data bus to the memory controller. The target MEC fetches the data from DRAM and returns it to MEC1 with the LVC entry ID, where it is inserted into the MEC1 LVC. Table 3 : 3Emulated SystemsSystem TL NUMA PCIe Ideal Processor One Intel Xeon E5-2640 Processors (6-core, 12-thread) 1 Local Memory 0-8GB 0-xGB 2 0-32GB Extended Memory 8-32GB x-32GB 2 - Shadow Memory 40-64GB - Extended Interface DDRx QPI PCIe - Access Mechanism TL-LF/TL-OoO cc-NUMA Swapping Ideal Table 4 : 4WorkloadsBenchmark Source Type Description Proportion in extended memory GUPS HPC Challenge [2] Micro-Benchmark Random access 100.00% Radix PARSEC3.0 [10] Kernel Integer sort 100.00% CG NPB2.3 [7] Scientific Computing Calculating conjugate gradient 99.43% FMM PARSEC3.0 [10] N-body simulation 94.39% BFS Graph500 [1] Graph Application Breadth-first search 99.79% BC SSCA2.2 [3] Calculating connection centrality 76.92% PageRank In-house implementation Calculating website ranks [16] 87.93% ScalParC NU-MineBench [8] Data Mining Parallel classification 94.48% StreamCluster PARSEC3.0 [10] Online clustering 92.93% Memcached Memcached-1.4.20 [5] Data Serving Key-value caching system 97.30% software to generate twin-loads to the extended and shadow memories for certain accesses. For the NUMA systems, we attach one DIMM to one processor to emulate local memory and attach seven DIMMs to the other processor to emulate extended memory. Programs execute on the former and access extended memory via QPI. For the PCIe system, we emulate extended memory as a host-memory RAM Disk configured as a swap partition. We emulate the remote page swapping with default Linux swapping to the RAM Disk. For the Ideal system, we emulate all local memory as host memory. No software change is required. Table 5 : 5Costs of various memory extension mechanismsCosts Baseline TL-OoO NUMA Cluster Processor 1 2×$1166/3 2×$1166/3 4×$3616/3 4×$1166/3 Memory 1 8×$175/3 16×$175/3 16×$175/3 16×$175/3 Motherboard and Disk 1 $1000/3 $1000/3 1.5×$1000/3 2 2×$1000/3 MEC 1 - 8*100/3 - - Server power 3 $252 1.3×$252 1.8×$252 2×$252 Other costs $1325 $1325 1.5×$1325 4 2×$1325 Total Costs $3154 $3963 $8696 $6308 Potential Speedup 1 x 2×x 2×x Correction Factor c = 1 c = 0.74 c 1 = 0.76 c 2 varies c varies CAS is named CMPXCHG in the x86 instruction set[28]. We have two processors, but use only one for program execution for all systems.2 We vary the cut-off point to emulate different local:swapped ratios. We assume a lifetime of three years.2 More processors require a larger motherboard.3 Processors and memory contribute 50% and 30%, respectively to server power.4 Servers with more processors take more space, increasing data center costs. 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[ "CONVERGENCE ANALYSIS OF THE NONOVERLAPPING ROBIN-ROBIN METHOD FOR NONLINEAR ELLIPTIC EQUATIONS ", "CONVERGENCE ANALYSIS OF THE NONOVERLAPPING ROBIN-ROBIN METHOD FOR NONLINEAR ELLIPTIC EQUATIONS " ]	[ "Emil Engström ", "Eskil Hansen " ]	[]	[]	We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L p -generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.	10.1137/21m1414942	[ "https://arxiv.org/pdf/2105.00649v1.pdf" ]	233,481,485	2105.00649	6483897ec353adfbe7ecbea09890b3c83bb5396d	CONVERGENCE ANALYSIS OF THE NONOVERLAPPING ROBIN-ROBIN METHOD FOR NONLINEAR ELLIPTIC EQUATIONS * Emil Engström Eskil Hansen CONVERGENCE ANALYSIS OF THE NONOVERLAPPING ROBIN-ROBIN METHOD FOR NONLINEAR ELLIPTIC EQUATIONS * Robin-Robin methodNonoverlapping domain decompositionNonlinear elliptic equationConvergenceSteklov-Poincaré operator AMS subject classifications 65N5565J1535J7047N20 We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L p -generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions. Introduction. Approximating the solution of an elliptic partial differential equation (PDE) typically demands large-scale computations requiring the usage of parallel and distributed hardware. In this context, a nonoverlapping domain decomposition method is a suitable choice, as it can be implemented in parallel with local communication. After decomposing the equation's spatial domain into nonoverlapping subdomains, the method consists of an iterative procedure that solves the equation on each subdomain and thereafter communicates the results via the boundaries to the adjacent subdomains. For a general introduction we refer to [18,23]. There is a vast amount of methods in the literature, employing different transmission conditions between the subdomains. The standard examples are based on the alternate use of Dirichlet and Neumann boundary conditions, but a competitive alternative is the Robin-Robin method, where the same type of Robin boundary condition is used for all subdomains. The Robin-Robin method was introduced in [13] together with a convergence proof when applied to linear elliptic equations. After applying a finite element discretization, convergence rates of the form 1 − O( √ h), with h denoting the mesh width, have been derived in various linear contexts [9,15,24]; also see [7]. For generalizations and further applications of the Robin-Robin method applied to linear PDEs we refer to [3,9] and references therein. When considering nonlinear elliptic PDEs the literature is more limited. Convergence studies relating to overlapping Schwarz methods are given in [6,21,22]. However, there are hardly any results dealing with nonoverlapping domain decomposition schemes. One exception is [2], where the convergence of the Dirichlet-Neumann and Robin-Robin methods are analyzed for a family of one-dimensional elliptic equations. A related study is [20], where the equivalence between a class of nonlinear elliptic equations and the corresponding transmission problems are proven for nonoverlapping decompositions with cross points, but no numerical scheme is considered. Apart from [22], all these nonlinear studies rely on frameworks similar to the linear case, e.g., assuming that the diffusion is uniformly positive. Hence, the aim of this paper is to derive a genuinely nonlinear extension of the linear convergence result given in [13] for the nonoverlapping Robin-Robin method. We will focus on nonlinear elliptic equations of the form (1.1) −∇ · α(∇u) + g(u) = f in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R d , d = 1, 2, . . . , with boundary ∂Ω. The functions α and g are assumed to have a p-structure; defined in section 2. This structure enables a clear-cut convergence analysis for a broad family of degenerate elliptic equations, i.e., α(∇u) may vanish for nonzero values of u. The latter typically prevents the existence of a strong solution in W 2,p (Ω). The archetypical examples of nonlinear elliptic equations with a p-structure are those governed by the p-Laplacian, where α(z) = \|z\| p−2 z. Examples include the computation of the nonlinear resolvent For sake of simplicity, we decompose the original domain Ω into two nonoverlapping subdomains {Ω i , for i = 1, 2}, with boundaries denoted by ∂Ω i , and separated by the interface Γ, i.e., Ω = Ω 1 ∪ Ω 2 , Ω 1 ∩ Ω 2 = ∅ and Γ = (∂Ω 1 ∩ ∂Ω 2 ) \ ∂Ω. Two examples of such decompositions are illustrated in Figures 1a and 1b, respectively. The analysis presented here can also, in a trivial fashion, be extended to the case when Ω i is a family of nonadjacent subdomains, e.g., the stripewise domain decomposition illustrated in Figure 1c. The strong form of the Robin-Robin method applied to (1.1) is then to find (u n 1 , u n 2 ), for n = 1, 2, . . . , such that (1.4)                        −∇ · α(∇u n+1 1 ) + g(u n+1 1 ) = f in Ω 1 , u n+1 1 = 0 on ∂Ω 1 \ Γ, α(∇u n+1 1 ) · ν 1 + su n+1 1 = α(∇u n 2 ) · ν 1 + su n 2 on Γ, −∇ · α(∇u n+1 2 ) + g(u n+1 2 ) = f in Ω 2 , u n+1 2 = 0 on ∂Ω 2 \ Γ, α(∇u n+1 2 ) · ν 2 + su n+1 2 = α(∇u n+1 1 ) · ν 2 + su n+1 1 on Γ, where ν i denotes the unit outward normal vector of ∂Ω i , u 0 2 is a given initial guess and s > 0 is a method parameter. Here, u n i and u n i \| Γ approximate the solution u restricted to Ω i and Γ, respectively. The convergence analysis is organized as follows. For linear elliptic equations, i.e., equations with a 2-structure, the analysis relies on the trace operator from H 1 (Ω i ) onto H 1/2 (∂Ω i ), and the Lions-Magenes spaces H 1/2 00 (Γ). We therefore start by introducing the generalized p-version of the trace operator, now given from W 1,p (Ω i ) onto W 1−1/p,p (∂Ω i ), and the corresponding Lions-Magenes spaces Λ i ; see sections 3 and 4. There is a surprising lack of proofs in the literature dealing with this generalized p-setting. We therefore make an effort to give precise definitions and proof references. With the correct function spaces in place, we prove that the nonlinear transmission problem is equivalent to the weak form of (1.1) in Theorem 5.2, and introduce the new nonlinear Steklov-Poincaré operators, as maps from Λ i to Λ * i , in section 6. The latter yields that the transmission problem can be stated as a problem on Γ and the Robin-Robin method reduces to the Peaceman-Rachford splitting. The main challenge is then to derive the fundamental properties of the nonlinear Steklov-Poincaré operators from the p-structure, which is achieved in section 7. By interpreting the nonlinear Steklov-Poincaré operators as unbounded, monotone maps on L 2 (Γ) we finally prove that the Robin-Robin method is well defined on W 1,p (Ω 1 ) × W 1,p (Ω 2 ); see Corollary 8.6, and convergent in the same space; see Theorem 8.9. The latter relies on the abstract theory of the Peaceman-Rachford splittings [14]. The continuous convergence analysis presented here also holds in the finite dimensional case obtained after a suitable spatial discretization, e.g., by employing finite elements. However, we will limit ourselves to the continuous case in this paper. Hence, important issues including convergence rates for the finite-dimensional case and the influence of the mesh width on the optimal choice of the method parameter s will be explored elsewhere. Finally, c i and C i will denote generic positive constants that assume different values at different occurrences. 2. Nonlinear elliptic equations with p-structure. Throughout the paper, we will consider the nonlinear elliptic equation (1.1) with f ∈ L 2 (Ω) and Ω being a bounded Lipschitz domain. The equation is assumed to have a p-structure of the form: Assumption 2.1. The parameters (p, r) and the functions α : R d → R d , g : R → R satisfy the properties below. • Let p ∈ [2, ∞) and r ∈ (1, ∞). If p < d then r ≤ dp/ 2(d − p) + 1. • The functions α and g are continuous and satisfy the growth conditions \|α(z)\| ≤ C 1 \|z\| p−1 and \|g(x)\| ≤ C 2 \|x\| r−1 , for all z ∈ R d , x ∈ R. • The function α is strictly monotone with the bound α(z) − α(z) · (z −z) ≥ c 1 \|z −z\| p , for all z,z ∈ R d . • The function α is coercive with the bound α(z) · z ≥ c 2 \|z\| p , for all z ∈ R d . • The function g is strictly monotone and coercive with the bounds Remark 2.4. Possible generalizations of Assumption 2.1, which we omit for sake of notational simplicity, include the cases: dependence on the spatial variable and first-order terms, e.g., α(∇u) = α(x, u, ∇u); the parameter choice p ∈ (2d/(d + 1), 2), which requires an additional set of embedding results for trace spaces; other variations similar to the p-Laplacian, e.g., g(x) − g(x) (x −x) ≥ c 3 \|x −x\| r and g(x)x ≥ c 4 \|x\| r , for all x,x ∈ R.α(z) = (\|z 1 \| p−2 z 1 , . . . , \|z d \| p−2 z d ), which only require slight reformulations of the monotonicity and coercivity conditions. Let V = W 1,p 0 (Ω) and define the form a : V × V → R by a(u, v) = Ω α(∇u) · ∇v + g(u)v dx. The weak form of (1.1) is to find u ∈ V such that (2.1) a(u, v) = (f, v) L 2 (Ω) , for all v ∈ V. The p-structure implies that there exist a unique weak solution of (2.1); see, e.g., [19,Theorem 2.36]. A central part of the existence proof, and our convergence analysis as well, is to observe that the p-structure directly implies that the form a is bounded, strictly monotone and coercive. Lemma 2.5. If Assumption 2.1 holds, then a : V × V → R is well defined and satisfies the upper bound \|a(u, v)\| ≤ C 1 ∇u p−1 L p (Ω) d ∇v L p (Ω) d + u r−1 L r (Ω) v L r (Ω) , the strict monotonicity bound a(u, u − v) − a(v, u − v) ≥ c 1 ∇(u − v) p L p (Ω) d + u − v r L r (Ω) and the coercivity bound a(u, u) ≥ c 2 ∇u p L p (Ω) d + u r L r (Ω) , for all u, v ∈ V . In order to conduct the convergence analysis, we also make the following additional regularity assumption on the weak solution. Assumption 2.6. The weak solution u ∈ V of (2.1) satisfies α(∇u) ∈ C(Ω) d . Note that the above regularity assumption does not imply that u is a strong solution in W 2,p (Ω). A possible generalization of Assumption 2.6 is discussed in Remark 8.3. Example 2.7. Consider the equations given by the p-Laplacian in Example 2.2. If p ≥ d then the weak solution u ∈ V is also in C(Ω). If in addition f ∈ L ∞ (Ω) and the boundary ∂Ω is C 1,β , then [12, Theorem 1] yields that u ∈ C 1,β (Ω). The latter implies that Assumption 2.6 is valid in this context. Finally, we will make frequent use of the fact that, under Assumption 2.1, the standard W 1,p (Ω)-norm is equivalent to the norm (2.2) u → ∇u L p (Ω) d + u L r (Ω) . For r ≥ p this follows directly by the Sobolev embedding theorem together with the assumed restrictions on (p, r). For r < p the equivalence follows by an additional bootstrap argument. 3. Function spaces and trace operators on Ω i . We start by considering a manifold M in R d , which will play the role of ∂Ω i or Γ. The manifold M is said to be Lipschitz if there exist open, overlapping sets Θ r such that M = m r=1 Θ r , where each Θ r can be described as the graph of a Lipschitz continuous function b r . More precisely, there exists (d − 1)-dimensional cubes θ r and local charts ψ r : Θ r → θ r that are bijective and Lipschitz continuous. The charts have the structure ψ −1 r = A −1 r • Q r , where A r : R d → R d is a coordinate transformation and Q r : θ r → R d : x r → x r , b r (x r ) for the Lipschitz continuous map b r : θ r → R. A function µ : M → R now has the local components µ • ψ −1 r . We refer to [10, Section 6.2] for further details. On a Lipschitz manifold we may introduce a measure [16,Chapter 3] and thus define the integral and the space L p (M); see, e.g., [4]. From [4, Chapters 3.4-3.5] it follows that L p (M) is a Banach space and that L 2 (M) is a Hilbert space with the inner product (η, µ) L 2 (M) = M ηµ dS. Let {ϕ r } be a partition of unity of M. The integral then satisfies [16,Theorem 3.9]. Seemingly obvious properties of the integral, including (3.1) M µ dS = m r=1 θr (µϕ r ) • ψ −1 r \|n r \| dx, where n r = (∂ 1 b r , ∂ 2 b r , . . . , ∂ d−1 b r , −1); seeM µ dS = M0 µ dS + M\M0 µ dS, relies heavily on the observation that the integral is independent of the representation (Θ r , A r , b r ) and the choice of partition of unity {ϕ r }; see [16, Theorems 3.5 and 3.7] and the comments thereafter. The equality (3.1) also shows that our integral and L p -spaces are equivalent to the ones used in [10]. Moreover, by [10, Lemma 6.3.5], the L p -norm used here is equivalent to the norm µ → m r=1 µ • ψ −1 r p L p (θr) 1/p . Finally, recall that for a Lipschitz manifold M the unit outward normal vector ν = (ν 1 , . . . , ν d ) is defined almost everywhere; see [10, Section 6.10.1]. The normal vector is given locally by ν • ψ −1 r = n r /\|n r \| and the Lipschitz continuity of b r yields that ν ∈ L ∞ (M). Assumption 3.1. The boundaries ∂Ω i and the interface Γ are all (d−1)-dimensional Lipschitz manifolds. We use the notation (Θ i r , θ i r , m i , ψ i r , b i r , φ i r , ν i ) for the quantities related to the local representations of ∂Ω i . For later use, we note that ν 1 = −ν 2 on Γ. Next, we define the fractional Sobolev spaces on the (d − 1)-dimensional cubes θ r . Let 0 < s < 1, then W s,p (θ r ) is defined as all u ∈ L p (θ r ) such that \|u\| s,θr = θr θr \|u(x) − u(y)\| p \|x − y\| d−1+sp dx dy 1/p < ∞. The corresponding norm is given by u W s,p (θr) = u L p (θr) + \|u\| s,θr . Having defined the fractional Sobolev space on θ i r we can also define them on ∂Ω i . For 0 < s < 1, introduce W s,p (∂Ω i ) = {µ ∈ L p (∂Ω i ) : µ • (ψ i r ) −1 ∈ W s,p (θ i r ), for r = 1, . . . , m i }, equipped with the norm µ W s,p (∂Ωi) = mi r=1 µ • (ψ i r ) −1 p W s,p (θ i r ) 1/p . By the definitions of the norms, it follows directly that µ L p (∂Ωi) ≤ C µ W s,W 1,p (Ω i ) → W 1−1/p,p (∂Ω i ) such that T ∂Ωi u = u\| ∂Ωi when u ∈ C ∞ (Ω i ). The operator T ∂Ωi has a bounded linear right inverse R ∂Ωi : W 1−1/p,p (∂Ω i ) → W 1,p (Ω i ). We can then define the Sobolev spaces on Ω i required for the domain decomposition, namely V 0 i = W 1,p 0 (Ω i ) and V i = {v ∈ W 1,p (Ω i ) : (T ∂Ωi v)\| ∂Ωi\Γ = 0}. The spaces are equipped with the norm v Vi = ∇v L p (Ωi) d + v L r (Ωi) . As for (2.2), this norm is equivalent to the standard W 1,p (Ω i )-norm under Assumption 2.1. Furthermore, the spaces V 0 i and V i are reflexive Banach spaces. 4. Function spaces and trace operators on Γ. The L p -form of the Lions-Magenes spaces can be defined as Λ i = {µ ∈ L p (Γ) : E i µ ∈ W 1−1/p,p (∂Ω i )}, with µ Λi = E i µ W 1−1/p,p (∂Ωi) . Here, E i µ denotes the extension by zero of µ to ∂Ω i . We also define the trace space Proof. Observe that E i is a linear isometry from Λ i onto Λ = {µ ∈ L p (Γ) : µ ∈ Λ i , for i = 1, 2}, with µ Λ = µ Λ1 + µ Λ2 .(4.1) R(E i ) = {µ ∈ W 1−1/p,p (∂Ω i ) : µ\| ∂Ω\Γ = 0}. Next, consider a sequence {µ k } ⊂ R(E i ) such that µ k → µ in W 1−1/p,p (∂Ω i ). Then µ k \| Ωi\Γ = 0 and µ k → µ in L p (∂Ω i ), which implies that (4.2) ∂Ωi\Γ \|µ\| p dS = ∂Ωi\Γ \|µ k − µ\| p dS ≤ ∂Ωi \|µ k − µ\| p dS → 0, as k → ∞. Hence, µ ∈ R(E i ) and consequently R(E i ) is a closed subset of W 1−1/p,p (∂Ω i ). The space Λ i is therefore isomorphic to a closed subset of the reflexive Banach space W 1−1/p,p (∂Ω i ), i.e., Λ i is complete and reflexive [11,Chapter 8,Theorem 15]. To prove that the same holds true for Λ introduce the reflexive Banach space X = W 1−1/p,p (∂Ω 1 ) × W 1−1/p,p (∂Ω 2 ), with the norm (µ 1 , µ 2 ) X = µ 1 W 1−1/p,p (∂Ω1) + µ 2 W 1−1/p,p (∂Ω2) , and the operator E : Λ → X defined by Eµ = (E 1 µ, E 2 µ). As E is a linear isometry from Λ onto R(E) = {(µ 1 , µ 2 ) ∈ X : µ 1 \| ∂Ω1\Γ = 0, µ 2 \| ∂Ω2\Γ = 0, µ 1 \| Γ = µ 2 \| Γ }, it is again sufficient to prove that R(E) is a closed subset of X. Let {(µ k 1 , µ k 2 )} ⊂ R(E) be a convergent sequence in X with the limit (µ 1 , µ 2 ). By the same argument as (4.2), we obtain that µ i \| Ωi\Γ = 0. As µ k 1 \| Γ = µ k 2 \| Γ , we also have the limit Γ \|µ 1 − µ 2 \| p dS ≤ 2 p−1 Γ \|µ k 1 − µ 1 \| p dS + Γ \|µ k 2 − µ 2 \| p dS → 0, as k → ∞, i.e., µ 1 \| Γ = µ 2 \| Γ in L p (Γ) and we obtain that (µ 1 , µ 2 ) ∈ R(E). Thus, R(E) is closed and Λ is therefore a reflexive Banach space. The proof of the lemma is almost identical to the proof of [10, Theorem 6.6.3] and is therefore left out. Remark 4.3. We conjecture that Λ 1 = Λ 2 . However, we will move on to a L 2 (Γ)framework for which it is not necessary to make this identification. Collecting these results yield the Gelfand triplets Λ i d → L 2 (Γ) ∼ = L 2 (Γ) * d → Λ * i and Λ d → L 2 (Γ) ∼ = L 2 (Γ) * d → Λ * . For future use, we introduce the Riesz isomorphism on L 2 (Γ) given by J : L 2 (Γ) → L 2 (Γ) * : µ → (µ, ·) L 2 (Γ) , which satisfies the relations Jη, µ i Λ * i ×Λi = (η, µ i ) L 2 (Γ) and Jη, µ Λ * ×Λ = (η, µ) L 2 (Γ) , for all η ∈ L 2 (Γ), µ i ∈ Λ i and µ ∈ Λ. Here, ·, · X * ×X denotes the dual pairing between a Banach space X and its dual X * . In the following we will drop the subscripts on the dual parings. In order to relate the spaces V i and Λ i , we observe that for v ∈ V i one has T ∂Ωi v ∈ R(E i ); see (4.1). Hence, the trace operator T i : V i → Λ i : v → T ∂Ωi v \| Γ is well defined. We also introduce the linear operator Proof. For v ∈ V i and µ ∈ Λ i we have, by Lemma 3.2, that R i : Λ i → V i : µ → R ∂Ωi E i µ.T i v Λi = E i (T ∂Ωi v)\| Γ W 1−1/p,p (∂Ωi) = T ∂Ωi v W 1−1/p,p (∂Ωi) ≤ C i v Vi and R i µ Vi = R ∂Ωi E i µ Vi ≤ C i E i µ W 1−1/p,p (∂Ωi) = C i µ Λi . Hence, the linear operators T i and R i are bounded. Furthermore, for every µ ∈ Λ i we have T i R i µ = T ∂Ωi R ∂Ωi E i µ \| Γ = (E i µ)\| Γ = µ, i.e., R i is a right inverse of T i . We continue by deriving a few useful properties related to the operator T i . Proof. Let v ∈ V . As C ∞ 0 (Ω) is dense in V , there exists a sequence {v k } ⊂ C ∞ 0 (Ω) such that v k → v in V . Set v i = v\| Ωi and v k i = v k \| Ωi . Clearly, T 1 v k 1 = T 2 v k 2 . Since v k → v in V , we also have that v k i → v i in V i . The continuity of T i then implies that T i v k i → T i v i in Λ i . Putting this together gives us Λ 1 T 1 v 1 = lim k→∞ T 1 v k 1 = lim k→∞ T 2 v k 2 = T 2 v 2 ∈ Λ 2 in L p (Γ). If we now define µ = T 1 v 1 = T 2 v 2 , then µ is an element in Λ = Λ 1 ∩ Λ 2 .v 1 ∈ V 1 and v 2 ∈ V 2 satisfies T 1 v 1 = T 2 v 2 , then v = {v 1 on Ω 1 ; v 2 on Ω 2 } is an element in V . Proof. It is clear that v ∈ L p (Ω). For each component 1 ≤ ≤ d, there exists a weak derivative ∂ v i ∈ L p (Ω i ) of v i ∈ V i ⊂ W 1,p (Ω i ). If we define z = {∂ v 1 on Ω 1 ; ∂ v 2 on Ω 2 }, then z ∈ L p (Ω). Let w ∈ C ∞ 0 (Ω) and set w i = w\| Ωi ∈ C ∞ (Ω i ). The W 1,p (Ω i )-version of Green's formula [17, Section 3.1.2] yields that Ω z w dx = 2 i=1 Ωi ∂ v i w i dx = 2 i=1 − Ωi v i ∂ w i dx + ∂Ωi (T ∂Ωi v i )w i ν i dS = − Ω v∂ w dx + 2 i=1 Γ (T i v i )wν i dS = − Ω v∂ w dx, i.e., z is the th weak partial derivative of v. By construction T ∂Ω v = 0, and v is therefore an element in V . Transmission problem and the Robin-Robin method. In order to state the Robin-Robin method we reformulate the nonlinear elliptic equation (2.1) on Ω into two equations on Ω i connected via Γ, i.e., we consider a nonlinear transmission problem. To this end, on each V i we define a i : V i × V i → R by a i (u i , v i ) = Ωi α(∇u i ) · ∇v i + g(u i )v i dx. We also define f i = f \| Ωi ∈ L 2 (Ω i ). Lemma 5.1. If the Assumptions 2.1 and 3.1 hold, then a i : V i × V i → R is well defined and satisfies the growth, strict monotonicity and coercivity bounds stated in Lemma 2.5, with the terms (a, V, Ω) replaced by (a i , V i , Ω i ). The weak form of the nonlinear transmission problem is then to find (u 1 , u 2 ) ∈ V 1 ×V 2 such that Proof. Assume that u ∈ V solves (2.1) and define (u 1 , u 2 ) = (u\| Ω1 , u\| Ω2 ) ∈ V 1 ×V 2 . (5.1)      a i (u i , v i ) = (f i , v i ) L 2 (Ωi) , for all v i ∈ V 0 i , i = 1, 2, T 1 u 1 = T 2 u 2 , 2 i=1 a i (u i , R i µ) − (f i , R i µ) L 2 (Ωi) = 0, for all µ ∈ Λ. For v i ∈ V 0 i we can extend by zero to w i ∈ V , by using Lemma 4.6. Then a i (u i , v i ) = a(u, w i ) = (f, w i ) L 2 (Ω) = (f i , v i ) L 2 (Ωi) . Moreover, T 1 u 1 = T 2 u 2 follows immediately from Lemma 4.5. For an arbitrary µ ∈ Λ let v i = R i µ. Then, by Lemma 4.6, v = {v 1 on Ω 1 ; v 2 on Ω 2 } is an element in V and a 1 (u 1 , R 1 µ) + a 2 (u 2 , R 2 µ) = a(u, v) = (f, v) L 2 (Ω) = (f 1 , R 1 µ) L 2 (Ω1) + (f 2 , R 2 µ) L 2 (Ω2) . This proves that (u 1 , u 2 ) solves (5.1). Conversely, let (u 1 , u 2 ) ∈ V 1 × V 2 be a solution to (5.1) and define u = {u 1 on Ω 1 ; u 2 on Ω 2 }. By Lemma 4.6, we have that u ∈ V . Next, consider v ∈ V and let v i = v\| Ωi ∈ V i . From Lemma 4.5 we have that µ = T i v i is well defined and µ ∈ Λ. The observation that v i − R i µ ∈ V 0 i , for i = 1, 2, then implies the equality a(u, v) = 2 i=1 a i (u i , v i − R i µ) + a i (u i , R i µ) = 2 i=1 (f i , v i − R i µ) L 2 (Ωi) + a i (u i , R i µ) = 2 i=1 (f i , v i ) L 2 (Ωi) + a i (u i , R i µ) − (f i , R i µ) L 2 (Ωi) = (f, v) L 2 (Ω) . As v can be chosen arbitrarily, u solves (2.1). Remark 5.3. As the nonlinear elliptic equation (2.1) has a unique weak solution, Theorem 5.2 implies that the same holds true for the nonlinear transmission problem (5.1). Remark 5.4. The equivalence can easily be generalized to more than two subdomains without cross points as in Figure 1c. However, for domain decompositions with cross points such as in Figure 1d, the situation is more delicate. A proof of the equivalence for quasilinear equations in H 1 (Ω) can be found in [20]. This result can most likely be generalized to our W 1,p (Ω)-setting, but we will study this aspects elsewhere. In order to approximate the weak solution (u n 1 , u n 2 ) ∈ V 1 × V 2 of the nonlinear transmission problem in a parallell fashion, we consider the Robin-Robin method. Multiplying by test functions and formally applying Green's formula to the equations (1.4) yield that the weak form of the method is given by finding (u n 1 , u n 2 ) ∈ V 1 × V 2 , for n = 1, 2, . . . , such that (5.2)                        a 1 (u n+1 1 , v 1 ) = (f 1 , v 1 ) L 2 (Ω1) , for all v 1 ∈ V 0 1 a 1 (u n+1 1 , R 1 µ) − (f 1 , R 1 µ) L 2 (Ω1) + a 2 (u n 2 , R 2 µ) − (f 2 , R 2 µ) L 2 (Ω2) = s(T 2 u n 2 − T 1 u n+1 1 , µ) L 2 (Γ) , for all µ ∈ Λ, a 2 (u n+1 2 , v 2 ) = (f 2 , v 2 ) L 2 (Ω2) , for all v 2 ∈ V 0 2 a 2 (u n+1 2 , R 2 µ) − (f 2 , R 2 µ) L 2 (Ω2) + a 1 (u n+1 1 , R 1 µ) − (f 1 , R 1 µ) L 2 (Ω1) = s(T 1 u n+1 1 − T 2 u n+1 2 , µ) L 2 (Γ) , for all µ ∈ Λ, where u 0 2 ∈ V 2 is an initial guess and s > 0 is the given method parameter. 6. Interface formulations. The ambition is now to reformulate the nonlinear transmission problem and the Robin-Robin method, which are all given on the domains Ω i , into problems and methods only stated on the interface Γ. As a preparation, we observe that nonlinear elliptic equations on Ω i with inhomogeneous Dirichlet conditions have unique weak solutions. Lemma 6.1. If the Assumptions 2.1 and 3.1 hold, then for each η ∈ Λ i there exists a unique u i ∈ W 1,p (Ω i ) such that (6.1) a i (u i , v i ) = (f i , v i ) L 2 (Ωi) , for all v i ∈ V 0 i , and T ∂Ωi u i = E i η in W 1−1/p,p (∂Ω i ). The proof can, e.g., be found in [19,Theorem 2.36]. With the notation of Lemma 6.1, consider the operator F i : η → u i , i.e., the map from a given boundary value on Γ to the corresponding weak solution of the nonlinear elliptic problem (6.1) on Ω i . From the statement of Lemma 6.1 we see that F i : Λ i → V i and T i F i η = η for η ∈ Λ i . In other words, the operator F i is a nonlinear right inverse to T i . This property will be frequently used as it, together with the boundedness and linearity of T i , gives rise to bounds of the forms η Λi ≤ C i F i η Vi and η − µ Λi ≤ C i F i η − F i µ Vi . We can now define the nonlinear Steklov-Poincaré operators as S i η, µ = a i (F i η, R i µ) − (f i , R i µ) L 2 (Ωi) , for all η, µ ∈ Λ i , and Sη, µ = 2 i=1 S i η, µ = 2 i=1 a i (F i η, R i µ) − (f i , R i µ) L 2 (Ωi) , for all η, µ ∈ Λ. Thus, we may restate the nonlinear transmission problem (5.1) as the Steklov-Poincaré equation, i.e., finding η ∈ Λ such that (6.2) Sη, µ = 0, for all µ ∈ Λ. That the reformulation is possible follows directly from the definitions of the operators F i and S, but we state this as a lemma for future reference. Lemma 6.2. Let the Assumptions 2.1 and 3.1 hold. If (u 1 , u 2 ) solves (5.1), then η = T 1 u 1 = T 2 u 2 solves (6.2). Conversely, if η solves (6.2), then (u 1 , u 2 ) = (F 1 η, F 2 η) solves (5.1). We next turn to the Robin-Robin method, which is equivalent to the Peaceman-Rachford splitting on the interface Γ. The weak form of the splitting is given by finding (η n 1 , η n 2 ) ∈ Λ 1 × Λ 2 , for n = 1, 2, . . . , such that (6.3) (sJ + S 1 )η n+1 1 , µ = (sJ − S 2 )η n 2 , µ , (sJ + S 2 )η n+1 2 , µ = (sJ − S 1 )η n+1 1 , µ , for all µ ∈ Λ, where η 0 2 ∈ Λ 2 is an initial guess. The observation that the Robin-Robin method and the Peaceman-Rachford splitting are equivalent for linear elliptic equations was made in [1]. The equivalence was also utilized in [5,Section 4.4.1] for the linear setting of the Stokes-Darcy coupling. 2 ) n≥1 = (T 1 u n 1 , T 2 u n 2 ) n≥1 is a weak Peaceman-Rachford approximation (6.3), with η 0 2 = T 2 u 0 2 . Conversely, if (η n 1 , η n 2 ) n≥1 is given by (6.3), then (u n 1 , u n 2 ) n≥1 = (F 1 η n 1 , F 2 η n 2 ) n≥1 fulfils (5.2), with u 0 2 = F 2 η 0 2 . Proof. First assume that (u n 1 , u n 2 ) n≥1 ⊂ V 1 × V 2 is a weak Robin-Robin approximation and define η n i = T i u n i ∈ Λ i . This definition, the existence of a unique solution of (6.1) and the first and third assertions of (5.2) yield the identification u n i = F i η n i . Inserting this into the second and fourth assertion of (5.2) gives us s(T 1 F 1 η n+1 1 , µ) L 2 (Γ) + a 1 (F 1 η n+1 1 , R 1 µ) − (f, R 1 µ) L 2 (Ω1) = s(T 2 F 2 η n 2 , µ) L 2 (Γ) − a 2 (F 2 η n 2 , R 2 µ) + (f, R 2 µ) L 2 (Ω2) , and s(T 2 F 2 η n+1 2 , µ) L 2 (Γ) + a 2 (F 2 η n+1 2 , R 2 µ) − (f, R 2 µ) L 2 (Ω2) = s(T 1 F 1 η n+1 1 , µ) L 2 (Γ) − a 1 (F 1 η n+1 1 , R 1 µ) + (f, R 1 µ) L 2 (Ω1) , for all µ ∈ Λ, which is precisely the weak form of the Peaceman-Rachford splitting (6.3), with η 0 2 = T 2 u 0 2 . Conversely, suppose that (η n 1 , η n 2 ) n≥1 ⊂ Λ 1 × Λ 2 is a weak Peaceman-Rachford approximation and define u n i = F i η n i ∈ V i . Inserting this into (6.3) directly gives that (u n 1 , u n 2 ) n≥1 , with u 0 2 = F 2 η 0 2 , is a weak Robin-Robin approximation (5.2). Remark 6.4. For now we do not know if the weak Robin-Robin and Peaceman-Rachford approximations actually exist, but we will return to this issue shortly. 7. Properties of the nonlinear Steklov-Poincaré operators. We proceed by deriving the central properties of the Steklov-Poincaré operators S i , S when interpreted as maps from Λ i , Λ into the corresponding dual spaces. Proof. The linearity of the functionals S i η and Sη follow by definition. As F i η ∈ V i we have, by Lemma 5.1, that \| S i η, µ \| ≤ \|a i (F i η, R i µ)\| + \|(f i , R i µ) L 2 (Ωi) \| ≤ c i ( ∇F i η p−1 L p (Ωi) d ∇R i µ L p (Ωi) d + F i η r−1 L r (Ωi) R i µ L r (Ωi) ) + f i L 2 (Ωi) R i µ L 2 (Ωi) ≤ C i ( ∇F i η Vi , f i L 2 (Ωi) ) R i µ Vi ≤ C i µ Λi , for all µ ∈ Λ i . Thus, S i η is a bounded functional on Λ i . The boundedness of Sη follows directly by summing up the bounds for S i . S i η − S i µ, η − µ ≥ c i ∇(F i η − F i µ) p L p (Ωi) d + F i η − F i µ r L r (Ωi) , for all η, µ ∈ Λ i , and Sη − Sµ, η − µ ≥ c 2 i=1 ∇(F i η − F i µ) p L p (Ωi) d + F i η − F i µ r L r (Ωi) , for all η, µ ∈ Λ, respectively. Proof. Since, w i = R i (η − µ) − (F i η − F i µ) ∈ V 0 i , for all η, µ ∈ Λ i , we have according to the definition of F i that (7.1) a i (F i η, w i ) − a i (F i µ, w i ) = 0. By this equality and Lemma 5.1, it follows that S i η − S i µ, η − µ = a i F i η, R i (η − µ) − a i F i µ, R i (η − µ) = a i (F i η, w i ) + a i (F i η, F i η − F i µ) − a i (F i µ, w i ) − a i (F i µ, F i η − F i µ) ≥ c i ∇(F i η − F i µ) p L p (Ωi) d + F i η − F i µ r L r (Ωi) , for all η, µ ∈ Λ i , which proves the monotonicity bound for S i . The bound for S follows directly by summing up the bounds for S i . lim η Λ i →∞ S i η, η η Λi = ∞ and lim η Λ →∞ Sη, η η Λ = ∞. Proof. As F i η Vi = ∇F i η L p (Ωi) d + F i η L r (Ωi) ≥ c i η Λi , we have that P ( ∇F i η L p (Ωi) d , F i η L r (Ωi) ) → ∞, as η Λi → ∞, where P (x, y) = (x p + y r )/(x + y). In particular, we assume from now on that P ( ∇F i η L p (Ωi) d , F i η L r (Ωi) ) ≥ f i L 2 (Ωi) . By observing that R i η − F i η ∈ V 0 i , Lemma 5.1 yields the lower bound S i η, η = a i (F i η, F i η) + a i (F i η, R i η − F i η) − (f i , R i η) L 2 (Ωi) = a i (F i η, F i η) + (f i , R i η − F i η) L 2 (Ωi) − (f i , R i η) L 2 (Ωi) ≥ c i ( ∇F i η p L p (Ωi) d + F i η r L r (Ωi) ) − (f i , F i η) L 2 (Ωi) ≥ c i P ( ∇F i η L p (Ωi) d , F i η L r (Ωi) ) F i η Vi − f i L 2 (Ωi) F i η Vi ≥ c i P ( ∇F i η L p (Ωi) d , F i η L r (Ωi) ) − f i L 2 (Ωi) η Λi , which implies that S i is coercive. For S we obtain that Sη, η η Λ ≥ 2 i=1 c i P ( ∇F i η L p (Ωi) d , F i η L r (Ωi) ) − f i L 2 η Λi η Λ1 + η Λ2 , which tends to infinity as η Λ tends to infinity. Thus, S is also coercive. In order to prove that the operators S i , S are demicontinuous, we first consider the continuity of the operators F i . Proof. Let η, µ be elements in Λ i . Using the equality (7.1) together with Lemma 5.1 gives us the bound (7.2) c i ( ∇(F i η − F i µ) p L p (Ωi) d + F i η − F i µ r L r (Ωi) ) ≤ a i (F i η, F i η − F i µ) − a i (F i µ, F i η − F i µ) = a i F i η, R i (η − µ) − a i (F i η, w i ) − a i F i µ, R i (η − µ) + a i (F i µ, w i ) ≤ C i ∇F i η p−1 L p (Ωi) d ∇R i (η − µ) L p (Ωi) d + F i η r−1 L r (Ωi) R i (η − µ) L r (Ωi) + ∇F i µ p−1 L p (Ωi) d ∇R i (η − µ) L p (Ωi) d + F i µ r−1 L r (Ωi) R i (η − µ) L r (Ωi) ≤ C i ∇F i η p−1 L p (Ωi) d + F i η r−1 L r (Ωi) + ∇F i µ p−1 L p (Ωi) d + F i µ r−1 L r (Ωi) η − µ Λi . Letting µ = 0 in (7.2) and employing the inequality \|x\| p − 2 p−1 \|y\| p ≤ 2 p−1 \|x − y\| p , twice, yield that c i ( ∇F i η p L p (Ωi) d + F i η r L r (Ωi) − 2 p−1 ∇F i 0 p L p (Ωi) d − 2 r−1 F i 0 r L r (Ωi) ) ≤ c i (2 p−1 ∇(F i η − F i 0) p L p (Ωi) d + 2 r−1 F i η − F i 0 r L r (Ωi) ) ≤ C i ∇F i η p−1 L p (Ωi) d + F i η r−1 L r (Ωi) + ∇F i 0 p−1 L p (Ωi) d + F i 0 r−1 L r (Ωi) η Λi . Thus, we have a bound of the form (7.3) ∇F i η p L p (Ωi) d + F i η r L r (Ωi) − c 1 ∇F i η p−1 L p (Ωi) d + F i η r−1 L r (Ωi) + c 2 ≤ C i η Λi , for every η ∈ Λ i , where c = c ( ∇F i 0 L p (Ωi) d , F i 0 L r (Ωi) ) ≥ 0. Assume that η k → η in Λ i . As η k is bounded in Λ i , the bound (7.3) implies that ∇F i η k and F i η k are bounded in L p (Ω i ) d and L r (Ω i ), respectively. By setting µ = η k in (7.2), we finally obtain that ∇F i η k → ∇F i η in in L p (Ω i ) d and F i η k → F i η in L r (Ω i ), i.e., F i η k → F i η in V i .k i → η i in Λ i and η k → η in Λ then S i η k i − S i η i , µ i → 0 and Sη k − Sη, µ → 0, for all µ i ∈ Λ i and µ ∈ Λ. Proof. Assume that η k i → η i in Λ i . Lemma 7.4 then implies that ∇F i η k → ∇F i η in L p (Ω i ) d and F i η k → F i η in L r (Ω i ). By the assumed continuity and boundedness of the functions α : z → α 1 (z), . . . , α d (z) and g, we also have that the corresponding Nemyckii operators α : L p (Ω i ) d → L p /(p−1) (Ω i ) and g : L r (Ω i ) → L r /(r−1) (Ω i ) are continuous [25,Proposition 26.6]. The demicontinuity of S i then holds, as \| S i η k i − S i η i , µ i \| ≤ d =1 α (∇F i η k i ) − α (∇F i η i ) L p /(p−1) (Ωi) + g(F i η k i ) − g(F i η i ) L r /(r−1) (Ωi) R i µ i Vi , for every µ i ∈ Λ i . The demicontinuity of S i directly implies the same property for S, as a convergent sequence {η k } in Λ is also convergent in Λ 1 and Λ 2 . 3) seems to be too general for a convergence analysis. To remedy this, we will restrict the domains of the operators S i , S such that the Steklov-Poincaré equation (6.2) and the Peaceman-Rachford splitting can be interpreted on L 2 (Γ) instead of on the dual spaces Λ * i , Λ * . This comes at the cost of requiring more regularity of the weak solution and of the initial guess η 0 2 . More precisely, we define the operators S i : D(S i ) ⊆ L 2 (Γ) → L 2 (Γ) as D(S i ) = {µ ∈ Λ i : S i µ ∈ L 2 (Γ) * } and S i µ = J −1 S i µ for µ ∈ D(S i ). Analogously, we introduce S : D(S) ⊆ L 2 (Γ) → L 2 (Γ) given by D(S) = {µ ∈ Λ : Sµ ∈ L 2 (Γ) * } and Sµ = J −1 Sµ for µ ∈ D(S). As the zero functional obviously is an element in L 2 (Γ) * , the unique solution η ∈ Λ of the Steklov-Poincaré equation is in D(S) and Sη = 0. However, the definition of the domains do not ensure that D(S) is equal to D(S 1 ) ∩ D(S 2 ), as (S 1 + S 2 )µ ∈ L 2 (Γ) * does not necessarily imply that S i µ ∈ L 2 (Γ) * . If the weak solution of the nonlinear elliptic equation (2.1) satisfies the additional regularity property stated in Assumption 2.6, then the corresponding solution of Sη = 0 is in fact an element in D(S 1 )∩D(S 2 ). This propagation of regularity will be crucial when proving convergence of the Peaceman-Rachford splitting. Proof. As u = {F 1 η on Ω 1 ; F 2 η on Ω 2 } is the weak solution of (2.1), we have that Ω α(∇u) · ∇v dx = − Ω g(u) − f v dx, for all v ∈ C ∞ 0 (Ω). The restrictions on r in Assumption 2.1 yield that u ∈ W 1,p (Ω) → L 2(r−1) (Ω). This together with the observation \|g(u)\| 2 ≤ C\|u\| 2(r−1) implies that g(u) − f ∈ L 2 (Ω), i.e., the distributional divergence of α(∇u) is in L 2 (Ω) d . By Assumption 2.6 and restricting to Ω i , we arrive at α(∇F i η) ∈ H(div, Ω i ) ∩ C(Ω i ) d , α(∇F i η) · ν i ∈ L ∞ (∂Ω i ) and ∇ · α(∇F i η) = g(F i η) − f i ∈ L 2 (Ω i ). The H(div, Ω i )-version of Green's formula [8, Chapter 1, Corollary 2.1] then gives us Ωi α(∇F i η) · ∇v dx = − Ωi ∇ · α(∇F i η)v dx + ∂Ωi α(∇F i η) · ν i T ∂Ωi v dS, for all v ∈ H 1 (Ω i ). Hence, S i η, µ = Ωi α(∇F i η) · ∇R i µ dx + Ωi g(F i η) − f i R i µ dx = ∂Ωi α(∇F i η) · ν i T ∂Ωi R i µ dS = α(∇F i η) · ν i , µ L 2 (Γ) , for all µ ∈ Λ i , which implies that η ∈ D(S i ), for i = 1, 2. Remark 8.3. From the proof it is clear that the regularity assumption α(∇u) ∈ C(Ω) d is stricter than necessary, and could be replaced by assuming that the normal component of α(∇u) on Γ can be interpreted as an element in L 2 (Γ) * . However, characterizing the spatial regularity of u required to satisfy this weaker assumption demands a more elaborate trace theory than the one considered in section 3. Proof. The monotonicity follows by Lemma 7.2, as (S i η − S i µ, η − µ) L 2 (Γ) = (J −1 S i η − J −1 S i µ, η − µ) L 2 (Γ) = S i η − S i µ, η − µ ≥ 0, for all η, µ ∈ D(S i ) ⊆ Λ i . For a fixed s > 0 and an arbitrary µ ∈ L 2 (Γ) we have, due to Corollary 7.7, that there exists a unique η ∈ Λ i such that (sJ + S i )η = Jµ in Λ * i , i.e., S i η = J(µ − sη) ∈ L 2 (Γ) * . Hence, η ∈ D(S i ) and (sI +S i )η = µ in L 2 (Γ). The operators sI +S i : D(S i ) → L 2 (Γ) are therefore bijective. which yields the representations η = µ + λ 2s , S 2 η = µ − λ 2 , S 1 η = λ − µ 2 , η n 2 = µ n + λ n 2s , S 2 η n 2 = µ n − λ n 2 , η n+1 1 = µ n+1 + λ n 2s , S 1 η n+1 1 = λ n − µ n+1 2 . The monotonicity of S i then gives the bounds 0 ≤ (S 2 η n 2 − S 2 η, η n 2 − η) L 2 (Γ) = 1 4s (µ n − µ) − (λ n − λ), (µ n − µ) + (λ n − λ) L 2 (Γ) = 1 4s µ n − µ 2 L 2 (Γ) − λ n − λ 2 L 2 (Γ) , and 0 ≤ (S 1 η n+1 1 − S 1 η, η n+1 1 − η) L 2 (Γ) = 1 4s (λ n − λ) − (µ n+1 − µ), (λ n − λ) + (µ n+1 − µ) L 2 (Γ) = 1 4s λ n − λ 2 L 2 (Γ) − µ n+1 − µ 2 L 2 (Γ) . Putting this together yields that µ n+1 − µ 2 L 2 (Γ) ≤ λ n − λ 2 L 2 (Γ) ≤ µ n − µ 2 L 2 (Γ) , and we obtain the telescopic sum 0 ≤ N n=0 µ n − µ 2 L 2 (Γ) − µ n+1 − µ 2 L 2 (Γ) ≤ µ 0 − µ 2 L 2 (Γ) − µ N +1 − µ 2 L 2 (Γ) , i.e., µ n − µ 2 L 2 (Γ) − µ n+1 − µ 2 L 2 (Γ) → 0 as n → ∞. The latter together with the bounds above imply the sought after limits (8.2). Theorem 8.9. Consider the Peaceman-Rachford approximation (η n 1 , η n 2 ) n≥1 , given by (8.1), of the Steklov-Poincaré equation Sη = 0 in L 2 (Γ), together with the corresponding Robin-Robin approximation (u n 1 , u n 2 ) n≥1 = (F 1 η n 1 , F 2 η n 2 ) n≥1 of the weak solution u = {F 1 η on Ω 1 ; F 2 η on Ω 2 } to the nonlinear elliptic equation (2.1). If η 0 2 ∈ D(S 2 ) and the Assumptions 2.1, 2.6 and 3.1 hold, then (8.3) η n 1 − η Λ1 + η n 2 − η Λ2 → 0, and u n 1 − u W 1,p (Ω1) + u n 2 − u W 1,p (Ω2) → 0, as n tends to infinity. Proof. By the monotonicity bound in Lemma 7.2, the property that η n i , η ∈ D(S i ) and Lemma 8.8, we have the limits c i ∇(F i η n i − F i η) p L p (Ωi) d + F i η n i − F i η r L r (Ωi) ≤ S i η n i − S i η, η n i − η = (S i η n i − S i η, η n i − η) L 2 (Γ) → 0, as n → ∞, for i = 1, 2. Hence, each of the terms ∇(F i η n i − F i η) L p (Ωi) d and F i η n i − F i η L r (Ωi) tend to zero, which yields that η n i − η Λi ≤ C F i η n i − F i η Vi → 0, as n → ∞, for i = 1, 2. The desired convergence (8.3) is then proven, as · Vi and · W 1,p (Ωi) are equivalent norms. · (\|∇u\| p−2 ∇u) + λu = f, arising in the context of an implicit Euler discretization of the parabolic p-Laplace equation, and the nonlinear reaction-diffusion problem (1.3) − ∇ · (\|∇u\| p−2 ∇u) + λ\|u\| p−2 u = f. Fig. 1 : 1Examples of different domain decompositions: (a) illustrates a domain decomposition with two intersection points; (b) a decomposition without intersection points; (c) a stripewise decomposition without cross points; (d) a full decomposition with cross points. Example 2. 2 . 2The equation (1.2) satisfies Assumption 2.1 with α(z) = \|z\| p−2 z, λ > 0, g(x) = λx and r=2. The same holds for equation (1.3) with g(x) = λ\|x\| p−2 x and r = p. Remark 2.3. The last assertion of Assumption 2.1 is made in order to ensure that the convergence analysis of the domain decomposition is valid without employing the Poincaré inequality, which allows decompositions where ∂Ω\∂Ω i = ∅; seeFigure 1b. If the latter setting is excluded, then the analysis is valid for a broader class of functions g, especially g = 0. Lemma 4 . 1 . 41If the Assumptions 2.1 and 3.1 hold, then Λ i and Λ are reflexive Banach spaces. Lemma 4. 2 . 2If the Assumptions 2.1 and 3.1 hold, then Λ i and Λ are dense in L 2 (Γ). Lemma 4. 4 . 4If the Assumptions 2.1 and 3.1 hold, then T i and R i are bounded, and R i is a right inverse of T i . Lemma 4. 5 . 5If the Assumptions 2.1 and 3.1 hold and v ∈ V , then µ = T 1 v\| Ω1 = T 2 v\| Ω2 is an element in Λ. Lemma 4. 6 . 6Let the Assumptions 2.1 and 3.1 hold. If two elements The framework given in section 4 enables us to prove equivalence between the nonlinear elliptic equation and the nonlinear transmission problem along the same lines as done for linear equations [18, Lemma 1.2.1]. Theorem 5.2. Let Assumptions 2.1 and 3.1 hold. If u ∈ V solves (2.1), then (u 1 , u 2 ) = (u\| Ω1 , u\| Ω2 ) solves (5.1). Conversely, if (u 1 , u 2 ) solves (5.1), then u = {u 1 on Ω 1 ; u 2 on Ω 2 } solves (2.1). Lemma 6 . 3 . 63Let the Assumptions 2.1 and 3.1 be valid. If (u n 1 , u n 2 ) n≥1 is a weak Robin-Robin approximation (5.2), then (η n 1 , η n Lemma 7 . 1 . 71If the Assumptions 2.1 and 3.1 hold, then S i : Λ i → Λ * i and S : Λ → Λ * are well defined. Lemma 7 . 2 . 72If the Assumptions 2.1 and 3.1 hold, then the operators S i : Λ i → Λ * i and S : Λ → Λ * are strictly monotone with Lemma 7. 3 . 3If the Assumptions 2.1 and 3.1 hold, then the operators S i : Λ i → Λ * i and S : Λ → Λ * are coercive, i.e, Lemma 7 . 4 . 74If the Assumptions 2.1 and 3.1 hold, then the nonlinear operators F i : Λ i → V i are continuous. Lemma 7 . 5 . 75If the Assumptions 2.1 and 3.1 hold, then the operators S i : Λ i → Λ * i and S : Λ → Λ * are demicontinuous, i.e., if η Theorem 7 . 6 . 76If the Assumptions 2.1 and 3.1 hold, then the nonlinear Steklov-Poincaré operators S i : Λ i → Λ * i and S : Λ → Λ * are bijective. Proof. The spaces Λ i and Λ are real, reflexive Banach spaces and, by Lemmas 7.2, 7.3 and 7.5, the operators S i : Λ i → Λ * i and S : Λ → Λ * are all strictly monotone, coercive and demicontinuous. With these properties, the Browder-Minty theorem; see, e.g., [25, Theorem 26.A(a,c,f)], implies that the operators are bijective.The next corollary follows by the same argumentation as for the bijectivity of S i . Corollary 7 . 7 . 77If the Assumptions 2.1 and 3.1 hold, then the operators sJ +S i : Λ i → Λ * i are bijective, for every s > 0. 8. Existence and convergence of the Robin-Robin method. The weak form of the Peaceman-Rachford splitting (6. Remark 8 . 1 . 81By the above construction, one obtains that D(S 1 ) ∩ D(S 2 ) ⊆ D(S) andSµ = S 1 µ + S 2 µ, for all µ ∈ D(S 1 ) ∩ D(S 2 ). Lemma 8. 2 . 2If the Assumptions 2.1, 2.6 and 3.1 hold and Sη = 0, then η ∈ D(S 1 ) ∩ D(S 2 ). Lemma 8 . 4 . 84If the Assumptions 2.1 and 3.1 hold, then the operators S i are monotone, i.e.,(S i η − S i µ, η − µ) L 2 (Γ) ≥ 0, for all η, µ ∈ D(S i ),and the operators sI + S i : D(S i ) → L 2 (Γ) are bijective for any s > 0. p (∂Ωi) .Furthermore, the space W s,p (∂Ω i ) is complete and reflexive; see [10, Definition 6.8.6] and the comment thereafter. Next, we recapitulate the trace theorem for W 1,pfunctions on Lipschitz domains; see, e.g., [10, Theorems 6.8.13 and 6.9.2].Lemma 3.2. If the Assumptions 2.1 and 3.1 are valid, then there exists a surjective bounded linear operator T ∂Ωi : and λ = (sI − S 2 )η, The Peaceman-Rachford splitting on L 2 (Γ) is now given by finding (η n 1 , η n 2 ) ∈ D(S 1 ) × D(S 2 ), for n = 1, 2, . . . , such that(8.1)(sI + S 1 )η n+1 1 = (sI − S 2 )η n 2 , (sI + S 2 )η n+1is an initial guess. Lemma 8.4 then directly yields the existence of the approximation.Corollary 8.5. If the Assumptions 2.1 and 3.1 hold and η 0 2 ∈ D(S 2 ), then there exists a unique Peaceman-Rachford approximation (η n 1 , η n 2 ) n≥1 ⊂ D(S 1 )×D(S 2 ) given by(8.1)in L 2 (Γ).Corollary 8.6. Let the Assumptions 2.1 and 3.1 hold, η 0 2 ∈ D(S 2 ) and set u 0This implies that (sJ + S 1 )η n+1 1 , µ = (sJ − S 2 )η n 2 , µ , for all µ ∈ Λ = Λ 1 ∩ Λ 2 , i.e., the first assertion of (6.3) holds. The same argumentation yields that the second assertion of (6.3) is valid. As (η n 1 , η n 2 ) n≥1 satisfies (6.3), Lemma 6.3 directly implies that (u n 1 , u n 2 ) n≥1 = (F 1 η n 1 , F 2 η n 2 ) n≥1 is a weak Robin-Robin approximation (5.2). Remark 8.7. At a first glance, finding an initial guess satisfying η 0 2 ∈ D(S 2 ) might seem limiting, as the domain is not explicitly given. However, such an initial guess can, e.g., be found by solving S 2 η 0 2 , µ = 0, for all µ ∈ Λ 2 . With this L 2 (Γ)-framework the key part of the convergence proof follows by the abstract result[14,Proposition 1]. For sake of completeness we state a simplified version of the short proof in the current notation.Proof. By the hypotheses and Lemma 8.2, we obtain that η ∈ D(S 1 ) ∩ D(S 2 ) and S 1 η = −S 2 η. Furthermore, η n+1 1 = (sI+S 1 ) −1 (sI−S 2 )η n 2 ∈ D(S 1 ) and η n+1 2 = (sI+S 2 ) −1 (sI−S 1 )η n+1 1 ∈ D(S 2 ).Next, we introduce the notation µ n = (sI + S 2 )η n 2 , µ = (sI + S 2 )η, λ n = (sI − S 2 )η n Variational algorithms of the domain decomposition method. V I Agoshkov, V I Lebedev, Soviet Journal of Numerical Analysis and Mathematical Modelling. 54Akad. Nauk SSSR, Otdel. Vychisl. Mat.V. I. Agoshkov and V. I. Lebedev, Variational algorithms of the domain decomposition method [translation of Preprint 54, Akad. Nauk SSSR, Otdel. Vychisl. Mat., Moscow, 1983], vol. 5, 1990, pp. 27-46. Soviet Journal of Numerical Analysis and Mathematical Modelling. Convergence behaviour of Dirichlet-Neumann and Robin methods for a nonlinear transmission problem. 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[]	[]	[]	[]	An (m, n)-branched twist spin is a fibered 2-knot in S 4 which is determined by a 1-knot K and coprime integers m and n. For a 1-knot, Lin proved that the number of irreducible SU (2, C)-metabelian representations of the knot group of a 1-knot up to conjugation is determined by the knot determinant of the 1-knot. In this paper, we prove that the number of irreducible SU (2, C)-metabelian representations of the knot group of an (m, n)-branched twist spin up to conjugation is determined by the determinant of a 1-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin's presentation of the knot group of the 1-knot.2010 Mathematics Subject Classification. Primary 57Q45; Secondary 57M60, 57M27.	10.1142/s021821651950007x	[ "https://arxiv.org/pdf/1704.08923v2.pdf" ]	119,308,314	1704.08923	6277b012063061535a75431e0278ccccab727a03	28 Apr 2017 28 Apr 2017arXiv:1704.08923v1 [math.GT] IRREDUCIBLE SU(2, C)-METABELIAN REPRESENTATIONS OF BRANCHED TWIST SPINS MIZUKI FUKUDAand phrases 2-knotscircle actionsrepresentations An (m, n)-branched twist spin is a fibered 2-knot in S 4 which is determined by a 1-knot K and coprime integers m and n. For a 1-knot, Lin proved that the number of irreducible SU (2, C)-metabelian representations of the knot group of a 1-knot up to conjugation is determined by the knot determinant of the 1-knot. In this paper, we prove that the number of irreducible SU (2, C)-metabelian representations of the knot group of an (m, n)-branched twist spin up to conjugation is determined by the determinant of a 1-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin's presentation of the knot group of the 1-knot.2010 Mathematics Subject Classification. Primary 57Q45; Secondary 57M60, 57M27. Introduction A 2-knot is a smoothly embedded 2-sphere in S 4 . A 2-knot is said to be fibered if its complement admits a fibration structure over the circle with some natural structure in a tubular neighborhood of the 2-knot. Although it is very difficult to see how the 2-knot is embedded in S 4 , the idea of admitting a fibration helps us to construct many examples of 2-knots, such as spun knots, twist spun knots, rolling, deformed spun knots, branched twist spins, fibered homotopy-ribbon knots, etc [1,2,5,7,9,16,17]. A branched twist spin is a 2-knot which admits an S 1 -action in its exterior. The terminology "branched twist spin" appears in the book of Hillman [5]. It is known by Pao and Plotnick that a fibered 2-knot is a branched twist spin if and only if its monodromy is periodic [14]. Therefore, this class has special importance among other known classes of fibered 2-knots. Note that spun knots and twist spun knots are included in the class of branched twist spins. We give here a short introduction of branched twist spins based on the classification of locally smooth S 1 -actions on the 4-sphere. Montgomery and Yang showed that effective locally smooth S 1 -actions are classified into four types [10] and Fintushel and Pao showed that there is a bijection between orbit data and weak equivalence classes of S 1 -actions on S 4 [3,13]. Suppose that S 1 acts locally smoothly and effectively on S 4 and the orbit space is S 3 . Then there are at most two types of exceptional orbits called Z m -type and Z n -type, where m, n are coprime positive integers. Let E m (resp. E n ) be the set of exceptional orbits of Z m -type (resp. Z n -type) and F be the fixed point set. The image of the orbit map of E n , denoted by E * n , is an open arc in the orbit space S 3 , and that of F , denoted by F * , is the two points in S 3 which are the end points of E * n . It is known that E * m ∪ E * n ∪ F * constitutes a 1-knot K in S 3 and E n ∪ F is diffeomorphic to the 2-sphere. The (m, n)-branched twist spin of K is defined as E n ∪ F . Note that the (m, 1)-branched twist spin is the m-twist spun knot and the (0, 1)-branched twist spin is the spun knot. If K is a torus knot or a hyperbolic knot then its (m, n)-branched twist spins with m > n and m ≥ 3 are non-trivial. This follows from the fact that K m,n is not reflexive known by Hillman and Plotnick [6]. An oriented k-knot K is said to be equivalent to another oriented k-knot K ′ , denoted by K ∼ K ′ , if there exists a smooth isotopy H t : S k+2 → S k+2 such that H 0 = id and H 1 (K) = K ′ as oriented k-knots. In [4], the author studied the elementary ideal of the fundamental group of the complement of a branched twist spin and gave a criterion to detect if two branched twist spins K m 1 ,n 1 1 and K m 2 ,n 2 2 are inequivalent by the knot determinants ∆ K 1 (−1) and ∆ K 2 (−1), where ∆ K (t) is the Alexander polynomial of a 1knot K in S 3 . Note that the definition of an (m, n)-branched twist spin is generalized to (m, n) ∈ Z × N by taking orientations into account, see Section 2.1. The knot determinant is related to the number of irreducible SU(2, C)-metabelian representations of the fundamental group of the knot complement [8,12]. The aim of this paper is to count the number of such representations for a branched twist spin. Similar to the results in [8,12], the number of such representations is given by the knot determinant as follows: Theorem 1.1. The number of irreducible SU(2, C)-metabelian representations of π 1 (S 4 \ intN(K m,n )) is    \|∆ K (−1)\| − 1 2 (m : even) 0 (m : odd), where N(K m,n ) is a compact tubular neighborhood of K m,n in S 4 . As an immediate corollary, we obtain the same criterion as in [4]. Corollary 1.2 (F. [4]). Branched twist spins K m 1 ,n 1 and K m 2 ,n 2 are inequivalent if one of the following holds: (1) m 1 and m 2 are even and \|∆ K 1 (−1)\| = \|∆ K 2 (−1)\|, (2) m 1 is even, m 2 is odd and \|∆ K 1 (−1)\| = 1. This paper is organized as follows: In Section 1, we define an (m, n)-branched twist spin K m,n as an oriented 2-knot and introduce Plotnick's presentation of π 1 (S 4 \ intN(K m,n )). In Section 2, we state the Lin's presentation of a 1-knot and the Nagasato-Yamaguchi's presentation of the m-fold cyclic branched cover of S 3 along K. In Section 3, we observe irreducible SU(2, C)-metabelian representations of π 1 (S 4 \ intN(K m,n )) and prove Theorem 1.1. [10]. The 3-ball and the 3-sphere in these notations represent the orbit spaces. In case (4), the union E * m ∪ E * n ∪ F * constitutes a 1-knot K in the orbit space S 3 and the union E n ∪F is diffeomorphic to the 2-sphere. This 2-sphere is embedded in S 4 , and is called the (m, n)-branched twist spin of K, denoted by K m,n . In case (3), for an arc A * in S 3 whose end points are F * , the preimage of A * is denoted by A. Then the union A ∪ F is diffeomorphic to the 2-sphere, and is called a twist spun knot. We may regard an m-twist spun knot as K m,1 , where K is A * ∪ E * m ∪ F * . We recall the definition of (m, n)-branched twist spins for (m, n) ∈ Z×N in [4]. First, we remark that the definition in [4] depends on the choice of the orientation of K. Actually, in the definition we fixed a preferred meridian-longitude (θ, φ) of S 3 \ intN(K), where N(K) is a compact tubular neighborhood of K, and replacing (θ, φ) by (−θ, −φ) may change the equivalence class of K m,n . We give the definition of K m,n . Let K be a 1-knot in S 3 and (m, n) be a pair of integers in (Z \ {0}) × N such that \|m\| and n are coprime. We decompose the orbit space S 3 into five pieces as follows: Figure 1. Considering the preimage of the orbit map, we decompose S 4 as follows: S 3 = (S 3 \ intN(K)) ∪ (E c * m × D 2 ) ∪ (E c * n × D 2 ) ∪ (D 3 * 1 ⊔ D 3 * 2 ), where D 3 * 1 ⊔ D 3 * 2 is a compact neighborhood of F * and E c * m and E c * n are the connected components of K \ int(D 3 * 1 ∪ D 3 * 2 ) such that E c * m ⊂ E * m and E c * n ⊂ E * n , see(2.1) S 4 = ( S 3 \ intN(K) × S 1 ) ∪ (E m × D 2 ) ∪ (E n × D 2 ) × (D 4 1 ⊔ D 4 2 ). Let p denote the orbit map. Choosing a point z * m in E c * m , let D 2 * z * m be a 2-disk in S 3 centered at z * m ∈ E c * m and transversal to E c * m . The preimage p −1 (D 2 * z * m ) is a solid torus V m whose core is the exceptional orbit of Z m -type. E * m E * n F * D 3 * 1 ⊔ D 3 * 2 E c * n E c * m Figure 1. Decomposition of S 3 Now we discuss the orientations of V m and E c * m . Let K be an oriented 1-knot in S 3 . First, fix the orientation of S 4 and those of orbits such that they coincide with the direction of the S 1 -action. These orientations determine the orientation of V m × E c * m . Let (θ, φ) be the preferred meridian-longitude pair of K such that the orientation of the longitude φ coincides the orientation of K. From the decomposition (2.1), we can see that φ is regarded as a coordinate of the second factor of V m × E c * m . We assign the orientation of V m so that the orientation of V m × E c * m coincides with the given orientation of S 4 . Finally, we choose the meridian and longitude pair (Θ, H) of V m ∼ = D 2 × S 1 such that H becomes the meridian of V n in the decomposition V m ∪ V n = p −1 (∂D 3 * i ) and the orbits of the S 1action are in the direction εnΘ + \|m\|H with n > 0, where ε = 1 if m ≥ 0 and ε = −1 if m < 0. Definition 2.1 (Branched twist spin). Let K be an oriented knot in S 3 . For each pair (m, n) ∈ Z × N with m = 0 such that \|m\| and n are coprime, let K m,n denote the 2-knot E n ∪ F . If (m, n) = (0, 1) then define K 0,1 to be the spun knot of K. The 2-knot K m,n is called an (m, n)-branched twist spin of K. Note that the branched twist spin K m,1 constructed from {(S 3 , K), m, 1} is an m-twist spun knot of K. Remark 2.2. Let −K be an oriented knot obtained from K by reversing the orientation of K. From the construction of K m,n , we see that K m,n is equivalent to −(−K) −m,n . Let K be a k-knot in S k+2 . The fundamental group of the knot complement S k+2 \ intN(K) is called the knot group of K, where N(K) is a compact tubular neighborhood of K. Lemma 2.3 ([4] ). Let K be an oriented 1-knot and K m,n be the (m, n)-branched twist spin of K with (m, n) ∈ Z × N, where \|m\| and n are coprime. Let y 1 , . . . , y s \| r 1 , . . . , r t be a presentation of the knot group of K such that y 1 is a meridian. Then the knot group of K m,n has the presentation Note that π 1 (S 4 \intN(K m,n )) is isomorphic to π 1 (S 4 \intN((−K) −m,n )) by Remark 2.2. (2.2) π 1 (S 4 \ intN(K m,n )) ∼ = y 1 , . . . , y s , h \| r 1 , . . . , r t , y i hy −1 i h −1 , y \|m\| 1 h β , 2.2. Plotnick's presentation. Assume that m = 0. We ignore the orientation of K m,n since we are interested in the fundamental group of its complement. By Remark 2.2, changing the orientation of K and the sign of m if necessary, we can assume that m is positive. Pao constructed the knot complement of K m,n as follows [13]: Let M K be the m-fold cyclic branched cover of S 3 along K and τ : M K → M K be the diffeomorphism associated with the canonical deck transformation of M K . Let M K × τ n S 1 be the manifold obtained from M K ×I by identifying M K ×{0} with M K ×{1} by (z, 1) → (τ n z, 0), where τ n means the n-th power of composite of τ . Note that M K × τ n S 1 has the natural S 1 -action ϕ s y, t = y, t+ s , where y, t denotes the image of (y, t) ∈ M K ×I by the identification. Let x be a branch point of M K . Then the orbit of x, 0 is a circle in M K × τ n S 1 . There is a neighborhood of the orbit which is invariant by the S 1 -action, denoted by T . It is known in [13] that the knot complement of K m,n is diffeomorphic to (M K × τ n S 1 ) \ intT , which is also diffeomorphic to punc(M K ) × τ n S 1 , where punc(M K ) = M K \ intD 3 with D 3 being a 3-ball in M K . Note that K m,n is regarded as the branch set of the n-fold cyclic branched cover of S 4 along the m-twist spun knot of K. The following lemma is shown by Plotnick in [15]. Lemma 2.4 (Plotnick [15]). Let K m,n be a branched twist spin of K. Then the following holds: π 1 (S 4 \ intN(K m,n )) ∼ = π 1 (punc(M K )) * η / η(τ n z)η −1 = z for all z ∈ π 1 (M K ) , where η is a meridian of K m,n . 2.3. Lin's presentation. Let K be a 1-knot in S 3 . A Seifert surface S of K is called free if S 3 = N(S) ∪ (S 3 \ intN(S)) gives a Heegaard splitting of S 3 . It is known that any 1-knot has a free Seifert surface. A presentation of π 1 (S 3 \ intN(K)) is obtained from the Heegaard splitting associated to a free Seifert surface as follows: Let S be a free Seifert surface of K of genus g and W be a spine of S. Then H 1 = S ×[−1, 1] and H 2 = S 3 \intH 1 is a Heegaard splitting of S 3 . Let K ′ be a simple closed curve obtained from K by pushing it into H 1 slightly. Choose a base point * in W ⊂ S × {0} such that * does not on K and K ′ . Since H 1 and H 2 are handlebodies with genus 2g, we may choose generators a 1 , . . . , a 2g of π 1 (H 1 ) and generators x 1 , . . . , x 2g of π 1 (H 2 ). Let a + 1 , . . . , a + 2g and a − 1 , . . . , a − 2g denote the loops a 1 × {1}, . . . , a 2g × {1} and a 1 × {−1}, . . . , a 2g × {−1}. Each a + i (resp. a − i ) is written in a word of x 1 , . . . , x 2g by the homeomorphism from ∂H 2 to ∂H 1 . The words of a + i (resp. a − i ) are denoted by α i (resp. β i ) for i = 1, . . . , 2g. There is a unique arc c, up to isotopy, such that ( * × [−1, 1]) ∪ c is a meridian of K ′ . The homotopy class of this loop is denoted by µ. From van Kampen theorem, the following theorem holds: Lemma 2.5 (Lin [8]). Let K be a 1-knot in S 3 and S be a free Seifert surface of K. Let S 3 = H 1 ∪ H 2 be the Heegaard splitting associated to S. For generators x 1 , . . . , x 2g of π 1 (H 2 ), π 1 (E K ) has the following presentation: (2.3) π 1 (S 3 \ intN(K)) ∼ = x 1 , . . . , x 2g , µ \| µα i µ −1 = β i , where g is the genus of S, and α i , β i are the words in x 1 , . . . , x 2g determined above. Let x 1 , . . . , x 2g , µ \| µα i µ −1 = β i be a Lin's presentation of π 1 (S 3 \ intN(K)). Denote the sum of indices of x j in α i by v ij and that in β i by u ij . Then the 2g × 2g matrix V = (v ij ) is defined. The matrix V is called a Seifert matrix and det(V + t V ) is called the knot determinant of K, which equals to ∆ K (−1). Note that all generators x 1 , . . . , x 2g are commutators of π 1 (S 3 \ intN(K)). Let ρ 0 : π 1 (S 3 \ intN(K)) → SU(2, C) be an irreducible SU(2, C)-metabelian representation. Since all x 1 , . . . , x 2g are commutators of π 1 (S 3 \ intN(K)), we can assume that (2.4) ρ 0 (x i ) = λ i 0 0 λ i , ρ 0 (µ) = 0 −1 1 0 (i = 1, . . . , 2g), up to conjugation (cf. [11]). Since α i and β i are written in words x 1 , . . . , x 2g , each ρ 0 (α i ) and ρ 0 (β i ) is a diagonal matrix. From (2.4), Lin checked directly the number of irreducible SU(2, C)-metabelian representations of π 1 (S 3 \ intN(K)). Theorem 2.6 (Lin [8]). The number of irreducible SU(2, C)-metabelian representations of π 1 (S 3 \ intN(K)) is \|∆ K (−1)\| − 1 2 . 2.4. Nagasato-Yamaguchi's presentation. Let M K be the m-fold cyclic branched cover of S 3 along K and τ be the canonical deck transformation on M K . The fundamental region of M K contains a free Seifert surface of K. Nagasato and Yamaguchi gave a presentation of π 1 (M K ) from the Lin's presentation of π 1 (S 3 \ intN(K)). Theorem 2.7 (Nagasato,Yamaguchi [12]). Let x 1 , . . . , x 2g , µ \| µα i µ −1 = β i be a Lin's presentation of a 1-knot K. Then π 1 (M K ) has the following presentation: π 1 (M K ) ∼ = τ 0x 1 , . . . , τ 0x 2g , . . . , τ m−1x 1 , . . . , τ m−1x 2g \|α (j) i =β (j−1) i , wherex i is the lift of x i to M K , andα (j) i ,β (j) i are the words obtained from α i , β i by replacing x 1 , . . . x 2g with τ jx 1 , . . . , τ jx 2g for i = 1, . . . , 2g and j ≡ 0, . . . , m − 1 (mod m). We can rewrite the presentation in Lemma 2.4 by applying a Nagasato-Yamaguchi's presentation to punc(M K ) as follows: τ 0x 1 , . . . , τ 0x 2g , . . . , τ m−1x 1 , . . . ,τ m−1x 2g , η \| α (j) i =β (j−1) i , ητ j+nx i η −1 = τ jx i . (2.5) Proof of Theorem 1.1 We first introduce a property of irreducible SU(2, C)-metabelian representations of π 1 (S 4 \ intN(K m,n )) from (2.5). Lemma 3.1. Let ρ be an irreducible SU(2, C)-metabelian representation of π 1 (S 4 \ intN(K m,n )). Then, up to conjugation, ρ is of the form ρ(τ jx i ) = λ (j) i 0 0 λ (j) i ρ(η) = 0 −1 1 0 , where i = 1, . . . 2g, j ≡ 0, . . . , m − 1 (mod m), and λ (j) i = λ (j) i for some i, j. Proof. Since ρ is a metabelian representation, ρ([π 1 (S 4 \intN(K m,n )), π 1 (S 4 \intN(K m,n ))]) is an abelian group. Up to conjugation of ρ, we can assume that ρ(x) is a diagonal matrix for any x ∈ [π 1 (S 4 \ intN(K m,n )), π 1 (S 4 \ intN(K m,n ))]. Since the generators τ jx i are on the Seifert surface of K m,n , all τ jx i are commutators in π 1 (S 4 \ intN(K m,n )). Then ρ(τ jx i ) are of the forms ρ(τ jx i ) = λ (j) i 0 0 λ (j) i (λ (j) i ∈ C, \|λ (j) i \| 2 = 1), see [11]. The matrix ρ(η) is determined by the relations ητ j+nx i η −1 = τ jx i as follows. Set ρ(η) = a b c d ∈ SU(2, C). Then ρ(ητ j+nx i ) and ρ(τ jx i η) are given as ρ(ητ j+nx i ) = a b c d λ (j+n) i 0 0 λ (j+n) i = aλ (j+n) i bλ (j+n) i cλ (j+n) i dλ (j+n) i , ρ(τ jx i η) = λ (j) i 0 0 λ (j) i a b c d = aλ (j) i bλ (j) i cλ (j) i dλ (j) i . These two matrices must be the same. Assume that λ (j+n) i = λ (j) i for all i, j. Since m and n are coprime, λ i for some i, then ρ(η) is a diagonal matrix and ρ(π 1 (S 4 \intN(K m,n ))) becomes an abelian group. It also contradicts the irreducibility of ρ. Therefore λ (j+n) i = λ (j) i for some i, j. In this case, a = d = 0 and ρ(η) = 0 b −b 0 . Set B = b 1 2 0 0 b 1 2 . Since Bρ(η)B −1 = 0 −1 1 0 and Bρ(τ jx i )B −1 = ρ(τ jx i ), we have ρ(η) = 0 −1 1 0 up to conjugation. Let S 0 be a free Seifert surface of K contained in a fundamental reagion of M K , where M K is a fiber of K m,n . Let S 1 , . . . , S m−1 be copies of S 0 by the deck transformations. We want to know relation between λ (j) i and λ (j+1) i for j ≡ 1, . . . , m − 1 (mod m). The relation ητ j+nx i η −1 = τ jx i means that the conjugation by η brings τ j+nx i on S j+n to τ jx i on S j . Let q be an integer such that nq ≡ 1(mod m) and take conjugation of τ jx i by η q . Then we obtain the relation (3.1) τ jx i = ητ j+nx i η −1 = η q τ j+nqx i η −q = η q τ j+1x i η −q , which brings τ j+1x i on S j+1 to τ jx i on S j , where we used nq ≡ 1 (mod m). Let ρ be an irreducible SU(2, C)-metabelian representation of π 1 (S 4 \ intN(K m,n )) in Lemma 3.1. From the relation (3.1) and Lemma 3.1 (3.2) λ (j+1) i 0 0 λ (j+1) i =            λ (j) i 0 0 λ (j) i (q : even) λ (j) i 0 0 λ (j) i (q : odd). Suppose that m is even. Then q is odd since m and q are coprime. We define the representation ρ by ρ(x) = ρ(ηxη −1 ) = ρ(η)ρ(x)ρ(η −1 ) for all x ∈ π 1 (S 4 \ intN(K m,n )). Note that ρ(η) = 0 −1 1 0 by Lemma 3.1. In particular, ρ(x) = ρ(x). By (3.2), ρ(τ j+1x i ) = ρ(τ jx i ) for all i, j. Sinceα (j) i andβ (j) i are words written in τ jx 1 , . . . τ jx 2g , we have (3.3) ρ(α (j+1) i ) = ρ(α (j) i ), ρ(β (j+1) i ) = ρ(β (j) i ). On the other hand, (3.4)β (j) i =α (j+1) i = η −qα (j) i η q holds, where the relation (3.1) is applied to the second equality. Since q is odd, ρ(η q xη −q ) = ρ(η q )ρ(x)ρ(η −q ) = ρ(η)ρ(x)ρ(η −1 ) = ρ(x). Hence, by (3.4), we have (3.5) ρ(β The second relations ητ j+nx i η −1 = τ jx i in (2.5) are equivalent to η q τ j+1x i η −q = τ jx i for all j as checked in (3.1). Therefore ρ(ητ j+nx i η −1 ) = ρ(τ jx i ) are equivalent to ρ(ητ j+1x i η −1 ) = ρ(τ jx i ). Hence the number of irreducible SU(2, C)-representations of the presentation (2.5) is equal to that of representations of the group presented by (3.6) τ 0x 1 , . . . , τ 0x 2g , . . . , τ m−1x 1 , . . . , τ m−1x 2g , η \| ηα (0) i η −1 =β (0) i , ητ j+1x i η −1 = τ jx i . Now, we reduce the generators τ 1x 1 , . . . , τ 1x 2g , . . . , τ m−1x 1 , . . . , τ m−1x 2g and the relations ητ j+1x i η −1 = τ jx i from the above presentation to simplify counting the number of irreducible SU(2, C)-metabelian representations of π 1 (S 4 \ intN(K m,n )). Lemma 3.2. Let m be an even integer. Then the number of irreducible SU(2, C)metabelian representations of π 1 (S 4 \intN(K m,n )) coincides that of the group G presented by (3.7) τ 0x 1 , . . . , τ 0x 2g , η \| ηα (0) i η −1 =β (0) i . Proof. A representation of (3.6) is a representation of (3.7). So, we prove the converse. The representation of τ jx i for j ≡ 1, . . . , m − 1 (mod m) is determined by the equality ρ(τ j+1x i ) = ρ(τ jx i ) obtained from (3.2). Hence, it is enough to prove that any irreducible SU(2, C)-metamerian representation ρ of (3.7) has the property ρ(ητ 0x i η −1 ) = ρ(τ 0x i ). Since the presentation in (3.7) is exactly of the same form as the Lin's presentation (2.3), all ρ(τ 0x i ) and ρ(η) are of the forms ρ(τ 0x i ) = S 1 1-actions are classified into the following four types: (1) {D 3 },(2) {S 3 }, (3) {S 3 , m}, (4) {(S 3 , K), m, n}, which are called orbit data where β is an integer such that nβ ≡ ε (mod m) if m is non-zero and β = 1 if m = 0. Recall that ε = 1 if m ≥ 0 and ε = −1 if m < 0. for any i, j. If λ (0) i = λ (0) i for any i, then ρ(τ (j)x i ) = ±1 0 0 ±1 for any i, j. Then ρ is not irreducible. If λ i ). From (3.3) and (3.5), one can see that the relations of representations of the first relations α (j) i =β (j−1) i in (2.5) are equivalent to ρ(β (0) i ) = ρ(ηα (0) i η −1 ). Acknowledgments. The author is grateful to his supervisor, Masaharu Ishikawa, for many helpful suggestions. Zur Isotopie zweidimensionalen Flachen im R 4. E Artin, Abh. Math. Sem. Univ. Hamburg. 4E. Artin, Zur Isotopie zweidimensionalen Flachen im R 4 , Abh. Math. Sem. Univ. Hamburg 4 (1926), 47-72. Ribbon knots in S 4. T Cochran, J. London Math. Soc. 2T. Cochran, Ribbon knots in S 4 , J. London Math. Soc. (2) 28 (1983), no.3, 563-576. Locally smooth circle actions on homotopy 4-spheres. R Fintushel, Duke Math. J. 43R. Fintushel, Locally smooth circle actions on homotopy 4-spheres, Duke Math. J. 43 (1976), 63-70. Branched twist spins and knot determinants, to appear in Osaka. M Fukuda, J. Math. M. Fukuda, Branched twist spins and knot determinants, to appear in Osaka J. Math. 2-knots and their groups. J A Hillman, Austr. Math. Soc. Lect. Ser. 5Cambridge Univ. PressJ. A. Hillman, 2-knots and their groups, Austr. Math. Soc. Lect. Ser. 5, Cambridge Univ. Press, 1989. Geometrically fibered two-knots. J A Hillman, S P Plotnick, Math. Ann. 287J. A. Hillman and S. P. Plotnick, Geometrically fibered two-knots, Math. Ann. 287 (1990), 259-273. Fibered ribbon disks. K Larson, J Meier, J. Knot Theory Ramifications. 241422K. Larson and J. 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On the geometry of the slice of trace-free SL(2, C)-characters of a knot group. F Nagasato, Y Yamaguchi, Math. Ann. 354F. Nagasato and Y. Yamaguchi, On the geometry of the slice of trace-free SL(2, C)-characters of a knot group, Math. Ann. 354 (2012), 967-1002. Non-linear circle actions on the 4-sphere and twisting spun knots. P S Pao, Topology. 17P. S. Pao, Non-linear circle actions on the 4-sphere and twisting spun knots, Topology 17 (1978), 291-296. Equivariant intersection forms, knots in S 4 , and rotations in 2-spheres. S Plotnick, Trans. Amer. Math. Soc. 2962S. Plotnick, Equivariant intersection forms, knots in S 4 , and rotations in 2-spheres, Trans. Amer. Math. Soc. 296 (1986), no.2, 543-575. The homotopy type of four-dimensional knot complements. S Plotnick, Math. Z. 183S. Plotnick, The homotopy type of four-dimensional knot complements, Math. Z. 183 (1983), 447- 471. Fibered knots in S 4 -twisting, spinning, rolling, surgery, and branching. S Plotnick, Contemp. Math. 5Amer. Math. SocS. Plotnick, Fibered knots in S 4 -twisting, spinning, rolling, surgery, and branching, Contemp. Math., vol. 5, Amer. Math. Soc., Providence, RI, 1984, pp.437-459. Twisting spun knots. E C Zeeman, Trans. Amer. Math. Soc. 115E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495.	[]
[ "Coverage centralities for temporal networks", "Coverage centralities for temporal networks" ]	[ "Taro Takaguchi \nNational Institute of Informatics\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan\n\nJST, ERATO\nKawarabayashi Large Graph Project\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan\n", "Yosuke Yano \nJST, ERATO\nKawarabayashi Large Graph Project\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan\n\nDepartment of Computer Science\nThe University of Tokyo\n3-7-1 Hongo, Bunkyo-ku113-8654TokyoJapan\n", "Yuichi Yoshida \nNational Institute of Informatics\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan\n\nPreferred Infrastructure, Inc\n2-40-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n" ]	[ "National Institute of Informatics\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan", "JST, ERATO\nKawarabayashi Large Graph Project\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan", "JST, ERATO\nKawarabayashi Large Graph Project\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan", "Department of Computer Science\nThe University of Tokyo\n3-7-1 Hongo, Bunkyo-ku113-8654TokyoJapan", "National Institute of Informatics\n2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan", "Preferred Infrastructure, Inc\n2-40-1 Hongo, Bunkyo-ku113-0033TokyoJapan" ]	[]	Structure of real networked systems, such as social relationship, can be modeled as temporal networks in which each edge appears only at the prescribed time. Understanding the structure of temporal networks requires quantifying the importance of a temporal vertex, which is a pair of vertex index and time. In this paper, we define two centrality measures of a temporal vertex by the proportion of (normal) vertex pairs, the quickest routes between which can (or should) use the temporal vertex. The definition is free from parameters and robust against the change in time scale on which we focus. In addition, we can efficiently compute these centrality values for all temporal vertices. Using the two centrality measures, we reveal that distributions of these centrality values of real-world temporal networks are heterogeneous. For various datasets, we also demonstrate that a majority of the highly central temporal vertices are located within a narrow time window around a particular time. In other words, there is a bottleneck time at which most information sent in the temporal network passes through a small number of temporal vertices, which suggests an important role of these temporal vertices in spreading phenomena.PACS. 89.75.Fb Structures and organization in complex systems -89.75.Hc Networks and genealogical trees -64.60.aq Networks arXiv:1506.07032v1 [physics.soc-ph]	10.1140/epjb/e2016-60498-7	[ "https://arxiv.org/pdf/1506.07032v1.pdf" ]	9,025,390	1506.07032	bd4f5e47fe7880bde141ad7851f09254b5ff4f37	Coverage centralities for temporal networks Taro Takaguchi National Institute of Informatics 2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan JST, ERATO Kawarabayashi Large Graph Project 2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan Yosuke Yano JST, ERATO Kawarabayashi Large Graph Project 2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan Department of Computer Science The University of Tokyo 3-7-1 Hongo, Bunkyo-ku113-8654TokyoJapan Yuichi Yoshida National Institute of Informatics 2-1-2 Hitotsubashi, Chiyoda-ku101-8430TokyoJapan Preferred Infrastructure, Inc 2-40-1 Hongo, Bunkyo-ku113-0033TokyoJapan Coverage centralities for temporal networks EPJ manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted later Structure of real networked systems, such as social relationship, can be modeled as temporal networks in which each edge appears only at the prescribed time. Understanding the structure of temporal networks requires quantifying the importance of a temporal vertex, which is a pair of vertex index and time. In this paper, we define two centrality measures of a temporal vertex by the proportion of (normal) vertex pairs, the quickest routes between which can (or should) use the temporal vertex. The definition is free from parameters and robust against the change in time scale on which we focus. In addition, we can efficiently compute these centrality values for all temporal vertices. Using the two centrality measures, we reveal that distributions of these centrality values of real-world temporal networks are heterogeneous. For various datasets, we also demonstrate that a majority of the highly central temporal vertices are located within a narrow time window around a particular time. In other words, there is a bottleneck time at which most information sent in the temporal network passes through a small number of temporal vertices, which suggests an important role of these temporal vertices in spreading phenomena.PACS. 89.75.Fb Structures and organization in complex systems -89.75.Hc Networks and genealogical trees -64.60.aq Networks arXiv:1506.07032v1 [physics.soc-ph] Introduction Complex networks such as social networks, information networks, and biological networks have been intensively studied in the past decade to understand their behavior under certain dynamics and develop efficient algorithms for them. See [1][2][3][4] for extensive surveys. However, many real-world networks are actually temporal networks [5,6], in which a vertex communicates with another vertex at specific time over finite duration. For example, social interaction between individuals, passenger flow between cities, and synaptic transmission between neurons can be represented as temporal networks. When we assume that the focal dynamical processes on networks, such as information propagation, occur on a time scale comparable to the change in network structure, a temporal-network representation gives us a precise way to capture the processes. We can describe the advantage of working with a temporal network using the example shown in Fig. 1. This temporal network consists of four vertices and eight edges, each of which has the time it appears. Let us assume that it takes unit time to send the information from the tail to the head of an edge. For example, suppose that the information starts to propagate from v 1 at time 1. a All the authors contributed equally to the work. b e-mail: t [email protected] Then, it reaches v 2 at time 2 through edge (v 1 , v 2 ), waits at v 2 till time 3, then reaches v 3 at time 4 through edge (v 2 , v 3 ). The information never reaches v 4 because the only edge incoming to v 4 is (v 2 , v 4 ) which appears at time 1, and v 2 does not have the information at that time. However, if we ignore the temporal information and regard the network as a static directed network, we mistakenly reach the conclusion that information in v 1 at time 1 can reach v 4 because there is a directed path from v 1 to v 4 . Therefore, we cannot dismiss temporal information to properly understand the structure of temporal networks. An important notion studied to understand the structure of (static) networks is vertex centrality, which measures the importance of a vertex. The following reasons motivate the study of centralities. First, we can use centralities to find important vertices in several applications such as suppressing the epidemics [7,8] or maximizing the spread of influence [9]. Second, we can use them to understand the structure of real-world networks by examining the difference between the distributions of the centrality values in such networks and in the randomized networks (e.g, [10,11]). Third, we can examine the validity of generative network models by investigating the distribution of centralities of the generated network (e.g., [12,13]). Hence, it is natural to study centralities for temporal networks. Since the most fundamental difference between a static network and a temporal network is that the latter involves time, we define the centrality of a vertex at a specific time. To distinguish from a vertex, we call the pair of a vertex and time a temporal vertex. In the literature, multiple centrality notions of temporal vertices based on time-respecting paths [5] have been proposed. Examples include the generalizations of the centrality notions to temporal networks, such as betweenness [15][16][17][18], closeness [16,17,19], communicability [20][21][22], efficiency [14], random-walk centrality [23], and win-lose score [24] (see Ref. [25] for a review of some of them). However, each previous centrality notion suffers from at least one of the following two issues: 1. We need to carefully set parameter values and (or) the time interval within which we consider time-respecting paths. 2. It is inefficient to compute the centrality. For the first issue, the time interval length especially requires careful tuning; if the time interval is too wide, then the centrality of a temporal vertex v becomes negligible because most of the paths finish before or start after v appears. By contrast, if the time interval is too narrow, again the centrality of v becomes negligible because paths can pass by only a tiny fraction of vertices in the time interval. For the second issue, even if we compromise to use an approximation, computing the approximated centrality value of a single temporal vertex requires computational time at least linear to network size [26]. In this paper, we propose two novel centrality notions for temporal networks that resolve these issues. The first one, called temporal coverage centrality (TCC), measures the fraction of pairs of (normal) vertices that can use the temporal vertex when sending information as quickly as possible. The second one, called temporal boundary coverage centrality (TBCC), measures the fraction of pairs of vertices that should use the temporal vertex when sending information as quickly as possible. Our centrality notions address the two issues described above in the following way. For the first issue, TCC and TBCC are free from setting of any parameters or time interval. To calculate the TCC or TBCC value of a temporal vertex v = (v, τ ), we only have to run over all pairs of vertices (u, w). Namely, we consider temporal vertices u = (u, τ u ) and w = (w, τ w ), where τ u is the latest time at which we can send information from u so that it reaches v at time τ , and τ w is the earliest time at which we can receive information at w that is sent from v at time τ . It should be noted that, if we fix focal temporal vertex v, τ u and τ w are uniquely determined by u and w, respectively, and that we thus do not have to care about the time interval around v. Then, we check whether the information sent from u = (u, τ u ) to w = (w, τ w ) can or should drop by v. For the second issue, although the definitions of TCC and TBCC might look complicated and hard to compute, this is not the case. Indeed, computing TCC and TBCC can be reduced to the problem of deciding whether or not there is a directed path between queried vertices in an associated directed network (see Section 2.2 for details). The latter problem is well studied in the database community [38][39][40][41][42], and it can be solved by constructing an index of the directed network, which computes the reachability between any pair of nodes by using information of the reachability between a fraction of node pairs. If it suffices to use approximations to the TCC and TBCC values, we only need to query the index at most O(log 2 N ) times, where N is the total number of vertices in the network (see Appendix A). Since we can efficiently process queries to the index in practice, this method is advantageous compared to the O(N ) time for approximating previous centrality notions. With the aid of our centrality notions, we are able to compute the centrality of all temporal vertices in a temporal network and analyze the statistics of the whole network. Using TBCC, we demonstrate that real-world temporal networks have a small number of temporal vertices without which information propagates more slowly. Surprisingly, we reveal that the temporal vertices of large centrality values form a narrow time region, and this time region seemingly corresponds to the beginning or the end of a time interval in which temporal edges occur in a bursty manner. In addition, by using TCC, we show that the remaining part of the temporal network is highly redundant in the sense that there are many ways to send information as quickly as possible. Although these properties are recognized in the network science community [28][29][30], we quantitatively confirm it for the first time using our centrality notions. We also demonstrate that the removal of temporal vertices according to their TBCC values is effective for hindering the propagation of information for both delaying and stopping it. The paper is organized as follows. In Section 2, we introduce basic notions of temporal networks and the directed network associated with a temporal network. Section 3 introduces our centrality notions for temporal vertices, and Section 4 explains detailed methods of computing our centrality notions. Section 5 is dedicated to demonstrating our experimental results. We give the conclusion in Section 6. Preliminaries about temporal networks 2.1 Basic notions We introduce the terminology and symbols to describe temporal network structure, which basically follow those used in Ref. [31]. For integer k, let [k] denote the set {1, 2, . . . , k}. We define R + as the set of non-negative real numbers. Let V be the set of vertices. A temporal edge is represented by quadruplet e = (u, v, τ, λ), where u, v ∈ V , τ ∈ R, and λ ∈ R + . For temporal edge e = (u, v, τ, λ), we refer to τ , λ, and τ + λ as the starting time, the duration, and the ending time of e, respectively. Temporal network G = (V, E) is a pair of set of vertices V and set of temporal edges E. When we study temporal networks, a vertex at a certain time is of interest. Therefore, we define a temporal vertex by a pair of vertex v ∈ V and time τ ∈ R. In the following, we always use bold symbols such as v to denote temporal vertices. For temporal vertex v = (v, τ ), we denote the time τ by τ (v). Temporal path P in temporal network G = (V, E) is defined as an alternating sequence of temporal vertices and edges P = v 1 , e 1 , v 2 , e 2 , . . . , e k−1 , v k satisfying the following properties. Let v i = (v i , τ i ) for each i ∈ [k]. Then for each i ∈ [k −1], the i-th temporal edge e i is of the form e i = (v i , v i+1 , τ, λ) such that τ i ≤ τ and τ + λ ≤ τ i+1 . We define the starting time, the duration, and the ending time of P as τ 1 , τ k − τ 1 , and τ k , respectively. For two temporal vertices u and v, relationship u v indicates that there is a temporal path from u to v. We define the earliest arrival time at vertex w when departing from temporal vertex v by the smallest τ ∈ R such that v (w, τ ), and we denote it by τ eat (v, w). If there is no such τ , we define τ eat (v, w) = ∞. Similarly, we define the latest departure time from a vertex u for arriving at v as the largest τ ∈ R such that (u, τ ) v, and we denote it by τ ldt (v, u). If there is no such τ , we define τ ldt (v, u) = −∞. A temporal shortest path from temporal vertex v to vertex w is a temporal path from v to (w, τ eat (v, w)), and a temporal shortest path from a vertex u to a temporal vertex v is a temporal path from (u, τ ldt (v, u)) to v. Directed acyclic graph representation A directed acyclic graph (DAG) is a directed network with no directed cycle. In this section, we describe the DAG representation of a temporal network, which is useful when solving problems related to temporal paths and describing the centrality notions we will introduce in Section 3. This DAG representation and its variants have been considered in the analysis of temporal networks [17,[32][33][34][35][36]. For temporal network G = (V, E), the DAG representation of G, denoted by G = ( V , E), is constructed as follows. A vertex in G represents a temporal vertex in G. For each v ∈ V , we first add to V two vertices corresponding to the temporal vertices (v, −∞) and (v, ∞). For each temporal edge (u, v, τ, λ) ∈ E, we add to V two vertices corresponding to temporal vertices u = (u, τ ) and v = (v, τ + λ) (if they do not exist in V ) and add edge (u, v) to E. Finally, for each pair of temporal vertices Figure 2 illustrates DAG representation G of temporal network G shown in Fig. 1. The vertex in the i-th row and the j-th column corresponds to the temporal vertex (v i , j). u = (u, τ ), u = (u, τ ) sharing the same vertex u, we add edge (u, u ) to E if there is no temporal vertex of the form (u, τ ) in V such that τ < τ < τ .For example, since there is temporal edge (v 1 , v 2 , 1, 1) in G, we have an edge from (v 1 , 1) to (v 2 , 2) in G. For the ith row, the leftmost and rightmost vertices correspond to the temporal vertices (v i , −∞) and (v i , ∞), respectively. From the construction of the DAG representation, we have the following useful properties: Lemma 1 Let G be a temporal network. Then, G is a DAG. Proof This is clear as we only add edges of the form ((u, τ ), (v, τ )), where τ < τ . Lemma 2 Let G be a temporal network. Suppose that temporal vertices u and v have corresponding vertices in G. Then, there is a temporal path from u to v in G if and only if there is a directed path from u to v in G. Proof Let P = v 1 , e 1 , v 2 , . . . , e k−1 , v k be a temporal path from v 1 = u to v k = v. Without loss of generality, we assume that the time of v i is equal to the starting time of e i or the ending time of v i−1 . Then, each v i has a corresponding vertex in G. Let v i = (v i , τ v i ) for each i ∈ [k] and e i = (v i , v i+1 , τ e i , λ e i ) for each i ∈ [k − 1]. Then, there is a directed path (v 1 , τ v 1 ), (v 1 , τ e 1 ), (v 2 , τ e 1 + λ e 1 ), (v 2 , τ v 2 ), (v 2 , τ e 1 ), (v 2 , τ e 2 + λ e 2 ), . . . , (v k , τ v k ) in G. The converse easily follows the correspondence explained above. Temporal coverage centralities In this section, we introduce the temporal coverage centrality and the temporal boundary coverage centrality. Algorithm 1 (The TCC value of v) 1: r ← 0. 2: for u ∈ V and w ∈ V do 3: u ← (u, τ ldt (v, u)). 4: w ← (w, τeat(v, w)). 5: if τeat(u, w) = τ (w) and τ ldt (w, u) = τ (u) then 6: r ← r + 1. 7: return r/\|V \| 2 . Temporal coverage centrality Before defining TCC, we define the notion of coverage in temporal networks by generalizing its original version in static networks [37] as follows. Let v be a temporal vertex and u, w be vertices. Let u = (u, τ ldt (v, u)) and w = (w, τ eat (v, w)). Then, we say that v covers node pair (u, w) if the following two conditions hold: 1. τ eat (u, w) = τ eat (v, w), 2. τ ldt (w, u) = τ ldt (v, u). In words, the earliest arrival time at w when departing from u does not change even if we drop by v (condition 1), and the latest departure time from u for arriving at w does not change even if we drop by v (condition 2). 9) are determined as shown in the figure. We observe that, if we depart from u and are not forced to drop by v, we can arrive at w = (v 2 , 8), which is earlier than w. Hence, node pair (u, w) is not covered by v but by w . v = (v 1 , 7). Then, temporal vertices u = (v 4 , τ ldt (v, v 4 )) = (v 4 , 4) and w = (v 2 , τ eat (v, v 2 )) = (v 2 , On the basis of this notion of coverage, the TCC value of v is defined as the fraction of pairs (u, w) ∈ V × V that are covered by v. By definition, the TCC value of a temporal vertex takes a real number in [0, 1]. If the TCC value is close to unity, the temporal vertex is said to be central in the sense that it covers many pairs of nodes. The formal definition is given in Algorithm 1 in an algorithmic manner. Temporal boundary coverage centrality Let v = (v, τ ) be a temporal vertex and u, w be vertices. Let u = (u, τ ldt (v, u)) and w = (w, τ eat (v, w)). Even if the TCC value of v is large, it does not always imply that Algorithm 2 (The TBCC value of v) 1: r ← 0. 2: for u ∈ V and w ∈ V do 3: u ← (u, τ ldt (v, u)). 4: w ← (w, τeat(v, w)). 5: if τeat(u, w) = τ (w) and τ ldt (w, u) = τ (u) then 6: if τeat(u, v) = τ (v) or τ ldt (w, v) = τ (v) then 7: r ← r + 1. return r/\|V \| 2 . removing the temporal edges involving v makes τ eat (u, w) larger or τ ldt (w, u) smaller. One particular reason for this is that sometimes we can reach v from u earlier than τ and can leave v later than τ to reach w (see temporal vertices v 2 and v 3 in Fig. 4). In some applications, we may want to regard such v as unimportant. To address this issue, we define TBCC by imposing additional criteria to the notion of coverage as follows. Note that, if focal temporal vertex v is an example of the situation stated in the previous paragraph, then τ eat (u, v) < τ or τ ldt (w, v) > τ should hold. Hence, we define that a pair (u, w) of vertices is covered at a boundary by temporal vertex v if the following hold: 1. (u, w) is covered by v, and 2. τ eat (u, v) = τ or τ ldt (w, v) = τ . We explain this definition using the example shown in Fig. 4. Let v i = (v, τ i ) for i ∈ [4]. Note that u = (u, τ ldt (v i , u)) and w = (w, τ eat (v i , w)) coincide for all i ∈ [4]. In addition, note that all v i cover (u, w). We can see that v 1 and v 4 cover (u, w) at the boundary because τ eat (u, v) = τ 1 and τ ldt (w, v) = τ 4 . By contrast, v 2 and v 3 do not cover (u, w) at the boundary. On the basis of this notion of coverage at the boundary, the TBCC value of v is defined as the fraction of pairs (u, w) that are covered at the boundary by v. Similar to TCC, the TBCC value of a temporal vertex takes a real number in [0, 1] by definition. The formal definition is given in Algorithm 2 in an algorithmic manner. Computing temporal coverage centralities We can straightforwardly calculate TCC and TBCC according to Algorithms 1 and 2. In this section, to manage large temporal networks, we give efficient methods for computing TCC and TBCC on the basis of a graphindexing technique developed recently in the database community [27]. The key idea is in how to speed up the computation of τ eat and τ ldt in Algorithms 1 and 2. We describe the exact computation of TCC and TBCC in this section, and we also give the algorithms to approximate the TCC and TBCC values whose running time is polylogarithmic in the total number of vertices in G (see Appendix A). In a directed network, we say that a vertex v t is reachable from v s if there is a directed path from v s to v t . With respect to Lemma 2, to enumerate the number of pairs (u, w) being covered by v (at the boundary, if needed), we want to efficiently answer reachability in the DAG representation G of given temporal network G. To this end, it is beneficial to construct an index of G that computes the reachability between any pair of nodes on the basis of information of the reachability between a fraction of node pairs. Such an index is often called a reachability oracle in the database community [38][39][40][41][42]. The basic idea of the construction of a reachability oracle for the present problem is the following. Naively, we want to compute a large table that stores the reachability of every pair of temporal vertices. If this were possible, we could answer reachability just by looking at that table. Unfortunately, however, perfecting this table requires O(\| V \| 2 ) computation time and O(\| V \| 2 ) space, which could be prohibitively slow and large. The reachability oracle overcomes this problem by carefully storing partial information of the network. Based on the information, it efficiently computes the reachability for the whole network. For example, the method proposed in Ref. [42], which we will use for the numerical experiments in Section 5, computes a small table for each temporal vertex that stores reachability from (and to) a small number of other certain temporal vertices. Then, we can answer the reachability from a temporal vertex u to a temporal vertex v by checking whether there is another temporal vertex w such that we can confirm the reachability from u to w and from w to v using the small tables of u and v. If there is such w, we indeed have a directed path from u to v. The challenging part of the construction lies in guaranteeing the other direction; if there is a directed path from u to v, then there is always such w. In addition, we need to be able to compute the small table for each vertex efficiently. This method resolves these issues, so that it can handle directed networks of millions of edges with the query time of less than a microsecond on average (see Ref. [42] for further technical details). With the aid of the reachability oracle, we can efficiently compute τ eat and τ ldt : Lemma 3 Let G be a temporal network and G be its DAG representation. We can compute τ eat and τ ldt with O(log \|E\|) queries to the reachability oracle of G. Proof We only consider τ eat as τ ldt can be computed similarly. Given temporal vertex v and vertex w, τ eat (v, w) is the minimum τ ∈ R such that there is a temporal path from v to (w, τ ). To find such τ , we perform a binary Table 1. Basic statistics of the datasets. Variables n, m, n, and τmax are the total number of vertices and temporal edges in G, the total number of vertices in G, and the maximum ending time of a temporal edge, respectively. The datasets are arranged in increasing order of m. Name n m n τmax Infectious [43] 410 17298 32218 1393 HT09 [43] 113 20187 48477 5246 Hospital [44] 75 32424 65296 9454 Irvine [45] 1899 59835 220772 58192 Email [46] 167 82927 254533 57843 search using the reachability oracle. Since the number of possible values for τ is O(\|E\|), the number of queries is O(log \|E\|). Lemma 4 Let G be a temporal network and G be its DAG representation. For any temporal vertex v, we can compute the TCC and TBCC values of v with O(\|V \| 2 log \|E\|) queries to the reachability oracle of G. Proof The proof is immediate from Lemma 3 and the algorithm definitions of TCC (Algorithm 1) and TBCC (Algorithm 2). Results The basic statistics of the datasets we use are summarized in Table 1. It should be noted that we do not use the actual time stamps in the datasets but define τ by the order of unique values of the time stamps. For example, if the dataset consists of two time stamps t = 1, 4, we translate them into τ = 1, 2. Although interactions in Irvine and Email are directed (i.e., from sender to receiver(s) of messages), we regard them as undirected. Figure 5 depicts the rank plots of the TCC and TBCC values of temporal vertices in the decreasing order. In all the datasets, at least 10% of temporal vertices have TCC values larger than 0.1 (Fig. 5(a)). This fact implies the redundancy of temporal networks in the sense that, when information flows between temporal vertices, it can drop by different vertices without increasing the total duration of the temporal paths. However, there are a smaller number of temporal vertices with large TBCC values ( Fig. 5(b)). This fact also implies the redundancy of temporal networks in a different sense such that, when information flows between temporal vertices, it is not forced to exist at a certain vertex at a certain time. Statistics of TCC and TBCC To see the impact of the structural peculiarity of temporal networks on these distributions, we computed the centrality values of temporal vertices in randomized temporal networks. We randomize an original temporal network by replacing the two ends of each temporal edge by vertices chosen uniformly at random (similar to the procedure called randomized edges with randomly permuted times in Ref. [5]). The resultant centrality values are shown in Fig. 6. We notice that more temporal vertices have sufficiently large centrality values (e.g., larger than 0.1) in real-world temporal networks (Fig. 5) than in randomized temporal networks (Fig. 6). The maximum centrality values are larger in the randomized than in the original networks for HT09 and Hospital, and vice versa for Infectious and Email. This fact implies that the way the flow concentrates upon temporal vertices depends on each dataset. Next, we examine how the centrality values change over time owing to the structural transformation of the temporal networks. Figure 7 depicts the change in the maximum TCC and TBCC values over temporal vertices at present and the number of temporal vertices at present for Infectious and Hospital. In both datasets shown in Fig. 7, we can see some periodic patterns in the number of temporal vertices. However, the maximum centrality values are not much affected by the patterns, which implies that these values are determined not by the mere activity level in the networks but by the structure of the temporal network. In addition, the fact that the maximum centrality values vary considerably throughout the observation periods suggests that we should carefully incorporate temporal structure to assess the importance of vertices. Generally, the maximum TCC values are larger than the maximum TBCC values, which makes sense according to their definitions (i.e., TBCC only counts the coverage of the temporal paths at the boundary but TCC does not impose this boundary criterion). When we focus on a particular vertex, two centrality values of it also vary in a different manner over time. Figure 8 depicts the change in the TCC and TBCC values of the vertex that are involved in the largest number of temporal edges in the two datasets, Infectious and Hospital. The TCC value of the vertex increases with time in Infectious ( Fig. 8(a)), simply because the number of present temporal vertices increases and thus the focal vertex can reach these vertices in this period (also see Fig. 7(a)). By contrast, the TBCC value does not exhibit such an increasing trend. This fact supports our original purpose of introducing TBCC, i.e., to discount the centrality values of the temporal vertices of the dispensable temporal paths. In addition, the plot of TBCC unveils that even the vertex with the largest number of temporal edges does not always bridge effective temporal paths. In Hospital (Fig. 8(b)), we can observe that the temporal edges associated with the focal vertex are partitioned into five time intervals, in each of which temporal edges occur in a bursty manner, and the centrality values of the vertex become larger at the beginning and the end of each of these time intervals. This observation makes sense because, at the endpoints of a time interval, a vertex tends to play the role as the gateway for information flowing into or out of the time interval. The computational efficiency of the two centralities enables us to draw a map of the centrality values of all the temporal vertices over time. This map reveals the existence of bottleneck time regions in the empirical temporal networks. Figures 9(a) and 9(b) depict the TCC values of temporal vertices as a heat map for Infectious and Hospital, respectively. In both datasets, most temporal vertices have non-negligible TCC values, and these results support the notion of redundancy of temporal networks (see Fig. 5(a)) such that all the vertices can belong to redundant temporal paths. In addition, the temporal vertices with the largest centrality values appear in the middle of the observation period, and the temporal vertices at the same time tend to have similar TCC values. We found the same phenomenon in all the datasets (see Electronic Supplementary Materials for the plots of the other datasets), and the existence of this bottleneck time period seems to be a common property of empirical temporal networks. If we are interested in when these bottleneck time periods begin and end, we can look at the heat map of the TBCC values. As an example, Fig. 9(c) magnifies a bottleneck time period in Infectious ( Fig. 9(a)) in which we observe many temporal vertices with the largest TCC values. However, the boundary of the bottleneck period is not clear in the figure. Figure 9(d) shows the heat map of the TBCC values in the same area as shown in Fig. 9(c). As we observe, the TBCC values indicate the boundaries at τ 660, 680, and 750. This boundary information should be meaningful, for example, when we narrow the candidates of the vertices to be vaccinated for epidemic spreading on temporal networks [47][48][49]. We finally stress again that it becomes possible to compute these statistics and analyze the structure of temporal networks in such detail because of the efficient computation of TCC and TBCC using the reachability oracle. Delay caused by removing a central temporal vertex In closing this section, to verify the relevance of the proposed centrality notions at the microscopic level, we briefly report that removing a temporal vertex with large TCC and TBCC values is effective in delaying the propagation of information. Let G = (V, E) be a temporal network, where V = {v 1 , v 2 , . . . , v n }. For a temporal vertex v = (v, τ ), let v i = (v i , τ eat (v, v i )) for each i ∈ [n] and τ be the (unique) time such that v has an edge to v = (v, τ ). We say that v i gets prolonged by removing v if τ eat (v, v i ) becomes larger by removing edges incident to v (and we keep edge (v, v )). In a similar manner, we say that v i becomes disconnected by removing v if we cannot reach v i from v after removing edges incident to v (where, again, we keep edge (v, v )). We investigate the fraction of prolonged or disconnected temporal vertices among v 1 , v 2 , . . . , v n , by removing one of the top 100 vertices with respect to the TCC or TBCC values. It should be noted that the fraction of temporal vertices becoming prolonged or disconnected is nontrivial because the definition of TCC and TBCC take into account temporal paths both before and after the focal temporal vertex. As a baseline for comparison, we also conduct the same test by removing a temporal vertex chosen randomly. For the random case, we randomly choose 100 temporal vertices without replacement and take the average of the fraction of prolonged or disconnected temporal vertices for these 100 trials. The results of the removal test of temporal vertices are summarized in Table 2 for the five datasets. As we expected, the removals according to the largest centrality values make more temporal vertices prolonged or disconnected than the random removals. The removals according to the largest TCC values tend to prolong a certain fraction of temporal vertices for all the datasets considered. However, it makes few temporal vertices disconnected. These outcomes make sense because the number of other temporal paths running alongside the temporal path going through the focal temporal vertex is not considered in TCC (also see Section 3.1). By contrast, the removals according to the largest TBCC values make a considerable fraction of temporal vertices prolonged and disconnected. Remarkably, 50.8% of the temporal vertices, on average, become disconnected from a removed temporal vertex in Irvine. There is no clear distinction between the results of the offline (i.e., Infectious, HT09, and Hospital) and online (i.e., Irvine and Email) networks. Conclusions We introduced two centrality notions for temporal networks-temporal coverage centrality and temporal boundary coverage centrality-to represent the importance of a temporal vertex by the fraction of vertex pairs that can or should use the temporal vertex when sending information as quickly as possible. Compared to centrality notions proposed in previous work, TCC and TBCC have two advantages: (i) Parameters or time windows do not need to be set and (ii) computation time is reasonable. Applying TCC and TBCC to multiple datasets of empirical temporal networks, we revealed that there tends to be particular bottleneck time periods that play a crucial role in propagating information quickly and that the rest of the networks is redundant in the sense that there are many temporal paths to send information with the same duration. Although such structural redundancy in temporal networks was suggested in some previous studies [28][29][30], our centrality notions enable us to clearly quantify and visualize this property. We believe that the centrality notions we proposed are useful for further studying the structure of temporal networks and verifying generative models of temporal networks. Datasets used in the numerical experiments, Infectious, HT09, and Hospital were originally collected and published by the SocioPatterns collaboration (http:// www.sociopatterns.org/). Datasets HT09 and Hospital were downloaded from the SocioPatterns website. Datasets Infectious, Irvine, and Email were downloaded from the Koblenz Network Collection (http://konect. uni-koblenz.de/). The authors thank Dr. James A Approximate computation of temporal coverage centralities By Lemma 4 (see Section 4), the number of queries to the reachability oracle for computing the TCC and TBCC values is (almost) quadratic in the number of vertices of a temporal network. However, in some applications, we may want to compute these centralities faster. Here, we introduce a standard technique that enables us to approximate these centrality values with a sublinear number of queries. Algorithm 3 (Approximation to the TCC value of v) 1: r ← 0. 2: for i = 1 to k := 1 2 2 log(2\|V \| 2 ) do 3: Sample vertices u, w ∈ V uniformly. 4: u ← (u, τ ldt (v, u)). 5: w ← (w, τeat(v, w)). 6: if τeat(u, w) = w and τ ldt (w, u) = u then 7: r ← r + 1. return r/k. We only explain the case of TCC; the case of TBCC is performed in a similar way. Algorithm 3 is an approximate method for computing the centrality value. The difference from Algorithm 1 is that, instead of enumerating all pairs (u, w), we only sample O(1/ 2 ) pairs of vertices and take the average over them, where is the parameter controlling the possible error in approximation. To show that Algorithm 3 gives a good approximation, we need to recall Hoeffding's inequality: Lemma 5 (Hoeffding's inequality [50]) Let X 1 , X 2 , . . . , X k be independent random variables in [0, 1] and X = (1/k) k i=1 X i . Then, for any positive real number t, Pr[\|X − E[X]\| ≥ t] ≤ 2 exp(−2t 2 k). Lemma 6 Let G be a temporal network and G be its DAG representation. For any temporal vertex v, with O(log 2 \|V \|/ 2 ) queries to the reachability oracle of G, we can compute the TCC value of v with additive error of with probability of at least 1 − 1/\|V \| 2 . Proof Consider Algorithm 3 and let C(v) denote its output. Algorithm 3 issues O(log 2 \|V \|/ 2 ) queries since τ ldt and τ eat can be computed with O(log \|V \|) queries (see Lemma 3). Let X i be the temporal edge at which we increment r in the i-th loop and X = (1/k) Hence, the lemma holds. Recalling that the query time of the reachability oracle is tiny, we find that the running time of Algorithms 3 can be seen as polylogarithmic in the input size. This is the great advantage of TCC and TBCC against other centrality notions. Electronic Supplementary Material Fig. 1 . 1Schematic of an example of temporal network. The number associated with each edge represents the time at which the edge appears. Fig. 2 . 2DAG representation of the temporal network shown inFig. 1. Fig. 3 . 3Schematic describing the concept of temporal coverage centrality. Figure 3 3explains condition 1. Let us focus on Fig. 4 . 4Schematic describing the concept of temporal boundary coverage centrality. Fig. 5 .Fig. 6 . 56Rank plots of the (a) TCC and (b) TBCC values. Rank plots of the (a) TCC and (b) TBCC values in randomized temporal networks. The curves for Irvine are not provided because the computation did not stop. Fig. 7 .Fig. 8 . 78Change in the maximum TCC and TBCC values over temporal vertices at present in (a) Infectious and (b) Hospital. For readability, we smoothed the curves by taking the average over a sliding window with a length of 100 units of time. Change in the TCC and TBCC values of the vertex with the largest number of temporal edges. (a) Vertex with label 195 in Infectious and (b) vertex with label 1115 in Hospital. Fig. 9 . 9Heat maps of the TCC values for (a) Infectious and (b) Hospital. (c) Heat map magnifying the area with 650 ≤ τ ≤ 800 and 100 ≤ ID ≤ 220 in (a). (d) Heat map of the TBCC values in the same area as shown in (c). Cheng for valuable discussions. Yuichi Yoshida is supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 26730009), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24106001), and JST, ERATO, Kawarabayashi Large Graph Project. k i=1 X i . Note that E[ C(v)] = E[X] = (1/k) k i=1 E[Xi] = C(v), where C(v)is the TCC value of v. Since X 1 , X 2 , . . . , X k are indepen-dent random variables in [0, 1], by Lemma 5, we havePr[\| C(v) − C(v)\| ≥ ] = Pr[\|X − C(v)\| ≥ ] Fig. S1 . S1Average (solid line) and 10 − 90% values (shaded areas) of TCC at each time for (a) Infectious, (b) HT09, (c) Hospital, (d) Irvine, and (e) Email. We consider only the temporal vertices involved in temporal edges with other vertices to calculate the statistics. 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[ "A Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm", "A Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm" ]	[ "Chihab Eddine Mokaddem ", "Houari Sahraoui ", "Eugene Syriani " ]	[]	[]	Maintaining software artifacts is among the hardest tasks an engineer faces. Like any other piece of code, model transformations developed by engineers are also subject to maintenance. To facilitate the comprehension of programs, software engineers rely on many techniques, such as design pattern detection. Therefore, detecting design patterns in model transformation implementations is of tremendous value for developers. In this paper, we propose a generic technique to detect design patterns and their variations in model transformation implementations automatically. It takes as input a set of model transformation rules and the participants of a model transformation design pattern to find occurrences of the latter in the former. The technique also detects certain kinds of degenerate forms of the pattern, thus indicating potential opportunities to improve the model transformation implementation.	10.1007/s10270-021-00936-4	[ "https://arxiv.org/pdf/2010.04759v1.pdf" ]	222,291,604	2010.04759	48610488cfcc158b7da7a8ddcd415c61f1b00c89	A Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm Chihab Eddine Mokaddem Houari Sahraoui Eugene Syriani A Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Design pattern · Model transformation · Pattern detection · String matching · Bit-vector · Model-driven engineering Maintaining software artifacts is among the hardest tasks an engineer faces. Like any other piece of code, model transformations developed by engineers are also subject to maintenance. To facilitate the comprehension of programs, software engineers rely on many techniques, such as design pattern detection. Therefore, detecting design patterns in model transformation implementations is of tremendous value for developers. In this paper, we propose a generic technique to detect design patterns and their variations in model transformation implementations automatically. It takes as input a set of model transformation rules and the participants of a model transformation design pattern to find occurrences of the latter in the former. The technique also detects certain kinds of degenerate forms of the pattern, thus indicating potential opportunities to improve the model transformation implementation. Introduction Model transformation is now the mainstream paradigm to manipulate models in model-driven software engineering (MDE) [6]. Designing model transformations is a tedious task. Moreover, like any other code artifact, model transformations evolve and should be maintained. To assist developers in writing and maintaining model transformations, several design patterns have been proposed [11,22]. These presumably facilitate the comprehension and manipulation of transformation programs [1]. Detecting instances of a pattern in a transformation provides valuable information to the developer, such as understanding high-level concepts used, and identifying refactoring and reuse opportunities. This helps, among others, improving the documentation. However, as for general programs, developers do not always implement perfectly a pattern in model transformations. Hence, design pattern detection should identify both complete and incomplete occurrences. Detecting design patterns in model transformations did not get much attention so far from the modeling community. To the best of our knowledge, only the work in [28] has attempted to automatically detect design patterns in model transformations. Preliminary results showed that this is an effective technique to find complete and approximate design pattern occurrences. However, this technique has performance limitations as it relies on a rule inference engine that is time and memory consuming. Another limitation of this technique is the need to specify a set of detection rules for each pattern. To find inspiration on how to detect patterns in transformations, we looked at the active community of design pattern detection in object-oriented programs. As reported in [1], there are dozens of detection approaches for this family of programs. However, as mentioned in [13], these approaches also suffer from performance problems, because detecting complete and incomplete occurrences is generally costly in time, due to the large search-space that includes all possible combinations of classes. These approaches are also prone to return many false positives, impeding program comprehension, and cluttering the maintainers' cognitive capabilities. To address the performance issues, the work by Kaczor et al. [20] uses a string matching technique inspired by pattern matching algorithms in bioinformatics to identify pattern occurrences in object-oriented programs. These algorithms allow to efficiently process a large amount of data if the problem to solve can be encoded as a string matching one. In this paper, we propose a generic technique to detect design pattern occurrences in model transformation implementations, without writing detection code for each design pattern, its variants, and approximations. Like in Kaczor et al., we rely on a bit-vector algorithm that has proven to be efficient for string matching problems [30]. The challenge we faced is how to encode model transformations, which are sets of rules linked by control schemes, as strings. The same challenge arises also in the encoding of the patterns as strings. We succeed to encode the participants of a patterns as strings, but had to complete our approach by a manual step to combine the identified participant instances to form pattern occurrences. Thus, the detection consists in an automated step that matches the participant strings of a pattern with rule strings of transformation, and a manual step to complete the occurrences. In addition to the performance, an advantage of using this approach is the fact both complete and incomplete occurrences can be detected. We evaluated our approach on a set of 18 transformations. Our results show that patterns are actually used in transformations and that our approach is able to detect them. Moreover, we found that patterns are not always used independently, but in combination with other patterns. The rest of the paper is structured as follows. In Section 2, we first introduce the basic notions used in our work, and then, discuss the related work. Section 3 details the different steps of our approach, whereas Section 4 describes the different forms of patterns that can be detected by our approach. We provide an evaluation of the detection approach in Section 5. Finally, we discuss the limits of our approach in Section 6, and conclude in Section 7. A survey on design pattern detection approaches. M Al-Obeidallah, M Petridis, S Kapetanakis, International Journal of Software Engineering. 7Al-Obeidallah, M., Petridis, M., Kapetanakis, S.: A survey on design pattern detection approaches. International Journal of Software Engineering 7, 41-59 (2016) Rule-based detection of design patterns in program code. A Alnusair, T Zhao, G Yan, 10.1007/s10009-013-0292-zSTTT. 163Alnusair, A., Zhao, T., Yan, G.: Rule-based detection of design patterns in program code. STTT 16(3), 315-334 (2014). DOI 10.1007/s10009-013-0292-z. 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[ "Quantile-based classifiers", "Quantile-based classifiers" ]	[ "Christian Hennig ", "Cinzia Viroli " ]	[]	[]	Quantile classifiers for potentially high-dimensional data are defined by classifying an observation according to a sum of appropriately weighted component-wise distances of the components of the observation to the within-class quantiles. An optimal percentage for the quantiles can be chosen by minimizing the misclassification error in the training sample.It is shown that this is consistent, for n → ∞, for the classification rule with asymptotically optimal quantile, and that, under some assumptions, for p → ∞ the probability of correct classification converges to one. The role of skewness of the involved variables is discussed, which leads to an improved classifier.The optimal quantile classifier performs very well in a comprehensive simulation study and a real data set from chemistry (classification of bioaerosols) compared to nine other classifiers, including the support vector machine and the recently proposed median-based classifier(Hall et al. (2009)), which inspired the quantile classifier.	10.1093/biomet/asw015	[ "https://arxiv.org/pdf/1303.1282v2.pdf" ]	33,980,160	1303.1282	a17f7c9fd726ceb6deb6e3c71ae74c62c2c1116e	Quantile-based classifiers 12 Nov 2013 Christian Hennig Cinzia Viroli Quantile-based classifiers 12 Nov 2013median-based classifierhigh-dimensional datamisclassification rateskewness Quantile classifiers for potentially high-dimensional data are defined by classifying an observation according to a sum of appropriately weighted component-wise distances of the components of the observation to the within-class quantiles. An optimal percentage for the quantiles can be chosen by minimizing the misclassification error in the training sample.It is shown that this is consistent, for n → ∞, for the classification rule with asymptotically optimal quantile, and that, under some assumptions, for p → ∞ the probability of correct classification converges to one. The role of skewness of the involved variables is discussed, which leads to an improved classifier.The optimal quantile classifier performs very well in a comprehensive simulation study and a real data set from chemistry (classification of bioaerosols) compared to nine other classifiers, including the support vector machine and the recently proposed median-based classifier(Hall et al. (2009)), which inspired the quantile classifier. Introduction Supervised classification is a major issue in statistics and has received a wide interest in the scientific literature of many disciplines. The "large microcosm" of classification methods (Hand, 1997) can be broadly divided into parametric methods, which make distributional assumptions about the data, and nonparametric methods, which alternatively concentrate on the local vicinity of the point to be classified, such as nearest neighbor methods (Cover and Hart, 1967) and kernel smoothing (Mika et al., 1999). Parametric methods use the estimated class conditional distributions for the construction of the classification rule. The traditional linear and quadratic discriminant analysis, mixture discriminant analysis (Hastie and Tibishirani, 1996), the naive Bayes probabilistic model (John and Langley, 1995;Hand and Yu, 2001), model-based discriminant analysis (Bensmail and Celeux, 1996;Fraley and Raftery, 2002) and nonlinear neural networks (Ripley, 1994) are examples of such methods. See also Friedman (1989); Guo et al. (2007); Cai and Liu (2011) and the references therein. Implementing such methods in high dimensional settings, which are very common nowadays, can be cumbersome and computationally demanding, because of the well-known curse of dimensionality (Bellman, 1961). A great deal of work, especially on distance-based methods, has been carried out to try to circumvent this problem. Distance-based classifiers only use partial information of the class conditional distributions, typically central moments. Centroid-based methods have been successfully used for gene expression data (Tibshirani et al., 2002;Dudoit et al., 2002;Dabney, 2005;Fan and Fan, 2008). Median-based classifiers (Jörnsten, 2004;Ghosh and Chaudhuri, 2005) represent a more robust alternative in problems where distributions have heavy tails. Hall et al. (2009) proposed a component-wise median based classifier which behaves well in high dimensional space. It assigns a new observed vector to the class having the smallest L 1 -distance from the class conditional component-wise median vectors of the training set. All these methods consider the distance from the "core" of a distribution as the major source of the discriminatory information. But tails may be important as well and may contain relevant information. It may therefore be fruitful to go beyond the central moments. In this work we define and explore a family of classifiers based on the quantiles of the class conditional distributions. The idea was originally inspired by the component-wise median classifier (Hall et al., 2009). More specifically, by using the natural distance for quantiles, we will obtain the componentwise quantile classifier as function of the θ-quantile, θ ∈ [0, 1]. The optimal θ chosen in the training set will define the empirically optimal quantile classifier. We will prove the consistency of this choice for the θ that yields the optimal true correct classification probability as n → ∞. We will also show under certain assumptions that the correct classification probability converges to one as p → ∞ together with the sample size, similarly to what Hall et al. (2009) did for the component-wise median classifier. The paper is organized as follows. In Section 2 we review the distance-based classifiers and define the proposed quantile classifier. The theoretical properties of the method are explored in Sections 3 and 4. A large simulation study and a real application are presented in Section 5. The classification rule 2.1 Distance-based classifiers We consider the problem of constructing a quantile distance-based discriminant rule for classifying new observations into one of g populations or classes. Without loss of generality we discuss the problem for g = 2. Generalization for g > 2 is straightforward. Let Π 0 and Π 1 be two populations with probability densities P 0 and P 1 on R p . Distance based classifiers (Jörnsten, 2004;Tibshirani et al., 2003;Hall et al., 2009) assign a new data value z = (z 1 , . . . , z p ) to the population from which it has lowest distance. More specifically, the decision rule allocates z to Π 0 if p j=1 {d(z j , Y j ) − d(z j , X j )} > 0,(1) where X = {X 1 , . . . , X p } and Y = {Y 1 , . . . , Y p } are p-variate random variables from populations Π 0 and Π 1 and d(·) denotes a specific distance measure. Expression (1) represents a rather general discriminant rule formulation that includes centroid classifiers (Tibshirani et al., 2002(Tibshirani et al., , 2003Wang and Zhu, 2007), the recent component-wise medianbased classifiers (Hall et al., 2009), and other variants by differently specifying the distance measure d(·). On the other hand, summing up component-wise differences means that correlation between variables is not taken into account. If p is small and there are many observations, this is rather restrictive. However, if p is large and the number of observations is rather low, it can be effective to avoid overfitting. By considering the Euclidean distance between z j and the expectations of X j and Y j , the component-wise centroid classifier assigns z to Π 0 if p j=1 {(z j − E(Y j )) 2 − (z j − E(X j )) 2 } > 0,(2) and to Π 1 otherwise. By taking the L 1 (Manhattan)-distance between z j and the medians of X j and Y j the component-wise median-based classification rule can be defined as p j=1 {\|z j − med(Y j )\| − \|z j − med(X j )\|} > 0.(3) Note that in realistic situations neither P 0 and P 1 nor their moments are known. We rather observe two sets x 1 , . . . , x n 0 and y 1 , . . . , y n 1 from Π 0 and Π 1 ; they represent the training data samples from which the desirable moments must be inferred. For instance, the sample version of the centroid classifier assigns z to Π 0 if p j=1 {(z j −ȳ j ) 2 − (z j −x j ) 2 } > 0,(4) whereȳ j andx j denote the jth component of the sample mean vectors. Analogously, the sample version of the discriminant rule (3) requires computing the empirical componentwise medians. Hall et al. (2009) stated that median classifiers are more robust against heavy tails of the data distribution than centroid classifiers, thanks to the metric L 1 instead of L 2 , and they provided a formal proof of the fact that asymptotically the correct decision is made by the rule with probability one, if the dimension as well as the numbers of observations in both classes tend to infinity under some further assumptions. The choice of the metric L 1 , instead of L 2 , in the median classifier addresses the need of consistency between metric and related minimizer moment; in fact, the mean vector (centroid) is the statistic that minimizes the sum of L 2 -distances of points to the centroid, whereas the median minimizes the sum of the corresponding L 1 -distances. Hybrid alternatives may exist, such as an L 1 -version of the centroid classifier. However, they look convincing from neither a theoretical nor a practical point of view. Not only does a hybrid alternative mismatch the relation between metric and related minimizer quantity, but it also seems to produce higher misclassification rates in practice (see, for instance, Hall et al. (2009)). The quantile classifier We introduce the family of the component-wise quantile classifiers that includes the median classifier as special case. By definition, the θ th quantile of a univariate random variable X with probability distribution function F X , denoted by q X (θ), is the solution to q X (θ) = F −1 X (θ) = inf{x : F X (x) ≥ θ}, with θ ∈ [0, 1]. Analogously to the roles of median and centroid with respect to the L 1 -and L 2 -metric, the θ th quantile of F X is the value q that minimizes the following population distance θ x>q \|x − q\|dF X (x) + (1 − θ) x<q \|x − q\|dF X (x).(5) This can be easily proven by observing that (5) is minimized for F X (q) = θ. Given a set of observations x 1 , x 2 , . . . , x n , the empirical θ th quantile of X can be found by minimizing the sample counterpart of (5): θ x i >q \|x i − q\| + (1 − θ) x i ≤q \|x i − q\| = x i θ + (1 − 2θ)1 [x i ≤q] \|x i − q\|.(6) The metric (6) is used to define the component-wise quantile-based new classifier. Given two sets of observations from the two populations Π 0 and Π 1 , x 1 , . . . , x n 0 and y 1 , . . . , y n 1 , (7) where q 0j (θ) and q 1j (θ) are the marginal quantile functions of the two class-distributions evaluated at a fixed value of θ. For j = 1, . . . , p and a new observation z = (z 1 , . . . , z p ) ∈ R p is assigned to Π 0 if p j=1 θ + (1 − 2θ)1 [z j ≤q 1j (θ)] \|z j − q 1j (θ)\| − θ + (1 − 2θ)1 [z j ≤q 0j (θ)] \|z j − q 0j (θ)\| > 0,k = 0, 1, let Φ j (z, θ, q) = θ + (1 − 2θ)1 [z j ≤q] \|z j − q\| and Φ kj (z, θ) = θ + (1 − 2θ)1 [ z j ≤q kj (θ) ] \|z j − q kj (θ)\|. Then, for fixed θ, the classification rule (7) is equivalent to assigning z to Π 0 if p j=1 Φ 0j (z, θ) < p j=1 Φ 1j (z, θ), and to Π 1 otherwise. Remark 1 The applicability of the decision rule (7) to more than g = 2 classes is straightforward. By definition, the quantile classifier rule for allocating an observation z to one of g populations Π 1 , . . . , Π g is to allocate z to the population which gives the lowest quantile distance p j=1 Φ kj (z, θ), with k = 1, . . . , g. Remark 2 Note that for θ = 0.5 the objective function in (6) (multiplied by 2) is the L 1 -distance between x and the median. Therefore decision rule (7) coincides with the component-wise median classifier when θ = 0.5. Given the two populations, Π 0 and Π 1 with prior probabilities π 0 and π 1 , respectively, the probability of correct classification of the quantile classifier is Ψ(θ) = π 0 1    p j=1 (Φ 1j (z, θ) − Φ 0j (z, θ)) > 0    dP 0 (z) + π 1 1    p j=1 (Φ 1j (z, θ) − Φ 0j (z, θ)) ≤ 0    dP 1 (z).(8) This quantity represents the theoretical rate of correct classification based on the true quantiles. This rate can be used to measure the performance of the discriminant rule with respect to the chosen value θ regardless of the sample size (we will later simulate such rates based on empirical quantiles, as relevant in real applications). The following lemma provides a useful formula to derive the theoretical rate of correct classification as function of θ for p = 1. Lemma 1 When p = 1, the probability of correct classification of the quantile classifier takes the following simple form. -If q 0 (θ) ≤ q 1 (θ), Ψ(θ) = π 0 F 0 (θ) + π 1 (1 − F 1 (θ)) (9) withθ = θq 0 (θ) + (1 − θ)q 1 (θ). -If q 0 (θ) > q 1 (θ), Ψ(θ) = π 1 F 1 (θ) + π 0 (1 − F 0 (θ)) (10) withθ = θq 1 (θ) + (1 − θ)q 0 (θ). where q 0 (θ) and q 1 (θ) are the true quantiles of the two populations. Proof of Lemma 1. Consider that in the univariate case Φ 0 (z, θ) and Φ 1 (z, θ) may be rewritten as Φ 0 (z, θ) = (1 − θ) (q 0 (θ) − z) 1 [z≤q 0 (θ)] + θ (z − q 0 (θ)) 1 [z>q 0 (θ)] Φ 1 (z, θ) = (1 − θ) (q 1 (θ) − z) 1 [z≤q 1 (θ)] + θ (z − q 1 (θ)) 1 [z>q 1 (θ)] For a fixed θ, the integral (8) can be easily solved by splitting it into four parts according to the possible disjoint regions of the domain of Z with respect to q 0 (θ) and q 1 (θ), namely: (a) z ≤ min(q 0 (θ), q 1 (θ)), (b) q 0 (θ) < z ≤ q 1 (θ), (c) q 1 (θ) ≤ z ≤ q 0 (θ) and (d) z > max(q 0 (θ), q 1 (θ)). If z ≤ min(q 0 (θ), q 1 (θ)) the integral becomes Ψ a (θ) = π 0 min(q 0 (θ),q 1 (θ)) −∞ 1 [(1−θ)(q 1 (θ)−q 0 (θ))>0] dP 0 (z) + π 1 min(q 0 (θ),q 1 (θ)) −∞ 1 [(1−θ)(q 1 (θ)−q 0 (θ))≤0] dP 1 (z) = π 0 q 0 (θ) −∞ dP 0 (z)1 [q 1 (θ)>q 0 (θ)] + π 1 q 1 (θ) −∞ dP 1 (z)1 [q 1 (θ)≤q 0 (θ)] = π 0 θ1 [q 1 (θ)>q 0 (θ)] + π 1 θ1 [q 1 (θ)≤q 0 (θ)] . In the second case the integral is Ψ b (θ) = π 0 q 1 (θ) q 0 (θ) 1 [(1−θ)(q 1 (θ)−z)−θ(z−q 0 (θ))>0] dP 0 (z) + π 1 q 1 (θ) q 0 (θ) 1 [(1−θ)(q 1 (θ)−z)−θ(z−q 0 (θ))≤0] dP 1 (z) = π 0 θq 0 (θ)+(1−θ)q 1 (θ) q 0 (θ) dP 0 (z)1 [q 0 (θ)≤q 1 (θ)] + π 1 q 1 (θ) θq 0 (θ)+(1−θ)q 1 (θ) dP 1 (z)1 [q 0 (θ)≤q 1 (θ)] . Similarly, for the cases (c) and (d) the integrals are Ψ c (θ) = π 0 q 0 (θ) θq 1 (θ)+(1−θ)q 0 (θ) dP 0 (z)1 [q 1 (θ)≤q 0 (θ)] + π 1 θq 1 (θ)+(1−θ)q 0 (θ) q 1 (θ) dP 1 (z)1 [q 1 (θ)≤q 0 (θ)] , and Ψ d (θ) = π 0 (1 − θ)1 [q 0 (θ)>q 1 (θ)] + π 1 (1 − θ)1 [q 0 (θ)≤q 1 (θ)] . Now, when q 0 (θ) ≤ q 1 (θ), Ψ(θ) is the sum of Ψ a (θ), Ψ b (θ) and Ψ d (θ) corresponding to disjoint domain regions of Z: Ψ(θ) = π 0 θ + π 0 θq 0 (θ)+(1−θ)q 1 (θ) q 0 (θ) dP 0 (z) + π 1 q 1 (θ) θq 0 (θ)+(1−θ)q 1 (θ) dP 1 (z) + π 1 (1 − θ) = π 0 θ + π 0 F 0 (θq 0 (θ) + (1 − θ)q 1 (θ)) − π 0 θ + π 1 θ −π 1 F 1 (θq 0 (θ) + (1 − θ)q 1 (θ)) + π 1 (1 − θ) = π 0 F 0 (θ) + π 1 (1 − F 1 (θ)). Analogously, when q 0 (θ) > q 1 (θ), Ψ(θ) is the sum of Ψ a (θ), Ψ c (θ) and Ψ d (θ) from which: Ψ(θ) = π 1 θ + π 0 q 0 (θ) θq 1 (θ)+(1−θ)q 0 (θ) dP 0 (z) + π 1 θq 1 (θ)+(1−θ)q 0 (θ) q 1 (θ) dP 1 (z) + π 0 (1 − θ) = π 1 F 1 (θ) + π 0 (1 − F 0 (θ)). Lemma 1 provides a direct formula to compute the probability of correct classificationanalytically or numerically -for given values of θ. Suppose the two populations Π 0 and Π 1 have exponential distributions but differ for a location shift c: X ∼ P 0 = Exp(λ) and Y ∼ P 1 = Exp(λ) + c, with c > 0. Then F 0 (x) = 1 − exp(−λx) and F 1 (y) = 1 −exp(−λ(y −c)). Since the probability distribution functions of the exponentials can be expressed in closed form, the two quantile functions can be analytically derived by solving F 0 (x) −1 and F 1 (y) −1 , from which q 0 (θ) = − ln(1−θ) λ and q 1 (θ) = − ln(1−θ) λ + c, respectively. Since c > 0, we have q 0 (θ) ≤ q 1 (θ) ∀θ ∈ [0, 1]. By applying (9), we get the rates of correct classification of the quantile classifier for two (varying-location) exponential distributions as a function of θ: Ψ(θ) = π 0 − (1 − θ)e cλθ (π 0 e −cλ − π 1 ). Figure 1 (second panel of third row) shows the theoretical misclassification rates, 1−Ψ(θ), of two exponential populations with λ = 1, c = 0.5 and π 0 = π 1 = 0.5. It is interesting to note that the minimum misclassification rate can be obtained for θ approaching zero. This particular choice for θ is related to the high level of skewness of the exponential distribution. To make this clearer, we also considered further scenarios, namely two location-shifted Gaussians, N (0, 1) and N (1, 1), and two location-shifted chi-squared distributions with 5 degrees of freedom and shift c = 2 (first and second rows of Figure 1). The theoretical misclassification rates, 1 − Ψ(θ), can be easily obtained numerically. In the Gaussian scenario the minimum value of 1 − Ψ(θ) is obtained for θ = 0.5. This is not surprising because of the symmetric shape of the Gaussian. But more asymmetric distributions (second and third rows in Figure 1) tend to yield an optimum θ far away from the midpoint 0.5, with positive skewness normally associated with the optimum being below 0.5 and negative skewness with an optimum above 0.5 (obviously, if skewness is reversed by multiplying a random variable by -1, the resulting optimal θ will be one minus the original optimum). This indicates that the best θ for one problem is not the best for another, and this choice is of crucial importance. For example, in the second case, the theoretical quantile function is minimized for θ = 0.236. The fourth row of Figure 1 shows the classification problem with two differently distributed populations, a Gaussian distribution with parameters 5 and 1 and a chi-squared distribution with 4 degrees of freedom. The optimal quantile classifier corresponds to θ = 0.162. Figure 2 shows the estimated misclassification rates obtained in the four scenarios by a simulation study with sample sizes of training set and test set equal to 500. The plotted line is the empirical curve of the misclassification rate obtained in the test set for different values of θ. It approximates the theoretical one well. The horizontal lines indicate the misclassification rates obtained by the centroid classifier, the median classifier and quantile classifier corresponding to the optimal value θ chosen in the training set. Unfortunately, Lemma 1 cannot easily be extended to the multivariate setting, unless some very restrictive conditions are assumed regarding independence of the variables and strict rules about the ranking of the p different quantiles q kj (θ) j = 1, . . . , p within each population. The empirically optimal quantile classifier In real applications the problem of the choice of the quantile value in the family of possible quantile classifiers can be addressed by selecting the optimum θ based on misclassification rates in the training sample. This leads to the definition of the empirically optimal quantile classifier. First, we introduce some notation. Let (Z 1 , C 1 ), (Z 2 , C 2 ), . . . be i.i.d. R p × {0, 1}-valued RV. Let Z 1 be distributed according to a 2-component mixture of distributions P 0 = L(Z 1 \|C 1 = 0) and P 1 = L(Z 1 \|C 1 = 1). Let π 0 = P {C 1 = 0}, π 1 = 1 − π 0 . Let P 01 , . . . , P 0p denote the marginal distributions of P 0 , analogously P 11 , . . . , P 1p . For arbitrarily small 0 < τ < 1 2 define T = [τ, 1 − τ ]. For θ ∈ [0, 1], j = 1, . . . , p, k = 0, 1 denote q kj (θ) the θ-quantile of P kj . For given (Z 1 , C 1 ), . . . , (Z n , C n ) let q kjn (θ) be the empirical θ-quantile for the subsample defined by C i = k, i = 1, . . . , n. For j = 1, . . . , p, k = 0, 1, z = (z 1 , . . . , z p ) ∈ R p , let Φ j (z, θ, q) = (θ + (1 − 2θ)1[z j ≤ q]) \|z j − q\| (in abuse of notation, assumption B2 of Theorem 2 will apply Φ j to infinite-dimensional z). Φ kj (z, θ) is used for Φ j (z, θ, q kj (θ)). Φ kjn (z, θ) is used for Φ j (z, θ, q kjn (θ)). The empirically optimal quantile classifier is defined by assigning Z to Π 0 if p j=1 (Φ 1jn (Z, θ n ) − Φ 0jn (Z, θ n )) > 0,(11) where θ n = arg max θ∈T Ψ n (θ) is the estimated optimal θ from (Z 1 , C 1 ), . . . , (Z n , C n ) (if the argmax is not unique, any maximizer can be chosen), and the observed rate of correct classification in data (z 1 , c 1 ), . . . , (z n , c n ) is Ψ n (θ) = 1 n       i: c i =0 1    p j=1 (Φ 1jn (z i , θ) − Φ 0jn (z i , θ)) > 0    + i: c i =1 1    p j=1 (Φ 1jn (z i , θ) − Φ 0jn (z i , θ)) ≤ 0          . Note that we look for the optimal value of θ in T , a closed interval not containing zero. In practice, a small nonzero τ needs to be chosen, and Ψ n (θ) is evaluated on a grid of equispaced values between τ and 1 − τ . T will in practice depend on the number of observations. τ should be chosen small but large enough that there is still a certain amount of information to estimate the τ -quantile. τ should not be seen as a crucial tuning parameter of the method; we recommend to choose it as small as possible in order to find the empirical optimum of θ, only making sure that the estimated τ -quantile still is of some use. In case of a tie (i.e., equal training set misclassification rates for different values of θ, which can easily happen for data sets with small n), we recommend to fit a square polynomial to the misclassification rate as function of θ and to choose the optimum θ according to this fit out of the empirically optimal ones. Remark 3 As well as a number of other classifiers, the quantile classifier depends on the scaling of the variables. This dependence can be removed by standardizing the variables. Straightforward ways of doing this would be standardization to unit variance, range, or interquartile range. Standardization can be seen as implicit reweighting of the variables. Optimally, variables are treated in such a way that their relative weights reflect their relative information contents for classification. This means that in practice, in some situations, standardizing is not advisable, namely where variables have the same measurement units and there are subject-matter reasons to expect that the information content of the variables for classification may be indicated by their variation. Section 5.2 presents an example for a situation in which the variability of variables is connected to their information content, and for a standardization scheme driven by subject knowledge. Where variables are standardized, standardization to unit pooled within-class variance (or range, or interquartile range) as estimated from the training data can be expected to improve matters compared with the plain variance, because the separation between classes contributes strongly to the plain variance. This means that variables with a strong separation between classes and hence a large amount of classification information will be implicitly downweighted, whereas standardization to unit pooled within-class variance will downweight variables for which the classes are heterogeneous and which are therefore not so useful for classification. Given enough data, one could use cross-validation to choose an optimal standardization scheme. In the next section, we will present some theoretical properties of the proposed classifier. Consistency of the quantile classifier The asymptotic probability of correct classification of the quantile classifier is defined in (8). Letθ = arg max θ∈T Ψ(θ) be the optimal θ regarding the true model. The theory needs the following assumptions: A1 For all j = 1, . . . , p, k = 0, 1 : q kj is a continuous function of θ ∈ T . A2 For all θ ∈ T , P p j=1 (Φ 1j (Z, θ) − Φ 0j (Z, θ)) = 0 = 0. If A1 and A2 are not fulfilled, there may be ambiguities regarding the optimal quantile or the classification of a set of points with nonzero probability. In case of violation of A2, the problem caused by this will affect a subset of the data space with at most the probability given in A2. A1 will probably only affect consistency if violation happens around the optimal θ, and probably only weakly so if the discontinuity is small. Theorem 1 Assume A1 and A2. Then, for any ǫ > 0, lim n→∞ P {\|Ψ(θ) − Ψ(θ n )\| > ǫ} = 0. This means that for n → ∞ the optimal true correct classification probability equals the true one corresponding to the empirically optimal θ n , i.e., the θ chosen for the quantile classifier, which is therefore asymptotically optimal (and therefore at least as good as 1 2 , which defines the median classifier). Theorem 1 is based on g = 2. This is for convenience of the proof only. Arguments carry over to g > 2 in a straightforward manner. Proof of Theorem 1. \|Φ j (z, θ 1 , q 1 ) − Φ j (z, θ 2 , q 2 )\| ≤ \|z j \|\|θ 2 − θ 1 \| + 4\|q 2 − q 1 \|(12) is proved below as Lemma 2 for j = 1, . . . , p, θ i -quantiles q i , i = 1, 2. Together with A1, this implies the continuity of Ψ, because for given z, Φ kj is a continuous function of θ, and the dominated convergence theorem makes the integrals of the indicator functions converge for θ n → θ. The proof of Theorem 1 is now based on \|Ψ(θ) − Ψ(θ n )\| ≤ \|Ψ(θ) − Ψ n (θ)\| + \|Ψ n (θ) − Ψ n (θ n )\| + \|Ψ n (θ n ) − Ψ(θ n )\|.(13) In order to show that all three terms on the right side are asymptotically small, the following result is proved below as Lemma 3: ∀ǫ > 0 : lim n→∞ P sup θ∈T \|Ψ n (θ) − Ψ(θ)\| > ǫ = 0.(14) (14) forces the first and third term on the right side of (13) to converge to zero in probability. Consider now the second term. By definition, Ψ n (θ n ) ≥ Ψ n (θ), Ψ(θ) ≥ Ψ(θ n ). Using (14) again, for large n both \|Ψ n (θ) − Ψ(θ)\| and \|Ψ n (θ n ) − Ψ(θ n )\| will be arbitrarily small with arbitrarily large probability, and this makes \|Ψ n (θ) − Ψ n (θ n )\| arbitrarily small, too. Altogether, this proves the theorem. Lemma 2 (12) holds for j ∈ {1, . . . , p}, θ 1 , θ 2 ∈ (0, 1), q 1 , q 2 ∈ R, assuming θ 1 ≤ θ 2 ⇒ q 1 ≤ q 2 and analogously for "≥" (as holds if q k is a quantile belonging to θ k ). Proof of Lemma 2: assume w.l.o.g. q 1 ≤ q 2 , 0 < θ 1 ≤ θ 2 < 1. Consider z j ≤ q 1 , q 1 < z j < q 2 , q 2 ≤ z j separately; first z j ≤ q 1 . By definition, \|Φ j (z, θ 1 , q 1 ) − Φ j (z, θ 2 , q 2 )\| = \|(1 − θ 1 )(q 1 − z j ) − (1 − θ 2 )(q 2 − z j )\| = \|(q 1 − q 2 ) + (θ 1 + θ 2 )(q 2 − q 1 ) − θ 1 q 2 + θ 2 q 1 + z j (θ 1 − θ 2 )\| ≤ \|q 2 − q 1 \| + \|θ 1 + θ 2 \|\|q 2 − q 1 \| + θ 2 \|q 2 − q 1 \| + \|z j (θ 1 − θ 2 )\| ≤ \|z j \|\|θ 2 − θ 1 \| + 4\|q 2 − q 1 \|. For q 1 < z j < q 2 : \|Φ j (z, θ 1 , q 1 ) − Φ j (z, θ 2 , q 2 )\| = \|θ 1 (z j − q 1 ) − (1 − θ 2 )(q 2 − z j )\| ≤ \|q 2 − q 1 \|. For q 2 ≤ z j : \|Φ j (z, θ 1 , q 1 ) − Φ j (z, θ 2 , q 2 )\| = \|θ 1 (z j − q 1 ) − θ 2 (z j − q 2 )\| and (12) follows along the lines of the first case. Lemma 3 (14) holds under the conditions of Theorem 1. Proof of Lemma 3: Suppose (14) were wrong. This means that there exist ǫ > 0, δ > 0, a subsequence M of (1, 2, . . .) and (θ * m ) m∈M such that ∀m ∈ M : P {\|Ψ m (θ * m ) − Ψ(θ * m )\| > ǫ} ≥ δ.(15) W.l.o.g. (because (θ m ) m∈M ∈ T M is bounded and at least a subsequence has a limit) there exists θ * = lim m→∞ θ * m . Consider \|Ψ m (θ * m ) − Ψ(θ * m )\| ≤ \|Ψ m (θ * m ) − Ψ m (θ * )\| + \|Ψ m (θ * ) − Ψ(θ * )\| + \|Ψ(θ * ) − Ψ(θ * m )\|. (16) Continuity of Ψ forces the third term of the right side of (16) to converge to 0. Regarding the second term, define a version of Ψ n using the true quantiles instead of the empirical ones: Ψ * n (θ) = 1 n i: C i =0 1 p j=1 (Φ j (Z i , θ, q 1j (θ)) − Φ j (Z i , θ, q 0j (θ))) > 0 + i: C i =1 1 p j=1 (Φ j (Z i , θ, q 1j (θ)) − Φ j (Z i , θ, q 1j (θ))) ≤ 0 . Consider \|Ψ m (θ * ) − Ψ(θ * )\| ≤ \|Ψ m (θ * ) − Ψ * m (θ * )\| + \|Ψ * m (θ * ) − Ψ(θ * )\|. Because of the strong law of large numbers, lim m→∞ \|Ψ * m (θ * ) − Ψ(θ * )\| = 0 a.s. For given z and θ, Φ j is continuous in q. Furthermore quantiles are strongly consistent, and therefore (12) will enforce lim m→∞ \|Ψ m (θ * ) − Ψ * m (θ * )\| = 0 a.s. Now consider the first term of the right side of (16). \|q kjm (θ * m )−q kjm (θ * )\| ≤ \|q kjm (θ * )−q kj (θ * )\|+\|q kjm (θ * m )−q kj (θ * m )\|+\|q kj (θ * m )−q kj (θ * )\|. (17) From Theorem 3 in Mason (1982), which assumes A1, lim m→∞ sup θ∈T \|q kj (θ)−q kjn (θ)\| = 0 a.s.. This enforces the first two terms on the left side of (17) to converge to zero a.s.. The last term converges to zero because of A1. Therefore \|q kjm (θ * m ) − q kjm (θ * )\| → 0 a.s. (18) Let D n (θ, z) = p j=1 (Φ 1jn (z, θ) − Φ 0jn (z, θ)), D(θ, z) = p j=1 (Φ 1j (z, θ) − Φ 0j (z, θ)). For ǫ > 0 define Z ǫ = {z : \|D(θ * , z)\| > ǫ} ∩ z : p j=1 \|z j \| ≤ 1 ǫ , so that \|Ψ m (θ * m ) − Ψ m (θ * )\| = 1 m i: C i =0, Z i ∈Zǫ [1(D m (θ * m , Z i ) > 0) − 1(D m (θ * , Z i ) > 0)] + i: C i =1, Z i ∈Zǫ [1(D m (θ * m , Z i ) ≤ 0) − 1(D m (θ * , Z i ) ≤ 0)] + i: C i =0, Z i ∈Zǫ [1(D m (θ * m , Z i ) > 0) − 1(D m (θ * , Z i ) > 0)] + i: C i =1, Z i ∈Zǫ [1(D m (θ * m , Z i ) ≤ 0) − 1(D m (θ * , Z i ) ≤ 0)] . Now for large m and arbitrarily small δ > 0, 1 m i: C i =0, Z i ∈Zǫ [1(D m (θ * m , Z i ) > 0) − 1(D m (θ * , Z i ) > 0)] + i: C i =1, Z i ∈Zǫ [1(D m (θ * m , Z i ) ≤ 0) − 1(D m (θ * , Z i ) ≤ 0)] ≤ 1 − P (Z ǫ ) + δ a.s. Furthermore, by (12), \|D m (θ * m , Z i ) − D m (θ * , Z i )\| ≤ p j=1 (2\|Z j \|\|θ * m − θ * \| + 8\|q kjm (θ * m ) − q kjm (θ * )\|) . Because \|θ * m − θ * \| → 0, by (18) and p j=1 \|Z j \| ≤ 1 ǫ for Z ∈ Z ǫ , this difference becomes arbitrarily small a.s. for large enough m, and therefore for Z i ∈ Z ǫ , D m (θ * m , Z i ) and D m (θ * , Z i ) will for large enough m be on the same side of zero and their "> 0" and "≤ 0"-indicators will therefore be the same, a.s. For ǫ ց 0, A2 enforces P (Z ǫ ) → 1. This forces the first term on the right side of (16) to zero for large m, a.s., in contradiction to (15), which in turn proves (14). A result for p → ∞ Theorem 1 refers to n → ∞ for fixed finite p. In many modern applications, p is so large and often larger than n that results for p → ∞ seem more appealing, although such results require n → ∞ as well and it is not entirely clear whether they give a better justification of a method for applications with given n and p. In any case they contribute to the exploration of a classifier's properties. Hall et al. (2009) prove under some conditions that the misclassification probability of the median classifier converges to zero for n, p → ∞. Unfortunately we were not able to prove a result ensuring that the quantile classifier is, asymptotically, always at least as good and sometimes better than the median classifier, as one would hope. Analyzing the proof in Hall et al. (2009), it can be seen that it adapts in a more or less straightforward manner to classifiers based on any fixed quantile other than the median. Despite the fact that one may expect the quantile classifier to do at least as good a job (because it incorporates finding the optimal quantile), this classifier is more difficult to handle theoretically. We present a result that requires stronger assumptions than those in Hall et al. (2009), namely considering them uniformly for a range of quantiles. The arguments in Hall et al. (2009) then ensure that the zero misclassification result carries over to classifiers based on whatever quantile selection rule is chosen, obviously including selecting the empirically optimal one. We restrict ourselves to applying this idea to Theorem 1 in Hall et al. (2009). Let again T = [τ, 1 − τ ] for arbitrarily small 0 < τ < 1 2 . Let U = (U 1 , U 2 , . . .) denote an infinite sequence of random variables, each U i with uniquely defined θ-quantiles q i (θ) for all θ ∈ T and median zero. For infinite sequences of constants (ν X1, 1 2 , ν X2, 1 2 , . . .), (ν Y 1, 1 2 , ν Y 2, 1 2 , . . .), assume that for each p, the p-vectors X 1 , . . . , X m are identically distributed as (ν X1, 1 2 + U 1 , . . . , ν Xp, 1 2 + U p ), and the p-vectors Y 1 , . . . , Y n are identically distributed as (ν Y 1, 1 2 + U 1 , . . . , ν Y p, 1 2 +U p ). Define for i ≥ 1 the quantiles ν Xi,θ = ν Xi, 1 2 +q i (θ), ν Y i,θ = ν Y i, 1 2 +q i (θ). Let C be a [0, 1]-valued RV and assume Z to be distributed as X 1 if C = 0 and as Y 1 if C = 1, and X 1 , . . . , X m , Y 1 , . . . , Y n and (Z, C) as totally independent. Assumptions: B1 lim λ→∞ sup k≥1 E{\|U k \|1(\|U k \| > λ)} = 0. B2 For each c > 0 : inf k≥1 inf \|x\|≥c inf θ∈T [EΦ k (U, θ, q k (θ) + x) − EΦ k (U, θ, q k (θ))] > 0. B3 For each ǫ > 0 : inf k≥1 inf θ∈T [min{θ − P [U k ≤ q k (θ) − ǫ], θ − P [U k ≥ q k (θ) + ǫ]}] > 0. B4 With B denoting the class of Borel subsets of the real line, lim k→∞ sup k 1 ,k 2 : \|k 1 −k 2 \|≥k sup B 1 ,B 2 ∈B \|P (U k 1 ∈ B 1 , U k 2 ∈ B 2 ) − P (U k 1 ∈ B 1 )P (U k 2 ∈ B 2 )\| = 0. B5 The differences \|ν Xk,θ − ν Y k,θ \| are uniformly bounded. B6 For sufficiently small ǫ > 0, the proportion of values k ∈ [1, p] for which \|ν Xk,θ − ν Y k,θ \| > ǫ ∀θ ∈ T is bounded away from zero as p diverges. The assumptions B1 and B4 are identical to (4.1) and (4.4) in Hall et al. (2009). B2, B3, B5 and B6 are (4.2), (4.3), (4.5), (4.6) in Hall et al. (2009) enforced to hold uniformly for all θ ∈ T . B4 and B6 enforce a steady flow of relevant information to be added by the data for increasing p. Note that both conditions together mean that at any stage an infinite amount of relevant information in new variables independent of what is already known is still waiting to be discovered. This may look unrealistic but such a thing is essentially needed for any theory for any method based on p → ∞ faster than n and m. B1 and B5 are needed, given B6, to prevent classification from being dominated by a single or a finite number of variables, B2 and B3 are about uniform continuity and well-definedness of the quantiles. See Hall et al. (2009) for further discussion of these assumptions. Let R : N → T any quantile selection rule. Let R m,n,i , i ∈ N be the sequence of {0, 1}- valued R(i)-quantile classifiers computed from [(X 1 , 0), . . . , (X m , 0), (Y 1 , 1), . . . , (Y n , 1)]. Theorem 2 Assume B1-B6 and that both n and m diverge as p → ∞. Then, with probability converging to 1 as p increases, the classifier R m,n,p makes the correct decision, i.e., P {R m,n,p (Z) = 1\|C = 0} + P {R m,n,p (Z) = 0\|C = 1} → 0. Proof of Theorem 2: In the proof of Theorem 1 in Hall et al. (2009), B2, B3, B5 and B6 enforce every statement to hold uniformly for θ ∈ T , after definitions have been adapted to general quantile classifiers (i.e., W k , D k , D(Z), S λ , d(Z), K ǫ and d k need to be defined as functions of θ with quantiles replacing medians, Φ k replacing the absolute value where B2 is applied and q k (θ) replacing zero where B3 is applied). Equations (A.1)-(A.6) in Hall et al. (2009) then hold uniformly over T . Remark 4 Similar arguments should be possible regarding Theorem 2 in Hall et al. (2009), which has different assumptions. Individual treatment of variables The empirically optimal quantile classifier as defined above is based on finding a single θ that is optimal looking at all variables simultaneously. One could wonder whether it would be better to choose different θ-values for each variable. Unfortunately, choosing different θ-values for different variables is not straightforward. We have tried choosing variablewise θ-values by looking at misclassification rates obtained from looking at p classification problems, each based on a single variable, and then we used the resulting variable-wise θ-values for a classification rule incorporating all variables. In most cases this yielded clearly worse results than selecting a single θ by looking at all variables together. There are two major reasons for this. Firstly, the misclassification rates based on a single variable are not very informative for the misclassification result based on all variables simultaneously. Secondly, using different values of θ for different variables results in different scale and distributional shape of the variable-wise contributions to (5), so that certain variables are implicitly up-and downweighted regardless of their information content for classification. Using a single optimal θ for all variables, on the other hand, gives variables with better discriminative power some more influence, because they tend to dominate the selection of the optimal θ, and this is beneficial. We tried to treat the first problem by defining a one-dimensional parameter governing convex combinations between the optimal variable-wise values of θ and the single optimal value. This parameter was chosen by optimizing the overall misclassification rate, but on independent test sets this did not lead to significant improvements compared to the single optimal θ. There is still some potential for methods finding individual variable-wise values for θ, but we leave this for further research. However, we found a simple method to increase adaptation to the individual variables, which led to a significant improvement in some situations while not making things significantly worse elsewhere. As previously observed in the univariate setting, θ will depend on the skewness of the involved distributions. In practice, a set of p > 1 measurements could be skewed in different directions, giving conflicting messages about what values of θ are to be preferred. In order to overcome this problem, we recommend to change the direction of skewness of variables by applying sign changes in order to unify the direction of skewness. More specifically, compute a skewness measure separately for each variable, such as the conventional third standardized empirical moment or, alternatively, a measure from the family of the robust quantile-based quantities (Hinkley, 1975): τ (u) = F −1 (u) + F −1 (1 − u) − 2F −1 (1/2) F −1 (u) − F −1 (1 − u) , where F denotes the marginal cumulative distribution function and u a fixed value in the interval [0.5,1]. When u = 3/4 the previous expression corresponds to Galton's measure of skewness, for u = 0.1 it corresponds to the less robust Kelley's measure of skewness. Evaluate the amount of skewness of each variable separately within classes, in order to avoid overall masking effects due to unbalanced populations, and then summarize by averaging all the within-class measures with equal weights. The signs of variables with negative skewness are then changed, so that finally the variables used for the quantile estimator all have the same (positive) direction of skewness. This approach takes into account the individuality of the variables in a rather rough way. Unfortunately in general the connection between skewness and optimal θ is not straightforward, so that there is little hope to employ skewness in a more sophisticated way. The approach recommended here has the advantage that the choice of θ is still governed by a one-dimensional optimization of the overall misclassification rate, and that there is no issue scaling variable-wise contributions to (5) against each other. The results in Sections 3 and 4 carry over if the skewness of all variables is estimated correctly with probability 1 for large enough n. Numerical results Simulation study We evaluated the performance of the component quantile classifier by a large simulation study comprising several simulated experiments with the aim of assessing the effect of the following factors: sample size, dimensionality, shape of the class-distributions and different level of relevance of the variables for classification. We generated p vectors from g = 2 populations in four different main scenarios. In the first scenario we considered symmetric Student's t-distributed variables W j (j = 1, . . . , p) with 3 degrees of freedom. We simulated two location-shifted populations from W j as X j = W j and Y j = W j + 0.5. In the second setting we tested the behavior of the classifiers in highly skewed data, by generating identically distributed vectors, W j with j = 1, . . . , p, from a multivariate Gaussian distribution, and transforming them using the exponential function, X j = exp(W j ) and Y j = exp(W j ) + 0.2. In the third scenario we considered differing distributions for the p variables. More specifically, we first generated W j from a multivariate Gaussian distribution and then we split p in 5 balanced blocks of different transformations: 1. X j = W j and Y j = W j + 0.2, 2. X j = exp(W j ) and Y j = exp(W j ) + 0.2, 3. X j = log(\|W j \|) and Y j = log(\|W j \|) + 0.2, 4. X j = W 2 j and Y j = W 2 j + 0.2, 5. X j = \|W j \| and Y j = \|W j \| + 0.2. In the fourth scenario we simulated differing distributional shapes and levels of skewness even for different classes within the same variable. Here, within each class, data was generated according to Beta distributions with parameters a and b in the interval (0.1,10) randomly generated for each class within each variable. Within each class data have been centered about 0, so that information about class differences is only in the distributional shape, not in the location. For each of the four scenarios we evaluated the combination of several factors: p = 50, 100, 500, n = 50, 100, 500, different percentage of relevant variables for classification (100%, 50%, and 10%) and independent or dependent variables (the latter except in the fourth scenario), for a total of 189 different settings. The dependence structure between the variables has been introduced by generating correlated variables W j (j = 1, . . . , p) from a Gaussian distribution with equicorrelated covariance matrix (ρ = 0.2). The irrelevant 'noise' variables have been generated independently of each other and the relevant variables by taking the same base distribution as for the informative variables and leaving out the additive constant (in the fourth scenario a new set of parameters was drawn at random for all observations of each noise variable). Variables were standardized to unit within-class pooled variance in the third scenario but not standardized in the three others, because in the third scenario the scales of the variables seem incompatible, whereas in reality for datasets like those from the other scenarios the reasons against standardization given in Remark 3 may apply. For each setting we simulated 100 data sets as training sets and 100 as test sets. The pairs of data sets were split into the two balanced populations with sample size n/2. The component-wise quantile based classifier has been implemented in the R package quantileDA, (the package will be available on CRAN R homepage soon). Data have been preprocessed according to the skewness correction discussed in Section 2.3 using the conventional skewness measure and the Galton's robust version. In each setting we have evaluated the classifier on a grid of equispaced values θ in T = [τ, 1 − τ ] with τ = 0.02. In general, τ could be tuned to the sample size n as, say, τ = 5/n. The optimal θ has been chosen in each training set. In order to see which θ-values were chosen depending on the model setup, an average value of these has been computed across all the 100 data sets. The mean of the misclassification rates and the standard error of these means were estimated from the classification results in the replicated test sets. Tables 1-7 show the obtained results of the quantile classifiers with data preprocessed according to the Galton and the Skewness corrections (QCG, QCS). The tables show the average misclassification errors and the average of the optimal θ values across all the 100 data sets in each considered setting. In brackets standard errors have been reported. We compared the quantile classifier misclassification rates with the ones obtained by nine other classifiers: the component-wise centroid and median classifier (CC, MC), Fisher's linear discriminant analysis (LDA), the k-nearest-neighbor classifier (k-NN; Cover and Hart (1967)), the naive Bayes classifier (Hand and Yu, 2001), the support vector machine (SVM; Cortes and Vapnik (1995); Wang et al. (2008)), the nearest-shrunken centroid method (Tibshirani et al., 2002), penalized logistic regression (Park and Hastie, 2008) and classification trees (rpart; Breiman et al. (1984)). We used the R package MASS to implement Fisher's LDA, the library CLASS for k-NN with k = 5, the library e1071 for the naive Bayes classifier and SVM (Support Vector Machine) with the default settings, the package pamr for the nearest-shrunken centroid with threshold set to 1, the package stepPlr for penalized logistic regression wit regularization parameter λ = 1, and the package rpart for implementing the classification trees. For all methods, the misclassification rates decrease as the sample size increases. With reference to the quantile classifier the larger the sample size is, the more consistent the choice of the optimal θ appears and consequently the discriminative power of the method increases. Not surprisingly, the classification performance worsens as the number of irrelevant variables increases. For fixed sample size and percentage of relevant variables, the methods seem to perform better as p increases, in almost all settings. To summarize and compare results of the different classifiers, we have computed the relative performance of each classifier with respect to the Galton quantile classifier misclassification rates taken as baseline. More specifically, we have transformed the misclassification rates of each classifier as error rate minus baseline error rate divided by the average error rate in the given setting. The distribution of these rescaled results (aggregated over the different choices of p, n, dependence/independence and the percentage of relevant variables) is represented in the boxplots of Figures 3 and 4. Results indicate that the quantile classifier performs very well in most situations compared to the other classifiers. The skewness correction according to the conventional third standardized moment seems to produce a slightly better classification performance in the asymmetric setups. However, the Galton skewness correction is preferable when analyzing real data more sensitive to outliers, as it will be shown in the next section. In the scenarios with equal distributional shapes and symmetric variables, the performance of the quantile classifiers is similar to the one of centroid and the median classifier, and this is consistent with the chosen optimal value of θ, which is on average close to the midpoint 0.5. stepPlr and SVM perform also very well in this scenario. In the scenarios with equal distributional shapes and asymmetric variables, the quantile classifiers outperform all other methods clearly and more or less uniformly. With differing distributions of variables, the quantile classifiers again show excellent results. The only method with a better median of the relative performance (see Figures 3 and 4) is rpart, which is the best method in the setups with few informative variables, due to its use of only a small number of variables. rparts relative performance in the setups with 100% informative variables and in setups with small n/largep is worse, though. The overall results of Centroid, SVM, NSC, and stepPlr are not much worse than those of the quantile classifier, but they are rarely significantly better and sometimes clearly worse. The fourth scenario with Beta distributions differing between variables and classes within variables is again generally dominated by the quantile classifiers, with nBayes achieving similar results overall and only rpart winning some settings with highest noise ratio. Overall, the methods that compete well with the quantile classifiers in one or two scenarios fall clearly behind in some others. Real data example For illustration, we apply the quantile classifier to a data set from chemistry. These data were collected testing a new method to detect bioaerosol particles based on gaseous plasma electrochemistry. The presence of such particles in air has a big impact on health, but monitoring bioaerosols poses great technical challenges. Sarantaridis et al. (2012) attempted to tell several different bioaerosols apart based on voltage changes over time on eight different electrodes when particles passed a premixed laminar hydrogen/oxygen/nitrogen flame. The resulting data are eight time series with 301 observations each for each particle. Details are given in Sarantaridis et al. (2012). Actually a seventh variable (time point of maximum change) was used there, which we omit here. Although in Sarantaridis et al. (2012) it contributed to the classification, the chemists (personal communication) suspected this to be an artifact because knowledge of the experiment suggests that this variable is caused by other experimental features than the type of the bioaerosol. We are therefore left with 48 variables (six for each of the eight electrodes). In the current example, we apply a scheme for variable standardization driven by subject knowledge, which is motivated by the expectation of the chemists that the size of variation in voltage and length of effect is informative and that electrodes and variables for which the electrode causes stronger variation are actually more important for discrimination (low variation often indicates that only noise was picked up by the electrode). Standardization of every variable would remove such information. Still, the variables 1-4 (voltages) on one hand and 6-7 (effect lengths) on the other hand do not have comparable measurement units. Therefore we computed one standard deviation from all 8 * 4 voltage variables and standardized all these variables by the same standard deviation, and the 8 * 2 effect length variables were also standardized by the standard deviation computed from all of them combined. We confine ourselves to the classification problem of distinguishing between two bioaerosols, namely Bermuda Smut Spores and Black Walnut Pollen. For each bioaerosol there were data from thirty particles. The quantile classifier has been applied on no-preprocessed data and on data with signs adjustments according to the conventional skewness and its robust Galton version. We used leave-one-out cross-validation to assess the performance of the classifier. Within each fold we selected the optimal θ in the training set. Table 8 contains the misclassification rates of the quantile classifier according to the different preprocessing strategies. We also evaluated other discriminant methods: the component-wise centroid and median classifiers, linear and quadratic discriminant analysis, the k-nearest-neighbor classifier with k = 5, the naive Bayes classifier, the support vector machine, the nearest-shrunken centroid method, penalized logistic regression and classification trees. It can be seen from these results that the quantile classifier with Galton skewness correction is particularly effective for classifying the two bioaerosols and outperforms the other methods. Only two particles are misclassified. It is worth noting that the sign adjustment preprocessing step is particularly relevant. If no sign adjustment is performed, the choice of the optimal quantile value is more variable across the cross validated sets (and closer to the midpoint on average) because of the possible different directions of skewness in the observed variables. In this case, when data are preprocessed according to the Galton skewness correction, the selected optimal θ across the cross validated sets is always extremely small with an average of 0.04. This means that more discriminant information between the two bioaerosols is contained in the left tail of the observed distributions rather than in their "core". Conclusion The idea of the componentwise quantile classifier was inspired by the componentwise median classifier in Hall et al. (2009). The simulations and the application show that the quantile classifier can compete with the median classifier in the (symmetric) situations where the median classifier is best, but is much better for asymmetric and mixed variables due to its larger flexibility. It also compares very favorable to all the other classifiers tested in the present work. Basic issues with the componentwise quantile classifier are that it ignores the correlation structure (which though does not seem to do much harm in the simulations with dependent variables) and that it requires scaling of the variables because it is not scale equivariant. As all distance-based classifiers, it does not require the classification information to be concentrated on a much lower dimensional space. First attempts to use different θ-values for different variables were not successful. This is an issue for future research. Figure 1 : 1Theoretical misclassification rates 1 − Ψ(θ) for four different scenarios. First row: probability density functions of two location-shifted Gaussians and corresponding misclassification function of θ. Second row: two location-shifted chi-squared distributions. Third row: two location-shifted exponentials. Last row: a Gaussian vs a chi-squared distribution. Figure 2 : 2Misclassification rates obtained in the test set of a simulation study. For comparative purposes, the horizontal lines indicate the misclassification rates of the median classifier, of the centroid classifier and of the optimal quantile classifier in the training set. Sarantaridis et al. (2012) discussed how the relevant information in every time series can be summarized in six characteristic features, namely 1. Maximum voltage in series. 2. Minimum voltage in series. 3. Maximum voltage change caused by electrode. 4. Difference between final and initial voltage. 5. Length of positive change caused by the electrode. 6. Length of negative change caused by the electrode. Figure 3 : 3Relative performance of the classifiers with respect to the quantile classifier with Galton skewness correction taken as baseline for all runs in scenarions 1Differing distributions of variables and classes within variables Figure 4 : 4Relative performance of the classifiers with respect to the quantile classifier with Galton skewness correction taken as baseline for all runs in scenarions 3-4. Table 1 : 1Simulation study: independent identically distributed symmetric variables.Mis- Table 2 : 2Simulation study: dependent identically distributed symmetric variables. Mis- classification rates (with standard deviations, i.e. 10standard errors, in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets. n = 50 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.27 (0.09) 0.32 (0.08) 0.42 (0.06) 0.24 (0.07) 0.27 (0.07) 0.41 (0.06) 0.21 (0.07) 0.21 (0.06) 0.35 (0.08) θ Galton 0.37 (0.25) 0.43 (0.23) 0.47 (0.18) 0.39 (0.30) 0.42 (0.23) 0.44 (0.16) 0.36 (0.31) 0.42 (0.19) 0.46 (0.09) QCS 0.27 (0.08) 0.31 (0.08) 0.43 (0.05) 0.24 (0.08) 0.27 (0.08) 0.41 (0.06) 0.22 (0.06) 0.22 (0.07) 0.36 (0.07) θ Skewn. 0.34 (0.25) 0.44 (0.25) 0.40 (0.19) 0.30 (0.26) 0.37 (0.23) 0.41 (0.14) 0.22 (0.22) 0.36 (0.18) 0.43 (0.09) CC 0.24 (0.07) 0.31 (0.07) 0.43 (0.05) 0.21 (0.07) 0.27 (0.08) 0.43 (0.06) 0.19 (0.06) 0.23 (0.09) 0.40 (0.08) MC 0.24 (0.06) 0.29 (0.06) 0.42 (0.05) 0.20 (0.06) 0.25 (0.06) 0.40 (0.06) 0.18 (0.05) 0.21 (0.06) 0.35 (0.07) LDA 0.43 (0.05) 0.41 (0.06) 0.43 (0.05) 0.32 (0.07) 0.34 (0.08) 0.43 (0.05) 0.22 (0.06) 0.33 (0.07) 0.43 (0.05) knn 0.27 (0.06) 0.33 (0.08) 0.43 (0.05) 0.25 (0.07) 0.31 (0.08) 0.44 (0.05) 0.24 (0.08) 0.30 (0.09) 0.44 (0.06) n-Bayes 0.35 (0.08) 0.41 (0.06) 0.45 (0.04) 0.34 (0.10) 0.40 (0.08) 0.45 (0.04) 0.34 (0.10) 0.37 (0.09) 0.44 (0.05) SVM 0.24 (0.06) 0.29 (0.07) 0.42 (0.05) 0.23 (0.07) 0.26 (0.07) 0.42 (0.06) 0.21 (0.06) 0.22 (0.07) 0.41 (0.07) NSC 0.32 (0.07) 0.37 (0.07) 0.43 (0.06) 0.29 (0.07) 0.33 (0.06) 0.43 (0.06) 0.22 (0.06) 0.25 (0.07) 0.39 (0.07) stepPlr 0.28 (0.07) 0.29 (0.06) 0.41 (0.06) 0.24 (0.07) 0.23 (0.07) 0.38 (0.07) 0.19 (0.05) 0.12 (0.06) 0.28 (0.07) rpart 0.39 (0.06) 0.41 (0.06) 0.43 (0.05) 0.39 (0.06) 0.41 (0.07) 0.44 (0.05) 0.40 (0.06) 0.40 (0.06) 0.43 (0.05) n = 100 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.26 (0.05) 0.30 (0.05) 0.43 (0.04) 0.23 (0.06) 0.26 (0.06) 0.40 (0.05) 0.20 (0.05) 0.21 (0.06) 0.31 (0.06) θ Galton 0.40 (0.24) 0.41 (0.21) 0.43 (0.17) 0.41 (0.30) 0.42 (0.22) 0.43 (0.14) 0.38 (0.33) 0.36 (0.25) 0.46 (0.12) QCS 0.26 (0.05) 0.30 (0.06) 0.43 (0.04) 0.24 (0.06) 0.25 (0.06) 0.40 (0.04) 0.22 (0.06) 0.21 (0.05) 0.30 (0.06) θ Skewn. 0.42 (0.24) 0.47 (0.23) 0.47 (0.21) 0.37 (0.29) 0.42 (0.23) 0.47 (0.17) 0.35 (0.34) 0.38 (0.28) 0.47 (0.14) CC 0.21 (0.04) 0.27 (0.04) 0.42 (0.05) 0.20 (0.04) 0.24 (0.05) 0.39 (0.05) 0.19 (0.06) 0.20 (0.05) 0.33 (0.09) MC 0.22 (0.05) 0.28 (0.04) 0.42 (0.04) 0.20 (0.04) 0.23 (0.04) 0.38 (0.05) 0.18 (0.04) 0.19 (0.04) 0.30 (0.06) LDA 0.32 (0.05) 0.31 (0.05) 0.42 (0.05) 0.44 (0.05) 0.42 (0.06) 0.45 (0.04) 0.23 (0.04) 0.26 (0.05) 0.41 (0.04) knn 0.24 (0.05) 0.31 (0.06) 0.44 (0.04) 0.22 (0.04) 0.27 (0.05) 0.43 (0.05) 0.21 (0.05) 0.24 (0.07) 0.42 (0.06) n-Bayes 0.35 (0.09) 0.41 (0.07) 0.46 (0.03) 0.35 (0.10) 0.40 (0.09) 0.46 (0.03) 0.34 (0.09) 0.36 (0.10) 0.46 (0.04) SVM 0.23 (0.04) 0.26 (0.05) 0.40 (0.05) 0.21 (0.04) 0.22 (0.04) 0.38 (0.05) 0.20 (0.04) 0.14 (0.04) 0.35 (0.07) NSC 0.28 (0.05) 0.32 (0.06) 0.42 (0.05) 0.24 (0.05) 0.29 (0.06) 0.40 (0.06) 0.20 (0.05) 0.22 (0.04) 0.30 (0.06) stepPlr 0.28 (0.05) 0.27 (0.05) 0.41 (0.05) 0.24 (0.04) 0.21 (0.04) 0.36 (0.05) 0.20 (0.04) 0.08 (0.03) 0.21 (0.05) rpart 0.39 (0.05) 0.40 (0.05) 0.45 (0.04) 0.38 (0.05) 0.40 (0.05) 0.44 (0.04) 0.38 (0.04) 0.40 (0.05) 0.43 (0.04) n = 500 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.22 (0.02) 0.25 (0.02) 0.38 (0.03) 0.20 (0.02) 0.22 (0.02) 0.34 (0.03) 0.18 (0.02) 0.19 (0.02) 0.24 (0.04) θ Galton 0.41 (0.15) 0.43 (0.11) 0.44 (0.12) 0.39 (0.13) 0.39 (0.13) 0.47 (0.14) 0.41 (0.25) 0.43 (0.25) 0.45 (0.15) QCS 0.22 (0.02) 0.26 (0.02) 0.38 (0.02) 0.20 (0.02) 0.23 (0.02) 0.34 (0.03) 0.18 (0.02) 0.19 (0.02) 0.24 (0.04) θ Skewn. 0.46 (0.15) 0.50 (0.13) 0.49 (0.13) 0.49 (0.19) 0.48 (0.18) 0.51 (0.13) 0.46 (0.25) 0.46 (0.28) 0.49 (0.18) CC 0.21 (0.02) 0.24 (0.02) 0.37 (0.03) 0.19 (0.02) 0.22 (0.03) 0.33 (0.03) 0.18 (0.02) 0.18 (0.02) 0.23 (0.05) MC 0.22 (0.02) 0.25 (0.02) 0.37 (0.02) 0.20 (0.02) 0.22 (0.02) 0.33 (0.03) 0.18 (0.02) 0.18 (0.02) 0.23 (0.04) LDA 0.25 (0.02) 0.23 (0.02) 0.36 (0.03) 0.26 (0.02) 0.18 (0.02) 0.32 (0.03) 0.47 (0.02) 0.41 (0.04) 0.45 (0.03) knn 0.23 (0.02) 0.26 (0.02) 0.41 (0.03) 0.21 (0.02) 0.23 (0.02) 0.39 (0.03) 0.19 (0.02) 0.18 (0.03) 0.34 (0.05) n-Bayes 0.30 (0.08) 0.38 (0.07) 0.47 (0.03) 0.32 (0.09) 0.38 (0.09) 0.46 (0.04) 0.31 (0.10) 0.37 (0.10) 0.47 (0.04) SVM 0.22 (0.02) 0.21 (0.02) 0.34 (0.02) 0.20 (0.02) 0.16 (0.02) 0.29 (0.02) 0.18 (0.02) 0.06 (0.01) 0.14 (0.02) NSC 0.22 (0.02) 0.25 (0.02) 0.34 (0.03) 0.20 (0.02) 0.22 (0.02) 0.31 (0.03) 0.18 (0.02) 0.18 (0.02) 0.22 (0.02) stepPlr 0.25 (0.02) 0.23 (0.02) 0.36 (0.03) 0.26 (0.02) 0.19 (0.02) 0.32 (0.03) 0.23 (0.02) 0.05 (0.01) 0.15 (0.02) rpart 0.36 (0.02) 0.37 (0.02) 0.42 (0.02) 0.36 (0.02) 0.37 (0.02) 0.40 (0.03) 0.36 (0.02) 0.36 (0.02) 0.39 (0.03) Table 3 : 3Simulation study: independent identically distributed asymmetric variables.Misclassification rates (with standard deviations, i.e. 10standard errors, in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets. n = 50 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.25 (0.09) 0.36 (0.08) 0.43 (0.05) 0.26 (0.10) 0.36 (0.09) 0.44 (0.05) 0.26 (0.07) 0.35 (0.07) 0.45 (0.04) θ Galton 0.18 (0.16) 0.28 (0.26) 0.46 (0.31) 0.35 (0.27) 0.44 (0.28) 0.60 (0.26) 0.48 (0.23) 0.52 (0.20) 0.61 (0.11) QCS 0.20 (0.07) 0.28 (0.08) 0.42 (0.05) 0.21 (0.08) 0.24 (0.07) 0.42 (0.06) 0.27 (0.10) 0.26 (0.07) 0.30 (0.09) θ Skewn. 0.06 (0.05) 0.08 (0.10) 0.29 (0.30) 0.06 (0.10) 0.10 (0.18) 0.38 (0.38) 0.06 (0.10) 0.05 (0.09) 0.15 (0.30) CC 0.43 (0.05) 0.44 (0.05) 0.46 (0.04) 0.43 (0.05) 0.43 (0.05) 0.44 (0.05) 0.39 (0.06) 0.43 (0.05) 0.45 (0.04) MC 0.38 (0.07) 0.43 (0.05) 0.44 (0.05) 0.34 (0.07) 0.40 (0.06) 0.45 (0.04) 0.17 (0.06) 0.30 (0.07) 0.43 (0.05) LDA 0.44 (0.05) 0.44 (0.04) 0.44 (0.05) 0.44 (0.04) 0.43 (0.04) 0.45 (0.04) 0.43 (0.05) 0.44 (0.05) 0.45 (0.04) knn 0.45 (0.05) 0.46 (0.03) 0.46 (0.04) 0.44 (0.04) 0.45 (0.04) 0.45 (0.04) 0.45 (0.05) 0.46 (0.04) 0.46 (0.03) n-Bayes 0.44 (0.04) 0.44 (0.05) 0.44 (0.05) 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.44 (0.05) 0.44 (0.05) 0.44 (0.05) SVM 0.43 (0.04) 0.44 (0.05) 0.44 (0.05) 0.43 (0.05) 0.43 (0.04) 0.45 (0.04) 0.39 (0.06) 0.43 (0.05) 0.45 (0.04) NSC 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.45 (0.05) 0.44 (0.05) 0.44 (0.04) 0.43 (0.06) 0.43 (0.05) 0.45 (0.04) stepPlr 0.43 (0.05) 0.44 (0.05) 0.44 (0.04) 0.44 (0.05) 0.43 (0.05) 0.44 (0.05) 0.38 (0.07) 0.42 (0.05) 0.45 (0.04) rpart 0.42 (0.05) 0.42 (0.06) 0.44 (0.04) 0.42 (0.06) 0.42 (0.06) 0.44 (0.05) 0.41 (0.06) 0.42 (0.06) 0.44 (0.05) n = 100 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.09 (0.04) 0.18 (0.05) 0.42 (0.05) 0.07 (0.04) 0.14 (0.05) 0.37 (0.06) 0.05 (0.07) 0.14 (0.11) 0.42 (0.06) θ Galton 0.04 (0.02) 0.04 (0.03) 0.19 (0.25) 0.05 (0.06) 0.06 (0.06) 0.17 (0.23) 0.27 (0.20) 0.29 (0.24) 0.56 (0.25) QCS 0.09 (0.03) 0.17 (0.04) 0.41 (0.05) 0.06 (0.03) 0.12 (0.04) 0.35 (0.05) 0.01 (0.02) 0.06 (0.07) 0.27 (0.10) θ Skewn. 0.04 (0.02) 0.03 (0.02) 0.13 (0.21) 0.03 (0.02) 0.04 (0.02) 0.10 (0.17) 0.17 (0.15) 0.11 (0.17) 0.17 (0.31) CC 0.43 (0.05) 0.45 (0.03) 0.46 (0.03) 0.41 (0.05) 0.44 (0.04) 0.46 (0.03) 0.33 (0.05) 0.40 (0.05) 0.46 (0.03) MC 0.34 (0.06) 0.41 (0.05) 0.46 (0.04) 0.30 (0.04) 0.38 (0.06) 0.45 (0.03) 0.11 (0.03) 0.25 (0.04) 0.43 (0.04) LDA 0.44 (0.04) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.04) 0.46 (0.03) 0.41 (0.05) 0.44 (0.04) 0.45 (0.03) knn 0.45 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.04) 0.46 (0.03) 0.47 (0.03) n-Bayes 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) SVM 0.42 (0.04) 0.45 (0.04) 0.46 (0.03) 0.41 (0.04) 0.44 (0.05) 0.46 (0.03) 0.34 (0.05) 0.41 (0.05) 0.46 (0.03) NSC 0.45 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.04) 0.46 (0.04) 0.46 (0.03) 0.42 (0.05) 0.44 (0.04) 0.46 (0.03) stepPlr 0.43 (0.04) 0.46 (0.03) 0.46 (0.03) 0.42 (0.05) 0.45 (0.04) 0.46 (0.03) 0.33 (0.05) 0.40 (0.05) 0.46 (0.03) rpart 0.36 (0.06) 0.39 (0.06) 0.44 (0.05) 0.37 (0.06) 0.38 (0.07) 0.43 (0.05) 0.37 (0.06) 0.38 (0.06) 0.43 (0.05) n = 500 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.02 (0.01) 0.09 (0.01) 0.34 (0.03) 0.00 (0.00) 0.03 (0.01) 0.26 (0.02) 0.00 (0.00) 0.00 (0.00) 0.06 (0.01) θ Galton 0.02 (0.01) 0.02 (0.01) 0.05 (0.03) 0.02 (0.01) 0.02 (0.01) 0.03 (0.02) 0.17 (0.04) 0.02 (0.00) 0.03 (0.01) QCS 0.02 (0.01) 0.09 (0.01) 0.34 (0.03) 0.00 (0.00) 0.03 (0.01) 0.26 (0.02) 0.00 (0.00) 0.00 (0.00) 0.06 (0.01) θ Skewn. 0.02 (0.01) 0.02 (0.01) 0.05 (0.03) 0.02 (0.01) 0.02 (0.01) 0.03 (0.02) 0.17 (0.04) 0.02 (0.00) 0.03 (0.01) CC 0.40 (0.02) 0.44 (0.02) 0.48 (0.02) 0.36 (0.02) 0.42 (0.02) 0.48 (0.02) 0.22 (0.02) 0.33 (0.02) 0.46 (0.02) MC 0.31 (0.02) 0.37 (0.02) 0.46 (0.02) 0.23 (0.02) 0.32 (0.02) 0.45 (0.02) 0.05 (0.01) 0.15 (0.02) 0.38 (0.02) LDA 0.41 (0.02) 0.44 (0.02) 0.48 (0.02) 0.37 (0.02) 0.42 (0.02) 0.48 (0.02) 0.47 (0.02) 0.48 (0.02) 0.48 (0.02) knn 0.46 (0.02) 0.47 (0.02) 0.48 (0.01) 0.45 (0.02) 0.47 (0.02) 0.48 (0.01) 0.43 (0.03) 0.46 (0.02) 0.48 (0.01) n-Bayes 0.47 (0.02) 0.48 (0.02) 0.48 (0.01) 0.47 (0.02) 0.48 (0.01) 0.48 (0.01) 0.47 (0.02) 0.48 (0.02) 0.48 (0.01) SVM 0.37 (0.02) 0.43 (0.02) 0.48 (0.02) 0.33 (0.02) 0.41 (0.02) 0.47 (0.02) 0.19 (0.02) 0.32 (0.02) 0.46 (0.02) NSC 0.45 (0.02) 0.46 (0.02) 0.48 (0.02) 0.43 (0.02) 0.45 (0.02) 0.48 (0.02) 0.36 (0.03) 0.40 (0.02) 0.46 (0.02) stepPlr 0.41 (0.02) 0.44 (0.02) 0.48 (0.02) 0.37 (0.02) 0.42 (0.03) 0.48 (0.02) 0.27 (0.02) 0.36 (0.02) 0.47 (0.02) rpart 0.21 (0.03) 0.22 (0.03) 0.36 (0.02) 0.22 (0.03) 0.21 (0.03) 0.28 (0.04) 0.23 (0.03) 0.23 (0.03) 0.22 (0.03) Table 4 : 4Simulation study: dependent identically distributed asymmetric variables. Mis- classification rates (with standard deviations, i.e. 10standard errors, in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets. n = 50 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.36 (0.10) 0.40 (0.08) 0.44 (0.04) 0.37 (0.09) 0.38 (0.08) 0.44 (0.04) 0.39 (0.08) 0.37 (0.07) 0.43 (0.04) θ Galton 0.30 (0.35) 0.34 (0.33) 0.48 (0.28) 0.44 (0.40) 0.38 (0.33) 0.53 (0.34) 0.53 (0.42) 0.34 (0.37) 0.34 (0.35) QCS 0.29 (0.11) 0.33 (0.10) 0.43 (0.05) 0.33 (0.11) 0.32 (0.11) 0.42 (0.06) 0.37 (0.12) 0.31 (0.13) 0.38 (0.08) θ Skewn. 0.18 (0.29) 0.19 (0.27) 0.33 (0.28) 0.34 (0.38) 0.28 (0.33) 0.33 (0.31) 0.54 (0.40) 0.35 (0.35) 0.30 (0.35) CC 0.45 (0.04) 0.44 (0.05) 0.45 (0.04) 0.44 (0.05) 0.45 (0.05) 0.45 (0.05) 0.44 (0.06) 0.44 (0.05) 0.45 (0.05) MC 0.41 (0.06) 0.43 (0.05) 0.44 (0.04) 0.41 (0.06) 0.43 (0.05) 0.45 (0.04) 0.41 (0.06) 0.41 (0.06) 0.44 (0.05) LDA 0.45 (0.04) 0.44 (0.04) 0.45 (0.04) 0.44 (0.05) 0.44 (0.05) 0.45 (0.04) 0.44 (0.05) 0.44 (0.04) 0.44 (0.04) knn 0.45 (0.04) 0.45 (0.04) 0.44 (0.04) 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.46 (0.04) 0.46 (0.04) 0.46 (0.04) n-Bayes 0.44 (0.04) 0.45 (0.05) 0.45 (0.04) 0.44 (0.04) 0.45 (0.05) 0.44 (0.05) 0.44 (0.05) 0.45 (0.05) 0.44 (0.04) SVM 0.43 (0.06) 0.44 (0.05) 0.44 (0.05) 0.43 (0.05) 0.45 (0.04) 0.44 (0.04) 0.43 (0.05) 0.44 (0.04) 0.44 (0.05) NSC 0.46 (0.03) 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.45 (0.04) 0.43 (0.06) 0.44 (0.05) 0.44 (0.05) stepPlr 0.44 (0.05) 0.44 (0.05) 0.44 (0.05) 0.44 (0.05) 0.44 (0.04) 0.44 (0.04) 0.44 (0.05) 0.42 (0.05) 0.44 (0.05) rpart 0.43 (0.06) 0.43 (0.05) 0.44 (0.05) 0.43 (0.05) 0.44 (0.05) 0.44 (0.04) 0.43 (0.05) 0.43 (0.06) 0.43 (0.05) n = 100 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.19 (0.05) 0.25 (0.07) 0.42 (0.05) 0.20 (0.09) 0.25 (0.09) 0.41 (0.06) 0.34 (0.14) 0.32 (0.11) 0.38 (0.08) θ Galton 0.03 (0.06) 0.05 (0.12) 0.27 (0.31) 0.07 (0.18) 0.10 (0.22) 0.30 (0.33) 0.51 (0.42) 0.42 (0.38) 0.32 (0.33) QCS 0.18 (0.05) 0.24 (0.06) 0.42 (0.05) 0.17 (0.07) 0.22 (0.08) 0.40 (0.06) 0.33 (0.16) 0.31 (0.14) 0.34 (0.10) θ Skewn. 0.03 (0.06) 0.05 (0.12) 0.24 (0.30) 0.05 (0.15) 0.09 (0.19) 0.23 (0.30) 0.51 (0.42) 0.45 (0.38) 0.31 (0.34) CC 0.44 (0.04) 0.45 (0.03) 0.46 (0.03) 0.45 (0.04) 0.45 (0.04) 0.46 (0.03) 0.44 (0.04) 0.45 (0.04) 0.46 (0.03) MC 0.41 (0.05) 0.43 (0.05) 0.46 (0.03) 0.41 (0.05) 0.43 (0.05) 0.46 (0.03) 0.41 (0.05) 0.41 (0.06) 0.45 (0.04) LDA 0.46 (0.03) 0.45 (0.04) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.03) 0.45 (0.03) 0.46 (0.04) 0.45 (0.03) knn 0.45 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.04) 0.46 (0.04) 0.46 (0.03) 0.46 (0.04) 0.46 (0.04) 0.46 (0.03) n-Bayes 0.46 (0.03) 0.46 (0.03) 0.47 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.03) 0.46 (0.03) 0.46 (0.03) SVM 0.44 (0.04) 0.45 (0.04) 0.45 (0.04) 0.43 (0.05) 0.45 (0.04) 0.46 (0.03) 0.42 (0.05) 0.43 (0.05) 0.46 (0.03) NSC 0.46 (0.03) 0.47 (0.03) 0.47 (0.03) 0.46 (0.03) 0.46 (0.03) 0.46 (0.03) 0.45 (0.04) 0.45 (0.04) 0.47 (0.03) stepPlr 0.46 (0.03) 0.45 (0.04) 0.46 (0.03) 0.45 (0.03) 0.44 (0.04) 0.46 (0.03) 0.45 (0.03) 0.40 (0.05) 0.45 (0.04) rpart 0.40 (0.05) 0.42 (0.04) 0.45 (0.04) 0.41 (0.05) 0.42 (0.05) 0.44 (0.04) 0.41 (0.05) 0.41 (0.05) 0.43 (0.05) n = 500 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.14 (0.01) 0.18 (0.02) 0.36 (0.03) 0.12 (0.01) 0.15 (0.02) 0.29 (0.02) 0.11 (0.01) 0.12 (0.05) 0.19 (0.07) θ Galton 0.02 (0.00) 0.02 (0.00) 0.06 (0.05) 0.02 (0.00) 0.02 (0.00) 0.04 (0.03) 0.02 (0.00) 0.04 (0.12) 0.10 (0.15) QCS 0.14 (0.01) 0.18 (0.02) 0.36 (0.03) 0.12 (0.01) 0.15 (0.02) 0.29 (0.02) 0.11 (0.01) 0.12 (0.05) 0.19 (0.07) θ Skewn. 0.02 (0.00) 0.02 (0.00) 0.06 (0.05) 0.02 (0.00) 0.02 (0.00) 0.04 (0.03) 0.02 (0.00) 0.04 (0.12) 0.10 (0.15) CC 0.46 (0.02) 0.46 (0.02) 0.48 (0.02) 0.45 (0.02) 0.45 (0.02) 0.48 (0.02) 0.45 (0.02) 0.44 (0.03) 0.47 (0.02) MC 0.42 (0.02) 0.43 (0.02) 0.47 (0.02) 0.42 (0.02) 0.41 (0.02) 0.46 (0.02) 0.41 (0.02) 0.40 (0.04) 0.44 (0.04) LDA 0.47 (0.02) 0.45 (0.02) 0.48 (0.01) 0.48 (0.02) 0.43 (0.02) 0.47 (0.02) 0.48 (0.01) 0.47 (0.02) 0.48 (0.01) knn 0.46 (0.02) 0.46 (0.02) 0.48 (0.02) 0.47 (0.02) 0.47 (0.02) 0.48 (0.01) 0.48 (0.01) 0.47 (0.02) 0.48 (0.02) n-Bayes 0.47 (0.02) 0.48 (0.01) 0.48 (0.01) 0.47 (0.02) 0.48 (0.02) 0.48 (0.01) 0.47 (0.02) 0.47 (0.02) 0.48 (0.01) SVM 0.44 (0.02) 0.43 (0.02) 0.47 (0.02) 0.43 (0.02) 0.42 (0.02) 0.47 (0.02) 0.41 (0.02) 0.36 (0.02) 0.45 (0.02) NSC 0.46 (0.02) 0.47 (0.02) 0.48 (0.02) 0.46 (0.02) 0.46 (0.02) 0.48 (0.02) 0.45 (0.02) 0.45 (0.02) 0.47 (0.02) stepPlr 0.47 (0.02) 0.45 (0.02) 0.47 (0.02) 0.48 (0.02) 0.43 (0.02) 0.47 (0.02) 0.48 (0.01) 0.35 (0.02) 0.44 (0.02) rpart 0.29 (0.03) 0.31 (0.03) 0.38 (0.02) 0.29 (0.03) 0.31 (0.03) 0.36 (0.03) 0.29 (0.03) 0.31 (0.03) 0.33 (0.03) Table 5 : 5Simulation study: independent not identically distributed variables. Misclassi- fication rates (with standard deviations, i.e. 10standard errors in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets. n = 50 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.25 (0.08) 0.36 (0.07) 0.43 (0.05) 0.19 (0.09) 0.33 (0.08) 0.43 (0.05) 0.06 (0.04) 0.17 (0.06) 0.43 (0.05) θ Galton 0.27 (0.28) 0.41 (0.32) 0.55 (0.32) 0.25 (0.26) 0.45 (0.34) 0.58 (0.30) 0.02 (0.04) 0.03 (0.07) 0.33 (0.18) QCS 0.22 (0.07) 0.33 (0.08) 0.43 (0.05) 0.15 (0.08) 0.27 (0.08) 0.44 (0.05) 0.03 (0.03) 0.12 (0.05) 0.43 (0.06) θ Skewn. 0.15 (0.20) 0.23 (0.28) 0.53 (0.32) 0.17 (0.23) 0.25 (0.26) 0.47 (0.35) 0.02 (0.04) 0.03 (0.06) 0.32 (0.13) CC 0.24 (0.07) 0.36 (0.08) 0.44 (0.05) 0.17 (0.05) 0.29 (0.07) 0.43 (0.05) 0.02 (0.02) 0.13 (0.05) 0.44 (0.05) MC 0.26 (0.06) 0.36 (0.07) 0.44 (0.05) 0.19 (0.07) 0.32 (0.07) 0.43 (0.05) 0.03 (0.02) 0.14 (0.05) 0.44 (0.05) LDA 0.41 (0.07) 0.43 (0.05) 0.45 (0.05) 0.32 (0.07) 0.38 (0.06) 0.44 (0.04) 0.25 (0.07) 0.36 (0.07) 0.45 (0.04) knn 0.34 (0.06) 0.41 (0.06) 0.44 (0.05) 0.30 (0.07) 0.38 (0.06) 0.44 (0.05) 0.13 (0.06) 0.28 (0.07) 0.44 (0.05) n-Bayes 0.40 (0.06) 0.43 (0.05) 0.45 (0.04) 0.37 (0.07) 0.43 (0.05) 0.45 (0.04) 0.34 (0.07) 0.41 (0.05) 0.45 (0.05) SVM 0.28 (0.06) 0.36 (0.07) 0.43 (0.05) 0.20 (0.06) 0.32 (0.07) 0.43 (0.05) 0.03 (0.03) 0.14 (0.06) 0.50 (0.01) NSC 0.32 (0.07) 0.37 (0.07) 0.43 (0.06) 0.24 (0.07) 0.33 (0.07) 0.42 (0.06) 0.07 (0.04) 0.16 (0.05) 0.44 (0.04) stepPlr 0.28 (0.07) 0.36 (0.08) 0.43 (0.05) 0.19 (0.05) 0.32 (0.07) 0.43 (0.05) 0.03 (0.03) 0.13 (0.05) 0.44 (0.05) rpart 0.33 (0.09) 0.35 (0.09) 0.41 (0.07) 0.32 (0.09) 0.34 (0.09) 0.42 (0.06) 0.31 (0.09) 0.32 (0.10) 0.41 (0.06) n = 100 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.18 (0.04) 0.31 (0.07) 0.44 (0.04) 0.11 (0.04) 0.25 (0.05) 0.45 (0.04) 0.02 (0.02) 0.10 (0.09) 0.40 (0.06) θ Galton 0.09 (0.10) 0.22 (0.27) 0.51 (0.33) 0.17 (0.14) 0.22 (0.21) 0.56 (0.32) 0.15 (0.18) 0.32 (0.30) 0.58 (0.33) QCS 0.17 (0.05) 0.29 (0.06) 0.44 (0.04) 0.11 (0.03) 0.23 (0.06) 0.44 (0.04) 0.01 (0.01) 0.09 (0.09) 0.38 (0.06) θ Skewn. 0.06 (0.10) 0.11 (0.18) 0.48 (0.36) 0.12 (0.12) 0.18 (0.20) 0.43 (0.35) 0.20 (0.18) 0.25 (0.25) 0.50 (0.37) CC 0.21 (0.04) 0.31 (0.06) 0.44 (0.04) 0.13 (0.04) 0.24 (0.05) 0.43 (0.05) 0.01 (0.01) 0.06 (0.03) 0.35 (0.05) MC 0.24 (0.04) 0.33 (0.05) 0.45 (0.04) 0.16 (0.04) 0.28 (0.04) 0.43 (0.04) 0.01 (0.01) 0.09 (0.03) 0.37 (0.05) LDA 0.28 (0.05) 0.36 (0.05) 0.45 (0.04) 0.40 (0.06) 0.44 (0.05) 0.46 (0.03) 0.16 (0.04) 0.28 (0.06) 0.44 (0.04) knn 0.32 (0.05) 0.40 (0.05) 0.46 (0.03) 0.27 (0.05) 0.37 (0.05) 0.46 (0.03) 0.10 (0.04) 0.25 (0.05) 0.43 (0.04) n-Bayes 0.35 (0.05) 0.42 (0.04) 0.46 (0.03) 0.32 (0.05) 0.40 (0.05) 0.45 (0.04) 0.28 (0.05) 0.37 (0.05) 0.46 (0.03) SVM 0.24 (0.04) 0.33 (0.06) 0.45 (0.04) 0.15 (0.04) 0.26 (0.05) 0.43 (0.04) 0.01 (0.01) 0.07 (0.03) 0.36 (0.05) NSC 0.25 (0.05) 0.32 (0.06) 0.42 (0.06) 0.18 (0.04) 0.26 (0.05) 0.39 (0.06) 0.02 (0.01) 0.07 (0.03) 0.29 (0.05) stepPlr 0.24 (0.05) 0.34 (0.05) 0.44 (0.04) 0.16 (0.04) 0.28 (0.05) 0.44 (0.04) 0.01 (0.01) 0.07 (0.03) 0.36 (0.05) rpart 0.18 (0.06) 0.21 (0.07) 0.39 (0.06) 0.17 (0.06) 0.19 (0.07) 0.31 (0.06) 0.17 (0.05) 0.17 (0.06) 0.20 (0.07) n = 500 p = 50 p = 100 p = 500 100% 50% 10% 100% 50% 10% 100% 50% 10% QCG 0.12 (0.03) 0.21 (0.04) 0.40 (0.03) 0.07 (0.01) 0.16 (0.01) 0.37 (0.03) 0.00 (0.00) 0.02 (0.01) 0.50 (0.01) θ Galton 0.06 (0.06) 0.08 (0.08) 0.22 (0.25) 0.09 (0.06) 0.07 (0.06) 0.22 (0.22) 0.31 (0.05) 0.17 (0.09) 0.23 (0.16) QCS 0.10 (0.02) 0.17 (0.02) 0.39 (0.03) 0.06 (0.01) 0.13 (0.02) 0.36 (0.03) 0.00 (0.00) 0.01 (0.01) 0.50 (0.00) θ Skewn. 0.02 (0.00) 0.02 (0.01) 0.11 (0.16) 0.05 (0.04) 0.03 (0.02) 0.10 (0.16) 0.33 (0.05) 0.13 (0.07) 0.16 (0.15) CC 0.17 (0.02) 0.26 (0.02) 0.41 (0.02) 0.09 (0.01) 0.18 (0.02) 0.38 (0.02) 0.00 (0.00) 0.02 (0.01) 0.50 (0.00) MC 0.21 (0.02) 0.29 (0.02) 0.42 (0.02) 0.12 (0.01) 0.22 (0.02) 0.40 (0.02) 0.01 (0.00) 0.04 (0.01) 0.50 (0.01) LDA 0.19 (0.02) 0.27 (0.02) 0.41 (0.02) 0.11 (0.01) 0.20 (0.02) 0.39 (0.03) 0.41 (0.05) 0.43 (0.04) 0.48 (0.01) knn 0.29 (0.02) 0.38 (0.02) 0.47 (0.02) 0.22 (0.02) 0.34 (0.03) 0.46 (0.02) 0.06 (0.02) 0.19 (0.02) 0.49 (0.01) n-Bayes 0.27 (0.03) 0.35 (0.03) 0.46 (0.02) 0.22 (0.02) 0.32 (0.03) 0.46 (0.02) 0.11 (0.02) 0.24 (0.02) 0.49 (0.01) SVM 0.19 (0.02) 0.27 (0.02) 0.42 (0.02) 0.10 (0.01) 0.20 (0.02) 0.38 (0.02) 0.00 (0.00) 0.03 (0.01) 0.50 (0.00) NSC 0.19 (0.02) 0.27 (0.02) 0.39 (0.02) 0.10 (0.01) 0.18 (0.02) 0.35 (0.02) 0.00 (0.00) 0.02 (0.01) 0.50 (0.00) stepPlr 0.19 (0.02) 0.27 (0.02) 0.41 (0.02) 0.12 (0.02) 0.21 (0.02) 0.39 (0.03) 0.00 (0.00) 0.03 (0.01) 0.50 (0.00) rpart 0.04 (0.01) 0.07 (0.01) 0.32 (0.03) 0.04 (0.01) 0.04 (0.01) 0.22 (0.03) 0.05 (0.01) 0.04 (0.01) 0.38 (0.03) Table 6 : 6Simulation study: dependent not identically distributed variables. Misclassification rates (with standard deviations, i.e. 10standard errors in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets.n = 50 Table 7 : 7Simulation study: Beta distributed variables, differing between classes. Misclassification rates (with standard deviations, i.e., 10standard errors in brackets) for different methods. Rows 2 and 4 contain the mean of the chosen values of θ in the training sets.n = 50 Table 8 : 8Leave-one-out cross-validated misclassification rates of the bioaerosol particles data. 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[ "Avouris, P. Nature Photon", "Avouris, P. Nature Photon" ]	[ "A K Geim ", "K S Novoselov ", "K S Mater ; 3. Novoselov ", "V Fal ", "L Colombo ", "P Gellert ", "M Schwab ", "K Kim ", "F Bonaccorso ", "Z Sun ", "T; Hasan ", "A C Ferrari ", "P Avouris ", "F Xia ", "F N Bull ; Xia ", "T Mueller ", "Y M Lin ", "A Valdes-Garcia ", "P Avouris ", "T Mueller ", "F N A Xia ", "J Li ", "L Niu ", "Z Zheng ", "F Yan ", "Z Sun ", "H Chang ", "C.-H Nano ; Liu ", "Y.-C Chang ", "T B Norris ", "Z Zhong ", "Y Shi ", "W Fang ", "K Zhang ", "W Zhang ", "L J Li ", "X Small ; Gan ", "R.-J Shiue ", "Y Gao ", "I Meric ", "T F Heinz ", "K Shepard ", "J Hone ", "S Assefa ", "D Englund ", "Z Z Cheng ", "K Xu ", "H Tsang ", "J B Xu ", "G Konstantatos ", "M Badioli ", "L Gaudreau ", "J Osmond ", "M Bernechea ", "F P Garcia De Arquer ", "F Gatti ", "F H Koppens ", "Z Liu ", "J Li ", "G A Tai ", "S P Lau ", "F Yan ", "Y Q Huang ", "R J Zhu ", "N Kang ", "J Du ", "H Q Xu " ]	[]	[ "C.; Lindefelt, U. J. Appl. Phys" ]	Due to its high charge carrier mobility, broadband light absorption, and ultrafast carrier dynamics, graphene is a promising material for the development of high-performance photodetectors. Graphene-based photodetectors have been demonstrated to date using monolayer graphene operating in conjunction with either metals or semiconductors. Most graphene devices are fabricated on doped Si substrates with SiO 2 dielectric used for back gating. Here, we demonstrate photodetection in graphene field effect phototransistors fabricated on undoped semiconductor (SiC) substrates. The photodetection mechanism relies on the high sensitivity of the graphene conductivity to the local change of the electric field that can result from the photoexcited charge carriers produced in the back-gated semiconductor substrate. We also modeled the device and simulated its operation using the finite element method to validate the existence of the field induced photoresponse mechanism and study its properties. Our graphene phototransistor possesses a room-temperature photoresponsivity as high as ~ 7.4 A/W, higher than the required photoresponsivity (1 A/W) in most practical applications. The light power-dependent photocurrent and photoresponsivity can be tuned by the source-drain bias voltage and back-gate voltage. Graphene phototransistors based on this simple and generic architecture can be fabricated by depositing graphene on a variety of undoped substrates, and are attractive for many applications in which photodetection or radiation detection is sought.	null	[ "https://arxiv.org/pdf/1409.5725v2.pdf" ]	117,962,994	1409.5725	e12767bacca7dace9ee9329ef7e1026921850529	Avouris, P. Nature Photon 2007. 2009. 2012. 2010. 2012. 2009. 2010. 2013. 2014. 2014. 2014. 2009. 2009. 2005-2011. 17. 2012. 2013. 2013. 2012. 2012. 2013. 1997. 2004 A K Geim K S Novoselov K S Mater ; 3. Novoselov V Fal L Colombo P Gellert M Schwab K Kim F Bonaccorso Z Sun T; Hasan A C Ferrari P Avouris F Xia F N Bull ; Xia T Mueller Y M Lin A Valdes-Garcia P Avouris T Mueller F N A Xia J Li L Niu Z Zheng F Yan Z Sun H Chang C.-H Nano ; Liu Y.-C Chang T B Norris Z Zhong Y Shi W Fang K Zhang W Zhang L J Li X Small ; Gan R.-J Shiue Y Gao I Meric T F Heinz K Shepard J Hone S Assefa D Englund Z Z Cheng K Xu H Tsang J B Xu G Konstantatos M Badioli L Gaudreau J Osmond M Bernechea F P Garcia De Arquer F Gatti F H Koppens Z Liu J Li G A Tai S P Lau F Yan Y Q Huang R J Zhu N Kang J Du H Q Xu Avouris, P. Nature Photon C.; Lindefelt, U. J. Appl. Phys Xu, X662007. 2009. 2012. 2010. 2012. 2009. 2010. 2013. 2014. 2014. 2014. 2009. 2009. 2005-2011. 17. 2012. 2013. 2013. 2012. 2012. 2013. 1997. 20041 Corresponding Author. Due to its high charge carrier mobility, broadband light absorption, and ultrafast carrier dynamics, graphene is a promising material for the development of high-performance photodetectors. Graphene-based photodetectors have been demonstrated to date using monolayer graphene operating in conjunction with either metals or semiconductors. Most graphene devices are fabricated on doped Si substrates with SiO 2 dielectric used for back gating. Here, we demonstrate photodetection in graphene field effect phototransistors fabricated on undoped semiconductor (SiC) substrates. The photodetection mechanism relies on the high sensitivity of the graphene conductivity to the local change of the electric field that can result from the photoexcited charge carriers produced in the back-gated semiconductor substrate. We also modeled the device and simulated its operation using the finite element method to validate the existence of the field induced photoresponse mechanism and study its properties. Our graphene phototransistor possesses a room-temperature photoresponsivity as high as ~ 7.4 A/W, higher than the required photoresponsivity (1 A/W) in most practical applications. The light power-dependent photocurrent and photoresponsivity can be tuned by the source-drain bias voltage and back-gate voltage. Graphene phototransistors based on this simple and generic architecture can be fabricated by depositing graphene on a variety of undoped substrates, and are attractive for many applications in which photodetection or radiation detection is sought. Graphene, a single layer of carbon atoms in a honeycomb lattice, is a fascinating new material with a potential for use in a variety of applications, including the next-generation electronic and optoelectronic devices. [1][2][3][4][5] In particular, graphene-based photodetectors have attracted significant attention due to their relatively wide absorption spectrum, high carrier mobility, low cost, and feasibility of their integration into flexible and transparent devices. [6][7][8][9][10][11][12][13][14] Over the last several years, a variety of graphene-based photodetectors have been reported, [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] which can be classified into two primary types (type I and type II). In type I graphene-based photodetectors, graphene is used to both generate and transport photoexcited carriers. The mechanisms of photodetection identified in these photodetectors include the photovoltaic effect, 6,7,15,18 photothermoelectric effect, 8,16,17,23,24 and bolometric effect. 18,21 Regardless of the photodetection mechanism, photoresponse of the type I graphene-based photodetectors is very fast (on the order of GHz). However, the photoresponsivity (photocurrent per unit power of incident light) is relatively low (on the order of mA/W), largely due to the weak light absorption by graphene. 6,7,12,13 Various techniques, such as integrating graphene with photonic nanostructures (e.g. microcavities, waveguides, plasmonic arrays, etc.), have been proposed to increase the photoresponsivity by increasing the light absorption, but the photoresponsivity has not been improved to more than a few tens of mA/W. 11,[25][26][27][28] Moreover, the fabrication procedures of these photodetectors are relatively complex. Recently, band structure engineering in graphene has enhanced photoresponsivity but only at temperatures below ~ 200 K, while the photoresponsivity at room temperature is still relatively low. 20 In the type II graphene-based photodetectors, a combination of graphene and other photoactive nanomaterials (such as semiconductor quantum dots) are used. [29][30][31] In this case, photo-carriers are generated in the photoactive materials and then transferred to and transported by graphene, which acts as a conducting channel. Using this approach, the photoresponsivity is increased significantly by the virtue of the much higher photo-absorption by the photoactive nanomaterials. 29,30 In this work, we demonstate a simple approach to graphene-based photodetection that utilizes a photo-actuated graphene field effect transistor (GFET) on a back-gated undoped semiconductor substrate (Fig. 1a). In such a graphene phototransistor, the photodetection mechanism relies on the high sensitivity of the conductivity of graphene to the local change of the electric field, resulting from photo-excited carriers produced in the underlying electrically gated, undoped semiconductor substrate. We validate this field effect based photodetection mechanism with the finite element method (FEM) simulations of the electric and potential field distribution within the GFET for different laser powers (Fig. 1d). We show that the GFETs fabricated on undoped silicon carbide (SiC) substrates exhibit a high photoresponsivity of ~7.4 A/W at room temperature. This is about three orders of magnitude higher than photoresponsivity of type-I photodetectors graphene photodetectors (typically fabricated on doped silicon substrates). 7,18 The photocurrent and photoresponsivity of the GFETs based on this novel architecture can be tuned by the gate voltage and source-drain bias voltage and is dependent on the incident optical power. The response time is reduced significantly with increasing incident optical power. The shortest response time measured in our device is ~ 1 s. The methodology presented here can provide a new and simple pathway for the development of high-responsitivity graphene photodetectors (particularly for applications where a high speed of response is not essential). A typical device architecture of the GFET on an undoped semiconductor substrate (SiC in our case) is shown in Fig. 1a. Our devices were fabricated by transferring mechanically exfoliated single layer graphene onto 416 μm thick SiC substrates, followed by electron beam lithography and deposition of Cr/Au contacts (see Methods and Supplementary Information for details of the fabrication process). An optical image (top view) of a part of a fabricated device is displayed in the inset of Fig. 1b. The presence of single layer graphene is confirmed by Raman spectroscopy (Inset of Fig. 1c, and Supplementary Information). The optoelectronic measurements were performed by illuminating the entire device with a laser beam (wavelength λ = 400 nm, corresponding photon energy = 3.1 eV). All measurements were performed at room temperature and atmospheric pressure. Figure 1b shows the measured drain-source current (I ds ) as a function of the back-gate voltage (V g ) of a representative device without ("dark") and with laser illumination ("light"). Without illumination, the effect of V g on the dark current (I dark ) is relatively small. When the device is illuminated, the field effect response is significantly enhanced, as demonstrated by a larger gate voltage modulation of the current under laser illumination (I light , Fig. 1b), suggesting that the (same) gate voltage now exerts a stronger electric field on graphene. In Fig. 1c, the photocurrent (I photo , defined as I light -I dark ) is displayed as a function of V g . The I photo -V g plot shows that the photocurrent is positive for sufficiently negative V g but undergoes a sign reversal near V g ~ 0V and becomes negative for positive V g . Thus, both the polarity and magnitude of photocurrent of our device can be tuned with the gate voltage. The observed gate voltage-dependent photocurrent of our GFETs can be qualitatively explained by the following mechanism. Under dark condition, the undoped SiC is highly insulating (bandgap ~3 eV) 32 and the applied V g drops uniformly across the SiC substrate. Due to the relatively large thickness (d ~ 416 μm) of our SiC substrate, the electric field (E = V g /d) experienced by graphene is relatively small, giving rise to a weak field effect. The observed finite small field effect without illumination may arise from the residual conductivity of the SiC due to impurities or trapped charges. When the SiC is illuminated, photo-excited charge carriers are generated in SiC, leading to an increased conductivity. While the SiC becomes more conductive under illumination, the experimentally observed leakage current between the back gate and graphene does not increase notably (the SiC does not form a shorted connection between the backgate and graphene). This can be due to the presence of a native oxide layer that often forms naturally on the SiC surface. 33,34 This native oxide (whose bandgap is much larger than our laser photon energy) remains insulating even under illumination. This could also arise from the spatially non-uniform distribution of the photogenerated carriers in SiC, where parts of the SiC may remain insulating under illumination. The enhanced field effect seen in Fig. 1b suggests that with increasing conductivity of SiC, the electric field at the graphene due to the applied back gate voltage increases. Such a photo-actuated change in the electric field is sensed by the change of graphene conductivity via field effect, allowing us to detect the light interacting with SiC. 35 This proposed mechanism is also consistent with the observation of near-zero photocurrent at V g ~ 0 V, where there is no electric field (and thus no field effect) to modulate the graphene conductivity. The small offset of zero crossing point of photocurrent away from V g = 0 V (Fig. 1c) may be related to gate hysteresis and trapped charges in the SiC. 36 To better understand the field effect based photodetection mechanism, we conducted FEM simulations of the electric field and potential distribution within the SiC substrate in the GFET using COMSOL Multiphysics. 35 The results of our simulation are presented in Fig. 1d. The architecture and thickness of the SiC substrate used in the modeled device closely match that of our experimental devices. To qualitatively capture the effect of the native oxide and the part of SiC substrate that remains insulating under illumination, we assumed the conductivity of the top 10 nm portion of our SiC substrate to be unaffected by illumination in our simulation. The laser illumination modifies the electric field within the SiC via the change in conductivity within SiC (except the top 10 nm). The conductivity of SiC affected by illumination is calculated for different incident laser power (see Supplementary Information) and is used as an input to the model which simulates the electric field. A representative result showing the simulated electric potential in the SiC for a laser power of 100 µW is displayed in the inset of Fig. 2d. The calculated electric field under graphene (at the SiC/graphene interface) for various laser powers is plotted in Fig. 2d. The model generally shows that for increasing illumination power a greater electric field exists in the vicinity of graphene (Fig. 1d), which results in the modulation of conductivity of graphene. This change of graphene conductivity is used to detect the light incident on the GFET. To further validate the proposed mechanism of photoresponse in our GFETs, we fabricated and measured two control devices. One is a SiC device fabricated by making contacts on top of a bare SiC substrate (no graphene in the channel), and another is a dummy device in which gold is used as the channel instead of graphene (see Supplementary Information). The SiC device shows a very small photocurrent (on the order of nA, ~ 0.1 % of graphene photocurrent), whereas the dummy device shows no change in current with change in gate-voltage, both in dark and under light illumination. These measurements confirm that the photocurrent of our GFETs does not result from the Schottky contact at the metal/SiC interface or the field effect from the SiC; rather, the photocurrent originates from the modulated charge carriers in the graphene due to graphene field effect. In addition to these control experiments, we also measured the gate leakage current of the GFET, finding that it is small (<1 nA at V g = ± 30 V), even with light illumination (see Supplementary Information), much lower than the measured photocurrent, which can reach many tens of µA. This further confirms that photocurrent of our GFET is not the result of collection of charge from the SiC. We further studied the dependence of photoresponse on the source-drain bias voltage (V ds ) at different illumination powers. Fig. 2a shows the I ds -V ds characteristics of a typical device without and with illumination for a series of incident laser power P in (varying from 1 to 184 µW) for a representative V g = -20 V. We found that all I ds -V ds curves pass through the origin, while the slope (indicating the conductance of graphene) of I ds -V ds curves increase with increasing laser power. Fig. 2b displays an enlarged view of the circled region shown in (a), showing that I light increases with increasing P in . Using the data in Fig. 2a, we calculated the photocurrent (I photo ) and plotted its dependence on V ds in Fig. 3a. For all P in , I photo increases linearly with increasing V ds , and a large photocurrent ~ 34 µA is observed for V ds = -0.5 V and P in = 184 µW. One of the most important figures of merit of a photodetector is its photoresponsivity (R), defined as the ratio of photocurrent and the incident laser power, R = I photo /P in . The plots of R as a function of V ds at different laser powers (Fig. 3b) show that R increases linearly with increasing V ds , suggesting that the device is in the linear response regime, and the photoresponsivity can be increased by applying a higher V ds . For V ds = -0.5 V, our device shows a high photoresponsivity of 7.4 A/W, which is more than three orders of magnitude higher than that previously measured in the ("type-I") graphene photodetectors (with a similar or higher V ds ). 7,18,[25][26][27][28] Although the photoresponsivity of our device is lower than that observed in the graphene-quantum dot hybrid phototransistors, 29,30 it is still larger than the required photoresponsivity (~1 A/W) for most practical applications. 3,14 Moreover, the simple device architecture and fabrication process required by this architecture may offer significant practical advantages. We also note that the photoresponsivity we report here (photoresponsivity is calculated using the total laser power incident on our entire device, not considering only the area of graphene) is likely to be underestimated, since a part of the laser beam incident on the SiC far away from the graphene may not contribute significantly to the observed photoresponse. We attribute this high photoresponsivity of our device to the unique device architecture, which supports an entirely different photodetection mechanism. Unlike the type-I graphene photodetectors reported to date, 6,7,15,18 in our devices the undoped SiC substrate is employed as a light absorber. In the presence of the back-gate voltage the photoexcited carriers in the SiC modulate the electric field, thus also inducing the charge carriers in graphene via a field effect. The highly sensitive field effect of graphene provides an efficient intrinsic amplification mechanism that (indirectly) converts the photon energy into a large electrical signal and hence leads to a high photoresponsivity. The number of modulated charge carriers (electrons or holes) in the graphene per incident photon in our device can reach as high as ~23 at a laser power of 1 µW (see Supplementary Information). More insight into the photoresponse characteristics of our device can be obtained from the dependence of photocurrent and photoresponsivity on the incident laser power P in . As shown in Fig. 3c, at lower P in (for example, below ~15 µW for V ds = -0.5 V), the photocurrent increases with increasing P in due to an increase in the modulated charge carriers. However, at higher P in , the photocurrent saturates (Fig. 3c), leading to a decrease in the photoresponsivity, as shown in Fig. 3d. One possible reason for this observed photocurrent saturation could be the saturation of graphene field effect itself at large (modulated) charge carrier densities (seen also in Fig. 1b) due to factors such as contact resistance and existence of charge trap states in graphene or at the graphene-SiC interface. The saturation might also result from decreased electric field modulation in the substrate at higher incident optical powers. We found that the decrease of R with increasing P in can be fitted by a power law, R ∞  in P with β ~ -0.8 (inset of Fig. 3d). We note that similar power law relations have been observed in phototransistors based on graphene-MoS 2 hybrid with β ~ -0.8, 37 and based on black phosphorus with β ~ -0.3, 38 (in the latter work this was attributed to the reduction of photogenerated carriers at the higher power due to the recombination/trap states). 38 We now turn our attention to the transient photoresponse of our devices. Time-dependent photocurrent for different representative gate voltages were measured as the laser was turned on and off (Fig. 4a). It is found that the sign of photocurrent changes from positive to negative as V g changes from -20 V to +20 V, and the photocurrent is almost zero for V g = 0 V. Both features are consistent with the field effect measurement shown in the Fig. 1c and confirm the gate tunability of our device's photoresponse. The gate-tunability is important for photodetection since it offers a convenient on-off switching control. In addition to the gate-tunability, our device maintains a long-term stability and a good reproducibility of the photoresponse for a series of repeated laser on/off switching, as shown in Fig. 4b. We found that the characteristics of time-dependent photocurrent curves vary significantly with increasing laser power (Fig. 4c). For a laser power of 184 µW, the photocurrent to dark-current ratio (I photo /I dark ) of our device can reach up to 10.3% (inset of Fig. 4d), which is higher than that of other recently reported graphene devices. 39,40 We calculated photocurrent response time (τ) by fitting the experimental data in Fig. 4c to an exponential function (see Supplementary Information) and plotted τ as a function of P in in Fig. 4d. We find that the response time for photocurrent rise (τ rise ) and fall (τ fall ) for each P in are similar. As the laser power increases, the response time decreases and can be fitted with a power law, τ  in P , with α ~ -0.7. The shortest response time of our device is ~ 1 s (measured at the highest P in = 184 µW), which is similar to the response time of a graphene-quantum dot hybrid photodetector. 30 The possible reasons for long response time could be due to the electrochemical doping of graphene. 36 In summary, we have demonstrated a novel and relatively simple approach to photodetection with a high photoresponsivity using a graphene phototransistor fabricated on undoped SiC substrate. The photoresponse characteristics of the device based on this new architecture show many distinct advantages, including strong and ambipolar gate voltage tunability, high photocurrent to dark-current ratio, and high photoresponsivity at room temperature. The high photoresponsivity (~7.4 A/W) of our device is not only superior to most other recently developed graphene photodetectors but also higher than the required photoresponsivity (1 A/W) in most practical applications. We anticipate that the photoresponsivity of devices based on the demonstrated approach can be further improved by optimizing the fabrication processes and measurement conditions (e.g., increasing source-drain bias voltage). In addition, our method may take advantage of a wide range of undoped semiconductors (differing in bandgaps and other electro-optical properties) as substrates for fabricating photodetectors. Our simple approach can also be generalized to other "beyondgraphene" 2D-semiconductors such as molybdenum disulfide (MoS 2 ), 41 or to higher-energy radiation. 35 Given the significant design flexibility and simplicity of our approach, this work provides a promising groundwork for the future development of graphene-based highperformance optoelectronic devices. Methods Device fabrication: Monolayer graphene was prepared by a micromechanical exfoliation method from highly ordered pyrolytic graphite (Momentive Performance Materials Inc.) and subsequently transferred (see details of the transfer process in Supplementary Information) onto an undoped 6H (Si-faced) SiC substrate (Pam-Ximan, with typical absorption coefficient ~40/cm at wavelength of 400 nm). The sourcedrain contacts with channel length of ~ 2 µm and channel width of ~ 2 µm were fabricated using electron beam lithography followed by deposition of Cr (5 nm)/Au (65 nm). The back-gate contact was fabricated by deposition of Cr (5 nm)/Au (65 nm) onto the back side of SiC wafer. Device characterization: The two-terminal dc transport measurements of the GFETs were performed using Keithley 2400 source meters controlled by a LabView program. The photoelectronic response was measured by illuminating the entire device by a laser with wavelength of 400 nm. The incident laser beam spot size on the device is ~ 2 mm. The laser power was tuned by controlling the laser drive current and was calibrated using a power meter. Figure S1: Schematic of the process to transfer an exfoliated monolayer graphene onto a SiC substrate. Transfer of the exfoliated graphene onto SiC substrate We transfer an exfoliated monolayer graphene onto a silicon carbide (SiC) substrate by the following processes. First, polyvinyl alcohol (PVA) solution is coated on a sacrificial substrate (here, Si/SiO 2 with dimension of 2 × 2 cm is used) at 3000 rpm for 45 s and baked on a hotplate at 90 0 C for 5 min. Then PMMA (polymethyl methacrylate) is coated onto the PVA film and similarly baked (Fig. S1a). Monolayer graphene was prepared by a micromechanical exfoliation technique and transferred onto the polymer (PMMA + PVA) films (Fig. S1b). The polymer films containing the graphene are then separated from the sacrificial substrate (Fig. S1 c) and transferred onto an undoped SiC substrate using a homemade transfer stage (Fig. S1d). Finally, the SiC substrate is submerged in acetone for a few hours to remove the polymer films (PMMA + PVA), then rinsed with IPA (isopropyl alcohol) and blown dry with nitrogen gas (Fig. S1e). Raman characterization of the graphene on SiC substrate We used Raman spectroscopy to confirm that the transferred exfoliated graphene on the SiC substrate is a monolayer. The Raman spectrum is measured using a Horiba Jobin Yvon Xplora confocal Raman microscope with a 532 nm excitation laser. Spectra were taken under the same experimental conditions on the same device at two different spots; one spot is on the graphene on the SiC substrate, and the other spot is on the SiC substrate (where no graphene is present) (inset in Fig. S2b). Since the intensity of Raman peaks varies slightly from spot to spot, the spectra were normalized by the strongest peaks. Since the Raman spectra of graphene and SiC have substantial overlap with each other, we subtracted the normalized SiC spectrum (Fig. S2a) from the normalized spectrum of graphene on the SiC (graphene + SiC) (Fig. S2b). The difference of the normalized spectrum of SiC, and graphene on the SiC is the graphene spectrum, which is shown in the inset of Fig. 1c in the main manuscript. The graphene spectrum shows no D peak, suggesting negligible defects in graphene. 1 The ratio of the 2D to G peaks intensity (I 2D /I G ) of the graphene spectrum is more than two, indicating a monolayer graphene in our device (inset, Fig. 1c). 1, 2 Figure S2: Raman spectrum of (a) SiC substrate (without graphene) and (b) graphene on SiC. Inset of (b): Optical image of a fabricated device. Spot 1 (SiC) and 2 (graphene on SiC) show where the Raman spectra were taken. The spectrum intensity is normalized by its strongest peak. The difference between the spectra (b) and (a) is extracted as the graphene Raman spectrum and shown in the inset of Fig. 1c. Calculation of Conductivity The conductivity of the SiC substrate increases by absorption of the incident light, whereby electrons and holes are produced in the SiC. The change in SiC conductivity due to light illumination can be calculated by Δσ = qμ= q'τμ, where q(=q'τ) is the number of steady state carriers (for both the electrons and holes) produced per unit volume and q' is the number of carriers produced per unit volume per unit time through light absorption, μ (= 400 cm 2 /V.s, given by manufacturer, PAM-Xiamen) is the mobility within SiC and τ is the carrier life time (recombination time), the mean time a conductive charge may exist within the substrate before recombination with an opposite charge. Here we assume that τ = 1 μs [ref. 3,4]. We consider the influence of penetration depth of the light in the SiC substrate. For simplicity, we divided the total thickness of SiC substrate into three parts and the profile of light absorption throughout the depth of the SiC substrate is used to calculate charge density produced per unit time for each part. For example, the time dependent number of change carrier per unit volume for the top 1/3 of SiC substrate is calculated to be q' = 1.04×10 3 C/m 3 /s for P in = 1 μW. The final SiC conductivity of the top 1/3 SiC substrate due to laser irradiation is given as σ = σ t + Δσ = 1×10 -3 + P in (4.2×10 -5 ) S/m, where P in is in μW and σ t is typical value of un-irradiated SiC conductivity, σ t ≈ 1×10 -3 S/m is given the manufacturer (PAM-Xiamen). Gate leakage current of device in dark and under laser illumination To confirm that gate leakage current (I g ) is not contributing to the photocurrent of our device, I g is monitored in both the dark and light illumination conditions. Figure S3a, b and c show the plots of I g as a function of gate-voltage (V g ), I g as a function of source-drain bias (V ds ) and I g as a function of time, respectively, measured in the dark and under illumination with various incident laser powers on the device (P in ). From these plots (Fig. S3a-c), we found that the device (i) leakage current is small (<1 nA) compared to the measured photocurrent (in the range of µA), and that (ii) leakage currents both in the dark and under laser illumination with a low laser power are almost similar. These features indicate that leakage current does not increase significantly with a low incident laser power. While leakage current does increase for a higher laser powers, it remains less than 1 nA. We therefore conclude that gate leakage current does not contribute to the measured photocurrent in our device. Control experiment 1: SiC device (without graphene) To confirm the photoresponse of our GFETs is not due to the photoresponse of the substrate (SiC) or Schottky contact at the SiC/metal interface, we fabricated SiC control devices without graphene (making a direct contact on top of SiC). Optical image of a fabricated SiC device (without graphene) is shown in the inset of Fig. S4a. The plots of I ds -V g and I photo -V g characteristics of a representative SiC device with and without illumination are shown in Figure S4a and b, respectively. We use same scale in the Fig.S4b and Fig. 1c (I photo -V g of GFET in the main figure) in order to clearly show the difference between the photocurrents and their gate dependence in the GFET and SiC devices. These plots show that both the current (in dark as well as under illumination) and photocurrent of SiC device are very small (of the order nA), at least three order lower than that observed photocurrent in the GFET (of the order µA). In addition, the photocurrent in the SiC device does not change significantly with the gate voltage, whereas the photocurrent in GFETs shows a strong gate-voltage dependence (Fig. 1c). From those measurements we can conclude that the photoresponse of our GFETs does not result from the Schottky contact of SiC or the collection of the charge from the SiC; rather it is the result of the modulation of charge carriers in the graphene via field effect. Figure S4. (a) Dependence of the drain-source current on the back-gate voltage (I ds -V g ) of a SiC device (without graphene) without and with laser illumination (λ = 400 nm) at incident laser power (P in ) of 86 µW. Inset: Optical image of a fabricated SiC device (without graphene). (b) The dependence of the photocurrent (I photo ) of the GFET on the gate voltage. The scales of Fig.4Sb and Fig. 1c (in the main paper) are kept the same to clearly show the differences in the gate dependence of photocurrent generation. Inset: the same Fig.4Sb plot, but on a nA scale. Control experiment 2: Dummy device (gold in the channel instead of graphene) We also fabricated dummy devices that contain no graphene, but use gold as a channel between the source-drain contacts (Inset of Fig. S5a). The I ds -V g plot of the dummy device shows no change in source-drain currents with the gate-voltage, with and without illumination. The I ds -t characteristics also show no change in current under light illumination (Fig. S5b). These observations further confirm that graphene is essential for the field effect photoresponse observed in the GFETs, and the photoresponse does not come from the SiC substrate or SiC/Au interface. Modulated charge carriers per incident photon The number of modulated charge carriers in the device per incident photon is calculated using the formula, 5 (I photo /P in )×(hc/eλ); where, I photo is the photocurrent, P in is the incident laser power on the device, h is Planck's constant, e is electron charge, and λ is the wavelength of incident light (400 nm). The number of modulated charge carriers per incident photon increases with increasing source-drain bias voltage because photocurrent increases with increasing the source-drain bias. We found that for a source-drain bias of -0.5 V, approximately 23 electrons (or holes) can be modulated in graphene by a single photon incident into the SiC substrate of our device. Photocurrent response time The photocurrent (I photo ) rise and fall response times (τ) for all laser powers was calculated by fitting the photocurrent vs. time data to an exponential function. Four representative fitted curves (for both rise and fall) for two different laser powers of 184 µW and 25 µW are shown in Fig. S6. The rise and fall times for P in = 184 µW are found to be 1.0 and 1.3 s, respectively, whereas for P in = 25 µW, the rise and fall time are 2.6 and 5.6 s, respectively. Figure 1 \| 1Gate voltage tunable photoresponse and operational principle of a graphene phototransistor on undoped semiconducting substrate. a. Schematic of a graphene field effect transistor (GFET) on an undoped semiconductor substrate. In this work, an undoped silicon carbide (SiC) is used as the substrate. A back-gate voltage is applied at the back of SiC substrate to produce an electric field acting on the graphene and modulating graphene conductivity via field effect. b. Source-drain current (I ds ) as a function of back-gate voltage (V g ) of a GFET on SiC substrate for a fixed V ds = -0.1 V, without and with illumination of a laser (wavelength λ = 400 nm, laser power incident on device P in = 86 µW). Inset: Optical microscope image of a representative GFET device (top view). c. The dependence of the photocurrent (I photo ) of the GFET on gate voltage. The photocurrent is extracted by subtracting the dark current (I dark ) from the light current (I light ), both shown inFig. 1b. Inset: Raman spectrum of exfoliated graphene on a SiC substrate, indicating a single layer of graphene. d. A plot of electric field (E) (simulated with COMSOL Multiphysics) under the graphene as a function of P in , showing an increase in E with increasing incident laser power P in . Inset: A representative simulation of the electric potential (color scale) and electric field lines in a GFET with a back gate voltage (V g ) of 20 V and P in = 100 µW. The SiC thickness (416 µm) in the modeled device is the same as the thickness of SiC in our experimental devices. The scale bar for the SiC is 100 µm. The graphene and back-gate electrode are not drawn to the scale. To qualitatively account for the effects of native oxide and spatially non-uniform generation of photo carriers in the substrate, we have assumed the conductivity of the top 10 nm of the SiC to be not affected by illumination. The stream lines represent the electric field lines, which direct the photogenerated carriers toward the location directly under the graphene. The electric field shown in the main panel (d) is calculated at the location under the graphene. The strength of the electric field under graphene increases with increasing P in . The change in the electric field is detected by the change of conductivity of graphene, allowing us to detect the light incident on the GFET. Figure 2 \| 2Dependence of current (I ds ) on source-drain bias voltage (V ds ) of a graphene phototransistor with increasing light power. (a) I ds -V ds characteristics of a typical GFET at V g = -20 V without and with illumination for a series of incident laser power P in (varying from 1 to 184 µW). All I ds -V ds curves pass through the origin, while the slope of I ds -V ds curves increase with increasing laser power. (b) Enlarged view of the circled region shown in (a), showing the increase of current (I lihgt ) under laser illumination with increasing laser power. Figure 3 \| 3Dependence of photocurrent and photoresponsivity on source-drain bias voltage and light power. (a) Photocurrent (I photo ) at various source-drain bias voltages V ds (from 0 to -0.5 V) for a series of incident laser powers P in (from 1 to 184 µW) and V g = -20 V. (b) Photoresponsivity (R) as a function of V ds for various P in as shown in (a). Both the photocurrent and photoresponsivity increase with increasing V ds. A photoresponsivity of 7.4 A/W is achieved for V ds = -0.5 V at P in = 1 µW. (c) Photocurrent as a function P in is shown for different V ds (from 0 to -0.5 V), indicating that photocurrents saturate for higher laser powers for all V ds . (d) The dependence of photoresponsivity on P in for the same V ds as shown in (c).Inset: A log-log plot of R versus P in for V ds = -0.5 V. The dashed line is a power law fit (R ∞  in P ) to the experimental data (filled circles) with a power  ~ -0.8. Figure 4 \| 4Dynamical photoresponse of a graphene phototransistor. (a) Time-dependent photocurrent of the GFET for V g of -20 V, 0 and 20 V, as the laser is turned on and off. A positive photocurrent is observed for V g = -20 V, whereas a negative photocurrent is observed for V g = 20 V. Photocurrent is nearly zero for V g = 0 V. The sign of the photocurrent is consistent with the field effect measurement in Fig. 1b & c. (b) Time-dependent photocurrent as the laser (P in = 184 µW) is repeatedly turned on and off at V ds = -0.5 V and V g = -20 V. (c) Photocurrent as a function of time at V ds = -0.5 V, V g = -20 V and various incident laser powers P in (from 1 to 184 µW). Shaded regions in (a-c) mark time intervals during which the laser is on. (d) The response time (τ) of the rise and fall of photocurrent dynamics is shown in (c) as a function of P in . The shortest response time of our device is ~ 1 s. Solid straight lines represent power law fits (τ ∞  in P ). Inset: The ratio of photocurrent to dark-current I photo /I dark (in %) as a function of P in . The maximum I photo /I dark of our GFET is ~ 10.5 %, measured for laser power of 184 μW. Figure S3 : S3(a) Gate leakage current (I g ) vs. gate voltage (V g ), (b) I g vs. source-drain bias voltage (V ds ), (c) I g vs. time, measured in the dark and under illumination with various incident laser powers (P in ) ranging from 1 to 184 µW. The shaded region in c labels the time interval when the laser is turned on. Figure S5 . S5(a) I ds -V g characteristics of a dummy device without and with laser illumination (λ = 400 nm). Inset: Optical image of a fabricated dummy device (gold in the channel instead of graphene) (b) I ds -t characteristics for different the gate voltages when the laser switches on and off. Shaded area indicates the time intervals during which the laser is on. Figure S6 : S6(a-d) Photocurrent (I photo ) vs. time for V g = -20 V and V ds = -0.5 V (a, b) for incident laser powers (P in ) of 184 µW and (c, d) P in = 25 µW. Red circles are experimental data; solid lines represent exponential fits to extract the time constants. Shaded regions in a-d label time intervals during which the laser is on. AcknowledgementsThe authors acknowledge partial support of this work from DHS (grant 2009-DN-077-ARI036), and DTRA (grant HDTRA1-09-1-0047). 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[ "OUTFLOW POSITIVITY LIMITING FOR HYPERBOLIC CONSERVATION LAWS. PART I: FRAMEWORK AND RECIPE", "OUTFLOW POSITIVITY LIMITING FOR HYPERBOLIC CONSERVATION LAWS. PART I: FRAMEWORK AND RECIPE" ]	[ "Evan Alexander Johnson ", "James A Rossmanith " ]	[]	[]	To support physically faithful simulation, numerical methods for hyperbolic conservation laws are needed that efficiently mimic the constraints satisfied by exact solutions, including material conservation and positivity, while also maintaining high-order accuracy and numerical stability. Finite volume methods such as discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) schemes allow efficient high-order accuracy while maintaining conservation. Positivity limiters developed byZhang andsummarized in [Proc. R. Soc. A 467, 2752 (2011)] ensure a minimum time step for which positivity of cell average quantities is maintained without sacrificing conservation or formal accuracy; this is achieved by linearly damping the deviation from the cell average just enough to enforce a cell positivity condition that requires positivity at boundary nodes and strategically chosen interior points.We assume that the set of positive states is convex; it follows that positivity is equivalent to scalar positivity of a collection of affine functionals. Based on this observation, we generalize the technique of Zhang and Shu to a framework that we call outflow positivity limiting: First, enforce positivity at boundary nodes. If wave speed desingularization is needed, cap wave speeds at physically justified maxima by using remapped states to calculate fluxes. Second, apply linear damping again to cap the boundary average of all positivity functionals at the maximum possible (relative to the cell average) for a scalar-valued representation positive in each mesh cell. This be done by enforcing positivity of the retentional, an affine combination of the cell average and the boundary average, in the same way that Zhang and Shu would enforce positivity at a single point (and with similar computational expense). Third, limit the time step so that cell outflow is less than the initial cell content.We show that enforcing positivity at the interior points in Zhang and Shu's method is actually a means of capping boundary averages at the maximum possible for a positive solution. Capping boundary averages allows computational interior points to be chosen without sacrificing the guaranteed positivity-preserving time step so as to optimize stabilization benefits relative to computational expense, e.g. by choosing points that coincide with nodal points of a DG scheme.	null	[ "https://arxiv.org/pdf/1212.4695v1.pdf" ]	118,999,401	1212.4695	1cc2159bcfd1f6e7074c3abd767768ed5115a872	OUTFLOW POSITIVITY LIMITING FOR HYPERBOLIC CONSERVATION LAWS. PART I: FRAMEWORK AND RECIPE Evan Alexander Johnson James A Rossmanith OUTFLOW POSITIVITY LIMITING FOR HYPERBOLIC CONSERVATION LAWS. PART I: FRAMEWORK AND RECIPE Nonlinear Hyperbolic Conservation LawsFinite Volume MethodsDiscontinuous Galerkin Finite Element MethodPositivity LimitersExplicit Time-Stepping AMS subject classifications 35L0265M6065M20 To support physically faithful simulation, numerical methods for hyperbolic conservation laws are needed that efficiently mimic the constraints satisfied by exact solutions, including material conservation and positivity, while also maintaining high-order accuracy and numerical stability. Finite volume methods such as discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) schemes allow efficient high-order accuracy while maintaining conservation. Positivity limiters developed byZhang andsummarized in [Proc. R. Soc. A 467, 2752 (2011)] ensure a minimum time step for which positivity of cell average quantities is maintained without sacrificing conservation or formal accuracy; this is achieved by linearly damping the deviation from the cell average just enough to enforce a cell positivity condition that requires positivity at boundary nodes and strategically chosen interior points.We assume that the set of positive states is convex; it follows that positivity is equivalent to scalar positivity of a collection of affine functionals. Based on this observation, we generalize the technique of Zhang and Shu to a framework that we call outflow positivity limiting: First, enforce positivity at boundary nodes. If wave speed desingularization is needed, cap wave speeds at physically justified maxima by using remapped states to calculate fluxes. Second, apply linear damping again to cap the boundary average of all positivity functionals at the maximum possible (relative to the cell average) for a scalar-valued representation positive in each mesh cell. This be done by enforcing positivity of the retentional, an affine combination of the cell average and the boundary average, in the same way that Zhang and Shu would enforce positivity at a single point (and with similar computational expense). Third, limit the time step so that cell outflow is less than the initial cell content.We show that enforcing positivity at the interior points in Zhang and Shu's method is actually a means of capping boundary averages at the maximum possible for a positive solution. Capping boundary averages allows computational interior points to be chosen without sacrificing the guaranteed positivity-preserving time step so as to optimize stabilization benefits relative to computational expense, e.g. by choosing points that coincide with nodal points of a DG scheme. 1. Introduction. Consider a variable-coefficient hyperbolic scalar partial differential equation (PDE) of the form ∂ t u(t, x) + ∇ ⋅ f (t, x, u) = 0 for x ∈ Ω, t ∈ R ≥0 , (1.1) where Ω ⊂ R D is the physical domain, t is the time coordinate, ∇ ⋅ represents the divergence with respect to x, u ∶ R ≥0 × R D → R is a conserved scalar quantity, and the flux function f ∶ R ≥0 × R D × R → R D is assumed differentiable. Assume that solutions of (1.1) are positivity-preserving: that is, if u(0, x) ≥ 0, then u(t, x) ≥ 0 for all t > 0. The context of this work will mostly be restricted to a single mesh cell in the interior of the domain, so we do not concern ourselves with boundary conditions for the PDE. A numerical method for equation (1.1) evolves a numerical solution U that approximates u and is represented by a finite number of degrees of freedom; we say that the method is accurate if the error U − u ∞ vanishes as the degrees of freedom are increased in an appropriate way. Assumed properties and requirements We assume a hyperbolic conservation law of the form ∂ t u(t, x) + ∇ ⋅ f (t, x, u) = 0, (1.2) where hyperbolicity means that the flux Jacobian matrix (i, j) ↦n ⋅ ∂fi ∂uj is diagonalizable with real eigenvalues and a full set of eigenvectors for any unit direction vector n ∈ R D , where n = 1. Each eigenvalue is a wave speed. Valid physical solutions violate the differential form (1.2) at shock discontinuities but still satisfy the integral form of conservation law (1.2), d t K u + ∮ ∂Kn ⋅ f = 0, (1.3) for arbitrary K. We assume that there is a convex set P (satisfying sP + (1 − s)P ⊂ P ∀s ∈ [0, 1]) of positive states such that P is an invariant domain: if the initial data u(0, x) is in P then the physical solution remains in P for all time: u(t, x) ∈ P ∀t ≥ 0. Furthermore, we assume that f → 0 on the boundary of P. Positivity limiters yield a numerical method that is high-order accurate for smooth solutions and satisfies a discrete local conservation law like (1.3) and a discrete local positivity condition ∫ K u ∈ P, where K is restricted to a union of mesh cells. Accuracy is not the only goal of a numerical method, however. One also seeks physicality of numerical solutions. If a physical condition is maintained by the solution, then it is desirable for a numerical method to mimic this by maintaining a discrete version of this condition. Such methods are referred to as mimetic or conforming. For example, in a problem where material is conserved, the solution u is restricted for all time to a manifold of solutions that have the same total amount of material. A numerical method that fails to satisfy a discrete material conservation law can over time drift from this manifold, resulting in unphysical behavior. Although one can correct for this by a global adjustment to the solution, such an approach still can yield locally incorrect physics. Specifically, solutions that fail to satisfy a discrete local conservation law result in simulated shocks that travel at incorrect speeds (see e.g. pages 237-239 of LeVeque [8]). Similarly, in a problem where the solution should remain positive, a numerical solution that fails to maintain a discrete positivity condition could drift from the set of positive solutions, resulting in an unphysical or even unstable solution. Again, one could globally damp the deviation from the global average. But to ensure local physicality and stability, positivity preservation should be enforced in a local manner. Finite volume methods are called conservative because they are designed to mimic the conservation property of conservation laws. Positivity-preserving methods are designed to mimic the positivity-preserving property. The challenge of positivity limiting is to design finite volume methods that are conservative, positivity-preserving, high-order accurate, and numerically stable. Generalizing to hyperbolic systems, the fundamental assumptions and require-ments for positivity limiting are summarized in Figure 1.1. Historical overview. Consider a 1D conservation law ∂ t u(t, x) + ∂ x f (u) = 0 which maintains the condition u ≥ 0. A finite volume method partitions the domain into intervals called mesh cells. On a mesh cell of width ∆x centered at x = x i we denote the numerical cell average at time t n by U n i : U n i ∶= 1 ∆x xi+ ∆x 2 xi− ∆x 2 U (t n , x) dx. (1.4) An Euler step for a method-of-lines finite volume method (e.g., DG or WENO) updates the cell average as follows: U n+1 i = U n i − ∆t ∆x h(U − i+1 2 , U + i+1 2 ) − h(U − i−1 2 , U + i−1 2 ) (1.5) where U − i−1 2 and U + i−1 2 are approximate solution values on the left and right of cell interface x i−1 2 , respectively, and the numerical flux h(U − , U + ) is consistent with physical flux: h(u, u) = f (u). In our abstract setting, what we know about f is that it is defined so that physical solutions maintain positivity. Therefore, to maintain positivity in the numerical scheme, the numerical flux is defined in terms of the solution to a physical problem. This insight lead to the first general positivity-preserving scheme: the Godunov scheme [5], which iterates the following: 1. In each mesh cell replace the solution with its average value. 2. Physically evolve the solution for a time step ∆t. If ∆t is sufficiently short, then the flux at each interface is given by the solution to a Riemann problem; this is guaranteed if ∆tλ < ∆x, where λ is the sum of left-going and right-going signal speeds propagating from each interface in the physical solution; see Figure 2.1. In this case, the Godunov scheme can be implemented by equation (1.5) if the numerical flux h(U − , U + ) is chosen to be the interface flux of the Riemann problem with initial states (U − , U + ) and the solution is assumed to be constant in each cell. A simpler choice of h which maintains positivity in update (1.5) is the HLL numerical flux function [6]. The HLL flux is defined to account for the transfer of material implied by the solution to the Riemann problem modified by averaging in an interval that contains the signals emanating from the interface. To show that HLL preserves positivity, Perthame and Shu [11] used a modified Godunov scheme: before averaging in each mesh cell, the physically evolved solution is first averaged in a set of nonoverlapping intervals each of which contains the signals emanating from one of the interfaces. The Godunov scheme is unfortunately only first-order accurate in space. To achieve high-order accuracy in space, the numerical flux function h used in update (1.5) needs to accurately approximate the physical flux f . This will be the case if h is consistent with the physical flux and if U − and U + are both high-order accurate approximations to the exact solution at the interface. Therefore, we assume in each cell a representation of the solution of the form U n i (x) ∶= U n i + dU n i ,(1.6) where U n i is the cell average and dU n i is a high-order correction that is a polynomial of degree at most k. The DG method evolves such a representation, and WENO Conservative local accuracy-preserving positivity limiting by linear damping of the deviation from the cell average x U U U The positivity limiting framework introduced by Liu and Osher [10] achieves highorder accuracy while maintaining positivity of the cell average U in each mesh cell. Prior to each time step the deviation from the cell average is linearly damped until a cell positivity condition is satisfied; a time step is then calculated that preserves positivity of the cell average. Zhang and Shu [16] reduce the cell positivity condition from positivity at every point in the mesh cell (i.e. for an infinite collection of point evaluation functionals) to positivity at a finite collection of positivity points. The limited solution isŨ ∶= U + θ(U − U ), where θ ∈ [0, 1] is just small enough so thatŨ is positive at each (black) interior point and at each (gray) boundary node. reconstructs such a representation with each time step. The flux function h remains unchanged (i.e. is defined using the Riemann problem with states U − and U + .) High-order accurate positivity limiters were developed in the seminal works of Zhang [15] and Zhang and Shu [16]. An in-depth review of the history of this method can be found in the excellent review article [18]. As depicted in Figure 1.2, in each mesh cell they linearly damp the high-order corrections just enough to enforce positivity at a set of strategically chosen positivity points. For concreteness, we describe their method in detail for the 1D scalar case. Let K = [x i − ∆x 2, x i + ∆x 2] denote a mesh cell, and let {s j }ñ j=0 denote the points of the Gauss-Lobatto quadrature rule that has just enough points to exactly integrate polynomials of degree at most k. As depicted in figure 1.2, to limit U n i , linearly damp dU n i just enough to ensure that the solution is positive at each quadrature point. In formulas: U min ∶= min j U n i (s j ), θ ∶= min 1, U n i U n i − U min ,Ũ n i (x) ∶= U n i + θ dU n i . (1.7) We note that θ ∈ [0, 1], and in particular, if U min ≥ 0, then θ = 1. Finally, the cell averages are updated by using the damped solution in (1.5): U n+1 i = U n i − ∆t ∆x h Ũ − i+1 2 ,Ũ + i+1 2 − h Ũ − i−1 2 ,Ũ + i−1 2 ,(1.8) whereŨ ± i−1 2 are the edge values computed from the limited solution (1.7). We will see that this algorithm maintains accuracy and guarantees the same positivity-preserving time step that is guaranteed if positivity is enforced everywhere in the mesh cell. Zhang and Shu have extended their positivity limiters to shallow water and gas dynamics [17], as well as to higher-dimensional problems on both Cartesian and unstructured grids [17,20]. As leveraged in [16], a framework for positivity limiting can also be used to implement a maximum-principle-satisfying scheme. Suppose that the exact (entropy) Outflow capping CU n+1 ∆t ∆t safe ∆t zero ∆t stable CU 30%CU − Bh For an Euler step, the updated cell content CU n+1 depends linearly on the time step ∆t. A stable time step that caps outflow at e.g. α = 70% can be directly calculated, and in the scalar case is ∆t safe = min αCU Bh , ∆t stable . solution u(t, x) remains bounded in the interval [U min , U max ]. By negation and translation of state space, one can use scalar positivity limiting to enforce that the affine functionals U n i − U min and U max − U n i are both positive. 1.2. Outflow positivity limiting framework. The main contribution of this work is an analysis and generalization of Zhang and Shu's positivity limiter based on limiting the potential and actual outflow from each cell. This yields a multi-stage framework we call outflow positivity limiting. 1. The first stage is point-wise positivity limiting, which consists of enforcing positivity at a set of positivity points consisting of boundary nodes and interior points. Enforcing positivity at boundary nodes ensures that numerical fluxes are defined with positive values. For shallow water and gas dynamics, when positivity of the depth or density is enforced at a boundary node, it is important to calculate fluxes with remapped states in order to desingularize wave speeds. Enforcing positivity at interior points may improve stability. 2. The second stage is boundary average limiting. Like Zhang and Shu, we maintain positivity of the cell average by after each time step linearly damping the deviation from the cell average just enough so that a cell positivity condition is satisfied. The essential difference is that our cell positivity condition directly caps the boundary average rather than requiring positivity at points in the cell interior. Specifically, the optimal version of our cell positivity condition requires the boundary average to be no greater than the maximum boundary average possible if the solution were everywhere positive and had the same cell average. Enforcing positivity at the optimal interior points of Zhang and Shu is a means of boundary average limiting. 3. The third stage is outflow capping. Rather than restricting the time step to the guaranteed positivity-preserving time step (and assuming that the estimated upper bound on wave speeds is legitimate), we directly calculate a maximal stable time step that caps cell outflow at e.g. 70%. The updated cell average varies linearly with the length of an Euler time step, and therefore outflow capping can be performed with the same limiting procedure that Zhang and Shu use to enforce positivity at a positivity point. See Figure 1.3. 1.3. Benefits of the outflow positivity limiting framework. This framework offers simplicity, insight, flexibility, and computational efficiency. Our framework is implied by a set of natural requirements, e.g. to maintain positivity of the cell average and to guarantee a stable, positivity-preserving time step. We thereby decouple requirements and arrive at a framework that is intrinsic to the requirements and ar-guably optimal. Outflow capping ensures that knowledge of the fastest wave speed is not needed to compute the positivity-preserving time step. A minimum positivitypreserving time step is guaranteed by positivity of the retentional, which is defined in terms of the data that actually determines the evolution of the cell average (i.e. boundary node data and the initial cell average), and has a physical meaning (cell content minus maximum possible loss). The proof that positivity is maintained is thus intuitively obvious and works for spatially varying flux functions and for arbitrary cell geometry and representation space. Boundary average limiting can be used to guarantee a positivity-preserving time step even if the data used at the boundary nodes is not interpolated by a polynomial. This is relevant in the gas-dynamics case, where it becomes necessary to modify states at boundary nodes before evaluating fluxes, e.g. to desingularize fluid velocity when enforcing positivity of the density or to enforce realizability or hyperbolicity in highermoment gas-dynamic models. Capping the boundary average is enforced via positivity of a single linear functional, the retentional, and is therefore no more expensive than enforcing positivity at a single point. For stabilization purposes, one may additionally enforce positivity at interior points, but the choice of these points can be optimized for requirements of efficiency, simplicity, or stability; if positivity points are instead optimized for maximizing the guaranteed positivity-preserving time step, then enforcing positivity at these points enforces the optimal cap on the boundary average (as illustrated in Figure 1.4.) While we show that such optimal positivity points always exist (see Figure 5.1), precisely isolating them can be non-trivial (in general requiring the solution of linear programming problems on a convex domain), and in the case of nodal DG it is inefficient to check positivity at these non-nodal points if they are relatively numerous. In contrast, outflow positivity limiting merely requires an estimate of the maximum boundary crowding M ⋆ realizable by a positive solution. Note that precisely isolating M ⋆ also requires solving linear programming problems on a convex domain. Parts. This work is conceived of as the first in a three-part series. This first part is analytical rather than computational and avoids making claims dependent on computational experiment; rather, we lay out a general framework that incorporates previous work and that will guide subsesquent computational investigation. The second part will give a detailed justification of the analytical results. The third part is planned to consider issues that require computational simulation. In particular, the stability consequences of enforcing positivity at Zhang and Shu's points, at all nodal points, or only at boundary nodes, will be a natural follow-up study, and results will likely be dependent on the details of the choice of oscillation-suppressing limiters. Since our framework generalizes that of Zhang and Shu, it has in a sense already been computationally tested by their work [14,15,16,17,18,19,20]. 1.5. Section summary. The outflow positivity limiting framework is developed in subsequent sections of this paper. In Section 2 we describe how DG (or WENO) updates the cell average. In Section 3 we show in the context of scalar conservation laws that enforcing positivity of the retentional guarantees that the cell average remains positive for a time step inversely proportional to the boundary crowding cap M chosen in the definition of the retentional. In Section 4 we show that the same statement holds for hyperbolic systems of conservation laws. In Section 5 we develop a theory of admissible (accuracy-preserving) and optimal cell positivity conditions that shows that, as long as the boundary crowding cap is admissible (i.e. no less than an optimal M ⋆ ), then enforcing positivity of the retentional by linearly damping the Positivity limiting caps the boundary average. deviation from the cell average preserves the order of accuracy and can be accomplished by enforcing positivity at strategically chosen interior points. In Section 6 we apply this theory of cell positivity conditions to compute optimal or near-optimal boundary crowding caps and optimal interior points for typical cell geometries and polynomial orders. In Section 7 we discuss practical implementation issues regarding efficiency, robustness, and generalization, with specific reference to shallow water and gas dynamics. These issues include efficient and correct positivity checks, wave speed desingularization, stability benefits of interior positivity points, stable optimal time stepping, multistage and local time stepping, and application to isoparametric mesh cells. In Section 8 we summarize the key results of this work. Boxed figures present a largely self-contained summary of the work. U X U R U −1 1 1/5 M U U BU BU X BU R −1 1 1/5 − 1/5 U U M U U BU Retentional 2. High-order discontinuous Galerkin schemes. For simplicity, we discuss positivity-preserving limiters with reference to the discontinuous Galerkin (DG) method. This does not entail loss of generality, since a WENO finite volume method can be thought of as a DG method that reconstructs the high-order component of the representation prior to each time step. Standard DG schemes discretize the weak form of hyperbolic conservation law (1.1): d dt K u ϕ = K f ⋅ ∇ϕ − ∮ ∂Kn ⋅ f ϕ, (2.1) where K is an open subset of R D , ∂K is the boundary of K,n is an outward pointing unit vector that at every point on ∂K is perpendicular to the boundary ∂K, and The Riemann problem with states (U − , U + ) and flux function f is defined to be the problem (PDE) ∂ t u(t, x) + ∂ x f (u) = 0, u(0, x) = U − if x < 0, U + if x > 0. Riemann problems (and thus their solutions) are by requirement self-similar, i.e. invariant under dilation of space-time: u(at, ax) = u(t, x) for any a > 0 and t ≥ 0. So we can write u(t, x) =∶û(x t) for t > 0. In hyperbolic problems information propagates at finite speed. Define the signals s − (t) ≤ 0 ≤ s + (t) emanating from the interface x = 0 so that σ ∶= [s − , s + ] is the smallest interval such that u(t, ⋅) agrees with u(0, ⋅) outside σ. By self-similarity, the signal speedsṡ ± are independent of t and define the right-going signal speedṡ + (U − , U + ) and left-going signal speeḋ s − (U − , U + ). The interface flux f 0 is defined to be the flux at x = 0 needed to account for the amount of material that u accumulates on each side of the interface; it is independent of t > 0 and equals f (û(0)) unlessũ happens to be discontinuous at 0; then subtracting ∫ 0 x − (PDE) from ∫ x + 0 (PDE), applying the fundamental theorem of calculus, and solving reveals that 2f 0 = f (U − ) + f (U + ) + d t ∫ x + 0 u(t, ⋅) − ∫ 0 x − u(t, ⋅) if x − < ts − and x + > ts + .ϕ ∈ C 1 0 is a test function. Assume that Ω is the union of a finite set Ω h of non-overlapping mesh cells. We assume that the solution, when restricted to any mesh cell K ∈ Ω h , belongs to a polynomial space V = V(K) and is otherwise unconstrained (e.g. by the requirement of continuity across mesh cell boundaries). Often V is P k D , the set of polynomials in D variables of degree at most k. More generally, we define deg(V) ≥ 0 to be the largest k such that V contains P k D . The discontinuous Galerkin method approximates the exact solution to (2.1) with a piecewise polynomial function: U K = N (k) i=0 U i ϕ i ∈ V,(2.2) where, on each cell K, {ϕ j } N (k) j=0 is a basis for V(K) and {ϕ j } N (k) j=0 is its co-basis. The basis and co-basis are mutually orthonormal in the inner product 1 K ∫ K ϕ i ϕ j = δ i j , where K is the measure of the mesh cell K. In nodal DG, the co-basis consists of a set of point-evaluation functionals that evaluate the solution at the nodes, and the solution is thus represented by nodal values. In modal DG, the basis is often chosen to be a sequence of orthonormal polynomials of increasing degree. As a consequence, the basis and co-basis are identical and the coefficients of higher-order basis functions decay: specifically, for a smooth function u, the L 2 -projection coefficients ∫ K u ϕ j decay like O(∆x k ′ +1 ), where ∆x is the mesh cell diameter and k ′ is largest such that P k ′ D ⊂ span ϕ i ∶ i < j . The DG method solves a discretization of equation (2.1) in each mesh cell K: d dt K U ϕ j = Q K f (t, x, U ) ⋅ ∇ϕ j − ∮ Q ∂K h(U − , U + ,n) ϕ j ,(2. 3) Framework of a method-of-lines finite volume Euler step Integrating the hyperbolic differential equation ∂ t u(t, x) + ∇ ⋅ f (t, x, u) = 0 over a mesh cell K gives the integral form d dt K u + ∮ ∂Kn ⋅ f = 0. Making the replacements u → U ∈ V, ∮ ∂K → B, andn ⋅ f → h, where V is a finite- dimensional representation space, h is a numerical flux, and B is a numerical boundary quadrature, gives the ordinary differential equation of a method-of-lines finite volume approximation d t CU = −Bh, where C ∶= ∫ K . An Euler step for this ODE is CU n+1 = CU − ∆tBh. Typically V when restricted to a mesh cell is a polynomial representation space containing all polynomials of degree at most k, ∮ Q ∂K h = ∑ x∈Q ω x h(x) is a numerical quadrature rule with points Q ⊂ ∂K and weights ω x > 0, and h(x) is a numerical interface flux for the Riemann problem with frozen flux function f ∶= u ↦n ⋅ f (t, x, u) and states (U − , U + ) (see Figure 2.1); here U − (x) is the value of U at x approached from within K and U + (x) is the value of U at x approached from outside K. DG and WENO are two ways of updating the deviation of U from the cell average. wheren is the outward pointing unit normal to ∂K and where h(U − , U + ,n) denotes a numerical flux function where U − represents the state at the boundary node approached from inside the cell and U + represents the state at the same boundary node approached from outside the cell. See Figure 2.2. In equation (2.3) we have replaced exact integration by quadrature rules; in particular, ∫ Q K denotes a quadrature rule exactly equal to ∫ K for polynomials of degree at most 2k − 1 and ∮ Q ∂K denotes a quadrature rule exactly equal to ∮ ∂K for polynomials of degree at most 2k. If K is a polytope then the boundary quadrature rule can be written as the sum over all faces of a Gaussian quadrature rule on each face. An important property of the DG scheme is that it conserves the total amount of U from time step to time step; indeed, taking ϕ = 1 ∈ V, equation (2.3) becomes the conservation law d dt K U = − ∮ Q ∂K h. (2.4) The entire context of this work is concerned with maintaining positivity in a single arbitrary mesh cell. We therefore adopt a simplified notation that omits explicit reference to the mesh cell K. We define the cell content to be the cell integral C ∶= U ↦ ∫ K U or its scale-invariant version, the cell average C ∶= U ↦ − ∫ K U . We define the boundary sum to be the boundary integral quadrature B ∶= U ↦ ∮ Q ∂K U − or its scale-invariant version, the boundary average quadrature B ∶= U ↦ − ∫ Q ∂K U − . We use V ∶= K to denote the "volume" of the mesh cell and A ∶= ∂K to denote the "area" of its boundary. With these conventions, the ODE (2.4) is expressed as d t CU = −Bh, i.e., d t CU = −Bh, where h ∶= A V h is a scale-invariant version of the numerical flux, and an explicit Euler step is expressed as CU n+1 = CU − ∆tBh, i.e., (2.5) CU n+1 = CU − ∆tBh. (2.6) This method-of-lines finite volume framework is summarized in Figure 2.2. 3. Framework for outflow positivity limiting. In this work positive means nonnegative unless qualified with the adjective strictly. A positive combination means a linear combination with positive coefficients, and a positive functional is defined to be a functional that is positive on positive functions. A simple algorithm that maintains positivity of the cell average is to repeat the following sequence: 1. Assume that the cell average is positive. 2. If necessary, linearly damp (rescale) the deviation from the cell average just enough so that a cell positivity condition is satisfied. (See Figure 1.2.) 3. Execute an Euler step for a stable time step that is just short enough so as to guarantee that positivity of the cell average is maintained (given that the cell positivity condition is satisfied). This algorithm was first introduced by Liu and Osher [10] and further developed by Zhang and Shu [16]. Assuming that numerical fluxes are calculated by evaluating the flux at the boundary node states (rather than at remapped states), the cell positivity condition should at least require the solution to be positive at the boundary nodes so that the Riemann problem used to define the numerical flux function is well-defined. To ensure that positivity limiting does not compromise order of accuracy, it will be enough to require that the cell positivity condition be satisfied if the initial data is already everywhere positive (as we will verify in Theorem 5.2). Given the constraints of this framework, we ask: What cell positivity condition and time step will guarantee that − ∫ K U n+1 ≥ 0? If we are not allowed to modify a positive solution, then the maximum time step for which we can hope to guarantee positivity is the maximum such time step if the initial data is positive. We therefore ask: What is the maximum time step for which it is guaranteed that − ∫ K U n+1 ≥ 0 if U ≥ 0 in K? We can assume that the outflow ∮ Q ∂K h is strictly positive, since otherwise the cell average will remain positive for any positive time step. In this case, setting CU n+1 ≥ 0 in equation (2.5) or (2.6) gives ∆t −1 ≥ ∆t −1 zero ∶= Bh CU = Bh CU (3.1) That is, the time ∆t zero until strict positivity of the cell average is violated is the ratio of the total integral of U in the cell to the net rate of flux out of the boundary. Clearly, to guarantee positivity we need a constraint on the flux out of the boundary. A constraint that extends naturally to systems is to specify a cap on wave speeds. Abstractly we impose that h ≤ λU − , (3.2) where we call λ the speed cap; equivalently, h ≤ λU − , where λ ∶= A V λ is a scaleinvariant version of the speed cap. This condition clearly holds for a scalar problem with a convex flux function (e.g. Burgers' equation) if λ is the wave speed, and in Theorem 3.7 we show that for general systems λ can be defined as a cap on the sum of incoming and outgoing signal speeds at each boundary node. Remark 3.1 (omitting bars). The form of the scale-invariant equations is identical to the form of the non-invariant equations, so we can choose whether to omit or retain bars when discussing general properties. 3.1. Boundary crowding caps and the retentional. Equations (3.1) and (3.2) imply that positivity is maintained if ∆t −1 ≥ λBU = λBU, (3.3) where we callBU ∶= BU CU or its scale-invariant versionBU ∶= BU CU the boundary crowding. Therefore, we can guarantee a minimum positivity-preserving time step by enforcing bounds on λ andBU . To maintain order of accuracy, these bounds need to be physically justified. Enforcing a cap on λ is briefly considered in Section 7.2. The focus of this paper is on enforcing a justified cap onBU . Given that M ≥BU , equation (3.3) says that positivity is maintained if ∆t −1 ≥ ∆t −1 pos ∶= λM . (3.4) To determine a justified M , we use that physical solutions satisfy positivity: Definition 3.2 (M ⋆ ). We define the positive solutions to be the set V + K ∶= {U ∈ V ∶ U ≥ 0 in K and CU > 0} of representations positive and somewhere nonzero in the mesh cell, and we define the optimal boundary crowding cap (or optimal interior weight) M ⋆ to be the maximum boundary crowding over all positive solutions: M ⋆ K (V) ∶= sup U ∈V + KB (U ). (3.5) We can enforce that M ≥BU in exactly the same way that positivity is enforced at positivity points if we define this condition in terms of positivity of a linear functional: ). Throughout this paper, any symbol involving the letter "W" represents the reciprocal of the corresponding symbol obtained by replacing "W" with "M". Definition 3.5 (admissible weights). If M ≥ M ⋆ , then we say that M and W are admissible weights, because in this case positivity of U implies positivity of R M , which means that enforcing positivity of R M respects accuracy. Formally, we have the following two theorems, which comprise the essence of the paper. 3.2. First fundamental theorem of outflow rate limiting. The first essential theorem of outflow rate limiting is displayed in Figure 3.1. It asserts that at each boundary node the numerical outgoing flux rate h is bounded by the product of the interior value U − at the node and a speed cap λ; it assumes that boundary node states are positive and that h preserves positivity for a one-cell problem with states (0, U − , U + ) and cell width ∆x = λ for any ∆t ≤ 1. Remark 3.8. Multiplying by cell area over cell volume, h(U − , U + ) ≤ λU − . Remark 3.9. Examples of such a positivity-preserving flux function h are the numerical flux defined by the exact Riemann solver or the Harten-Lax-van Leer (HLL) [6] or local Lax-Friedrichs (LLF) [13] approximate Riemann solvers used with numerical signal speeds that are truly upper bounds for physical signal speeds. See Remark 3.10. If outflow capping is used, then it is not necessary to compute the speed cap λ; it is sufficient to know that such a finite cap exists. Remark 3.11. The proof assumes that Riemann problems are well-defined for vacuum states and involve finite wave speeds. Riemann problems involving vacuum states were considered for gas dynamics in [9]. For the proof it is enough that there exist a sequence of strictly positive states Z m approaching vacuum such that the lim sup of the signal speeds of the Riemann problems with states (Z m , U − ) is finite. In systems such as shallow water and gas dynamics for which any state can be connected to the vacuum state without shocks, one can choose states Z m so thatṡ + (Z m , U − ) = λ max (U − ) for all Z m ; that is,ṡ + (0, U − ) = λ max (U − ). 3.3. Second fundamental theorem of outflow rate limiting. The second essential theorem is displayed in Figure 3.2 and states that if the retentional R W = C − W B is positive then an Euler step maintains positivity of the cell average for any time step whose nondimensionalized version λ∆t is no greater than W , where λ is a uniform upper bound on the speed cap over all nodes. The first fundamental theorem can be invoked to satisfy the first hypothesis (h ≤ λU − ) of the second fundamental theorem if at each boundary node the numerical flux function is defined using a 1D Riemann problem with frozen flux (see Figure 2.2). The proof of Theorem 3.7 assumes that physical solutions to the two auxiliary Riemann problems are well-defined and maintain positivity. For simplicity, we can impose this assumption rather than assume that the full conservation law (1.2) of Figure 1.1 maintains positivity. But in fact, omitting careful justification, it is an implication of this work that, with appropriate regularity and convergence assumptions, we have: I: h(U − , U + ) ≤ λU − for a positivity-preserving numerical flux. x u(0, x) u(∆t, x) u HLL (∆t, x) (U − ) n+1 HLL 0 ∆x s + 0 s − 0 = S + S − s − s + U − U + Definition 3.6. Define the one-cell problem with states (Z, U − , U + ), cell width ∆x, and flux function f by ∂ t u + ∂ x f (u) = 0, u(0, x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Z if x < 0, U − if 0 < x < ∆x, U + if ∆x < x; (3.6) it is implied that Z, U − , U + ≥ 0. Define the speed cap to be the sum of the signal speeds entering the cell: To see how to define λ and h so that the Euler update (3.7) maintains positivity, for 0 ≤ t ≤ ∆t, let s − 0 (t) ≤ 0 ≤ s + 0 (t) be the signals emanating from x = 0 and let s − (t) ≤ ∆x ≤ s + (t) be the signals emanating from x = ∆x (see Figure 2.1). For the HLL flux, the numerical signal speedsṠ − and S + are required to satisfyṠ − ≤ṡ − ≤ 0 ≤ṡ + ≤Ṡ + . Assume that λ is an upper bound on the sum of the signal speedsṡ + 0 and −Ṡ − . Since ∆x = λ∆t, the signals s + 0 and S ± (t) ∶= ∆x + tṠ ± do not cross (i.e., λ(Z, U − , U + ) ∶= s + (Z, U − ) − s − (U − , U + ).λ ∶=ṡ + (Z, U − )−Ṡ − (U − , U + ), wherė S − (U − , U + ) <ṡ − (U − , U + ) ≤ 0. Let h 0 (A, B) be(U − ) n+1 = U − − ∆t ∆x [h(U − , U + ) − h 0 (Z, U − )] ≥ 0 (3.7) if λ∆t ≤ ∆x, i.e.,s + 0 (∆t) ≤ S − (∆t) ≤ s − (∆t) ). An exact Riemann solver uses the flux of the exact solution at the ∆x interface. The HLL solution u HLL (t, x) equals u(t, x) except in the interval S(t) ∶= [S − (t), S + (t)], where it equals the average value of u in S(t), U * (U − , U + ) ∶= f (U − )−f (U + )+Ṡ + U + −Ṡ − U − S + −Ṡ − . The flux needed at x = ∆x to account for the amount of material that u HLL accumulates on each side of the interface defines the can be approximated arbitrarily well by the solutionũ(t, x) to a multidimensional problem (1.1) with initial dataũ 0 (x) equal to B ifn ⋅ x > 0, else equal to A. In particular,û(ξ) = lim t↘0ũ (t, tξn). We omit a careful justification of this limit, and simply note that f evaluated atn ⋅ x approximatesn ⋅ f arbitrarily well for sufficiently small t and x; here we rely on the fact that f is differentiable (hence continuous) and thatũ by definition depends continuously on f whereverũ is well-defined. Sinceũ satisfies positivity, so doû and u. HLL numerical flux h(U − , U + ) ∶= 1 2 f (U + ) + f (U − ) + (Ṡ + +Ṡ − )U * −Ṡ + U + −Ṡ − U − ( That 2 implies 3 follows from the definition of HLL in terms of averagings of physical solutions (see Figure 3.1). That 3 implies 1 follows from the assumption that the positivity-preserving algorithm described in this work converges at least in the first-order case of a representation space that is constant in each cell. We remark that condition 3 gives a practical means of verifying that a physical system maintains positivity. U + i−1 2 +U − i+1 2 2 is the boundary average; we assume that the frozen flux function h i+1 2 used at any interface is positivity-preserving, in the sense that if it is used at all interfaces then it maintains positivity of cell averages for data constant in each cell if λ ∆t ∆x ≤ 1 (i.e., ∆t is short enough that signals from cell interfaces cannot cross in Godunov's method). To facilitate comparison with Theorem 2.1 of [17], we offer a direct proof. Proof. Substituting U n i = RU i + W BU i (a boundary-weighted quadrature rule), U n+1 i = U n i − ∆t ∆x h i+1 2 (U − i+1 2 , U + i+1 2 ) − h i−1 2 (U − i−1 2 , U − i+1 2 ) ≥ RU i + W 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ U − i+1 2 − 2∆t W ∆x h i+1 2 (U − i+1 2 , U + i+1 2 ) − h i+1 2 (Z + , U − i+1 2 ) +U + i−1 2 − 2∆t W ∆x h i−1 2 (U + i−1 2 , Z − ) − h i−1 2 (U − i−1 2 , U + i−1 2 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , which is a positive combination of positive quantities if the Euler steps in brackets satisfy 2λ∆t W ∆x ≤ 1; here we take Z + and Z − to be the vacuum state and we use that material cannot flow out of a vacuum: cell under a coordinate map. In the case of isoparametric mesh cells, the coordinate map is a diffeomorphism, and it is important to work in canonical coordinates. If the coordinate map is affine, however, then we can avoid making a distinction between physical and canonical coordinates if we adopt an affine-invariant formulation. h i+1 2 (Z + , U − i+1 2 ) ≤ 0 and h i−1 2 (U + i−1 2 , Z − ) ≥ In canonical coordinates, the mesh cell is almost always a cube or simplex and the representation space almost always consists of polynomials. This remains true for isoparametric mesh cells. With an affine-invariant formulation, our results are generally applicable, independent of whether the canonical simplex is defined to be regular or the corner of a box. In Section 6, we tabulate values or estimates of M ⋆ for regular canonical polytopes. Thus, when using our tabulated weights to apply the outflow positivity limiting framework, one should take wave speeds to be in terms of their values in the coordinates of a regular canonical mesh cell. A natural way to do this is to use affine invariants. An affine-invariant formulation of the Euler step (2.5) that agrees with the scaleinvariant formulation (2.6) in the case of a regular canonical mesh cell is CU n+1 = CU − ∆tB F h F , where B F is an affine-invariant version of the boundary average that computes the arithmetic average over all faces of the average on each face; here h F e ∶= If one is computing with a non-regular canonical polytopeK (e.g. the standard orthogonal simplex defined as the corner of a box) then an affine-invariant definition should be used in the definition of the retentional: R W (U ) ∶= CU − W B F U. (3.8) If outflow capping is used, then computing wave speeds is not necessary, and use of the boundary face average B F in the definition (3.8) of the retentional is the only modification needed to implement affine-invariant positivity-limiting. Systems versus scalar case. In the systems case of Figure 1.1, the set of positive states P is a convex set P ⊂ R N . A corollary of Theorem 3.4 in [12] is that any open or closed convex set is an intersection of half-planes. But any half-plane is the set on which some affine functional A is positive. Therefore, we assume that there exists a set P * of affine functionals such that a state u ∈ R N is positive if for all A ∈ P * A(u) ≥ 0. We call A a state positivity functional. Any such affine functional A decomposes uniquely as A =∶ s + Λ, where s = A(0) is a scalar shift and Λ ∶= A − s is its linear part. Applying A reduces the systems case to the scalar case: ∂ t A(u) + ∇ ⋅ (Λf ) = 0. After applying A to states and Λ to fluxes, all statements and reasoning of Theorems 3.7 and 3.12 remain valid. For example, the inequality h ≤ λU − becomes the inequality (Λh) ≤ λA(U − ). Let R ∶ V → R be a retentional. Applied component-wise, the retentional defines a state-valued linear mapR ∶ V → R N , which we identify with R. Applied point-wise, a state positivity functional A = s + Λ defines a mapÃ = s +Λ ∶ V N → V that we identify with A. Observe that RΛ = ΛR. It is desirable that the retentional commute with all state positivity functionals: RÃ = AR; then enforcing positivity ofR is the same as enforcing RÃ ≥ 0 for all A ∈ P * . This will hold if Rs = s, which holds if s is always 0 (that is, if P is a convex cone) or if the retentional is rescaled so that R1 = 1. Therefore, for any retentional R we define its unital retentional to bê R ∶= R R1 . ThenR1 = 1. To be concrete:R M = M C−B M −1 ; we remark thatR is an affine combination of state values and therefore is invariant under translation of state space. For typical systems such as shallow water and gas dynamics, the set of positive states is a convex cone (i.e. invariant under rescaling), so s = 0 and A = Λ and there is no need to rescale the retentional to its unital version. We remark that by adding the trivial equation ∂ t u extra = 0 to the system, where u extra is a scalar taken to be 1 for physical solutions, defining the set of positive states to be {(ru, r) ∶ u ∈ P, r ≥ 0}, and extending each positivity functional A = s+Λ to the linear functionalΛ ∶= (u, u extra ) ↦ su extra + Λu, the set of positive states can be assumed without loss of generality to be a convex cone. E v p = 0 v p (θ = 0) (θ = θ p ) (θ = 1) u u + θ p du u + du (θ = θ s ) The damping coefficient θ P is defined so that u p ∶= u + θ P du ∈ ∂P. There exists a state positivity functional Λ p which is zero at u p . Here we take Λ p = E vp , the energy in the reference frame of v p . The damping coefficient θ P can be calculated by finding a root of the quadratic polynomial ρp [17]. By concavity of the pressure, positivity is also assured by using the secant rule (i.e., by pretending that pressure is linear); see [14]. Note that, before enforcing p > 0, enforcing ρ > 0 may be needed [17]. While we can assume the scalar case without loss of generality when it comes to the positivity-preserving results of Section 3, the accuracy result of Theorem 5.2 does not go through e.g. to the gas-dynamics case (see Section 4.2). State positivity indicators. Outflow positivity requires the solution to the following problems: 1. To cap outflow, determine the largest value of θ ∈ [0, 1] for which u(θ) ∶= u + θdu satisfies positivity, where u = CU and du = ∆t max Bh. Definition 4.1. Let P be a closed convex set of states deemed positive, and let P * be a collection of affine functionals such that P = u ∈ R N ∶ (∀A ∈ P * ) Au ≥ 0 . Let A ∈ P * . Let u, u ∈ R N , where u ∈ P. Define du ∶= u − u. Define θ A (u, du) to be the largest θ ∈ [0, 1] such that A(θu + θu) ≥ 0, where we define θ ∶= 1 − θ. Define θ P (u, u) to be the largest θ ∈ [0, 1] such that θu + θu ∈ P. Observe that θ A (u, u) equals 1 if A(u) ≥ 0 and else equals Au Au−Au = s+Λu −Λdu . Observe that θ P (u, du) = min A∈P * θ A (u, du). If P * is a finite set, then it can be used directly to calculate θ P , as illustrated in Figure 1.3 for the case of scalar outflow capping. In the example of Euler gas dynamics, however, the state is positive if the energy in an arbitrary reference frame (a linear functional of conserved variables) is positive. But such energies comprise an infinite collection P * of linear functionals. Instead, one can use the density ρ (a linear functional), and pressure p (a concave nonlinear functional, see section 3.1 of [14]) or ρp (a quadratic functional of conserved variables, see [17]) as a finite set of positivity indicators. See In the case of shallow water, positive states are characterized by positivity of a single linear functional (the depth). Therefore, the proof of accuracy for the scalar case can be invoked to conclude that positivity limiting does not compromise accuracy of the water depth. Assuming that fluid velocities are bounded, this in turn implies that positivity limiting does not compromise accuracy of the solution as a whole. But in the case of gas dynamics, one must also enforce positivity of the pressure. There exist physical solutions with arbitrarily large and rapid variation in density for which the pressure remains arbitrarily close to zero. Therefore, if positivity is enforced by damping the deviation of the cell average by a scalar value of θ ∈ [0, 1], then a given amount of damping needed to enforce positivity of the pressure can entail damping of the density variation that is arbitrarily large in magnitude. On the other hand, if the pressure of the exact solution is strictly bounded away from zero, then positivity limiting respects accuracy for the simple reason that positivity limiters are not triggered if the mesh is sufficiently fine, if the exact solution is smooth, and if M > 1. In this case, positivity limiting is really about practical robustness. The study of accuracy and stability for gas-dynamic problems (or sequences of problems) for which the pressure is not bounded away from zero is delicate and is left to future work. Admissible weights and cell positivity functionals. In this section we show that positivity of the retentional R M (U ) can be enforced without loss of accuracy as long as the interior weight M is greater than or equal to an optimal value M ⋆ K and that this holds precisely when positivity of R M (U ) can be enforced by enforcing positivity at appropriately chosen interior points in the mesh cell K (see Figure 1.4). In Section 6 we use this theory (and interior points in particular) to compute (or isolate) the optimal value M ⋆ for important mesh cell geometries and representation spaces. As elsewhere, in this section the following context is assumed: Context 5.1. K is a compact mesh cell (in canonical coordinates) and V is a finite-dimensional polynomial space which contains the set P k D of all polynomials in D variables of degree at most k ≥ 0. We always assume that U ∈ V. Recall from equation The following results will be proved in detail in Part II [7]. We summarize these results in Figure 5.1 and illustrate them in Figure 1 Proof (sketch). Since physical solutions are positive, damping the variation from the cell average just enough to enforce positivity everywhere in the mesh cell preserves accuracy; a formal proof uses that the maximum n max ∶= dU ↦ max(dU (K)) and the magnitude of the minimum n min ∶= dU ↦ max(−dU (K)) are (equivalent) asymmetric norms on the finite-dimensional linear space of polynomials dU ∈ V whose cell average is zero (see [18]). If M is an admissible weight, then the retentional R M (U ) is positive if U is positive everywhere in the mesh cell, so enforcing positivity of R M (U ) also retains accuracy. ◻ Theorem 5.3 (Enforcing positivity of the points of a nodal scheme is sufficient to enforce a positive retentional for some M < ∞). Let X be a set of points capable of representing the solution and for which the cell average can be represented as a strictly positive combination of point values. Then M ⋆ X < ∞, where M ⋆ X ∶= supB(V + X ) andV + X ∶= {U ∈ V ∶ CU = 1 and (∀x ∈ X) U (x) ≥ 0}. Proof (sketch). If U is positive on X then CU > 0, soB(U ) is continuous on V + X . Using that n max and n min are equivalent asymmetric norms, one can show that V + K ∶= {U ∈ V + K ∶ CU = 1} is bounded and hence compact. Therefore,B(V + X ) has a finite maximum. Proof (sketch). If X is a subset of K, then R M is positive onV + X if M ≥ M ⋆ X ∶= supB(V + X ) and thus (if X is a finite set) is representable as a positive combination of values at a subsetX ⊂ X of points which comprise linearly independent point evaluation functionals; a compactness argument extends this statement from finite X to X = K. ◻ Theorem 5.7. An optimizer U ⋆ ∈ V + K exists such that M ⋆ =B(U ⋆ ). Proof (sketch). By definition, M ⋆ is defined to be the supremum of the boundary crowdingB(U ) ∶= B(U ) C(U ) over all nonzero U positive in the mesh cell K. Thus, any U positive on K gives a lower boundB(U ) on M ⋆ . The question is whether an arg max ofB exists. SinceB is continuousV + K andV + K is compact (see the proof of Theorem 5.2), an arg max U ⋆ ofB exists. ◻ Thus, it is possible to determine the optimal weight M ⋆ and a set of optimal retentional points simply by guessing an optimizer and an optimal boundary-weighted quadrature rule and using them to confirm one another. We now identify properties that reduce the set of guesses that we have to consider. Theorem 5.8 (Invariance under isometries can be required of retentional points and optimizer candidates). Proof (sketch). A typical canonical mesh cell has a group of isometries; by averaging over an orbit, for any solution U there exists aŨ invariant under the action of the group of isometries of the mesh cell with the same values of C, B, and B. Similar averaging shows that every boundary-weighted quadrature rule C(U ) = Optimal accuracy-respecting cell positivity functionals Assume Context 5.1: K is a compact mesh cell, V is a finite-dimensional polynomial space which contains the set P k D of all polynomials in D variables of degree at most k ≥ 0, CU ∶= − ∫ K U, BU ∶= − ∫ ∂K U,BU ∶= CU BU, V + K ∶= U ∈ V ∶ inf U (K) ≥ 0, CU > 0 , M ⋆ ∶= supB(V + K ), and R M ∶= M C − B. Then the following results hold: Corollary 5.11 (existence of an invariant optimizer and of invariant retentional points which isolate the optimal weight). Linearly damping the deviation from the cell average just enough to enforce positivity of the retentional R M retains order-k local accuracy as long as M is admissible, i.e., M ≥ M ⋆ (Theorem 5.2). Any solution U ∈ V positive in K and any correct boundary-weighted quadrature rule C(U ) = ∑ x∈X w x U (x) + W B(U ) give a bracketB(U ) ≤ M ⋆ ≤ W −1 for the optimal weight (Theorem 5.4). Furthermore, an optimizer U ⋆ exists (Theorem 5.7) and an optimal boundary-weighted quadrature rule exists (Theorem 5.6) such that the bracket is an equality:B(U ⋆ ) = M ⋆ = W −1 . Enforcing positivity at the set X of retentional points enforces positivity of the retentional R M (see Remark 5.5). The zero-set of U ⋆ is nonempty if M ⋆ ≠ 1 (Theorem 5.10) and must contain X if X is optimal (by Theorem 5.6). All these statements continue to hold if the optimizer U ⋆ or the quadrature rule (i.e. the retentional points X and the quadrature weights w x ) is required to be invariant under the action of the isometries of the mesh cell (Theorem 5.8). Proof. We have that M ⋆ =BU ⋆ . Let U min ∶= min U ⋆ (K). Assume that U min > 0. LetŨ ∶= U ⋆ − U min . SinceB(U min ) = 1, by Lemma 5.9,B(U ⋆ ) =B(Ũ + U min ) = aB(Ũ ) + (1 − a), for some 0 < a < 1. IfB(Ũ ) < 1 thenB(U ⋆ ) < 1 =B(U min ), contradicting the maximality ofB(U ⋆ ). IfB(Ũ ) > 1 thenB(U ⋆ ) <B(Ũ ), again contradicting the maximality ofB(U ⋆ ). Therefore,B(U ⋆ ) = 1. ButB(1) = 1, so 1 is an optimizer. ◻ 6. Calculated weights (functional analysis calculations). The previous section developed a general theory of cell positivity functionals. In this section, we use the results of Section 5 summarized in Figure 5.1 to determine retentional points and lower and upper bounds for the optimal weight M ⋆ K for the two standard mesh cell geometries (box and simplex) and for polynomial representation spaces of varying order. The most important results of this section are summarized in Figure 6.1. Remark 6.1 (retaining bars when computing weights). In accordance with Remark 3.1, we dropped bars in the previous section without affecting the validity of the general theorems. In this section, however, we obtain concrete numerical values for the interior weight M ; since we want these results to be independent of the scale of the canonical mesh cell, we retain bars in this section. 6.1. Definitions and summary of results. Recall that M ⋆ K (V) denotes the optimal interior weight for mesh cell K with representation space V. We tabulate values or upper and lower bounds on the optimal interior weights for standard canonical mesh cells: the unit interval [0, 1], the unit square [0, 1] 2 , the unit cube [0, 1] 3 , an equilateral triangle △ 2 , and a regular tetrahedron △ 3 . Each upper bound M constitutes an admissible weight and is calculating using a quadrature rule for the retentional R M whose quadrature points can be used as positivity points. Definition 6.2 (canonical mesh cells). In R D , define canonical mesh cells: [0, 1] D = regular box, D = sphere, △ D = regular simplex, D ∶ "star-regular" polytope. We define a star-regular polytope to be any star-convex mesh cell that can be centered on the origin to have a constant value ofn ⋅ x, wheren is the outward unit normal and x is position on the cell boundary. We use D as a generic designation for a star-regular polytope. Regular polytopes are star-regular. Since most often the representation space is P k D , we define M k K ∶= M ⋆ K (P k D ). For practical use, we summarize our calculation of interior weights in Figure 6.1. Additional definitions for regular polytopes. To calculate quadrature rules that yield admissible boundary weights for higher-dimensional mesh cells, we try to reduce to a one-dimensional problem using the following symmetry framework. Definition 6.3 (notation and conventions). Let C = − ∫ K denote the cell average. With star-regular polytopes in mind, assume that the canonical mesh cell is centered on the origin. Define the radius r to ben ⋅ x, which for a regular polytope is the distance from the origin to the closest point on the boundary of the mesh cell. We will assume unless stated otherwise (and without loss of generality) that the radius of K is 1. Let B r denote the average over the boundary of the mesh cell rK rescaled to have radius r. Proposition 6.4 (The cell average is a weighted average of boundary averages over rescaled cells). Let K be a star-regular polytope in D-dimensional space. Then C = ∫ 1 0 r D−1 B r dr ∫ 1 0 r D−1 dr = D 1 0 r D−1 B r dr. (6.1) Furthermore, if the polynomial representation space V is a subset of P k D then B r is a polynomial in r of degree at most k. Proof. Equation (6.1) equates weighted averages and is justified by observing that the thickness of the infinitesimal shell [r, r + dr] ⋅ K is dr and its area is proportional to r D−1 . Since any polynomial is a sum of homogeneous polynomials, to justify the final statement it is enough to observe that the statement is correct for homogeneous polynomials; indeed, if U is homogeneous of degree k ′ then so is B r . ◻ Tabulated results of Section 6 6.1.1. Boxes and even star-regular mesh cells. If the mesh cell K is even (i.e., there exists K ∈ K such that K ⊂ K − K), then for a regular polygon the optimal interior weight for polynomials of degree at most k is bounded above by the optimal interior weight M k D for a D-dimensional sphere. Based on equations (6.4) and (6.7), we tabulate exact values or bounding intervals for the optimal interior weight for an interval (M k [0,1] ), a square (M k [0,1] 2 ), and a cube (M k [0,1] 3 ). Note that for a tensor product polynomial space the optimal weight is M k [0,1] independent of the dimension D of the box (see Remark 6.14). Note also that for k ≤ 3, enforcing positivity at the cell center enforces positivity of the retentional for the optimal weight (see Theorems 6.6 and 6.7). 6.1.2. Simplices and arbitrary star-regular mesh cells. If the mesh cell is not necessarily even, then for a regular polygon the optimal interior weight is still bounded above by M k,+ D . Based on equations (6.5) and (6.8), as well as Remark 6.10 and Theorem 6.9, we tabulate exact values or bounding intervals for optimal interior weights for the triangle (M k △ 2 ) and tetrahedron (M k △ 3 ). Note that to enforce positivity of the retentional for the optimal weight, for a quadratic representation space it is sufficient to enforce positivity at the cell center (see Theorem 6.6), and, in the case of a simplex, for a cubic representation space it is sufficient to also enforce positivity at the center of each face (see Theorem 6.4). When using these weights, take the boundary average as the arithmetic average of face averages ( §3.5). 6.3. Linear and quadratic representation spaces. Theorem 6.5 (linear representation space). Let K be a star-regular polytope and let V = P 1 D . Assume that the origin is the only point whose orbit under the isometries of K is a singleton. Then the optimal interior weight is M ⋆ = 1 and the positivity-preserving time step is guaranteed simply by enforcing positivity at the boundary nodes. k (V = P k D ) ∶ 0, 1 2, 3 4, 5 6, 7 8, 9 10, 11 n = ⌊k 2⌋ ∶ 0 1 2 3 4 5 m = ⌊n 2⌋ ∶ 0 0 1 1 2 2 M k [0,1] = (n+1)(n+2) 2 ∶ 1 3 6 10 15 21 M k 2 = (m + 1) ⋅ ⌊ n+3 2 ⌋ 1 2 4 6 9 12 M k 3 = m+1 3 ⋅ 3 + 2⌊ n+1 2 ⌋ 1 1.6 3.k (V = P k D ) ∶ 0 1 2 3 4 5 6 7 n = ⌊k 2⌋ ∶ 0 0 1 1 2 2 3 3 M k [0,1] = (n+1)(n+2) 2 ∶ 1 1 3 3 6 6 10 10 M k,+ 2 = n+1 2 (⌊ k+1 2 ⌋+2 Proof. If U is a linear function thenB(U ) = 1, as can be seen by averaging over the orbit of U under the group of isometries of the polytope. ◻ Theorem 6.6 (quadratic representation space). Let K be a star-regular polytope, and let V = P 2 D . Assume that the origin is the only point whose orbit under the isometries of K is a singleton. Then the optimal time step is guaranteed simply by enforcing positivity at the boundary nodes and at the cell center. Enforcing positivity at the cell center is equivalent to enforcing positivity of the retentional R ⋆ = M ⋆ C − B, where the optimal interior weight is M ⋆ = D+2 D . Proof. As seen from equation (6.1), for any homogeneous quadratic U ,BU = D+2 D . If U is a constant-valued function, thenBU = 1. So the optimizer cannot be constant. So by Theorem 5.10 its zero set must be nonempty. We can require the optimizer to be invariant under the isometries of the polytope. So it must be a positive rotationally invariant quadratic that has a zero at the origin, i.e. a homogeneous quadratic. The support of the optimal interior sum is therefore restricted to the origin. So the retentional is proportional to the value at the cell center. Proof. The isometries of an even polytope by definition include negation x ↦ −x. Therefore an optimizer U ⋆ ∈ P 2k+1 D symmetric under the isometries of the polytope lacks odd-degree terms, so U ⋆ ∈ P 2k D . ◻ Theorem 6.8 (cubic representation space for simplices: optimal retentional points). Assume that V = P 3 D . For any simplex, optimal retentional points are located at the cell center and at the centers of the faces. Proof. By Figure 5.1, the set of optimal retentional points must be zeros of an optimizer and can be required to be invariant under the isometries of K. Since V is cubic, this means that optimal retentional points can exist only at the center of the cell and at the center of its faces. (Note that an optimizer candidate uniformly zero on the boundary could not have a boundary crowding exceeding zero.) ◻ Theorem 6.9 (cubic representation space: optimal weight for triangle and tetrahedron). Assume that V = P 3 D . For a triangle the optimal interior weight is M 3 △ 2 = 20 9, and the optimal retentional points are located at the cell center and at the centers of the edges. For a tetrahedron the optimal interior weight is M 3 △ 3 = 11 6, Remark 6.10. In [20], Zhang, Xia, and Shu construct a boundary-weighted quadrature rule for triangles. For representation spaces P 2n D and P 2n+1 For quadratic and cubic polynomial representation spaces, the set of optimal interior points for a box is simply the cell center (Theorems 6.6 and 6.7). V = P 2 (x) · P 2 (y) M = 3 V = P 3 (x) · P 3 (y) M = 3 For tensor product polynomial spaces, the optimal interior points for a box are indicated in Remark 6.14 (see [17]). k = 2, M = 2 M = 1.6 k = 2 For quadratic polynomials in a simplex, the set of optimal interior points is simply the cell center (Theorem 6.6; contrast with Figure 1 in [18]). k = 3, M = 2.2 M = 1.83 k = 3 For cubic polynomials in a simplex, the set of optimal "interior" points consists of the cell center and the centers of the faces (Theorem 6.4). Positivity is also enforced at boundary nodes, but for a different purpose: to ensure that Riemann problems are solved with physical states. For a linear representation space, the set of retentional points is empty, and it is enough to enforce positivity at the boundary nodes. cell and at the centers of its faces. Both for a triangle and for a tetrahedron, up to multiplication by a nonzero scalar there exists a unique such cubic polynomial. This polynomial is definite (i.e. positive after negating if necessary), and evaluatingB for this polynomial yields the values 20 9 for a triangle and 11 6 for a tetrahedron. ◻ Results for quadratic and cubic polynomials are summarized in Figure 6.2. 6.5. High-order representation spaces. Theorem 6.11 (weights for 1D interval). M 2n [0,1] = M 2n+1 [0,1] = W −1 ∶= (n + 1)(n + 2) 2 . (6.2) Note that the optimal boundary-weighted quadrature is the Gauss-Lobatto quadrature with n interior points and end weight W . Proof. For the canonical mesh cell K = [0, 1], the Gauss-Lobatto quadrature 2 − Q [0,1] U = W (U (0) + U (1)) + n i=1 w i U (x i ) (6.3) with n interior points is exact for polynomials of degree at most 2n + 1. The function U ⋆ (x) = ∏ n i=1 (x − x i ) 2 is zero at all interior points and is of degree 2n. So for the representation spaces P 2n D and P 2n+1 D , the quadrature rule is exact, U ⋆ is a positive function in the representation space, and U ⋆ is zero on the interior points. Therefore the hypotheses of Theorem 5.11 are satisfied and the conclusion follows. ◻ Remark 6.12. For 1D mesh cells, Zhang and Shu enforce positivity at the Gauss-Lobatto quadrature points, which of course implies positivity of the retentional [17]. The previous theorem is a special case of the following theorem. Theorem 6.13 (optimal interior weight for spherical mesh cells). M 2n D = M 2n+1 D = (⌊ n 2 ⌋ + 1)(2⌊ n+1 2 ⌋ + D) D . (6.4) Furthermore, M k [0,1] D ≤ M k D , with equality precisely when D = 1 or k < 4. Remark 6.14. For boxes one can also construct a boundary-weighted quadrature rule by taking the tensor product of a Gauss-Lobatto quadrature rule with a quadrature rule used to integrate over a face and averaging over all D such quadrature rules, as is done in [17] for the case D = 2. The resulting interior weight is 1] independent of D. This weight and set of quadrature points is optimal for a tensor product polynomial space; indeed, invoking Corollary 5.11, a confirming optimizer is 1] is a unit-interval optimizer. But for P k D , in case D > 1 and k > 1, the boundary weight of this theorem is an improvement over M k [0,1] . Proof (sketch). Recall from (6.1) that C = D ∫ 1 0 r D−1 B r dr, and observe that, for boxes and spheres, B r is an even polynomial in r of degree less than or equal to the degree k of the representation space. So we can write B r = B(r 2 ), where deg B = ⌊k 2⌋. We seek a quadrature rule that maximizes the weight on the point r = 1. Making the substitution t = 1 − 2r 2 shows that C is given by an integral with a Jacobi weight function. M ⋆ = M k [0,U ⋆ (x) ∶= ∏ D i=1 U k,⋆ [0,1] (x i ), where U k,⋆ [0, In case k = 4m (or k = 4m + 1) we use a Gauss-Radau quadrature rule for Jacobi weight (GRJ) with m interior points, which is exact for polynomials of degree at most 2m (see [3]). In the case of a spherical mesh cell, an optimizer that confirms that the boundary weight is maximal is U ⋆ (r 2 ) = ∏ m i=1 (r 2 − r 2 i ) 2 , where the r i are the interior quadrature points corresponding to the interior quadrature points t i of the GRJ quadrature rule. In case k = 4m + 2 (or k = 4m + 3), we use a Gauss-Lobatto quadrature rule for Jacobi weight (GLJ) with m interior points, which is exact for polynomials of degree at most 2m+1 (see [4]). In the case of a spherical mesh cell, an optimizer that confirms that the boundary weight is maximal is U ⋆ (r 2 ) = r 2 ∏ m i=1 (r 2 − r 2 i ) 2 , where the r i are the interior quadrature points corresponding to the interior quadrature points t i of the GLJ quadrature rule. ◻ Theorem 6.15 (admissible interior weight for any star-regular mesh cell). M k △ D ≤ M k,+ D ∶= (⌊ k 2 ⌋ + 1)(⌊ k+1 2 ⌋ + D) D . (6.5) Remark 6.16. Equality holds only for the case M 0 △ D = 1. Recall from Remark 6.10 that M 2n △ 2 ≤ M 2n+1 △ 2 ≤ (n+1)(n+2) 2 , which agrees with estimate (6.5) for evendimensional representation spaces and for odd-dimensional representation spaces is slightly better than estimate (6.5). Proof (sketch). We want a quadrature rule with such weight on the boundary average for C = D ∫ 1 0 r D−1 B r dr. Unlike for boxes and spheres, we cannot assume that B r is an even polynomial. For P 2n D we use a GRJ quadrature (see [3]) with n points to a get a quadrature rule for C, and for P 2n+1 D we use a GLJ quadrature (see [4]) with n points to get a quadrature rule for C; in each case, the resulting end weight is the reciprocal of the interior weight appearing in (6.5). ◻ 6.6. Lower bounds for interior weights. In the previous sections we computed exact values and upper bounds for optimal interior weights. In the case of higher-order representation spaces where we have merely computed upper bounds, we would also like to have a lower bound that suggests how much the estimate might be improved. In particular, we will obtain bounding intervals that prove that M k D = Ω(k 2 ), M k [0,1] D = Ω(k 2 ), and M k △ D = Ω(k 2 ). (6.6) Recall that for 1D intervals the optimal weight is given by equation (6.2). For boxes and for simplices, we now bootstrap from this result to obtain lower bounds for the interior weight. as will be shown in detail in Part II of this work [7]; we can take M k,− [0,1] ∶= M k [0,1] . By induction on D, the recurrence relation (6.7) implies that M k [0,1] D = O(k 2 ), so by Theorem 6.13, M k [0,1] D = Ω(k 2 ). 6.6.2. Simplices. Let K be a regular simplex of dimension D. Suppose that U is a polynomial of degree at most k defined on the face F of K not containing vertex V , and suppose that U is positive when restricted to F . Then U extends uniquely to Prescriptions for positivity limiting Check positivity efficiently. One can use interval arithmetic to inexpensively confirm cell positivity in the vast majority of mesh cells. For nodal DG, choose positivity points to be nodal points to avoid additional computational expense. See §7.1. Pad inequalities. In practice, error in machine arithmetic makes it problematic to enforce exact inequalities such as ρ ≥ 0 or p ≥ 0. Instead, enforce ρ ≥ ρ or p ≥ p for some small ρ > 0 and p > 0. See [14]. Use positivity points to improve stability. Enforcing positivity at additional interior points may improve stability (likely depending on the choice of oscillationsuppressing limiters). Since values used in volume and boundary quadratures must be calculated (or available) anyway, positivity of these values can be efficiently enforced, and enforcing positivity at the points guarantees that state values used in the numerical method are always positive. For modal DG, positivity at any finite set of points can be efficiently checked, and it makes sense to include the optimal retentional points if they are known. Use wave speed desingularization for systems, especially for shallow water. When enforcing positivity in shallow water, fluxes need to be calculated with remapped states in order to desingularize wave speeds. See remarks in Algorithm 1 and Algorithm 2 and Section 7.2. For gas dynamics, wave speed desingularization is needed when enforcing positivity of the density (although more often it is positivity of the pressure that needs to be enforced). Estimate an optimal time step after enforcing the cell positivity condition. The factor by which the time step must be shortened determines the expense of positivity limiting ( §7.3). Enforcing the cell positivity condition guarantees a minimum time step ∆t pos that preserves positivity of the cell average; see equation (3.4). One can directly calculate the maximum one-stage time step ∆t zero that maintains positivity of the cell average ( §4.1) in addition to calculating the maximum stable time step ∆t stable to determine an optimal safe time step. For multistage and local time stepping, one can use ∆t stable , ∆t zero , and optionally ∆t pos to maintain and iteratively adjust an estimate of a safe time step that is both stable and positivity-preserving. a polynomial that is homogeneous of degree k when expanded about V . This yields a recurrence relation for a lower bound M k,− △ D ≤ M k △ D , M k,− △ D = D + k D ⎛ ⎝ 1 + D D−1 D+k−1 M k,− △ D−1 1 + D ⎞ ⎠ ,(6.8) as will be shown in detail in Part II of this work [7]. By induction on D, recurrence relation (6.8) implies that M k △ D = O(k 2 ). Thus, by Theorem 6.15, M k △ D = Ω(k 2 ). 7. Practical application of positivity limiting. We briefly consider the most prominent issues that arise when enforcing positivity. Detailed treatment of practical issues is deferred to Part II of this work [7]. We summarize in Figure 7.1. Efficient implementation of positivity checks. For almost all problems, in the large majority of cells one can confirm that the solution is positive and will remain so for any stable time step by using interval arithmetic. Thus, one can enforce positivity with little additional computational expense per time step. For example, in the case of gas dynamics, one can quickly confirm positivity of the pressure in the vast majority of cells by checking that P min ∶= ρ min E min − m 2 max 2 is positive, where ρ min and E min are lower bounds on density and energy in the cell and m max is an upper bound on momentum. The question is how to obtain a tuple of intervals [u min , u max ] for the components of the conserved variables. In modal DG one can obtain lower and upper bounds that hold globally in the mesh cell by using lower and upper bounds on the polynomial basis functions ϕ j . This works well, because for smooth solutions the coefficients of higher-order modes decay rapidly as the mesh is refined. Nodal DG represents the solution using values at nodes; therefore, one can easily obtain an interval [u min , u max ] that applies to all nodal values. Thus, even for nodal DG, checking positivity at all nodes is usually no more expensive than checking positivity at a point. By Theorem 5.3, positivity at the nodes of a nodal DG scheme implies positivity of a retentional with weight W ≤ W ⋆ , because the cell average is a positively weighted sum of values at the nodes (including boundary nodes) and so is a boundary-weighted quadrature. 7.2. Wave speed desingularization is needed when limiting density. Let ρ denote depth in the shallow water case or density in the gas dynamics case. After enforcing positivity of ρ at boundary nodes, wave speeds need to be de-singularized. This can be done by modifying states used to compute fluxes when the wave speed exceeds an estimate of the maximum wave speed in the physical solution. In both shallow water and gas dynamics, desingularization of momentum (i.e. fluid velocity) is necessary. Let u cap be a fluid speed cap that, for a sufficiently refined mesh, is guaranteed to exceed the fluid speed that arises in the exact solution. Then an accuracy-respecting desingularization map of the momentum (or fluid velocity) is the rescaling u ← u ⋅ R 1 (u cap u ), where R 1 (x) = 1 if x ≥ 1 and equals x (or the spline x(2 − x)) if 0 ≤ x ≤ 1 (see Section 4.4 of [1]). For shallow water, enforcing positivity means enforcing positivity of ρ and therefore essentially always entails wave speed desingularization. For gas dynamics, in addition to desingularizing the fluid speed u, it may also be necessary to desingularize the sound speed c = √ γθ, e.g. by capping the pseudo-temperature θ ∶= p ρ. More careful study of wave speed desingularization is left to future work. 7.3. Cost of positivity limiting. For most problems, for the vast majority of cell updates, a quick check ( §7.1) will confirm that positivity limiting is unnecessary. If local time stepping is used, then the total cost of positivity limiting will be marginal. But if global time stepping is used, then the expense of positivity limiting is measured by the factor by which the time step must be shortened to maintain positivity. This suggests (1) a careful consideration of how to obtain tight and reliable wave speed bounds for use in wave speed desingularization and (2) a comparison of stable and positivity-preserving time steps. For (2), in the case of one-dimensional mesh cells with kth order polynomial space, values are known: an approximate value for the maximum stable time step is given by ∆t −1 stable ≈ (k + 1 2)λ ∆x if an SSP-RK time-stepping method of order k is used (see Table 2.2 in [2]); in comparison, capping boundary crowding by M ⋆ = (n + 1)(n + 2) 2 gives ∆t −1 pos = λM ⋆ = (n + 1)(n + 2)λ ∆x, where k equals 2n or 2n + 1. Isoparametric mesh cells. For a typical isoparametric mesh cell, in canonical coordinates the mesh cell is a box or simplex and the representation space is a polynomial space (e.g. P k D ). Each physical mesh cell is the image of such a canonical mesh cell under a diffeomorphism φ, and the physical representation space is the push-forward under φ of the canonical representation space. The outflow positivity limiting framework is defined in canonical coordinates and can be applied without modification for isoparametric mesh cells: under reasonable regularity assumptions on φ, direct application of positivity limiters in canonical coordinates respects accuracy and guarantees a minimum positivity-preserving time step [7]. 8. Conclusion. This work consists of two main results: a framework and algorithm for positivity limiting, and calculations of optimal weights and positivity points needed by this framework. 8.1. Outflow positivity limiting framework. We have developed a framework for preserving positivity of each cell average based on limiting the rate and amount of material that can flow out of the cell. This is achieved by linear damping of high-order corrections, remapping boundary node states to limit wave speeds, and limiting the time step. In each cell, the high-order corrections are linearly damped just enough to enforce a physically justified cap M > 1 on the boundary crowdinĝ BA(U ) ∶= BA(U ) CA(U ) for all affine functionals A in a set of state positivity functionals P * , where C is the cell average and B is the arithmetic average over all faces of the average over each face (see Section 3.8); this condition is satisfied precisely when the unital retentionalR (U ) = M CU − BU M − 1 satisfies positivity. One way to enforce positivity of the unital retentional is to enforce that the solution is positive at appropriately chosen quadrature points in the mesh cell, as Zhang and Shu have done [16,18]. Directly enforcing positivity of the unital retentional is computationally inexpensive and makes it unnecessary to determine and enforce positivity at these points. For concreteness, we display positivity-preserving algorithms for scalar conservation laws and shallow water (Algorithm 1) and for gas dynamics (Algorithm 2). 8.2. Optimal weights and positivity points. Just as point-wise positivity limiting prompts the search for optimal positivity points, retentional positivity limiting prompts the need for a reasonably close upper bound M on the optimal interior weight M ⋆ K (V). In Section 5, we developed a general framework to estimate upper and lower bounds on M ⋆ K (V). In Section 6, we applied this framework to tabulate interior weights for boxes and simplices for polynomial representation spaces. In practice, the canonical mesh cell is almost always a cube or a simplex, and, for problems that are likely to entail positivity limiting, the representation space is likely to be a space of polynomials of at most cubic degree. Assuming these conditions, the set of optimal interior positivity points is very small, and the essential takehome result of this work is that to guarantee the same positivity-preserving time step as if positivity were enforced everywhere in the mesh cell, in addition to enforcing positivity (and, for shallow water, desingularizing wave speeds) at boundary nodes, it is sufficient to enforce positivity at the cell center, except that for a simplex with a cubic representation space one must also enforce positivity at the centers of the faces. See Fig. 1 . 1 : 11Assumptions and requirements of positivity limiting (Section 1). Fig. 1 . 2 : 12Positivity limiting illustrated for 1D mesh cells with cubic polynomials. Fig. 1 . 3 : 13Outflow capping in the scalar case. positivity limiting caps the boundary average by the maximum boundary average attainable by a positive solution. The original solution U = U + dU has the same cell average U as the retentional-positivity-limited solution U R = U + θ R dU and the pointwise-positivity-limited solution U X = U + θ X dU have. The optimizer U ⋆ shown has the maximum boundary average BU ⋆ = M ⋆ U out of all positive quartics with cell average U (Theorem 5.7). The interior (black) positivity points X = {± 1 5} are optimal and are thus zeros of U ⋆ (by Theorem 5.6). At each positivity point, U X is positive. For U R , the boundary average equals (at most) BU ⋆ and the optimal retentional R M ⋆ U ∶= M ⋆ CU − BU , proportional to the sum (in general a weighted sum) of the values at the interior points, is (at least) zero. See Figure 5.11. Fig. 1 . 4 : 14Retentional positivity limiting for a one-dimensional mesh cell with a quartic polynomial basis (M ⋆ = 6), compared with pointwise positivity limiting. Fig. 2 . 1 : 21Definition of the Riemann problem used to define numerical flux Fig. 2 . 2 : 22The context of this work as outlined in Section 2. . Enforcing a cap M onBU = BU CU is equivalent to enforcing positivity of the retentional 1 R M U ∶= M CU − BU, or one of its rescaled versions such as R W ∶= CU −W BU , where W ∶= M −1 . We call M the interior weight (or boundary crowding cap) and we call W the boundary weight (or positivity CFL cap -see Theorem 3.12). Figure 3 . 31. LLF is the special case of HLL where the left-going and right-going numerical signal speeds are equal:Ṡ − +Ṡ + = 0. Proposition 3 . 13 . 313The following are equivalent: 1. The full system (1.2) maintains positivity. 2. Any Riemann problem with positive states and frozen flux function maintains positivity. Theorem 3.7 (outflow rate is bounded by speed cap times interior value). Consider the one-cell problem with states (Z, U − , U + ), cell width ∆x, and positivity-preserving flux function f . Define the numerical speed cap the interface flux of the Riemann problem with states (A, B) and let h(A, B) be a consistent numerical flux function that preserves positivity for the Euler update if incoming signals do not cross. Assume that Z, Zλ(Z, Z, U − ), and f (Z) all equal or approach 0. Then h(U − , U + ) ≤ λU − .Proof. Choose ∆t = ∆x λ. Then (3.7) says that h(U − , U + ) ≤ λU − + h 0 (Z, U − ). But since material cannot flow out of a vacuum, h 0 (Z, U − ) ≤ 0; to verify this rigorously, make the replacements U − → Z, U + → U − , h → h 0 , and λ → λ(Z, Z, U − ), and use that h 0 (Z, Z) = f (Z), which approaches 0. Therefore, h(U − , U + ) ≤ λU − . ◻ an instance of the formula at the bottom of Figure 2.1). Fig. 3 . 31: A signal speed cap and boundary node value bound the outgoing flux rate. II: A speed cap and positive retentional guarantee a positivity-preserving time step. Theorem 3.12. Suppose that: h ≤ λU − (numerical flux is bounded via the speed cap λ), λ∆t ≤ W (positivity CFL number is bounded by W ), and W BU ≤ CU (the boundary crowding is capped by M = W −1 ). Then the updated cell average CU − ∆tBh is positive. Proof. ∆tBh ≤ λ∆tBU ≤ W BU ≤ CU. ◻ Remarks. The proof says that the loss is at most the cell content. The retentional R W U ∶= CU − W BU represents the content retained if the maximum possible loss occurs and is positive precisely when the boundary crowding BU CU is capped by M . Theorem 3.12 is sharp: a smooth example with spatially varying flux function satisfying f =nλu on ∂K shows that loss can equal cell content. Enforcing positivity of R M maintains accuracy for scalar positivity limiting if M ≥ M ⋆ . Fig. 3. 2 : 2A speed cap and boundary average cap bound the outgoing flux. 3. The one-cell Euler update (3.7) of Figure 3.1 with an HLL numerical flux function maintains positivity for sufficiently large λ. Justification. To see that 1 implies 2, assume without loss of generality that the flux function is frozen at t 0 = 0 and x 0 = 0 and in the directionn. The self-similar solution u(t, x) =û(x t) to the auxiliary Riemann problem with positive states (A, B) ◻ 3 . 4 . 34Direct proof for the 1D case. See Figure 3.3.3.5. Affine-invariant definitions of wave speed and retentional. We have used bars in Sections 2-3 to indicate the scale-invariant formulation. Note that all definitions and statements remain valid if bars are dropped. Alternatively, we can redefine barred quantities using affine-invariant definitions.Heretofore we have made no distinction between physical and canonical mesh cell coordinates. In general, each physical mesh cell is the image of a canonical meshComparison with Zhang and Shu (1D case)A corollary of the theorems inFigures 3.1 and 3.2 is the 1D case:Corollary 3.14. The Euler update (1.8) maintains positivity of the cell average if the retentional RU n i ∶= U n i −W BU n i is positive in each cell and if the time step satisfies 2∆tλ ∆x ≤ W , where λ 2 is an upper bound on signal speeds and BU i = Fig. 3. 3 : 30. Zhang and Shu instead assume a spatially invariant flux function (h i+1 2 = h i−1 2 ) and choose Z + = U + i−1 2 and Z − = U − i+1 2 . They write the retentional as a weighted sum over interior positivity points (see Theorem 5.6) and choose W = W ⋆ (see Theorem 6.2).◻ Direct proof for the 1D case. F dAe V h e is an affine-invariant version of the numerical flux, where dA e is the area of the face of boundary node e and F is the number of faces of the mesh cell. Similarly, since wave speeds scale like fluxes, if λ e is a wave speed at boundary node e in the canonical coordinates of mesh cell K, then the quantity λ F e ∶= F dAe V λ e is the affineinvariant version that agrees with the scale-invariant version λ e = A V λ e in the case of a canonical regular polytope. In the case of a regular canonical polytope the affineinvariant quantities h F e , λ F e , and B F agree with their scale invariant equivalents h e , λ e , and B, and the theory of Sections 3-5 goes through. Fig. 4 . 1 : 41Depiction of positivity limiting of a state value for Euler gas dynamics. Positivity limiting requires determining where a linear path in state space intersects the boundary of positive states. 2 . 2To cap boundary crowding at M , determine the largest value of θ ∈ [0, 1] for which u(θ) ∶= u + θdu satisfies positivity, where u =RU , du =RdU , R ∶= (R M 1) −1 R M , and R M ∶= M C − B. 3. To enforce positivity at x 0 , determine the largest value of θ ∈ [0, 1] for which u(θ) ∶= u + θdu satisfies positivity, where u = U and du = dU (x 0 ). All these problems seek the intersection of a line segment with the boundary of a convex set of states designated positive and require computing a damping coefficient: Figure 4 4of positivity limiting in the systems case. For the case of scalar conservation laws, accuracy of pointwise positivity limiting via linear damping is established in Section 5 based on Theorem 5.2. (3. 5 ) 5that the optimal interior weight M ⋆ = M ⋆ K (V) is defined as the supremum of the boundary crowdingB(U ) ∶= B(U ) C(U ) over the set V + K of all nonzero solutions U ∈ V positive in the mesh cell K, where C(U ) ∶= ∫ K U and B(U ) ∶= ∮ ∂K U . The retentional R W (U ) ∶= C(U ) − W B(U ), or R M (U ) ∶= M C(U ) − B(U ), is positive for all such positive U precisely when M ∶= W −1 is admissible, i.e., when M ≥ M ⋆ . . 4 . 4Theorem 5.2 (An admissible weight preserves accuracy). Let M ≥ M ⋆ . Then linearly damping the deviation from the cell average just enough to enforce positivity of the retentional R M (U ) retains order-k local accuracy. ◻ Theorem 5.4 (A boundary-weighted quadrature rule gives an upper bound for the optimal weight). Suppose that the cell content is given by a boundary-weighted quadrature rule, C(U ) = R(U )+W B(U ), where the boundary weight W is strictly positive and R(U ) represents a quadrature rule∑ x∈X w x U (x) for the retentional R W (U ) with positive weights w x that is exact for U ∈ V. Then M ⋆ ≤ M ∶= W −1 .Remark 5.5. Enforcing positivity at the points X enforces positivity of R W ; we call X a set of retentional (or interior) points for the weight M . Proof. Indeed, R = C − W B is positive for all U positive in K, so for any positive U , W −1 ≥ B(U ) C(U ) =B(U ), so W −1 is an upper bound on M ⋆ . ◻ Theorem 5.6 (An optimal boundary-weighted quadrature rule exists). An optimal quadrature rule C(U ) = ∑ x w x U (x) + WB(U ) exists such that M ⋆ = W −1 . Fig. 5 . 1 : 51Summary of the results of Section 5. ∑ x∈X w x U (x) + W B(U ) has an invariant version C(U ) = ∑ x∈Xw x U (x) + W B(U ). ◻ Lemma 5.9 (The boundary crowding of a positive combination of two functions with positive cell average is a convex combination of their boundary crowding values). Let U, V ∈ V + K . Let s > 0 and t > 0. ThenB(sU + tV ) = aB(U ) + (1 − a)B(V ) for some a ∈ (0, 1).Proof.B(sU + tV ) = sBU +tBV sCU +tCV = aB(U ) + (1 − a)B(V ), where a = sCU sCU +tCV . ◻ Theorem 5.10 (A non-constant optimizer has a zero). An optimizer U ⋆ has a zero in K unless U = 1 is an optimizer (in which case M ⋆ = 1). Fig. 6 . 1 : 61Essential results of Section 6. A bounding interval is given where the exact value is unknown. ◻ 6 . 4 . 64Cubic representation spaces. Theorem 6.7 (odd-degree representation spaces for boxes). For boxes, M 2n+1 [0,1] D = M 2n [0,1] D , and likewise for other even polytopes and for spheres. In particular, for a cubic representation space, M ⋆ = D+2 D , and positivity of the interior sum can be enforced simply by enforcing positivity of the value at the origin. 2 ≤ 3 = M 3 [0,1] . Proof (sketch). By Theorem 6.4 andFigure 5.1, we may demand an optimizer that is invariant under the isometries of K and that is zero at the center of the mesh Fig. 6 . 2 : 62Optimal interior points for standard mesh cells and low-order polynomials. Black positivity points are retentional (i.e. "interior") points. Gray points (when shown) are boundary nodes and depend on one's choice of a correct boundary quadrature. The black points are determined by mesh cell geometry and the representation space and are thus independent of the choice of gray points. Enforcing positivity at the black points automatically enforces positivity of the optimal retentional R ⋆ ∶= M ⋆ C−B and guarantees that positivity of the cell average is maintained for the same minimum time step that one would guarantee by enforcing positivity everywhere in the mesh cell (assuming that the same wave speed caps are enforced, see Section 7.2). 6.6.1. Boxes. First consider the case of the box [0, 1] D . It has 2D faces, each of which is a box of dimension D − 1. Each face has 2(D − 1) sub-faces, each of which is a box of dimension D − 2. Suppose that U is defined on the x = 1 face of the box and has interior weight M k,− [0,1] D−1 when restricted to this face. Then U can also be thought of as a function on [0, 1] D and has boundary weight given by the Fig. 7 . 1 : 71Key points of Section 7. Figure 6 PFig. 8 . 1 : 681* ⊂ A ∶ R N → R affine posit. functionals K ⊂ R D canonical mesh cell P D ⊂ f ∶ R D → R polynomials in D vars. U ) ∶= U ∶= 1 V ∫K U cell average Q = ∂ Q K ⊂ ∂Kboundary quadrature pts. Key definitions for cell positivity. An abbreviated form of retention functional, in the admittedly hokey mathematical tradition of using "adjectivals" as substantives and thus nouns. Acknowledgements. This work was supported in part by NSF grant DMS-1016202.Algorithm 1 Positivity-preserving Euler time-step for scalar conservation laws and shallow water.For shallow water, U , h, and f represent the first component of U, h, and f , respectively. For the canonical mesh cell K and representation space V determine an admissible weight M such thatis the boundary average (see §3.5). For efficiency, precompute the values B(ϕ j ).1. ∀K rescale the deviation dU from the cell average U by a percentage θ Y , damping just enough so that U > 0 at a set of positivity points X that includes all quadrature points in ∮ Q ∂K and so that R M (U ) = M U − BU > 0:If X is sufficiently rich (seeFigure 5.1) then θ X ≤ θ R M and computing θ R M is unnecessary.2. Compute a safe time-step ∆t:where ∆t stable is the maximum stable time-step and 0 < α z < 1 is a safety factor.3. (for shallow water, not scalar case.) Modify the boundary node states U − Q if necessary in order to desingularize wave speeds at boundary nodes (see §7.2).∀K update the solution:Algorithm 2 Positivity-preserving Euler time-step for gas dynamics (cf. §3.2 of[16]).Choose large enough values ρ > 0 and p > 0 that vanish with machine epsilon.1. ∀K evaluate the solution U n =∶ U = U + dU at a set of positivity points X which includes the set of boundary nodes Q (i.e. the quadrature points of ∮ Q ∂K ) to determine the nodal states U X ⊃ U − Q .2. ∀K linearly damp the high-order corrections dU by a percentage θ ρ , damping just enough so that the density ρ is positive at the points X and so that the density retentional R M (ρ) = M ρ − Bρ is positive:3. ∀K linearly damp the solution just enough so that the pressure is positive at the points X:4. Modify the boundary node states U − Q if necessary in order to desingularize wave speeds at boundary nodes (see §7.2).5.∀K and for some admissible interior weight M linearly damp dU so that the retentional6. Compute a stable, positivity-preserving time-step ∆t:where ∆t stable is the maximum stable time-step and 0 < α z < 1 is a safety factor.7. ∀K update the solution: Medvid'ová, Finite volume evolution Galerkin methods for the shallow water equations with dry beds. A Bollermann, S Noelle, M Lukáčová, Communications in Computational Physics. 10A. Bollermann, S. Noelle, and M. Lukáčová-Medvid'ová, Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Communications in Computational Physics 10 (2011), 371-404. Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. B Cockburn, C.-W Shu, J. Sci. Comput. 163B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection- dominated problems, J. Sci. 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Outflow positivity limiting for hyperbolic conservation laws. Part II: Analysis and extension. E A Johnson, manuscript in preparationE.A. Johnson, Outflow positivity limiting for hyperbolic conservation laws. Part II: Analysis and extension, manuscript in preparation. Finite volume methods for hyperbolic problems. R J Leveque, Cambridge University PressR.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, 2002. On the vacuum state for the isentropic gas dynamics equations. T.-P Liu, J A Smoller, Advances in Applied Mathematics. 1T.-P. Liu and J. A. Smoller, On the vacuum state for the isentropic gas dynamics equations, Advances in Applied Mathematics 1 (1980), 345-359. Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes. X.-D Liu, S Osher, SIAM J. Numer. Anal. 332X.-D. Liu and S. Osher, Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes, SIAM J. Numer. Anal. 33 (1996), no. 2, 760-779. On positivity preserving finite volume schemes for Euler equations. B Perthame, C.-W Shu, Numer. Math. 73B. Perthame and C.-W. Shu, On positivity preserving finite volume schemes for Euler equa- tions, Numer. Math. 73 (1996), 118-130. W Rudin, Functional analysis. McGraw-Hillsecond ed.W. Rudin, Functional analysis, second ed., McGraw-Hill, 1991. Calculation of interaction of non-steady shock waves with obstacles. V V Rusanov, J. Comp. Math. Phys. USSR. 1V.V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, J. Comp. Math. Phys. USSR 1 (1961), 267-279. Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. C Wang, X Zhang, C.-W Shu, J Ning, J. Comp. Phys. C. Wang, X. Zhang, C.-W. Shu, and J. Ning, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, J. Comp. Phys. (2011), 653-665. Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws. X Zhang, Providence, Rhode IslandBrown UniversityPh.D. thesisX. Zhang, Maximum-principle-satisfying and positivity-preserving high order schemes for con- servation laws, Ph.D. thesis, Brown University, Providence, Rhode Island, May 2011. On maximum-principle-satisfying high order schemes for scalar conservation laws. X Zhang, C.-W Shu, J. Comp. Phys. 229X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comp. Phys. 229 (2010), 3091-3120. On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comp. Phys. 229, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comp. Phys. 229 (2010), 8918-8934. Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: Survey and new developments. Proc. R. Soc. A. 467, Maximum-principle-satisfying and positivity-preserving high-order schemes for con- servation laws: Survey and new developments, Proc. R. Soc. A 467 (2011), 2752-2776. Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comp. Phys. 230, Positivity-preserving high order discontinuous Galerkin schemes for compressible Eu- ler equations with source terms, J. Comp. Phys. 230 (2011), 1238-1248. Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. X Zhang, Y Xia, C.-W Shu, J. Sci. Comput. 501X. Zhang, Y. Xia, and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. 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[ "COMMUTING CONJUGATES OF FINITE-ORDER MAPPING CLASSES", "COMMUTING CONJUGATES OF FINITE-ORDER MAPPING CLASSES" ]	[ "Neeraj K Dhanwani ", "Kashyap Rajeevsarathy " ]	[]	[]	Let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g ≥ 2. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in Mod(Sg). As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of Sg. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most 2. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of Mod(Sg) as isometry groups.2000 Mathematics Subject Classification. Primary 57M60; Secondary 57M50, 57M99.	10.1007/s10711-020-00523-9	[ "https://arxiv.org/pdf/1901.11314v1.pdf" ]	119,581,545	1901.11314	c400b337ad86c34ee6ce62fbf2a3203c205b8128	COMMUTING CONJUGATES OF FINITE-ORDER MAPPING CLASSES 31 Jan 2019 Neeraj K Dhanwani Kashyap Rajeevsarathy COMMUTING CONJUGATES OF FINITE-ORDER MAPPING CLASSES 31 Jan 2019 Let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g ≥ 2. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in Mod(Sg). As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of Sg. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most 2. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of Mod(Sg) as isometry groups.2000 Mathematics Subject Classification. Primary 57M60; Secondary 57M50, 57M99. introduction Let S g denote closed orientable surface of genus g ≥ 0, and let Mod(S g ) denote the mapping class group of S g . Given two finite-order mapping classes in Mod(S g ), for g ≥ 2, a natural question that arises is whether there exist representatives of their respective conjugacy classes that commute in Mod(S g ). (When two finite-order mapping classes satisfy this condition, we say that they weakly commute.) While finite abelian groups and their conjugacy classes in Mod(S g ) have been widely studied [2,4,8], our pursuit can be motivated with the following example. Consider the six involutions in Mod(S 8 ) shown in Figure 1 below, where each involution is realized as a π-rotation about an axis (passing through the origin) under a suitable isometric embedding S 8 ֒→ R 3 . Though all of these involutions are conjugate in Mod(S g ), note that each of the two pairs of involutions indicated in the first two subfigures clearly generate distinct subgroups of Mod(S 8 ) isomorphic to Z 2 ⊕ Z 2 , while the pair of involutions appearing in the third figure can be shown to generate a subgroup isomorphic to D 16 . As the main result of this paper (see Theorem 4.8), in Section 4, we derive necessary and sufficient conditions under which two finite-order mapping classes will have commuting conjugates in Mod(S g ). We appeal to Thurston's orbifold theory [16], and the classical theory [4,5,7] of group actions on surfaces for proving this result. A key ingredient in our proof is understanding the factors that determine whether a given Z n -action on S g induces a Z n -action on the quotient orbifold of another cyclic action, and also analyzing the properties of such an induced action. In this connection, we also provide an abstract tuple of integers called an "abelian data set" which corresponds to a two-generator finite abelian subgroup up to a notion of equivalence that we call "weak conjugacy", which, as the term suggests, is weaker than conjugacy (see Section 4). Let F ∈ Mod(S g ) be of order n. By the Nielsen-Kerckhoff theorem [6,11], F has a representative F ∈ Homeo + (S g ) such that F n = 1. We call the quotient orbifold S g / F the corresponding orbifold for F . For an m-sheeted cover p : S m(g−1)+1 → S g , let LMod p (S g ) denote the subgroup of Mod(S g ) of liftable mapping classes under p. As a first application of our main result, in Section 5, we derive conditions under which a finite-order mapping classes weakly commute with mapping classes represented by generators of certain free cyclic actions on S g (see Corollary 5.5). A direct consequence of this result is the following: Corollary 1. Let p : S m(g−1)+1 → S g be an m-sheeted cover whose deck transformation group is Z m . Let F ∈ Mod(S g ) be a finite-order mapping class whose corresponding orbifold is not a sphere. Then the conjugacy class of F has a representative F ′ such that F ′ ∈ LMod p (S g ). We also derive an analog of this corollary for certain finite-order mapping classes whose corresponding orbifolds are spheres (see Corollary 5.7). It is known [4,18] that an F ∈ Mod(S g ) of finite order with \|F \| > 2g + 1 is primitive. Using our theory, we give conditions that determine the primitivity (see Theorem 5.8) of an arbitrary finite-order mapping class. These conditions further lead to a characterization of the primitivity of certain surface rotations. Corollary 2. Let F ∈ Mod(S g ) be a finite-order mapping class. (i) If \|F \| = g − 1 and F is represented by the generator of a free action, then a nontrivial root G of F exists if, and, only if 2 ∤ (g−1). Moreover, G has degree 2. (ii) If 6 \| g and F is represented by a rotation of order g, then F is primitive. It is known [3] that a finite-order mapping class is irreducible if, and only if, its corresponding orbifold is a sphere with 3 cone points. Following the nomenclature from [14], we say an irreducible order n mapping class is of Type 1 if its corresponding orbifold has a cone point of order n, otherwise we say such a mapping class is of Type 2. In this connection, we prove the following: Corollary 3. Suppose that a finite abelian subgroup A of Mod(S g ) contains an irreducible finite-order mapping class F . (i) If F is of Type 2, then A = F . (ii) If F is of Type 1, then either A = G , where G is a root of F , or A ∼ = Z 2 ⊕ Z 2g+2 . Let c be a simple closed curve in S g for g ≥ 2, and let t c ∈ Mod(S g ) denote the left-handed Dehn twist about c. Let F ∈ Mod(S g ) be either a root of t c of degree n, or an order-n mapping class that preserves the isotopy class of c. Then we may assume up to isotopy that F (c) = c, and that F preserves a closed annular neighborhood N of c. Further, it is known [9,12,14] that F induces an order-n map F c on the surface obtained by capping off the components of S g \ N . As another application of our main result, we obtain the following characterization of weak commutativity of finite-order mapping classes with roots of Dehn twists about nonseparating curves. Corollary 4. Let F ∈ Mod(S g ) be a root of t c , where c is nonseparating, and G ∈ Mod(S g ) be of finite order. Then F and G have commuting conjugates if, and only if G(c) = c, and F c and G c have commuting conjugates. In particular, if F c is primitive, then F and G cannot commute in Mod(S g ). We also state an analog of this result (see Corollary 5.15) for the roots of Dehn twists about separating curves. Given a weak conjugacy class of a two-generator finite abelian group (encoded by an abelian data set), in Section 6, we provide an algorithm for determining the conjugacy classes of its generators. We indicate how this algorithm, along with theory developed in [14], leads to a procedure for determining the explicit hyperbolic structures that realize a two-generator abelian subgroup as a group of isometries. Finally, we classify the weak conjugacy classes of two-generator finite abelian subgroups of Mod(S 3 ). We conclude the paper by providing some non-trivial geometric realizations of some of these subgroups. Preliminaries A Fuchsian group [5] Γ is a discrete subgroup of Isom + (H) = PSL 2 (R). If H/Γ is a compact orbifold, then Γ has a presentation of the form α 1 , β 1 , . . . , α g 0 , β g 0 , ξ 1 , . . . , ξ ℓ \| ξ n 1 1 = · · · = ξ n ℓ ℓ = ℓ i=1 ξ i g i=1 [α i , β i ] = 1 . We represent Γ by a tuple (g 0 ; n 1 , n 2 , . . . , n ℓ ) which is called its signature, and we write Γ(g 0 ; n 1 , n 2 , . . . , n ℓ ) := Γ. Let Homeo + (S g ) denote the group of orientation-preserving homeomorphisms on S g . Given a finite group H < Homeo + (S g ), a faithful properly discontinuous H-action on S g induces a branched covering S g → O H := S g /H, which has ℓ branched points (or cone points) x 1 , . . . , x ℓ in the quotient orbifold O H ≈ S g 0 of orders n 1 , . . . , n ℓ , respectively. Thus, O H has a signature given by Γ(O H ) := (g 0 ; n 1 , n 2 , . . . , n ℓ ), and its orbifold fundamental group is given by π orb 1 (O H ) := Γ(g 0 ; n 1 , n 2 , . . . , n ℓ ). From orbifold covering space theory, the orbifold covering map S g → O H corresponds to an exact sequence 1 → π 1 (S g ) → π orb 1 (O H ) φ H − − → H → 1. This leads us to the following result [4] due to Harvey. For g ≥ 1, let H = F be a finite cyclic subgroup of Mod(S g ) of order n. By the Nielsen-Kerckhoff theorem [6,11], we may also regard H as a finite cyclic subgroup of Homeo + (S g ) generated by an F of order n. We call F a standard representative of the mapping class F . For notational simplicity, we will also denote the standard representative F by F . We refer to both F and the group it generates, interchangeably, as a Z n -action on S g . Moreover, F corresponds to an orbifold O H ≈ S g /H (called the corresponding orbifold), where for each i, the cone point x i lifts to an orbit of size n/n i on S g . The local rotation induced by F around the points in the orbit is given by (i) 2g − 2 \|H\| = 2g 0 − 2 + ℓ i=1 1 − 1 n i ,2πc −1 i /n i , where c i c −1 i ≡ 1 (mod n i ). We denote a typical point in O H by [x], where x is a lift of [x] under the branched cover S g → O H . We see that each cone point [x] ∈ O H corresponds to a unique pair in the multiset {(c 1 , n 1 ), . . . , (c ℓ , n ℓ )}, which we denote by (c x , n x ). So, we define P [x] := (c x , n x ), if [x] is a cone point of O H , and (0, 1), otherwise. Definition 2.2. We will now define a tuple of integers that will encode the conjugacy class of a Z n -action on S g . A data set of degree n is a tuple D = (n, g 0 , r; (c 1 , n 1 ), . . . , (c ℓ , n ℓ )), where n ≥ 2, g 0 ≥ 0, and 0 ≤ r ≤ n − 1 are integers, and each c i is a residue class modulo n i such that: (i) r > 0 if, and only if ℓ = 0, and when r > 0, we have gcd(r, n) = 1, (ii) each n i \| n, (iii) for each i, gcd(c i , n i ) = 1, (iv) lcm(n 1 , . . . n i , . . . , n ℓ ) = N , for 1 ≤ i ≤ r, where N = n, if g 0 = 0, and (v) ℓ j=1 n n j c j ≡ 0 (mod n). The number g determined by the Riemann-Hurwitz equation 2 − 2g n = 2 − 2g 0 + ℓ j=1 1 n j − 1 is called the genus of the data set. The following lemma is a consequence of [15,Theorem 3.8] and the results in [4]. Lemma 2.3. For g ≥ 1 and n ≥ 2, data sets of degree n and genus g correspond to conjugacy classes of Z n -actions on S g . The quantity r associated with a data set D will be non-zero if, and only if, D represents a free rotation of S g by 2πr/n. We will avoid writing r in the notation of a data set, whenever r = 0. From here on, we will use data sets to denote the conjugacy classes of cyclic actions on S g . Given a finite-order mapping class F , we define the data set associated with its conjugacy class by D F . Further, for convenience of notation, we also write the data set D as D = (n, g 0 , r; ((d 1 , m 1 ), α 1 ), . . . , ((d r , m r ), α r )), where (d i , m i ) are the distinct pairs in the multiset S = {(c 1 , n 1 ), . . . , (c ℓ , n ℓ )}, and the α i denote the multiplicity of the pair (d i , m i ) in S. Induced automorphisms on quotient orbifolds Consider a finite group H < Homeo + (S g ), and a subgroup H ′ ⊳ H. Then it is known [17] that the actions of H and H ′ on S g induces an action of H/H ′ on O H ′ . In this section, we analyze this induced action for the case when H is a two-generator finite abelian group, and H ′ is one of its cyclic factor subgroups. We will derive several properties of these induced actions, which will form the core of the theory that we develop in this paper. Definition 3.1. Let H < Homeo + (S g ) be a finite cyclic group. We say aF ∈ Homeo + (O H ) is an automorphism of O H ifF ([x]) = [y], for some [x], [y] ∈ O H , then P [x] = P [y] . We denote the group of automorphisms of O H by Aut(O H ). We derive three technical lemmas, which give necessary conditions under which a given orbifold automorphism is induced by a finite-order map. These lemmas will be used extensively in subsequent sections. G m ([x]) = [G m (x)] = [x], for [x] ∈ S g / F . To prove (iii), we first assume that t := \|Ḡ\| < m. Suppose we assume on the contrary that F l = G k , for any 1 ≤ l < n and 1 ≤ k < m. Then G t ([x]) = [x] ⇔ [G t (x)] = [x], for all [x] ∈ O H . Thus, for each [x] ∈ O H , there exists 1 ≤ l x ≤ n such that G t F lx (y) = y, for all y ∈ S g in the preimage of [x] under the branched cover S g → O H . If t < m, then for each l x , G t F lx is a non-trivial homeomorphism, which shows that every point of S g is fixed by some element of the abelian group F, G of order mn, which is impossible. The converse follows directly from the definition ofḠ. We call the mapḠ in Lemma 3.2 the induced map on O F by G. For an action of a group G on a set X, we denote the stabilizer of a point x ∈ X by Stab G (x). We will also need the following well known result [10, Proposition 3.1] from the theory of finite group actions on surfaces. Lemma 3.4. Let F, G ∈ Homeo + (S g ) be of orders n, m, respectively, and letF ∈ Homeo + (O G ) be induced by F as in Lemma 3.2. Suppose that F G = GF , and F p = G q , for any 1 < q < n and 1 < p < m. If for some x ∈ S g , G k (x) = x andF l ([x]) = [x] , for some 1 ≤ k < m and 1 ≤ l < n, then \|F l \| = ba, where gcd(b, m) = 1 and a \| m \|G k \| . Proof. It suffices to establish the result for the case when \|G k \| = m, that is, for k = 1. Suppose we assume on the contrary that \|F l \| = b, where gcd(m, b) = α = 1. Then there exists 1 ≤ t ≤ m such that G t F l (x) = x. Thus, we have that G m α , F lb α ∈ Stab A (x), where A = F, G . Since Stab A (x) is cyclic and \|G m α \| = \|F lb α \| = α, we have G m α ∈ F lb α , which is impossible. Hence, our assertion follows. Lemma 3.5. Let G, F ∈ Homeo + (S g ) be commuting homeomorphisms of orders m, n, respectively. LetF be the induced map on S g / G as in Lemma G m/nx (y ′ ) = G m/nx (F (x ′ )) = F (G m/nx (x ′ )) = F (x ′ ) = y ′ , where P [x] = (c x , n x ) . By a similar argument, we can show that G ny (x ′ ) = x ′ , and so it follows that n x = n y . To show that c x = c y , it now suffices to show that ifF ([x]) = [y], where n x = n y = m, then c x = c y . Without loss of generality, we assume that c y = 1. Now, there exists an G-invariant disk D 2 around y that G rotates by 2π/m, and there exists a is maximum number cone points of order n in O G i F , which completes the argument. G-invariant disk D 1 around x that G rotates by 2πc −1 x /m. So, we must have F GF −1 = G cx , which is impossible, as F and G commute. (ii) Suppose that F has m fixed points {x 1 , . . . x m } that form an orbit under the action of G on S g . Then, it is clear thatF ([x 1 ]) = [x 1 ], from which the assertion follows. (iii) If F (x) = x, then by definition,F ([x]) = [x], and so we have F (G i (x)) = G i (x), for each i. If The necessary conditions that appear in lemmas above, under which a given orbifold automorphism is induced, are summarized in the following two definitions. Definition 3.6. Let F, G ∈ Homeo + (S g ) be of orders n and m respectively, and let H = G . We say a mapF ∈ Aut(O H ) satisfies the induced map property (IMP) with respect to (F, G), if the following conditions hold. (i) For [x], [y] ∈ O H , ifF ([x]) = ([y]), we have P x = P y . (ii) For each orbit O of size \|F \| induced by the action of F on O G , there exists a point [x(O)] ∈ O Ḡ such that P [x(O)] = P [y] , where [y] ∈ O. (iii) Let F have β fixed points in S g . Ifβ denotes the number of fixed points ofF , then β m ≤β ≤ (m − 1)(2g − 2 + 2n) m(n − 1) + β m . (iv) If [x] is a cone point of order n ′ in O H , thenF l ([x]) = [x], only if \|F l \|= ba, where gcd(b, m) = 1 and a \| m n ′ . Definition 3.7. Let F, G ∈ Homeo + (S g ) be finite-order maps with D F = (n, g 1 , r 1 ; ((c 1 , n 1 ), α 1 ), . . . , ((c r , n r ), α r )) and D G = (m, g 2 , r 2 ; ((d 1 , m 1 ), β 1 ), . . . , ((d k , m k ), β k )), where m \| n. Then (G, F ) are said to form an essential pair if the following three conditions hold. (i) There exists a F ∈ Homeo + (S g 2 ) with D F = (n, g 0 , r o 1 ; (c o 1 , n o 1 ), . . . , (c o s , n o s )) on S g 2 which induces anF ∈ Aut(O G ) that satisfies the IMP with respect to (F, G). (ii) There exists a G ∈ Homeo + (S g 1 ) with D G = (m, g 0 , r o 2 ; (d o 1 , m o 1 ), . . . , (d o t , m o t )), which induces aḠ ∈ Aut(O F ) that satisfies the IMP with respect to (G, F ). (iii) Γ(O G / F ) = Γ(O F / Ḡ ). The number mn (written as m · n) is called the order of the essential pair (G, F ). Example 3.8. Let F, G ∈ Homeo + (S 7 ) with D F = D G = (6, 2, 1; ). Then (G, F ) is an essential pair of order 6 · 6, as F, G induceF ,Ḡ ∈ Homeo + (S 2 ) (resp.) with DF = DḠ = (6, 0; ((1, 2), 2), (1, 3), (2, 3)), and Γ(O G / F ) = Γ(O F / Ḡ ) = (0; 2, 2, 3, 3). Given a quotient orbifold O H , where H = F , we now state a set of necessary conditions (as we will show later in Theorem 4.8) for a given G ∈ Aut(O H ) to be induced by a finite-order map G such that G, F forms a two-generator abelian group. Definition 3.9. For finite-order maps F, G ∈ Homeo + (S g ), let (G, F ) form an essential pair of order m · n as in Definition 3.7. Then (G, F ) is said to be a weakly abelian pair of order m · n if the following conditions hold. ( 6), 2), (2, 3)), D G = (2, 0; ((1, 2), 6)), respectively. Then (G, F ) is an essential pair of order 2 · 6, with DF = (6, 0; (1, 6), (5, 6)) and DḠ = (2, 0; ((1, 2), 2)), where Γ(O G / F ) = Γ(O F / Ḡ ) = (0; 2, 6, 6). It is easy to check that (G, F ) is also a weak abelian pair of order 2 · 6. i) If Γ(O G / F ) = Γ(O F / Ḡ ) = (g 0 ; m ′ 1 n ′ 1 , . . . , m ′ l n ′ l ) such that for each i, m ′ i n ′ i = 1 and m ′ i n ′ i \| n. (ii) If g 0 = 0 in condition (iii), then there exist a sub-multiset A = {n 11 , . . . , n l1 } of the multiset B = {m ′ 1 n ′ 1 , . . . , m ′ l n ′ l } such that lcm( A) = lcm({n 11 , . . . , n i1 , . . . , n l1 }) = n and m \| lcm(B \ A). (iii) (a) Denoting lcm({m ′ k n ′ k : m ′ k = 1}) = B 1 , if n ′ i =1 n gcd(n, n ′ i m ′ i ) c i ≡ −δ 2 (mod n), where m ′ i ∈ {1, m o 1 , . . . , m o t } and n ′ i ∈ {1, n 1 , . . . , n r }, then n B 1 \|δ 2 . (b) Denoting lcm({m ′ l n ′ l : n ′ l = 1}) =B 2 , and gcd(B 2 , m) = B 2 , if m ′ i =1 m gcd(m, m ′ i n ′ i ) d i ≡ −δ 1 (mod m), where m ′ i ∈ {1, m 1 , . . . , m k } and n ′ i ∈ {1, n o 1 , . . . , n o s }, then m B 2 \|δ 1 . Example 3.10. Let F, G ∈ Homeo + (S 2 ) with D F = (6, 0; ((1, Given a finite set S of positive integers, we denote the least common multiple of the integers in S by lcm(S). In order to improve the clarity of exposition, we will divide the proof our main result into four subcases, of which the first two cases (that will form bulk of our proof) assume the following condition on the quotient orbifolds (of the cyclic factor subgroups). Definition 3.11. Let H < Homeo + (S g ) be a finite cyclic group, and let Γ(O H ) = (g 0 ; n 1 , . . . , n ℓ ). We say the action of H on S g satisfies the lcm condition if lcm({n 1 , . . . , n ℓ }) = \|H\|. We conclude this section with another lemma that will be used in one of the subcases of our main result. Lemma 3.12. Let F, G ∈ Homeo + (S g ) be of orders n and m, respectively. If F G = GF and S g / F, G ≈ S 0 , then there exists a F ′ ∈ F, G of order n such that the action of F ′ on S g satisfies the lcm condition. Proof. Let H = G . Consider the mapF ∈ Aut(O H ) induced by F . Since O H / F = S g / F, G , the action ofF on O H satisfies the lcm condition. Let DF = (n, 0; (c ′ 1 , n ′ 1 ), . . . , (c ′ s , n ′ s )). Consider a minimal subset {n 11 , . . . , n 1l } of the multiset {n ′ 1 , n ′ 2 , . . . , n ′ s } with the property lcm({n 11 , . . . , n 1l }) = n. Now, for each n 1i , there exists l i such that G l i F nc 1i n 1i (x i ) = x i , for some x i ∈ S g . It is apparent that \|G l i F nc 1i n 1i \| ≥ n 1i . For each 1 ≤ i ≤ l, we choose an appropriate power of G l i F nc 1i n 1i that we denote by F ′ i , so that gcd(\|F ′ i \|, \|F ′ j \|) = 1, when i = j, and lcm({\|F ′ 1 \|, . . . , \|F ′ l \|} = n. Thus, the assertion follows by choosing F ′ = F ′ 1 F ′ 2 . . . F ′ l . Main theorem By a two-generator finite abelian action of order mn (written as m · n), we mean a tuple (H, (G, F )), where m \| n and H < Homeo + (S g ), and H = G, F \| G m = F n = 1, [F, G] = 1 . Definition 4.1. Two finite abelian actions (H 1 , (G 1 , F 1 )) and (H 2 , (G 2 , F 2 )) or order m · n are said to be weakly conjugate if there exists an isomorphism, ψ : π orb 1 (O H 1 ) ∼ = π orb 1 (O H 2 ) and an isomorphism χ : H 1 → H 2 such that (i) χ((G 1 , F 1 )) = (G 2 , F 2 ), (ii) (χ•φ H 1 )(g) = (φ H 2 •ψ)(g), whenever g ∈ π orb 1 (O H 1 ) is of finite-order, and (iii) the pair (G 1 , F 1 ) is conjugate (component-wise) to the pair (G 2 , F 2 ) in Homeo + (S g ). The notion of weak conjugacy induces an equivalence relation on the twogenerator finite abelian subgroups of Homeo + (S g ), and we will call the equivalence classes as weak conjugacy classes. We will now define an abstract tuple of integers that encode, as we will see shortly in Proposition 4.3, the weak conjugacy class of a two-generator finite abelian action. where m, n ≥ 2, g 0 ≥ 0, and g ≥ 2 are integers satisfying the following conditions: (i) m \| n, (ii) 2g − 2 mn = 2g 0 − 2 + r i=1 1 − 1 n i , (iii) lcm(n 1 , . . . , n r ) = lcm(n 1 , . . . , n k , . . . , n r ) = N, and if g 0 = 0, then N = n, (iv) for each i, n i1 \|m, n i2 \|n, and lcm(n i1 , n i2 ) = n i , (v) for each i, j, either (c ij , n ij ) = 1, or c ij = 0, and c ij = 0, if, and only if, n ij = 1, . . , n r ). Let the presentation of Γ and Z m ⊕ Z n be given by (vi) r i=1 m n i1 c i1 ≡ 0 (mod m) and r i=1 n n i2 c i2 ≡ 0 (mod n), and (vii) when g 0 = 0, there exists (ℓ 1 , . . . , ℓ r ), (k 1 , . . . , k r ) ∈ Z r such that (a) r i=1 n n i1 c i1 ℓ i ≡ 0 (mod n) and r i=1 m n i2 c i2 ℓ i ≡ 1 (mod m), and (b) r i=1 n n i1 c i1 k i ≡ 1 (mod n) and r i=1 m n i2 c i2 k i ≡ 0 (mod m).α 1 , β 1 , . . . , α g 0 , β g 0 , ξ 1 , . . . , ξ ℓ \| ξ n 1 1 = · · · = ξ n ℓ ℓ = ℓ i=1 ξ i g i=1 [α i , β i ] = 1 and Z m ⊕ Z n = x, y \| x m = y n = [x, y] = 1 . First, we show the result for the case when g 0 = 0. We consider the map ξ i → x m n i1 c i1 y n n i2 c i2 , for 1 ≤ i ≤ r. Since \|x m n i1 c i1 \| = n i1 and \|y n n i2 c i2 \| = n i2 , condition (iv) implies that φ is an order-preserving map. Moreover, condition (vi) implies that φ satisfies the long relation r i=1 ξ i = 1. In order to show that φ is surjective, we establish that φ(Γ) generates the group Z m ⊕ Z n . But condition (vii) ensures that {φ(ξ i ) : 1 ≤ i ≤ r} generates Z m ⊕ Z n , and hence it follows that D determines a Z m ⊕ Z n -action on S g . When g 0 > 0, π orb 1 (O H ) also has hyperbolic generators (i.e. the α i and the β i ), which can be mapped surjectively to the generators of Z m ⊕ Z n . Conversely, suppose that there is a Z m ⊕ Z n -action D on S g such that O D had genus g 0 . Then by Theorem 2.1, there exists a surjective homomorphism φ : Γ → Z m ⊕ Z n : ξ i → x m n i1 c i1 y n n i2 c i2 , for 1 ≤ i ≤ r, that is order-preserving on the torsion elements. This yields an abelian data set of degree m·n and genus g as in Definition 4.2, and the result follows. where the suffix 5 in the second data set denotes the multiplicity of the subtuple [(1, 2), (1, 2), 2]. We will discuss such actions in more detail in Section 5. To each F ∈ Mod(S g ) of order n, we may associate a standard representative F ∈ Homeo + (S g ) of the same order whose conjugacy class we denote by D F . For a group G, if g, h ∈ G weakly commute, then we denote it by g, h = 1. It is clear from Definition 4.5 that if g, h = 1, then g and h cannot commute in G. Remark 4.6. It follows immediately from Definition 4.5 and the Nielsen-Kerckhoff theorem that given F, G ∈ Homeo + (S g ) of finite-order, F, G = 1 if, and only if, as mapping classes, they satisfy F, G = 1 in Mod(S g ). The proof of the main theorem we will also require the following elementary number-theoretic lemma. We will now state the main result in the paper. Proof. Let \|F \| = n and \|G\| = m, where m \| n, and let H = F . Let D F = (n, g 1 , r 1 ; ((c 1 , n 1 ), α 1 ), . . . , ((c r , n r ), α r )) and D G = (m, g 2 , r 2 ; ((d 1 , m 1 ), β 1 ), . . . , ((d k , m k ), β k )), respectively. First, we assume that F, G = 1, and show that (G, F ) form a weakly abelian pair of order m · n. Without loss of generality, we may assume that F and G commute in Mod(S g ). Further, by the Nielsen-Kerckhoff theorem, we may assume up to isotopy that F and G commute in Homeo + (S g ). Then by Lemma 3.2, it follows that (G, F ) forms an essential pair of order m · n. It remains to show that (G, F ) is a weakly abelian pair as in Definition 3.9. Condition (i) in this definition is a consequence of Proposition 3. c i , for 1 ≤ i ≤ r. The group relation r+l i=1 ξ i = g 0 j=1 [α j , β j ] of π orb 1 (O H / Ḡ ) would now imply that r+l i=1 φ(ξ i ) = 1. Thus, either r i=1 n n i c i = 0, or if r i=1 n n i c i = 0, then condition (iii) is necessary. Conversely, suppose that (G, F ) forms a weakly abelian pair of degree m·n as in Definition 3.9. By Remark 4.6, it suffices to show that our assumption yields an abelian data set as desired. Case 1: Let lcm({n 1 , . . . , n r }) = n. We further assume that m ′ i n ′ i = B 1 , where m ′ i = 1, for some i. We may assume, without loss of generality, that i = 1. Then we show that the tuple (m * n, g 0 ; (d ′ 1 , m ′ 1 ) , αc ′ 1 + δ ακ , m ′ 1 n ′ 1 κ , m ′ 1 n ′ 1 , (d ′ 2 , m ′ 2 ) , c ′ 2 κ 2 , m ′ 2 n ′ 2 κ 2 , m ′ 2 n ′ 2 , . . . , (d ′ l , m ′ l ) , c ′ l κ l , m ′ l n ′ l κ l , m ′ l n ′ l ), where gcd(c ′ j , m ′ j ) = κ j , κ = gcd(c 1 + δ α , m ′ 1 n ′ 1 ), α = n m ′ t 1 n ′ t 1 , d ′ t i = 0, if m ′ i / ∈l i=1 n m ′ i n ′ i c ′ i + δ 2 ≡ 0 (mod n) and l i=1 m m ′ i d ′ i ≡ 0 (mod m), which yields condition (vi). It now remains to show (vii), when g 0 = 0. (m * n, g0; d ′ 1 , m ′ 1 , α1c ′ 1 + δ ′ 1 α1ξ ′ 1 , m ′ 1 n ′ 1 ξ ′ 1 , m ′ 1 n ′ 1 , . . . , d ′ p , m ′ p , αpc ′ p + δ ′ p αpξ ′ p , m ′ p n ′ p ξ ′ p , m ′ p n ′ p , d ′ p+1 , m ′ p+1 , c ′ p+1 ξp+1 , m ′ p+1 n ′ p+1 ξp+1 , m ′ p+1 n ′ p+1 , . . . , d ′ l , m ′ l , c ′ l ξ l , m ′ l n ′ l ξ l , m ′ l n ′ l ), () where ξ ′ j = gcd({c j + δ ′ j α j , m ′ j n ′ j : 1 ≤ j ≤ p}) and gcd(c ′ i , m ′ i ) = ξ i , for p + 1 ≤ i ≤ l. As before, this tuple will satisfy all the conditions of an abelian data set. . . , n r }) < n, and g 0 > 1. Then the abelian data set and the representation φ from Case 1 also works for this case. Case 4: Let lcm({m 1 , . . . , m k }) < m, lcm({n 1 , . . . , n r }) < n, and g 0 = 0. Then by Lemma 3.12, it follows that there exists an F ′ ∈ F, G such that \|F ′ \| = n and D F ′ = (g ′ 0 ; (c 1 , n 1 ), . . . , (c r , n r )) satisfies lcm({n 1 , . . . , n r }) = n. Since (G, F ) is a weakly abelian pair, so is (G, F ′ ), and hence this case reduces to Case 1. Applications In this section, we derive several applications of the theory developed in the earlier section. Weak commutativity of involutions. It is well known that the conjugacy class of an involution F ∈ Mod(S g ) is represented by D F = (2, g 0 ; ((1, 2), k)), where k = 2(g − 2g 0 + 1), if F is a non-free action on S g , ans D F = (2, (g + 1)/2, 1; ), otherwise. In this subsection, we will derive conditions under which two involutions in Mod(S g ) will weakly commute. 2), 2k ′ )) and D G = (2, g ′′ 0 , r ′′ ; ((1, 2), 2k ′′ )), respectively. Then F, G = 1 if, and only if, the following conditions hold. (a) There existsḠ ∈ Homeo + (S g ′ 0 ) with DḠ = (2, g 0 , r 1 ; ((1, 2), 2s ′′ )) such Corollary 5.1. Let F, G ∈ Mod(S g ) be involutions such that D F = (2, g ′ 0 , r ′ ; ((1,that g + k ′′ + 1 ≥ 2s ′′ ≥ k ′′ . (b) There existsF ∈ Homeo + (S g ′′ 0 ) with DF = (2, g 0 , r 2 ; ((1, 2), 2s ′ )) such that g + k ′ + 1 ≥ 2s ′ ≥ k ′ . Proof. It suffices to show that conditions (a) -(b) mentioned above hold true if, and only if, (G, F ) is a weakly abelian pair. If (G, F ) is a weakly abelian pair, then it is apparent that (a) -(b) hold. Conversely, it is easy to see that conditions (a) -(b) imply that (G, F ) is an essential pair. It remains to show that conditions (i) -(iii) of Definition 3.9 hold true. A simple application of the Riemann-Hurwitz equation to the four data sets that appear in the statement above leads to a system of (four) linear equations, which can be simplified to yield the condition: 2s ′ − k ′ = 2s ′′ − k ′′ , from which (i)-(ii) follow. When g is odd, 4 \| r i=1 α i n n i c i , and so each δ i appearing in (iii) is 0. If g is even, then as no involution generates a free action, we have B i = 2. Thus, condition (iii) is satisfied, and the assertion follows. Let the conjugacy classes D F = (2, g ′ 0 , r ′ ; ((1, 2), 2k ′ )) and D G = (2, g ′′ 0 , r ′′ ; ((1, 2), 2k ′′ )), be represented by involutions F and G, which commute. Then, by Corollary 5.1, we have D F G = (2, g 0 , r ′′′ ; ((1, 2), 2k)), where k = 2s ′ − k ′ = 2s ′′ − k ′′ . Using this idea, one can obtain a geometric realization of a Klein 4-subgroup K 4 of Mod(S g ) by obtaining an isometric embedding of ι : S g ֒→ R 3 that is symmetric about origin such that ι(S g ) intersects, the x-axis at 2k ′ points, the y-axis at 2k ′′ points, and the z-axis at 2k points. It is now apparent that under this embedding the non-trivial elements of K 4 are realized as π-rotations about the three coordinates axes. This property is illustrated in the following example. In fact, all Klein 4-subgroups of Mod(S g ) can be realized in an analogous manner. 5.2. Finite abelain groups with irreducible finite-order mapping classes. We say a Z n -action is irreducible if it is irreducible as a mapping class. By a result of Gilman [3], this is equivalent to requiring that the corresponding orbifold of the action is a sphere with 3 cone points. Following the nomenclature in [1] and [14], a Z n -action on S g is said to be rotational if it can be realized as a rotation about an axis under a suitable isometric embedding of S g ֒→ R 3 . A non-rotational action is said of be of Type 1 if its quotient orbifold has signature (g 0 ; n 1 , n 2 , n), otherwise, it is called a Type 2 action. The following corollary characterizes the weak commutativity of Type 2 actions with finite-order maps. We now give a similar characterization for the weak commutativity of Type 1 actions . Corollary 5.4. Suppose that there exists a finite non-cyclic abelian subgroup A of Mod(S g ) that contains an irreducible Type 1 action. Then A ∼ = Z 2 ⊕ Z 2g+2 . Proof. Let F be an irreducible Type 1 action with Γ(O F ) = (0; n 1 , n 2 , n 3 ). Since F is of Type 1, there exists atleast one n i (say n 1 ) such that n 1 = n, and the following cases arise. Case 1: n 2 = n 3 and n 2 , n 3 < n. By an argument analogous to the one used in the proof of Corollary 5.3, it follows that F does not commute with any other finite-order element of Mod(S g ). Case 2 : n i = n, for 1 ≤ i ≤ 3. Then by the Riemann-Hurwitz equation, we have that n = 2g + 1. By applying a result of Maclachlan [8] that bounds the order of a finite abelian subgroup of Mod(S g ) by 4g + 4, it follows that only an involution can commute with F . When such an involution G does commute with F , it follows immediately that F, G ∼ = Z 4g+2 . Case 3: n 1 = n 2 = n = n 3 . Once again, by similar arguments as above, we can conclude that F cannot commute with any other finite-order G ∈ Mod(S g ) with \|G\| ≥ 3. When F commutes with an involution G, the induced mapḠ ∈ Aut(O F ) fixes the cone point of order n 3 in O F and permutes the remaining 2 cone points. Consequently, we have F, G ∼ = Z 2 ⊕ Z n . By the Riemann-Hurwitz equation, it follows that n ≥ 2g + 1, and hence n = 2g + 2, as 2n ≤ 4g + 4. 5.3. Weak commutativity with free cyclic actions. Any non-trivial finite m-sheeted cover of S g , for g ≥ 2, has the form p : S m(g−1)+1 → S g , where p is a covering map. Given such a cover p, let LMod p (S g ) be the subgroup of Mod(S g ) of mapping classes that lift under the cover. It is natural question to ask whether a given F ∈ Mod(S g ) of finite-order will have a conjugate F ′ such that F ′ ∈ LMod p (S g ). In this subsection, we answer this question for certain types of finite-order maps. We begin by determining when certain types of free cyclic actions weakly commute with other cyclic actions. Corollary 5.5. Let F, G ∈ Mod(S g ) with D F = (n, g 1 , r; ) and D G = (m, g 0 , r ′ ; ((d 1 , m 1 ), β 1 ), . . . , ((d k , m k ), β k )), respectively. Suppose that F induces a free action on O G . Then F, G = 1 if, and only if: (i) β j = 0 (mod n), for 1 ≤ j ≤ k, (ii) n\|(g 0 − 1), and (iii) k i=1 β i n m m i c i ≡ 0 (mod m) . Proof. We show that conditions (i) -(iii) are sufficient, as it follows directly from Theorem 4.8 that they are necessary. By conditions (i) -(ii) of our hypothesis, it follows that there exists a free action on S g 0 , which induces anF ∈ Aut(O G ). The Riemann-Hurwitz equation and Lemma 2.1 imply that there exists aḠ ∈ Aut(O F ) with DḠ = (m, (g 0 − 1) n + 1, r ′′ ; ((d 1 , m 1 ), β 1 n ), . . . , ((d k , m k ), β k n )). Hence, it follows that (G, F ) forms an essential pair, and the fact that they form an abelian pair follows directly from our hypothesis. In the following result, we show that a finite-order mapping class whose corresponding orbifold has genus > 0 has a conjugate that is liftable under a finite-sheeted cover of S g . Corollary 5.6. Consider an F ∈ Mod(S g ) of finite-order such that O F ≈ S 0 . Let p : S m(g−1)+1 → S g be an m-sheeted cover whose deck transformation group is given by G ∼ = Z m . Then there exists a conjugate F ′ of F such that F ′ ∈ LMod p (S g ). Proof. Let D F = (n, g 0 , r; (c 1 , n 1 ), . . . , (c r , n r )). Then by Corollary 5.5, we have thatF ∈ Mod(S m(g−1)+1 ) with DF = (n, m(g 0 − 1) + 1,r; ((c 1 , n 1 ), m), . . . , ((c r , n r ), m)) such that G,F = 1. Without loss of generality, we may assume that G andF commute in Homeo + (S g ). By the IMP, it now follows thatF induces F ′ ∈ Mod(S g ) that is conjugate to F . In the following corollary, we provide conditions under which certain finiteorder mapping class whose corresponding orbifolds are spheres have conjugates that lift under a finite cover of S g . Corollary 5.7. Let F ∈ Mod(S g ) with D F = (n, 0; (c 1 , n 1 ), . . . , (c r , n r )). Let p : S m(g−1)+1 → S g be an m-sheeted cover whose deck transformation group is given by G ∼ = Z m . Then there exists a conjugate F ′′ of F such that F ′′ ∈ LMod p (S g ), if the following conditions hold. Proof. Consider an F ′ ∈ Mod(S m(g−1)+1 ) with D F ′ = D. It is straightforward to check that (G, F ′ ) forms a weakly abelian pair. Thus, by Theorem 4.8, we have that F ′ , G = 1. So, F ′ induces F ′′ ∈ Mod(S g ) that is conjugate to F . 5.4. Primitivity of finite-order mapping classes. Let G be group, we say an x ∈ G has root of degree n if there exists y ∈ G such that y n = x. If a g ∈ G has no root of any degree greater than one, then g is said to be primitive in G. It is known [18] that the order a finite cyclic subgroup of Mod(S g ) is bounded above by 4g + 2. This would imply that no finite-order mapping class with order > 2g + 1 can have a nontrivial root. The following proposition gives conditions under which an arbitrary finite-order mapping class can have a root. {n 1 , . . . , n r } ∪ {m i \| gcd(m i , n) = 1} ∪ {n j m i \| gcd(m i , n) = 1} ∪ {nm j }. (iii) There exist a F ′ ∈ Mod(S g ) with D F ′ = (mn, g ′ , r ′′ ; (c ′ 1 , n ′ 1 ), . . . , (c ′ l , n ′ l )) such that for each i, c ′ i ≡ c j , if n ′ i = n j , and c j (mod n j ), if n ′ i = n j m i . Proof. First, we note that the conjugacy of (F ′ ) m is represented by D F . Thus, we have that (F ′ ) m and F are conjugate. So, we can find a conjugate of F ′ , say F , such that F m = F. Hence, F has a root of order m. Conversely, suppose that F has a root F ′ of order m. Then we show that conditions (i) -(iii) hold. Since Note that, t i ∈ {n 1 , . . . , n r } ∪ {m 1 , . . . , m k } ∪ {n i m j \|1 ≤ i ≤ r, 1 ≤ j ≤ k}. So, it remains to prove if t i = m j , then gcd(m j , n) = 1, and if t i = m j n k then either n k = n or gcd(m j , n) = 1. However, this follows directly from the structures of D F ′ and D F . A consequence of this theorem is the following corollary, which pertains to the roots of a mapping class of order g − 1. Corollary 5.9. Let F ∈ Mod(S g ) be represented by the generator of a free cyclic action on S g , and let F ′ be a nontrivial root of F (if it exists). Then: By arguments similar to those in Corollary 5.9, we can show that: (i) O F ′ ≈ S 0 , Corollary 5.10. If 6 \| g, then an F ∈ Mod(S g ) with D F = (g, 1; (c, g), (g − c, g)) is primitive. 5.5. Weak commutativity of finite-order maps with the roots of Dehn twists. Let c be a simple closed curve in S g , for g ≥ 2, and let t c ∈ Mod(S g ) denote the left-handed Dehn twist about c. A root of t c of degree n is an F ∈ Mod(S g ) such that F n = t c . Consider an F ∈ Mod(S g ) that is either an order-n mapping class that preserves c, or a root of t c of degree n. Then up to isotopy, we can assume that F (c) = c, and that F preserves a closed annular neighborhood N of c. Let S g (c) denote the surface obtained by capping off the components of S g \ N . Then by the theory developed in [9,12,14], it follows that F induces an order-n map F c ∈ Homeo + ( S g (c)) by coning. The following remark describes the construction of a root of a t c , when c is nonseparating. Remark 5.11. When c is nonseparating, it is well known [9] that (up to conjugacy) a root F of t c of degree n determines a Z n -action F c on S g−1 , which has two (distinguished) fixed points on S g (c), where it induces rotation angles add up to 2π/n (mod 2π). (We will call such an action a nonseparating root-realizing Z n -action.) Conversely, consider a Z n -action on S g−1 , which has two (distinguished) fixed points, where it induces rotation angles that add up to 2π/n (mod 2π). Then we can remove invariant disks around the fixed points and attach a 1-handle N with an 1/n th twist connecting the resulting boundary components to obtain a root of Dehn twist about the nonseparating curve in N . Moreover, it was shown in [9,12] that no root of t c can switch the two sides of c. Remark 5.12. Suppose that a Z m -action G ∈ Mod(S g ) preserves a curve c. Then G induces an order-m map G c on S g (c). In particular S g (c) ≈ S g−1 , if c is nonseparating, and S g (c) ≈ S g 1 ⊔S g 2 (in symbols S g = S g 1 # c S g 2 ), where g 1 + g 2 = g, when c is separating. Let N be a closed annular neighborhood of c such that G(N ) = N . Then the two distinguished points P, Q that lie at the center of the capping disks (of the two boundary components of the surface S g \ N ) are either fixed under the action of G c , or form an orbit of size 2. Conversely, given a Z m -action G c on a surface (≈ S g (c)) with two distinguished points P, Q, which are either fixed with locally induced rotation angles (around P and Q) adding up to 0 (mod 2π), or form a orbit of size 2, we may reverse the above process to obtain Z m -action on S g . Note that by [14] P, Q is an orbit of size 2, only when \| G c \| = 2. This leads us to the following characterization of weak commutativity of finite-order maps with roots of Dehn twists about nonseparating curves. Proof. Suppose that F, G = 1. Then up to conjugacy, we assume that F commutes with G, and so we have t c = Gt c G −1 = t G(c) . Hence, we may assume up to isotopy that G(c) = c, and both G and F preserve the same annular neighborhood N of c. Thus, F c and G c , which are induced by F and G, respectively, must commute as maps on S g−1 , and so it follows that F c , G c ′ = 1. Conversely, let us assume conditions (i) -(ii) hold true. Then F c and G c share the same set of two distinguished points P and Q (as in Remark 5.11) that are either fixed or form an orbit of size 2, under their actions. By Remarks 5.11-5.12, we construct maps F and G, which commute in Homeo + (S g ). Therefore, as mapping classes they satisfy F, G = 1. Let H = F c , G c . To show the final part of the assertion, we first observe that Stab H (P ) = H, when \|G\| > 2. Since H is cyclic (by Lemma 3.3), it follows that F c has a root of degree \|G\|. Further, it was shown in [9] that F is always a root of odd degree. So, when \|G\| = 2, it is apparent that H is cyclic. Therefore, if F c is primitive, then F and G cannot commute in Mod(S g ). Note that the conditions gcd(\| F c \|, \| G c \|) = 1 and \| F c \|\| G c \| ≤ (4(g − 1) + 2) determine an upper bound for \|G\|. Remark 5.14. Let c is a separating curve in S g so that S g = S g 1 # c S g 2 . It is known [13] that (up to conjugacy) a root F of t c of degree n corresponds to a pair F c = ( F 1,c , F 2,c ) of finite order maps, where F i,c ∈ Homeo + (S g i ) with \| F i,c \| = n i , for i = 1, 2, with distinguished fixed points P i ∈ S g i around which the locally induced rotational angles θ i , which satisfy θ 1 + θ 2 ≡ 2π/n (mod 2π), where n = lcm(n 1 , n 2 ). Further, if G is a finite-order map with G(c) = c and \|G\| > 2, then there is a decomposition of G c into a pair of actions ( G 1,c , G 2,c ), where G i,c is a Z m -action on S g i , for i = 1, 2. However, when \|G\| = 2, G c is either a single action on S g (c) that permutes the components S g i (in which case g 1 = g 2 ), or it decomposes into a pair of actions ( G 1,c , G 2,c ) as before. The ideas in Remarks 5.12 and 5.14 lead to the following analog of Corollary 5.13 for the roots of Dehn twists about separating curves. Corollary 5.15. Let c is a separating curve in S g so that S g = S g 1 # c S g 2 . Let F ∈ Mod(S g ) be a root of t c so that F c = ( F 1,c , F 2,c ). Then a G ∈ Mod(S g ) of finite order satisfies F, G = 1 if, and only if: (i) G(c) = c, and (ii) either G c = ( G 1,c , G 2,c ) and F i,c , G i,c = 1, for i = 1, 2, or F 1,c is conjugate with F 2,c . Hyperbolic structures realizing abelian actions In [1] and [14], a procedure to obtain the hyperbolic structures that realize cyclic subgroups of Mod(S g ) as isometries was described. In this section, we use this procedure, and theory developed in Sections 3-4 to give an algorithm for obtaining the hyperbolic structures that realize a given twogenerator finite abelian subgroup of Mod(S g ) as an isometry group. Given a finite subgroup H < Mod(S g ), let Fix(H) denote the subspace of fixed points in the Teichmuller space Teich(S g ) under the action of H. With this notation in place, we have the following elementary lemma. Lemma 6.1. Let F, G ∈ Mod(S g ) be commuting finite-order mapping classes. Then Fix( F, G ) = Fix( F ) ∩ Fix( G ). Proof. Suppose that x ∈ Fix( F, G ). Then x ∈ Fix( F ) and x ∈ Fix( G ), and so x ∈ Fix( F ) ∩ Fix( G ) Conversely, given x ∈ Fix( F ) ∩ Fix( G ), thus F (x) = G(x) = x so F l G k (x) = x, for all l, k, which implies that x ∈ Fix( F, G ). In [1,14], it was shown that: Theorem 6.2. For g ≥ 2, consider a F ∈ Mod(S g ) with D F = (n, g 0 ; (c 1 , n 1 ), (c 2 , n 2 ), (c 3 , n)). Then F can be realized explicitly as the rotation θ F of a hyperbolic polygon P F with a suitable side-pairing W (P F ), where P F is a hyperbolic k(F )-gon with k(F ) := 2n(1 + 2g 0 ), if n 1 , n 2 = 2, and n(1 + 4g 0 ), otherwise, and for 0 ≤ m ≤ n − 1, W (P F ) =          n i=1 Q i a 2i−1 a 2i with a −1 2m+1 ∼ a 2z , if k(h) = 2n, and n i=1 Q i a i with a −1 m+1 ∼ a z , otherwise, where z ≡ m + qj (mod n), q = (n/n 2 )c −1 , j = n 2 − c 2 , and Q r = g 0 s=1 [x r,s , y r,s ], 1 ≤ r ≤ n. Further, when g 0 = 0, this structure is unique. Suppose that a Z m -action on S g induces a pair of orbits of size r, where the induced rotation angles add up to 0 (mod 2π). Then we can remove cyclically permuted Z m -invariant disks around points in the orbits and then identifying the resultant boundary components to obtain a Z m -action on S g+m−1 . This construction is called a self r-compatibility, and we say that G as above admits a self r-compatibility. Conversely given a Z m -action on S g that admits a self r-compatibility, we can reverse the construction described above to recover the Z m -action on S g . Further, it was shown that a non-rotational Type 2 action can be realized from finitely many pairwise r-compatibilities between Type 1 actions. Given a weak conjugacy class of an abelian action (H, (G, F )) represented by (m · n, g 0 ; [(c 11 , n 11 ), (c 12 , n 12 ), n 1 ], . . . , [(c r1 , n r1 ), (c r2 , n r2 ), n r ]), we will now describe an algorithmic procedure for obtaining the conjugacy classes of its generators. Let H 1 = F and H 2 = G by applying . Step 1. It follows directly from our theory that the data sets DḠ = (m, g 0 ; (c 11 , n 11 ), . . . , (c r1 , n r1 )) and DF = (n, g 0 ; (c 12 , n 12 ), . . . , (c r2 , n r2 )) represent the conjugacy classes of the actionsḠ andF induced on the orbifolds O H 1 and O H 2 by the actions of H 1 and H 2 on S g , respectively. Step 2. We now note that the orbifold signatures Γ(O H i ) have the form Γ(O H 1 ) = (n, g(DḠ); ( with the understanding that if n i /n ij = 1, for some 1 ≤ i ≤ r and j = 1, 2, then we exclude it from the signatures. Step 3. We choose conjugacy classes D F = (n, g 1 ; ((c 1 , n 1 n 11 ), m n 11 ), . . . , ((c r , n r n r1 ), m n r1 )) and D G = (m, g 2 ; ((d 1 , n 1 n 12 ), n n 12 ), . . . , ((d r , n r n r2 ), m n r2 )), where c i ≡ c i2 (mod n i /n i1 ) and d i ≡ c i1 (mod n i /n i2 ). Step 4. Finally, using Lemma 6.1, Theorem 6.2, and the subsequent discussion on the theory developed in [1,14], we can obtain the hyperbolic structures that realize F, G as group of isometries. In Table 1 at then end of this section, we give a complete classification of weak conjugacy classes of two-generator finite abelian subgroups of Mod(S 3 ). Using the algorithm described above, in Figure 5, we provide a geometric realization of the weak conjugacy classes in S.Nos 10-12. The pairs of integers labeled in each subfigure are the pairs P [x] , which correspond to cone points [x] in the quotient orbifold O F . 4), 2)). D F can be realized as a 2compatibility between two actions F ′ and F ′′ , where D F ′ = (4, 0; (1, 2), ((3, 4), 2)) and D F ′′ = (4, 0; (1, 2), ((1, 4), 2)). Note that F ′ and F ′′ are realized rotations of the polygons P F ′ and P F ′′ described in Theorem 6.2. Table 1, with D G = (2, 2, 1; ) and D F = (4, 0; ((1, 4), 2), ((3, 4), 2)). Here, D F can be realized as a 2-compatibility between two actions F ′ and F ′′ (realized as before), where D F ′ = (4, 0; (1, 2), ((3, 4), 2)) and D F ′′ = (4, 0; (1, 2), ((1, 4), 2)). 4), 2)). Here, D F can be realized by 1-compatibilities of the two actions F ′ and F ′′ with F ′′′ , where D F ′ = (4, 0; (1, 2), ((1, 4), 2)), D F ′′ = (4, 0; (1, 2), ((1, 4), 2)), and D F ′′′ = (4, 0; (1, 2), ((3, 4), 2). Again F ′ , F ′′ and F ′′′ are irreducible Type 1 actions realized as rotations of polygons described in Theorem 6.2. Table 1. The weak conjugacy classes of two-generator finite abelian subgroups of Mod(S 3 ). (The suffix refers to the multiplicity of the tuple in the abelian data set.) Note that the actions S.Nos 17-24 in Table 1 have irreducible Type 1 actions as one of their generators. As the structure realizing such an action is unique, by lemma 6.1, the abelian groups representing these weak conjugacy classes are realized as isometry groups by a unique structure. Figure 1 . 1Six conjugate involutions in Mod(S 8 ) . Lemma 3 . 2 . 32Let G, F ∈ Homeo + (S g ) be commuting maps of order m, n, respectively, and let H = F . Then:(i) G induces aḠ ∈ Homeo + (O H ) such that O H / Ḡ = S g / F, G ,(ii) \|Ḡ\| \| \|G\|, and (iii) \|Ḡ\| < m if, and only if, F l = G k , for some 0 < k < m and 0 < l < n.Proof. DefiningḠ[x] = [G(x)], for [x]∈ S g / F , we see the (i) follows immediately. The assertion in (ii) follows from the fact that Lemma 3. 3 . 3Let H < Homeo + (S g ) be finite. Then Stab H (x) is a cyclic group, for every x ∈ S g . F has β fixed points, then there exist atleast β m distinct orbits which contain points fixed by F . Hence, the lower bound follows. To show the upper bound, we observe that ifF ([x]) = [x], then by definition, there exist 0 ≤ i ≤ m − 1 such that G i F (x) = x. When i = 0, by a direct application of the Riemann-Hurwitz equation, it follows that (2g−2+2n) (n−1) Definition 4 . 2 . 42An abelian data set of degree m · n and genus g is a tuple (m · n, g 0 ; [(c 11 , n 11 ), (c 12 , n 12 ), n 1 ], . . . , [(c r1 , n r1 ), (c r2 , n r2 ), n r ]), Proposition 4. 3 . 3For m, n, g ≥ 2 and m \| n, abelian data sets of degree m·n and genus g correspond to the weak conjugacy classes of Z m ⊕ Z n -actions on S g .Proof. Let D be an abelian data set of degree m · n and genus g as inDefinition 4.2. By Lemma 2.1, it suffices to show there exists a surjective map φ : π orb 1 (O H ) → H that preserves the order of torsion elements, where H = Z m ⊕ Z n and Γ(O H ) = (g 0 ; n 1 , . Definition 4. 5 . 5Two elements of a group G are said to weakly commute if there exists representatives in their respective conjugacy classes that commute. Lemma 4. 7 . 7Let δ ∈ Z n , and k 1 , . . . , k r are positive integers such that lcm({k 1 , . . . , k r }) = β \| n. If n β \|δ, then there exists δ 1 , . . . , δ r ∈ Z n such that n k i \|δ i and r i=1 δ i ≡ δ (mod n). Proof. Since lcm({k 1 , . . . , k r }) = β we have gcd({ n k 1 , . . . , n kr })\| n β . So, there exists integers c i such that c = r i=1 c i n k i . For some integer t, if δ = ct, where c = gcd({ n k 1 , . . . , n kr }), then δ = r i=1 tc i n k i . Taking δ i = tc i n k i , the assertion follows. Theorem 4. 8 ( 8Main Theorem). Let F, G ∈ M od(S g ) be finite-order maps. Then F, G = 1 and their commuting conjugates form a two-generator abelian group, if, and only if (G, F ) is a weakly abelian pair of order \|G\|·\|F \|. 5, while condition (ii) is a direct consequence of condition (vii) of Definition 4.2. To show condition (iii), it suffices to consider the case when D F = (n, g 1 ; ((c 1 , n 1 ), m), . . . , ((c r , n r ), m)), as all other cases follow from similar arguments. First, we note that G induces aḠ ∈ Aut(O H ) which does not fix any cone point of O H . Let Γ(O H / Ḡ ) = (g 0 ; n 1 , . . . , n r , n r+1 , . . . , n r+l ). Following the notation in the proof of Theorem 4.3, we map ξ i → F n n i {m o 1 1, m 2 o , . . . , m o t }, and c ′ i = 0, if n ′ i / ∈ {n 1 , n 2 , . . . , n r }, forms an abelian data set. Conditions (i) -(iii) of Definition 4.2 follow directly from our hypothesis. Moreover, for each i, we have gcd(d ′ i , m ′ i ) choice of κ i , we have lcm(m ′ i , m ′ i n ′ i κ i ) = m ′ i n ′ i, from which conditions (iv) and (v) follow. Furthermore, our choice of c ′ i and δ 2 ensures that Following the notation used in the proof of Theorem 4.3, we show that the generators y, x (of Z m ⊕ Z n ) can be expressed as products of elements in the set {φ(ξ i ) : 1 ≤ i ≤ l}. Consider the set S = {φ(ξ i ) m i : 1 ≤ i ≤ l}. Then by our choice of the map φ, each element of S equals some power of x, and \|φ(ξ i ) m i \| = n i . Since lcm(n 1 , . . . , n l ) = n, we have S = x . Now consider the set T = {φ(ξ r ) : φ(ξ r ) = y a x b , a = 0}. Since (G, F ) is an essential pair, yx t is a product of elements in T , and the assertion follows. Now suppose that lcm({m ′ k n ′ k : m ′ k = 1}) = B 1 , where no m ′ k n ′ k equals B 1 . Without loss of generality, we may assume that lcm({m ′ k n ′ k : m ′ k = 1 and 1 ≤ k ≤ p}) = B 1 . Then by Lemma 4.7, there exists δ ′ i , for 1 ≤ i ≤ p, such that p i=1 δ ′ i ≡ δ 2 (mod n). For each δ ′ i , we choose α i = n m ′ i n ′ i and consider the tuple Case 2 : 2Let lcm({m 1 , . . . , m k }) = m and lcm({n 1 , . . . , n r }) < n. By an argument analogous to Case 1, we obtain a representation φ : Γ → Z m ⊕ Z n such that the generators y, x (of Z m ⊕ Z n ) can be expressed as products of elements in the set {φ(ξ i ) : 1 ≤ i ≤ l}. Consider the set S = {φ(ξ i ) n i : 1 ≤ i ≤ l}. Then by our choice of φ and Proposition 3.4, it follows that each element of S equals some power of y and \|φ(ξ i ) n i \| = m i . Since lcm(m 1 , . . . , m l ) = m, we have S = y . Now consider the set T = {φ(ξ r ) : φ(ξ r ) = y a x b , b = 0}. As (G, F ) forms an essential pair, xy t is a product of elements in T , and the assertion follows. Case 3: Let lcm({m 1 , . . . , m k }) < m, lcm({n 1 , . Example 5 . 2 . 52Consider F, G ∈ Mod(S 7 ) whose conjugacy classes given by D F = (2, 4, 1; ), D G = (2, 3; ((1, 2), 4)), respectively. By the preceding discussion, there exist three possible choices for the conjugacy class of F G, namely: (a) D F G = (2, 4, 1; ) (b) D F G = (2, 2; ((1, 2), 8)) (c) D F G = (2, 0; ((1, 2), 16)) The realization of the group {1, F, G, F G} in each case is given in Figure 2 below. Corollary 5 . 3 .Figure 2 . 532There exists no finite non-cyclic abelian subgroup of Mod(S g ) that contains an irreducible Type 2 action.Proof. By Remark 4.6, it suffices to show that an irreducible Type 2 Z naction F does not commute with any other Z m -action. Since F is a Type Realizations of three distinct Klein 4-subgroups of Mod(S 7 ).2 action, we have Γ(O F ) = (0; n 1 , n 2 , n 3 ), where n i = n j and n i < n, for 1 ≤ i = j ≤ 3. In view of Theorem 4.8, if some G ∈ Mod(S g ) such that F, G = 1, then there existsḠ : O F → O F which satisfies the IMP with respect to (G, F ). This would imply thatḠ fixes all three cone points of O F . This is impossible, as any homeomorphism on a sphere can fix at most two points, and the assertion follows. (i) m \| n 1 and m \| n 2 . (ii) For k = 1, 2, there exists residue classes c ′ k modulo (n k /m) such that gcd(c ′ k , n k /m) = 1 and the tuple D = (n, 0; (c ′ 1 , n 1 /m), (c ′ 2 , n 2 /m), ((c 3 , n 3 ), m), . . . , ((c r , n r ), m)) forms a data set. Proposition 5 . 8 . 58Let F ∈ Mod(S g ) with D F = (n, g 0 , r; (c 1 , n 1 ), . . . , (c r , n r )), and let H = F . Then F has a root of degree m if, and only if the following conditions hold. (i) There exists a G ∈ Homeo + (S g ) with D G = (m, g ′ , r ′ ; (d 1 , m 1 ), . . . , (d k , m k )), which induces aḠ ∈ Aut(O H ). (ii) Γ(O H / Ḡ ) = (g ′ ; n ′ 1 , . . . , n ′ l ), where for each i, n ′ i belongs to the following union of multisets F ′ commutes with (F ′ ) m , the map F ([x]) = [F ′ (x)] defines an automorphism of O H . Furthermore, we have that Γ(O H / F ) = Γ(S g / F ′ ) = (g ′ ; t 1 , t 2 , . . . , t l ). and (ii) when \|F \| = g − 1, F ′ exists if, and only if, 2 ∤ (g − 1). Moreover, F ′ is a root of degree 2. Proof. (i) Suppose that O F ′ ≈ S 0 . Then, as discussed in the proof of Proposition 5.8, all its powers of prime order have a fixed point. (ii) Let F define a free action on S g , and H = F . Then O H ≈ S 2 , and by condition (i) of Theorem 5.8, F ′ induces anF ∈ Aut(O H ) of order n.In view of (i), it is clear that n ≤ 2, and further, by condition (ii) of Theorem 5.8, this is only possible when 2 ∤ (g − 1). If 2 ∤ (g − 1), then it easy check that F ′ with D F ′ = (2g − 2, 1; (1, 2), (1, 2)) is a root of F of degree 2. Corollary 5 . 13 . 513Let F ∈ Mod(S g ) be a root of t c , where c is nonseparating, and G ∈ Mod(S g ) be of finite order. Then F, G = 1 if, and only if G(c) = c and F c , G c = 1. In particular, if F c is primitive, then F and G cannot commute in Mod(S g ). Figure 3 . 3A realization of the action in S.No.10 of Table 1, with D G = (2, 0; ((1, 2), 8)) and D F = (4, 0; ((1, 4), 2), ((3, Figure 4 . 4A realization of the action in S.No.11 of Figure 5 . 5A realization of action in S.No.12 of Table 1, with D G = ( Lemma 2.1. A finite group H acts faithfully on S g with Γ(O H ) = (g 0 ; n 1 , . . . , n ℓ ) if, and only if, it satisfies the following two conditions: and (ii) there exists a surjective homomorphism φ H : π orb 1 (O H ) → H such that preserves the orders of all torsion elements of Γ. For [x], [y] ∈ O G , ifF ([x]) = ([y]), then P x = P y . (ii) For each orbit O of size \|F \| induced by the action of F on O G3.2. Then: (i) , there exists a point [x(O)] ∈ O Ḡ such that P [x(O)] = P [y] , where [y] ∈ O. (iii) Let F have β fixed points in S g . Ifβ denotes the number of fixed points ofF , then β m ≤β ≤ (m − 1)(2g − 2 + 2n) m(n − 1) + β m . Proof. (i) Suppose thatF ([x]) = [y]. Then there exists x ′ , y ′ ∈ S g in the pre-images of [x], [y] (under the branched cover) such that F (x ′ ) = y ′ . Then AcknowledgementsThe first author was supported by a UGC-JRF fellowship. The authors would like to thank Dheeraj Kulkarni and Siddhartha Sarkar for some helpful discussions. Abelian Data Cyclic factors. S No, S.No. Abelian Data Cyclic factors [D G ; Geometric realizations of cyclic actions on surfaces-ii. Atreyee Bhattacharya, Shiv Parsad, Kashyap Rajeevsarathy, arXiv:1803.00328arXiv preprintAtreyee Bhattacharya, Shiv Parsad, and Kashyap Rajeevsarathy. Geometric realiza- tions of cyclic actions on surfaces-ii. arXiv preprint arXiv:1803.00328, 2018. Finite abelian subgroups of the mapping class group. 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E-mail address: [email protected] Department of Mathematics, Indian Institute of Science Education and Research Bhopal. A Wiman, Ueber die hyperelliptischen curven und diejenigen vom geschlechte p= 3, welche eindeutigen transformationen in sich zulassen and ueber die algebraischen curven von den geschlechtern p= 4, 5 und 6 welche eindeutigen transformationen in sich besitzen. Svenska Vetenskaps-Akademiens Hadlingar. Stockholm; Bhopal Bypass Road, Bhauri, Bhopal; Madhya Pradesh, India; Bhauri, Bhopal; Madhya Pradesh9666Department of Mathematics, Indian Institute of Science Education and Research Bhopal. India E-mail address: [email protected] URLA Wiman. Ueber die hyperelliptischen curven und diejenigen vom geschlechte p= 3, welche eindeutigen transformationen in sich zulassen and ueber die algebraischen curven von den geschlechtern p= 4, 5 und 6 welche eindeutigen transformationen in sich besitzen. Svenska Vetenskaps-Akademiens Hadlingar, Stockholm, 96, 1895. Department of Mathematics, Indian Institute of Science Education and Re- search Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India E-mail address: [email protected] Department of Mathematics, Indian Institute of Science Education and Re- search Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India E-mail address: [email protected] URL: https://home.iiserb.ac.in/ kashyap/	[]
[ "NEARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES", "NEARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES" ]	[ "Jörg Schürmann " ]	[]	[]	In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-and motivic Chernclasses. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.	10.4171/118	[ "https://arxiv.org/pdf/1003.2343v2.pdf" ]	115,176,549	1003.2343	8fd9857ec525dcaf6427b9e6f55738577e213f3c	NEARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 4 May 2010 Jörg Schürmann NEARBY CYCLES AND CHARACTERISTIC CLASSES OF SINGULAR SPACES 4 May 2010 In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-and motivic Chernclasses. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface. A natural problem in complex geometry is the relation between invariants of a singular complex hypersurface X (like Euler characteristic and Hodge numbers) and the geometry of the singularities of the hypersurface (like the local Milnor fibrations). For the Euler characteristic this is for example a special case of the difference between the Fulton-and MacPherson-Chern classes of X, whose differences are the now well studied Milnor classes of X ( [1,6,7,8,27,29,34,35,45]). Their degrees are related to Donaldson-Thomas invariants of the singular locus ( [3]). A very powerful approach to this type of questions is by the theory of the nearby and vanishing cycle functors. For example a classical result of Verdier [44] says that the MacPherson Chern class transformation [25,22] commutes with specialization, which for constructible functions means the corresponding nearby cycles. Here we explain the corresponding result for our motivic Chern-and Hirzebruch class transformations as introduced in our joint work with J.-P. Brasselet and S. Yokura [5], i.e. they also commute with specialization defined in terms of nearby cycles. Here one can work either in the motivic context with relative Grothendieck group of varieties [4,19], or in the Hodge context with Grothendieck groups of M. Saito's mixed Hodge modules [30,31]. The key underlying specialization result [36] is about the filtered de Rham complex of the underlying filtered D-module in terms of the Malgrange-Kashiwara V -filtration. But here we focus on the geometric motivations and applications as given in our joint work with S.E. Cappell, L. Maxim and J.L. Shaneson [13]. In this paper we work (for simplicity) in the complex algebraic context, since this allows us to switch easily between an algebraic geometric language and an underlying topological picture. Many results are also true in the complex analytic or algebraic context over base field of characteristic zero. First we introduce the virtual characteristic classes and numbers of hypersurfaces and local complete intersetions in smooth ambient manifolds. Next we recall some of the theories of functorial characteristic classes for singular spaces [25,2,10,5,37]. Finally we explain the relation to nearby and vanishing cycles following our earlier results [34,35] about different Chern classes for singular spaces. of vector bundles on X, with ∪ given by the cup-or tensor-product. Such a characteristic class cl * corresponds by the "splitting principle" to a unique formal power series f (z) ∈ R[[z]] with cl * (L) = f (c 1 (L)) for any line bundle L on X. Here c 1 (L) ∈ H 1 (X) is the nilpotent first Chern class of L, which in the case H * (X) = K 0 (X) is given by c 1 (L) := 1 − [L ∨ ] ∈ K 0 (X) (with (·) ∨ the dual bundle). Finally cl * should be stable in the sense that f (0) ∈ R is a unit so that cl * induces a functorial group homomorphism cl * : K 0 (X), ⊕ → (H * (X) ⊗ R, ∪) . Let us now switch to smooth manifolds, which will be an important intermediate step on the way to characteristic classes of singular spaces. For a complex algebraic manifold M its tangent bundle T M is available and a characteristic class cl * (T M ) of the tangent bundle T M is called a characteristic cohomology class cl * (M ) of the manifold M . We also use the notation Example 1.1 (Hirzebruch '54). The famous Hirzebruch χ y -genus is the characteristic number whose associated characteristic class can be given in two versions (see [20]): (1) The cohomological version, with R = Q[y], is given by the Hirzebruch class cl * = T * y corresponding to the normalized power series f (z) := Q y (z) := z(1 + y) 1 − e −z(1+y) − zy ∈ Q[y][[z]] . (2) The K-theoretical version, with R = Z[y], is given by the dual total Lambda-class cl * = Λ ∨ y , with Λ ∨ y (·) := Λ y (·) ∨ = i≥0 Λ i (·) ∨ · y i corresponding to the unnormalized power series f (z) = 1 + y − yz ∈ Z[y][[z]] . So the χ y -genus of the compact complex algebraic manifold M is given by χ y (M ) := p≥0 χ(M, Λ p T * M ) · y p = p≥0   i≥0 (−1) i dim C H i (M, Λ p T * M )   · y p , with T * M the algebraic cotangent bundle of M . The equality (gHRR) χ y (M ) = deg T * y (T M ) ∩ [M ] ∈ Q[y], is (called) the generalized Hirzebruch Riemann-Roch theorem [20]. The corresponding power series Q y (z) (as above) specializes to Q y (z) =      1 + z for y = −1, z 1−e −z for y = 0, z tanh z for y = 1. Therefore the Hirzebruch class T * y (T M ) unifies the following important (total) characteristic cohomology classes of T M : which are, respectively, the Poincaré-Hopf or Gauss-Bonnet theorem, the Hirzebruch Riemann-Roch theorem and the Hirzebruch signature theorem. (1.2) T * y (T M ) =      c * (T M ) If X is a singular complex algebraic variety, then the algebraic tangent bundle of X doesn't exist so that a characteristic (co)homology class of X can't be defined as before. But if X can be realized as a local complete intersection inside a complex algebraic manifold M , then a substitute for T X is available. Indeed this just means that the closed inclusion i : X → M is a regular embedding into the smooth algebraic manifold M , so that the normal cone N X M → X is an algebraic vector bundle over X (compare [16]). Then the virtual tangent bundle of X (1.4) T vir X := [i * T M − N X M ] ∈ K 0 (X), is independent of the embedding in M (e.g., see [16][Ex.4.2.6]), so it is a well-defined element in the Grothendieck group of vector bundles on X. Of course T vir X = [T X] ∈ K 0 (X) in case X is a smooth algebraic submanifold. If cl * : K 0 (X) → H * (X) ⊗ R denotes a characteristic cohomology class as before, then one can associate to X an intrinsic homology class (i.e., independent of the embedding X ֒→ M ) defined as: (1.5) cl vir * (X) := cl * (T vir X) ∩ [X] ∈ H * (X) ⊗ R . Here [X] ∈ H * (X) is again the fundamental class (or the class of the structure sheaf) of X in H * (X) :=      H BM 2 * (X) , the Borel-Moore homology in even degrees. CH * (X) , the Chow group. G 0 (X) , the Grothendieck group of coherent sheaves. Here ∩ in the K-theoretical context comes from the tensor product with the coherent locally free sheaf of sections of the vector bundle. Moreover, for the class cl * = Λ ∨ y one has to take R := Z[y, (1 + y) −1 ] to make it a stable characteristic class defined on K 0 (X). Let i : X → M be a regular embedding of (locally constant) codimension r between possible singular complex algebraic varieties. Using the famous deformation to the normal cone, one gets functorial Gysin homomorphisms (compare [16,43,44]) (1.6) i ! : H * (M ) → H * −r (X) and (1.7) i ! : G 0 (M ) → G 0 (X) . Note that i is of finite tor-dimension, so that the last i ! can also be described as i ! = Li * : G 0 (M ) ≃ K 0 (D b coh (M )) → K 0 (D b coh (X)) ≃ G 0 (X) coming from the derived pullback Li * between the bounded derived categories with coherent cohomology sheaves. If M is also smooth, then one gets easily the following important relation between the virtual characteristic classes cl vir * (X) of X and the Gysin homomorphisms: (1.8) i ! (cl * (M )) = i ! (cl * (T M ) ∩ [M ]) = cl * (N X M ) ∩ cl vir * (X) . From now on we assume that X = {f = 0} = {f i = 0 \| i = 1, . . . , n} is a global complete intersection in the complex algebraic manifold M coming from a cartesian diagram (1.9) {f = 0} X i − −−− → M f 0   f =   (f 1 ,...,fn) {0} i 0 − −−− → C n . Then N X M ≃ f * N {0} C n = X × C n is a trivial vector bundle of rank n on X so that (1.10) cl * (N X M ) = 1 for cl * = T * y , c * , td * or L * . (1 + y) n for cl * = Λ ∨ y . Assume now that f is proper so that X is compact. Since the Gysin homomorphisms i ! commute with proper pushdown (compare [16,43,44]), one gets by the projection formula ♯ vir (X) := f 0 * cl vir * (X) = f 0 * cl * (N X M ) −1 ∩ i ! cl * (M ) = cl * (N {0} C n ) −1 ∩ i ! 0 (f * cl * (M )) . Taking a (small) regular value 0 = t ∈ C n , in the same way from the cartesian diagram (1.11) {f = 0} X i − −−− → M i ′ ← −−− − X t {f = t} f 0   f     ft {0} i 0 − −−− → C n it ← −−− − {t} for the "nearby" smooth submanifold X t = {f = t}, one gets the equality ♯(X t ) := f t * (cl * (X t )) = cl * (N {t} C n ) −1 ∩ i ! t (f * cl * (M )) . Note that the set of critical values of f is a proper algebraic subset of C n , as can be seen by "generic smoothness" or from an adapted stratification of the proper algebraic map f . Now N {0} C n ≃ C n ≃ N {t} C n and the smooth pullback π * for the (vector bundle) projection π : C n → {pt} is an isomorphism [16,43,44]), so that the "virtual characteristic number" π * : R = H * ({pt}) ⊗ R ≃ H * +n (C n ) ⊗ R with inverse i ! 0 and i ! t (see(1.12) ♯ vir (X) := f 0 * cl vir * (X) = ♯(X t ) ∈ R is the corresponding characteristic number of a "nearby" smooth fiber X t . Functorial characteristic classes of singular spaces For a more general singular complex algebraic variety X its "virtual tangent bundle" is not available any longer, so characteristic classes for singular varieties have to be defined in a different way. For an introduction to this subject compare with our survey paper [38] (and see also [37,46]). The theory of characteristic classes of vector bundles is a natural transformation of contravariant functorial theories. This naturality is an important guide for developing various theories of characteristic classes for singular varieties. Almost all known characteristic classes for singular spaces are formulated as natural transformations cl * : A(X) → H * (X) ⊗ R of covariant functorial theories. Here A is a suitable theory (depending on the choice of cl * ), which is covariant functorial for proper algebraic morphisms. There is always a distinguished element I X ∈ A(X) such that the corresponding characteristic class of the singular space X is defined as cl * (X) := cl * (I X ) . Finally one has the normalization cl * (I M ) = cl * (T M ) ∩ [M ] ∈ H * (M ) ⊗ R for M a smooth manifold, with cl * (T M ) the corresponding characteristic cohomology class of M . This justifies the notation cl * for this homology class transformation, which should be seen as a homology class version of the following characteristic number of the singular space X: ♯(X) := cl * (k * I X ) = deg (cl * (I X )) ∈ H * ({pt}) ⊗ R ≃ R , with k : X → {pt} a constant map. Note that the normalization implies that for M smooth: ♯(M ) = deg (cl * (M )) = deg (cl * (T M ) ∩ [M ]) so that this is consistent with the notion of characteristic number of the smooth manifold M as used before. But only few characteristic numbers and classes have been extended in this way to singular spaces. For example the three characteristic numbers (1.3) and classes (1.2) have been generalized to a singular complex algebraic variety X in the following way (where the characteristic numbers are only defined for X compact): (y = −1) e(X) = deg (c * (X)) , with c * : F (X) → H * (X) the Chern class transformation of MacPherson [25,22] from the abelian group F (X) of complex algebraically constructible functions to homology, where one can use the Chow group CH * (·) or the Borel-Moore homology group H BM 2 * (·, Z) (in even degrees). Here e(X) is the (topological) Euler characteristic of X, and the distinguished element I X := 1 X ∈ F (X) is simply given by the characteristic function of X. Then c * (X) := c * (1 X ) agrees by [9] via "Alexander duality" for compact X embeddable into a complex manifold with the Schwartz class of X as introduced before by M.-H. Schwartz [39]. Finally for compact X one also has (y = 0) χ(X) = deg (td * (X)) , with td * : G 0 (X) → H * (X) ⊗ Q(y = 1) sign(X) = deg (L * (X)) , with L * : Ω(X) → H 2 * (X, Q) the homology L-class transformation of Cappell-Shaneson [10] as formulated in [5]. Here Ω(X) is the abelian group of cobordism classes of selfdual constructible complexes. Then L * (X) := L * ([IC X ]) is the homology L-class of Goresky-MacPherson [18], with the distinguished element I X := [IC X ] the class of their intersection cohomology complex. So sign(X) is the intersection cohomology signature of X. For a rational PL-homology manifold X, these L-classes are due to Thom [42]. So all these theories have the same formalism, but they are defined on completely different theories. Nevertheless, it is natural to ask for another theory of characteristic homology classes of singular complex algebraic varieties, which unifies the above characteristic homology class transformations. Of course in the smooth case, this is done by the Hirzebruch class T * y (T M ) ∩ [M ] of the tangent bundle. An answer to this question was given in [5] (together with some improvements in [37]). Using Saito's deep theory of algebraic mixed Hodge modules [30,31], we introduced in [5] the motivic Chern class transformations as natural transformations (commuting with proper push down) fitting into a commutative diagram: G 0 (X)[y] − −−− → G 0 (X)[y, y −1 ] G 0 (X)[y, y −1 ] mCy   mCy     M HCy K 0 (var/X) − −−− → M(var/X) χ Hdg − −−− → K 0 (M HM (X)) . Here K 0 (M HM (X)) is the Grothendieck group of algebraic mixed Hodge modules on X, and K 0 (var/X) (resp. M(var/X) : = K 0 (var/X)[L −1 ]) is the (localization of the) relative Grothendieck group of complex algebraic varieties over X (with respect to the class of the affine line L, compare e.g. [4,19]). The distinguished element is given by the constant Hodge module (complex) resp. by the class of the identity arrow I X := [Q H X ] ∈ K 0 (M HM (X)) resp. I X := [id X ] ∈ K 0 (var/X) , and the canonical "Hodge realization" homomorphism χ Hdg is given by (2.1) χ Hdg : K 0 (var/X) → K 0 (M HM (X)); [f : Y → X] → [f ! Q H Y ] . The motivic Chern class transformations mC y , M HC y capture information about the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. The corresponding characteristic class of the space X, mC y (X) = M HC y (X) ∈ G 0 (X)[y] , can also be defined with the help of the (filtered) Du Bois complex of X [15], and satisfies for M smooth the normalization condition (2.2) mC y (M ) = M HC y (M ) = Λ ∨ y (T M ) ∩ [M ] ∈ G 0 (M )[y] . The motivic Chern class transformations are a K-theoretical refinement of the Hirzebruch class transformations T y * , M HT y * , which can be defined by the (functorial) commutative diagram : M(var/X) χ Hdg − −−− → K 0 (M HM (X)) M HCy −−−−→ G 0 (X)[y, y −1 ] Ty *   M HTy *     td * H * (X) ⊗ Q[y, y −1 ] − −−− → H * (X) ⊗ Q loc (1+y) − * · ← −−−−− − H * (X) ⊗ Q[y, y −1 ] , with td * : G 0 (X) → H * (X) ⊗ Q the Todd class transformation of Baum-Fulton-MacPherson [2,16] and (1+y) − * · the renormalization given in degree i by the multiplication (1+y) −i · : H i (−)⊗Q[y, y −1 ] → H i (−)⊗Q[y, y −1 , (1+y) −1 ] =: H * (−)⊗Q loc . This renormalization is needed to get for M smooth the normalization condition (2.3) T y * (M ) = M HT y * (M ) = T * y (T M ) ∩ [M ] ∈ H * (M ) ⊗ Q[y] . It is the Hirzebruch class transformation T y * , which unifies the (rationalized) Chern class transformation c * ⊗ Q, Todd class transformation td * and L-class transformation L * (compare [5]). The corresponding characteristic number χ y (X) := deg (M HT y * (X)) ∈ Z[y] for a singular (compact) algebraic variety X captures information about the Hodge filtration of Deligne's ( [14]) mixed Hodge structure on the rational cohomology (with compact support) H * (c) (X; Q) of X. In fact, by M. Saito's work [31] one has an equivalence M HM ({pt}) ≃ mHs p between mixed Hodge modules on a point space, and rational (graded) polarizable mixed Hodge structures. Moreover, the corresponding mixed Hodge structure on rational cohomology with compact support H * c (X; Q) = H * ({pt}; k ! Q H X ) (with k : X → {pt} a constant map) agrees with Deligne's one by another deep theorem of M. Saito [32]. Therefore the transformations M HC y and M HT y * can be seen as a characteristic class version of the ring homomorphism χ y : K 0 (mHs p ) → Z[y, y −1 ] defined on the Grothendieck group of (graded) polarizable mixed Hodge structures by (2.4) χ y ([H]) := p dim Gr p F (H ⊗ C) · (−y) p , for F the Hodge filtration of H ∈ mHs p . Note that χ y ([L]) = −y. These characteristic class transformations are motivic refinements of the (rationalization of the) Chern class transformation c * ⊗ Q of MacPherson. M HT y * factorizes by [37] as M HT y * : K 0 (M HM (X)) → H * (X) ⊗ Q[y, y −1 ] ⊂ H * (X) ⊗ Q loc , fitting into a (functorial) commutative diagram F (X) χ stalk ← −−− − K 0 (D b c (X)) rat ← −−− − K 0 (M HM (X)) c * ⊗Q   c * ⊗Q     M HTy * H * (X) ⊗ Q H * (X) ⊗ Q y=−1 ← −−− − H * (X) ⊗ Q[y, y −1 ] . Here D b c (X) is the derived category of algebraically constructible sheaves on X (viewed as a complex analytic space), with rat associating to a (complex of) mixed Hodge module(s) the underlying perverse (constructible) sheaf complex, and χ stalk is given by the Euler characteristic of the stalks. Let us go back to the case when X is a local complete intersection in some ambient smooth algebraic manifold. Then it is natural to compare cl * (X) for a functorial homology characteristic class theory cl * as above with the corresponding virtual characteristic class cl vir * (X). If M is smooth, then clearly we have that cl vir * (M ) = cl * (T M ) ∩ [M ] = cl * (M ) . However, if X is singular, the difference between the homology classes cl vir * (X) and cl * (X) depends in general on the singularities of X. This motivates the following Problem 2.1. Describe the difference cl vir * (X) − cl * (X) in terms of the geometry of singular locus of X. The above problem is usually studied in order to understand the complicated homology classes cl * (X) in terms of the simpler virtual classes cl vir * (X), with the difference terms measuring the complexity of the singularities of X. This question was first studied for the Todd class transformation td * , where this difference term is vanishing. More precisely one has the Theorem 2.2 (Verdier '76). Assume that i : X → Y is regular embedding of (locally constant) codimension n. Then the Todd class transformation td * commutes with specialization (see [43]), i.e. (2.5) i ! • td * = td * • i ! : G 0 (Y ) → H * −n (X) . Note that Y need not be smooth. Corollary 2.3. Assume that X can be realized as a local complete intersection in some ambient smooth algebraic manifold. Then td vir * (X) = td * (X). Especially, if X is a global complete intersection given as the zero-fiber X = {f = 0} of a proper morphism f : M → C n on the algebraic manifold M , then the arithmetic genus (2.6) χ(X) = χ vir (X) = χ(X t ) of X agrees with that of a nearby smooth fiber X t for 0 = t small and generic. The next case studied in the literature is the L-class transformation L * for X a compact global complex hypersurface. Theorem 2.4 (Cappell-Shaneson '91). Assume X is a global compact hypersurface X = {f = 0} for a proper complex algebraic function f : M → C on a complex algebraic manifold M . Fix a complex Whitney stratification of X and let V 0 be the set of strata V with dimV < dimX. Assume for simplicity, that all V ∈ V 0 are simply-connected (otherwise one has to use suitable twisted L-classes, see [11,12]). Then (2.7) L vir * (X) − L * (X) = V ∈V 0 σ(lk(V )) · L * (V ) , where σ(lk(V )) ∈ Z is a certain signature invariant associated to the link pair of the stratum V in (M, X). This result is in fact of topological nature, and holds more generally for a suitable compact stratified pseudomanifold X, which is PL-embedded into a manifold M in real codimension two (see [11,12] for details). If cl * = c * is the Chern class transformation, the problem amounts to comparing the Fulton-Johnson class c F J * (X) := c vir * (X) (e.g., see [16,17]) with the homology Chern class c * (X) of MacPherson. The difference between these two classes is measured by the so-called Milnor class M * (X) of X, which is studied in many references like [1,6,7,8,27,29,34,35,45]. This is a homology class supported on the singular locus of X, and for a global hypersurface it was computed in [29] (see also [34,35,45,27]) as a weighted sum in the Chern-MacPherson classes of closures of singular strata of X, the weights depending only on the normal information to the strata. For example, if X has only isolated singularities, the Milnor class equals (up to a sign) the sum of the local Milnor numbers attached to the singular points. In the following section we explain our approach [34,35] through nearby and vanishing cycles (for constructible functions), which recently was adapted to the motivic Hirzebruch and Chern class transformations [13,36]. Nearby and vanishing cycles Let us start to explain some basic constructions for constructible functions in the complex algebraic context (compare [33,34,35]). Here we work in the classical topology on the complex analytic space X associated to a separated scheme of finite type over Spec(C). (1) α is a finite sum α = j n j · 1 Z j , with n j ∈ Z and 1 Z j the characteristic function of the closed complex algebraic subset Z j of X. (2) α is (locally) constant on the strata of a complex algebraic Whitney b-regular stratification of X. This notion is closely related to the much more sophisticated notion of (algebraically) constructible (complexes of) sheaves on X. A sheaf F of (rational) vector-spaces on X with finite dimensional stalks is (algebraically) constructible, if there exists a complex algebraic Whitney b-regular stratification as above such that the restriction of F to all strata is locally constant. Similarly, a bounded complex of sheaves is constructible, if all it cohomology sheaves have this property, and we denote by D b c (X) the corresponding derived category of bounded constructible complexes on X. The Grothendieck group of the triangulated category D b c (X) is denoted by K 0 (D b c (X)). Since we assume that all stalks of a constructible complex are finite dimensional, by taking stalkwise the Euler characteristic we get a natural group homomorphism (3.1) χ stalk : K 0 (D b c (X)) → F (X); [F] → (x → χ(F x )) . Here F (X) is the group of (algebraically) constructible functions on X. It is easy to show that natural transformation χ stalk is surjective. As is well known (and explained in detail in [33]), all the usual functors in sheaf theory, which respect the corresponding category of constructible complexes of sheaves, induce by the epimorphism χ stalk well-defined group homomorphisms on the level of constructible functions. We just recall these, which are important for later applications or definitions. Definition 3.2. Let f : X → Y be an algebraic map of complex spaces associated to separated schemes of finite type over Spec(C). Then one has the following transformations: (1) pullback: f * : F (Y ) → F (X); α → α • f , which corresponds to the usual pullback of sheaves f * : D b c (Y ) → D b c (X) . (2) exterior product: α ⊠ β ∈ F (X × Y ) for α ∈ F (X) and β ∈ F (Y ), given by α ⊠ β((x, y)) := α(x) · β(y). This corresponds on the sheaf level to the exterior product Then one has ψ f : F (X) → F (X 0 ), corresponding to Deligne's nearby cycle functor ⊠ L : D b c (X) × D b c (Y ) → D b c (X × Y ) .ψ f : D b c (X) → D b c (X 0 ) . This was first introduced in [44] by using resolution of singularities (compare with [33] for another approach using stratification theory). By linearity, ψ f is uniquely defined by the convention that for a closed complex algebraic subspace Z ⊂ X the value (3.4) ψ f (1 Z )(x) := χ H * (F f \| Z ,x ; Q) is just the Euler-characteristic of a local Milnor fiber F f \| Z ,x of f \| Z at x. Here this local Milnor fiber at x is given by (3.5) F f \| Z ,x := Z ∩ B ǫ (x) ∩ {f = y} , with 0 < \|y\| << ǫ << 1 and B ǫ (x) an open (or closed) ball of radius ǫ around x (in some local coordinates). Here we use the theory of a Milnor fibration of a function f on the singular space Z (compare [24,33]). (6) vanishing cycles: Assume Y = C and let i : X 0 := {f = 0} ֒→ X be the inclusion of the zero-fiber. Then one has φ f : F (X) → F (X 0 ); φ f := ψ f − i * , corresponding to Deligne's vanishing cycle functor φ f : D b c (X) → D b c (X 0 ) . By linearity, φ f is uniquely defined by the convention that for a closed complex algebraic subspace Z ⊂ X the value (3.6) φ f (1 Z )(x) := χ H * (F f \| Z ,x ; Q) − 1 = χ H * (F f \| Z ,x ; Q) is just the reduced Euler-characteristic of a local Milnor fiber F f \| Z ,x of f \| Z at x.supp (φ f (1 M )) ⊂ X sing . And φ f (1 M )\|X sing is (up to a sign) the Behrend function of X sing (see [3]), an intrinsic constructible function of the singular locus appearing in relation to Donaldson-Thomas invariants. A beautiful result of Verdier [44,23] shows that for a global hypersurface MacPherson's Chern class transformation c * commutes with specialization, if one uses the nearby cycle functor ψ f on the level of constructible functions (as opposed to the pullback functor i * for the corresponding inclusion i : X = {f = 0} → Y ). Theorem 3.4 (Verdier '81). Assume that X = {f = 0} is a global hypersurface (of codimension one) in Y given by the zero-fiber of a complex algebraic function f : Y → C. Then the MacPherson Chern class transformation c * commutes with specialization (see [44,22]), i.e. (3.7) i ! • c * = c * • ψ f : F (Y ) → H * −1 (X) for the closed inclusion i : X = {f = 0} → Y . Note that Y need not be smooth. As an immediate application one gets by (1.8) and (1.10) the following important result (compare [34,35]): 3.8) c vir * (X) − c * (X) = c * (ψ f (1 M )) − c * (1 X ) = c * (φ f (1 M )) ∈ H * (X sing ) , since supp(φ f (1 M )) ⊂ X sing . Here we also use the naturality of c * for the closed inclusion X sing → X to view this difference term as a localized class in H * (X sing ). In particular: (1) c vir i (X) = c i (X) ∈ H i (X) for all i > dim X sing . (2) If X has only isolated singularities (i.e. dim X sing = 0), then c vir * (X) − c * (X) = x∈X sing χ H * (F x ; Q) , where F x is the local Milnor fiber of the isolated hypersurface singularity (X, x). (3) If f : M → C is proper, then deg (c * (φ f (1 M ))) = deg c vir * (X) − c * (X) = χ(X t ) − χ(X) is the difference between the Euler characteristic of a global nearby smooth fiber X t = {f = t} (for 0 = \|t\| small enough) and of the special fiber X = {f = 0}. For a general local complete intersection X in some ambient smooth algebraic manifold (e.g. a local hypersurface of codimension one), one doesn't have global equations so that the theory of nearby and vanishing cycles can't be applied directly. Instead one has to combine them with the deformation to the normal cone leading to Verdier's theory of specialization functors (compare [34,35]). But even if X = {f = 0} is a global complete intersection inside the ambient smooth algebraic manifold M , given by the zero-fiber of a complex algebraic map f : M → C n , one doesn't have a theory of nearby and vanishing cycles, because a local theory of Milnor fibers for f is missing (if n > 1). But if one fixes an ordering of the components of f (or of the coordinates on C n ), then a corresponding local Milnor fibration exists for any ordered tuple (3.9) ψ (f ) := ψ f 1 • · · · • ψ fn : F (Y ) → F (X) . By linearity, ψ (f ) is uniquely defined by the convention that for a closed complex algebraic subspace Z ⊂ Y the value (3.10) ψ (f ) (1 Z )(x) := χ H * (F (f )\| Z ,x ; Q) is just the Euler-characteristic of a local Milnor fiber F (f )\| Z ,x of (f )\| Z at x. Here this local Milnor fiber of (f ) at x is given by (3.11) F (f )\| Z ,x := Z ∩ B ǫ (x) ∩ {f 1 = y 1 , . . . , f n = y n } , with 0 < \|y n \| << · · · << \|y 1 \| << ǫ << 1 and B ǫ (x) an open (or closed) ball of radius ǫ around x (in some local coordinates, compare [28]). The corresponding vanishing cycles of (f ) are defined by (3.12) φ (f ) := ψ (f ) − i * : F (Y ) → F (X) , with i : X → Y the closed inclusion. By linearity, φ (f ) is uniquely defined by the convention that for a closed complex algebraic subspace Z ⊂ X the value (3.13) φ (f ) (1 Z )(x) := χ H * (F (f )\| Z ,x ; Q) − 1 = χ H * (F (f )\| Z ,x ; Q) is just the reduced Euler-characteristic of a local Milnor fiber F (f )\| Z ,x of (f )\| Z at x. Note that again supp φ (f ) (1 M ) ⊂ X sing in case the ambient space Y = M is a smooth algebraic manifold. Assume moreover that X is of codimension n so that the regular embedding i : X → Y factorizes into n regular embeddings of codimension one i = i n • · · · • i 1 : (3.14) X = {f 1 = 0, . . . , f n = 0} i 1 − −−− → {f 2 = 0, . . . , f n = 0} i 2 − −−− → · · · · · · {f n−1 = 0, f n = 0} i n−1 − −−− → {f n = 0} in − −−− → Y . By the functoriality of the Gysin homomorphisms one gets i ! = i ! 1 • · · · • i ! n : H * (Y ) → H * −n (X) . Since in Verdier's spezialisation theorem (3.4) the ambient space need not be smooth, we can apply it inductively to all embeddings i j (for j = n, . . . , 1) above. 3.15) c vir * (X) − c * (X) = c * (φ (f ) (1 M )) ∈ H * (X sing ) , since supp(φ (f ) (1 M )) ⊂ X sing . Here we also use the naturality of c * for the closed inclusion X sing → X to view this difference term as a localized class in H * (X sing ). In particular: (1) c vir i (X) = c i (X) ∈ H i (X) for all i > dim X sing . (2) If X has only isolated singularities (i.e. dim X sing = 0), then c vir * (X) − c * (X) = x∈X sing χ H * (F x ; Q) , where F x is the local Milnor fiber of the ordered n-tuple (f ) at the isolated singularity x. (3) If (f ) = (f 1 , . . . , f n ) : M → C n is proper, then deg c * (φ (f ) (1 M )) = deg c vir * (X) − c * (X) = χ(X t ) − χ(X) is the difference between the Euler characteristic of a global nearby smooth fiber X t = {f 1 = t 1 , . . . , f n = t n } (for t = (t 1 , . . . , t n ) with 0 < \|t n \| << · · · << \|t 1 \| small enough) and of the special fiber X = {f = 0}. As explained in section 2, the motivic Hirzebruch and Chern class transformations T y * , M HT y * and mC y , M HC y can be seen as "motivic or Hodge theoretical liftings" of the (rationalized) Chern class transformation c * under the comparison maps K 0 (var/Y ) χ Hdg − −−− → K 0 (M HM (Y )) rat − −−− → K 0 (D b c (Y )) χ stalk − −−− → F (Y ) . Here these Grothendieck groups have the same calculus as for constructible functions in definition 3.2(1-4), respected by these comparison maps. So it is natural to try to extend known results about MacPherson's Chern class transformation c * to these transformations. In the "motivic" (resp. " Hodge theoretical") context this has been worked out in [5] (resp. [37]) for (1) the functorialty under push down for proper algebraic morphism. (2) the functorialty under exterior products. (3) the functorialty under smooth pullback given by a related Verdier-Riemann-Roch theorem. And recently we could also prove the "counterpart" of Verdier's specialization theorem (3.4). Let X = {f = 0} be a global hypersurface in Y given by the zero-fiber of a complex algebraic function f on Y : (see [4,19]), or on the Hodge-theoretical level of algebraic mixed Hodge modules ( [30,31]), "lifting" the corresponding functors on the level of algebraically constructible sheaves ( [34]) and algebraically constructible functions as introduced before, so that the following diagram commutes: X := {f = 0} i − −−− → Y f − −−− → C .(3.16) M(var/Y ) ψ m f ,φ m f −−−−→ M(var/X) χ Hdg   χ Hdg   K 0 (MHM(Y )) ψ ′H f ,φ ′H f − −−−− → K 0 (MHM(X)) rat   rat   K 0 (D b c (Y )) ψ f ,φ f − −−− → K 0 (D b c (X)) χ stalk   χ stalk   F (Y ) ψ f ,φ f − −−− → F (X) . We also use the notation ψ ′H f := ψ H f [1] and φ ′H f := φ H f [1] for the shifted functors, with ψ H f , φ φ m f = ψ m f − i * and φ ′H f = ψ ′H f − i * . Remark 3.8. The motivic nearby and vanishing cycles functors of [4,19] take values in a refined equivariant localized Grothendieck group Mμ(var/X) of equivariant algebraic varieties over X with a "good" action of the profinite groupμ = lim µ n of roots of unity. For mixed Hodge modules this corresponds to an action of the semi-simple part of the monodromy. But in the following applications we don't need to take this action into account. Also note that for the commutativity of diagram ( Again the smoothness of Y is not needed. The appearence of the factor (1 + y) should not be a surprise, as it can already be seen in the case of a smooth hypersurface X inside a smooth ambient manifold Y , (1 + y) · M HC y (X) = i ! M HC y (Y ) , if one recalls (1.8), (1.10) and the normalization condition (2.2), with Q H X = i * Q H Y ≃ ψ ′H f (Q H Y ) in this special case. But the proof of this theorem given in [36] is far away from the geometric applications described here. In fact it uses the algebraic theory of nearby and vanishing cycles in the context of D-modules given by the V -filtration of Malgrange-Kashiwara, together with a specialization result about the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Using Verdier's result that the Todd class transformation td * commutes with specialization (see theorem 2.2), one gets ( [36]): Corollary 3.10. Assume that X = {f = 0} is a global hypersurface of codimension one given by the zero-fiber of a complex algebraic function Corollary 3.12. Assume that X = {f = 0} is a global hypersurface (of codimension one) in some ambient smooth algebraic manifold M , given by the zero-fiber of a complex algebraic function f : M → C. Then mC vir y (X) − mC y (X) = mC y φ m f ([id M ]) = M HC y * φ ′H f ( Q H M ) ∈ G 0 (X sing )[y] ,(3. 22) and T vir y * (X) − T y * (X) = T y * φ m f ([id M ]) = M HT y * φ ′H f ( Q H M ) ∈ H * (X sing ) ⊗ Q[y] . (3.23) Here we use supp φ ′H f Q H M ⊂ X sing and the naturality of our characteristic class transformations for the closed inclusion X sing → X. In particular: (1) T vir y,i (X) = T y,i (X) ∈ H i (X) ⊗ Q[y] for all i > dim X sing . (2) If X has only isolated singularities (i.e. dim X sing = 0), then mC vir y (X) − mC y (X) = x∈X sing χ y H * (F x ; Q) = T vir y * (X) − T y * (X) ,(3. 24) where F x is the Milnor fiber of the isolated hypersurface singularity (X, x). (3) If f : M → C is proper, then deg M HC y * φ ′H f ( Q H M ) = χ y (H * (X t ; Q)) − χ y (H * (X; Q)) = deg M HT y * φ ′H f ( Q H M ) (3.25) is the difference between the χ y -characteristics of a global nearby smooth fiber X t = {f = t} (for 0 = \|t\| small enough) and of the special fiber X = {f = 0}. Remark 3.13. (Hodge polynomials vs. Hodge spectrum) Let us explain the precise relationship between the Hodge spectrum and the less-studied χ ypolynomial of the Milnor fiber of a hypersurface singularity. Here we follow notations and sign conventions similar to those in [19]. Denote by mHs mon the abelian category of mixed Hodge structures endowed with an automorphism of finite order, and by K mon 0 (mHs) the corresponding Grothendieck ring. There is a natural linear map called the Hodge spectrum, for any mixed Hodge structure H with an automorphism T of finite order, where H C is the underlying complex vector space of H, H C,α is the eigenspace of T with eigenvalue exp(2πiα), and F is the Hodge filtration on H C . It is now easy to see that the χ y -polynomial χ y ([H]) of H is obtained from hsp([H]) by substituting t = 1 in t α for α ∈ Q ∩ [0, 1) and t = −y in t p for p ∈ Z. hsp : K mon 0 (mHs) → Z[Q] ≃ n≥1 Z[t 1/n , t −1/n ] , As already explained before, Corollary 3.12 reduces for the value y = −1 of the parameter to the (rationalized version of) Corollary 3.5. Since the ambient space in Theorem 3.9 and Corollary 3.10 need not be smooth, one can generalize in the same way the Corollary 3.7 for a global complete intersection X = {f = 0} = {f 1 = 0, . . . , f n = 0} (of codimension n) in some ambient smooth algebraic manifold M , given by the zero-fiber of an ordered n-tuple of complex algebraic function (f ) := (f 1 , . . . , f n ) : M → C n . Here we leave the details to the reader. It is also very interesting to look at the other specializations of Corollary 3.12 for y = 0 and y = 1. Let us first consider the case when y = 0. Note that in general T 0 * (X) = td * (X) for a singular complex algebraic variety (see [5]). But if X has only Du Bois singularities (e.g., rational singularities, cf. [32]), then by [5] we have T 0 * (X) = td * (X). So if a global hypersurface X = {f = 0} has only Du Bois singularities, then by Corollaries 2.3 and 3.12 we get: M HT 0 * φ ′H f ( Q H M ) = 0 ∈ H * (X) ⊗ Q . This vanishing (which is in fact a class version of Steenbrink's cohomological insignificance of X [40]) imposes interesting geometric identities on the corresponding Todd-type invariants of the singular locus. For example, we obtain the following Corollary 3.14. If the global hypersurface X has only isolated Du Bois singularities, then (3.27) dim C Gr 0 F H n (F x ; C) = 0 for all x ∈ X sing , with n = dim X. It should be pointed out that in this setting a result of Ishii [21] implies that (3.27) is in fact equivalent to x ∈ X sing being an isolated Du Bois hypersurface singularity. Also note that in the arbitrary singularity case, the Milnor-Todd class Finally, if y = 1, the formula (3.23) should be compared to the Cappell-Shaneson topological result of (2.7). While it can be shown (compare with [26]) that the normal contribution σ(lk(V )) in (2.7) for a singular stratum V ∈ V 0 is in fact the signature σ(F v ) (v ∈ V ) of the Milnor fiber (as a manifold with boundary) of the singularity in a transversal slice to V in v, the precise relation between σ(F v ) and χ 1 (F v ) is in general very difficult to understand. For X a rational homology manifold, one would like to have a "local Hodge index formula" σ(F v ) ? = χ 1 (F v ) , which is presently not available. But if the hypersurface X is a rational homology manifold with only isolated singularities, then this expected equality follows from [41][Thm.11]. One therefore gets in this case (by a comparison of the different specialization results for L * and T 1 * ) the following conjectural interpretation of L-classes from [5] (see [13] for more details): Theorem 3.15. Let X be a compact complex algebraic variety with only isolated singularities, which is moreover a rational homology manifold and can be realized as a global hypersurface (of codimension one) in a complex algebraic manifold. Then cl * (M ) := cl * (T M ) ∩ [M ] ∈ H * (M ) ⊗ R for the corresponding characteristic homology class of the manifold M , with [M ] ∈ H * (M ) the fundamental class (or the class of the structure sheaf) in H * (M ) 2 * (M ) , the Borel-Moore homology in even degrees. CH * (M ) , the Chow group. G 0 (M ) , the Grothendieck group of coherent sheaves. If M is moreover compact, i.e. the constant map k : M → {pt} is proper, one gets the corresponding characteristic number M ) := k * (cl * (M )) =: deg(cl * (M )) ∈ R . the Chern class for y = −1, td * (T M ) the Todd class for y = 0, L * (T M ) the Thom-Hirzebruch L-class for y = 1. The gHHR-theorem specializes to the calculation of the following important invariants: χ −1 (M ) = e(M ) = deg (c * (T M ) ∩ [M ]) the Euler characteristic, χ 0 (M ) = χ(M ) = deg (td * (T M ) ∩ [M ]) the arithmetic genus, χ 1 (M ) = sign(M ) = deg (L * (T M ) ∩ [M ]) the signature, (1.3) the Todd transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson [2] (for Borel-Moore homology) or Fulton [16] (for Chow groups). Here G 0 (X) is the Grothendieck group of coherent sheaves, with χ(X) the arithmetic genus (or holomorphic Euler characteristic) of X. Then td * (X) := td * ([O X ]), with the distinguished element I X := [O X ] the class of the structure sheaf. Definition 3.1. A function α : X → Z is called (algebraically) constructible,if it satisfies one of the following two equivalent properties: ( 3 ) 3Euler characteristic: Suppose X is compact and Y = {pt} is a point. Then one has χ : F (X) → Z, corresponding to RΓ(X, ·) = k * : D b c (X) → D b c ({pt}) on the level of constructible complexes of sheaves, with k : X → {pt} the constant proper map. By linearity it is characterized by the convention that for a compact complex algebraic subspace Z ⊂ X (3.2) χ(1 Z ) := χ (H * (Z; Q)) is just the usual Euler characteristic of Z. (4) proper pushdown: Suppose f is proper. Then one has f * = f ! : F (X) → F (Y ), corresponding to Rf * = Rf ! : D b c (X) → D b c (Y ) on the level of constructible complexes of sheaves. Explicitly it is given by * (α)(y) := χ(α\| {f =y} ) , and in this form it goes back to the paper [25] of MacPherson. (5) nearby cycles: Assume Y = C and let X 0 := {f = 0} be the zero fiber. Remark 3. 3 . 3Let the global hypersurface X = {f = 0} be the zero-fiber of an algebraic function f : M → C on the complex algebraic manifold M . Then the support of φ f (1 M ) is contained in the singular locus X sing of X: Corollary 3. 5 . 5Assume that X = {f = 0} is a global hypersurface (of codimension one) in some ambient smooth algebraic manifold M , given by the zero-fiber of a complex algebraic function f : M → C. Then ( (f ) := (f 1 , . . . , f n ) : Z → C n of complex algebraic functions on the singular algebraic variety Z (as observed in[28]). Definition 3 . 6 ( 36Nearby and vanishing cycles for an ordered tuple). Let (f ) := (f 1 , . . . , f n ) : Y → C n be an ordered n-tuple of complex algebraic functions on Y , with X := {f = 0} = {f 1 = 0, . . . , f n = 0} the zero-fiber of (f ). Then nearby cycles of (f ) := (f 1 , . . . , f n ) are defined by iteration as Corollary 3 . 7 . 37Assume that X = {f = 0} = {f 1 = 0, . . . , f n = 0} is a global complete intersection (of codimension n) in some ambient smooth algebraic manifold M , given by the zero-fiber of an ordered n-tuple of complex algebraic function (f ) := (f 1 , . . . , f n ) : M → C n . Then ( First note that one can use the nearby and vanishing cycle functors ψ f and φ f either on the motivic level of localized relative Grothendieck groups M(var/−) := K 0 (var/−)[L −1 ] H f : M HM (Y ) → M HM (X) and ψ f [−1], φ f [−1] : P erv(Y ) → P erv(X) preserving mixed Hodge modules and perverse sheaves, respectively. On the level of Grothendieck groups one simply has Theorem 3. 9 ( 9Schürmann '09). Assume that X = {f = 0} is a global hypersurface of codimension one given by the zero-fiber of a complex algebraic function f : Y → C. Then the motivic Hodge-Chern class transformation M HC y commutes with specialization in the following sense: y) · M HC y ( ψ ′H f (−) ) = i ! M HC y (−) as transformations K 0 (M HM (Y )) → G 0 (X)[y, y −1 ]. dim(Gr p F H C,α )t p   T 0 * 0φ m f ([id M ]) = M HT 0 * φ ′H f ( Q H M ) ∈ H * (X sing )⊗ Q carries interesting non-trivial information about the singularities of the hypersurface X. ( 3 . 328) L * (X) = T 1 * (X) ∈ H 2 * (X; Q) . 3.16) one has to use ψ ′H f , φ ′H Now we are ready to formulate the main new result from[36].f (as opposed to ψ H f , φ H f ). Acknowledgements: This paper is an extended version of some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. Here I would like to thank the organizers for the invitation to these conferences. I also would like to thank Sylvain Cappell, Laurentiu Maxim and Shoji Yokura for the discussions on our joint work related to the subject of this paper.f : Y → C. Then the motivic Hirzebruch class transformation M HT y * commutes with specialization, that is:Again the smoothness of Y is not needed here, but only the fact that X = {f = 0} is a global hypersurface (of codimension one) is needed. Also the factor (1 + y) in theorem 3.9 cancelled out by the renormalization factor (1 + y) −i · on H i (−) used in the definition of M HT y * , since the Gysin map i ! : H * (Y ) → H * −1 (X) shifts this degree by one.By the definition of ψ m f in[4,19]one has that. Together with[37][Prop.5.2.1] one therefore gets the following commutative diagram:As before one gets the following result from Theorem 3.9 and Corollary 3.10 together with (1.8) and (1.10): Chern classes for singular hypersurfaces. P Aluffi, Trans. Amer. Math. Soc. 35110Aluffi, P., Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc. 351 (1999), no. 10, 3989-4026. Riemann-Roch for singular varieties. P Baum, W Fulton, R Macpherson, Publ. Math. 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[ "Relaxation Time for Strange Quark Spin in Rotating Quark-Gluon Plasma", "Relaxation Time for Strange Quark Spin in Rotating Quark-Gluon Plasma" ]	[ "Joseph I Kapusta \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n", "Ermal Rrapaj \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n\nDepartment of Physics\nUniversity of California\n94720BerkeleyCAUSA\n", "Serge Rudaz \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n" ]	[ "School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA", "School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA", "Department of Physics\nUniversity of California\n94720BerkeleyCAUSA", "School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA" ]	[]	Experiments at the Relativistic Heavy Ion Collider (RHIC) have measured the net polarization of Λ andΛ hyperons and attributed it to a coupling between their spin and the vorticity of the fluid created in heavy ion collisions. Equipartition of energy is generally assumed, but the dynamical mechanism which polarizes them has yet to be determined. We consider two such mechanisms: vorticity fluctuations and helicity flip in scatterings between strange quarks and light quarks and gluons. With reasonable parameters both mechanisms lead to equilibration times orders of magnitude too large to be relevant to heavy ion collisions. Our conclusion is that strange quark spin or helicity is unchanged from the time they are created to the time they hadronize. A corollary is that vorticity fluctuations do not affect the hyperon spin either.	10.1103/physrevc.101.024907	[ "https://arxiv.org/pdf/1907.10750v1.pdf" ]	198,897,578	1907.10750	90505a9530937acd9c5ff9a6a74b5a4d11d0cce7	Relaxation Time for Strange Quark Spin in Rotating Quark-Gluon Plasma Joseph I Kapusta School of Physics and Astronomy University of Minnesota 55455MinneapolisMinnesotaUSA Ermal Rrapaj School of Physics and Astronomy University of Minnesota 55455MinneapolisMinnesotaUSA Department of Physics University of California 94720BerkeleyCAUSA Serge Rudaz School of Physics and Astronomy University of Minnesota 55455MinneapolisMinnesotaUSA Relaxation Time for Strange Quark Spin in Rotating Quark-Gluon Plasma (Dated: July 26, 2019) Experiments at the Relativistic Heavy Ion Collider (RHIC) have measured the net polarization of Λ andΛ hyperons and attributed it to a coupling between their spin and the vorticity of the fluid created in heavy ion collisions. Equipartition of energy is generally assumed, but the dynamical mechanism which polarizes them has yet to be determined. We consider two such mechanisms: vorticity fluctuations and helicity flip in scatterings between strange quarks and light quarks and gluons. With reasonable parameters both mechanisms lead to equilibration times orders of magnitude too large to be relevant to heavy ion collisions. Our conclusion is that strange quark spin or helicity is unchanged from the time they are created to the time they hadronize. A corollary is that vorticity fluctuations do not affect the hyperon spin either. I. INTRODUCTION Experiments at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC) have provided an abundance of data on the hot, dense matter created in heavy ion collisions [1]. Among these data are the coefficients of a Fourier expansion in the azimuthal angle for a variety of physical observables. The data provide strong evidence for collective expansion of the hot, dense matter and provide information on transport coefficients such as the shear viscosity [2]. In addition, the polarization of Λ andΛ hyperons was proposed as yet another observable that provides information on collective flow, in particular vorticity [3,4]. The vorticity arises in noncentral heavy ion collisions where the produced matter has considerable angular momentum. The spins of the Λ andΛ couple to the vorticity, resulting in a splitting in energy between particles with spin parallel and antiparallel to the vorticity. The decay products of these hyperons are used to infer their polarizations. Measurements of the polarizations have been made by the STAR collaboration from the lowest to the highest beam energies at RHIC [5][6][7], noting that RHIC produces matter with the highest vorticity ever observed. The standard picture of Λ andΛ polarization in non-central heavy ion collisions assumes equipartition of energy [8,9]. The spin-vorticity coupling is the same for baryons and antibaryons, which is approximately what is observed. The relatively small difference was studied in Ref. [10] and will not be addressed here. Similar to the difficult question of how the quarks and gluons come to thermal equilibrium is the dynamical mechanism by which the hyperons become polarized. Within the quark model the spin of the Λ is carried by the strange quark [11,12]. One possibility is that the s ands quarks become polarized in the quark-gluon plasma phase and pass that poalrization on to the Λ andΛ during hadronization. We shall estimate the relaxation rates and times for the strange quark spin to come to equilibrium with the vorticity in an idealized situation of a rotating quark-gluon plasma. Suppose that the strange quark spin is not in equilibrium. We consider two mechanisms by which it would be brought back to equilibrium. The first mechanism recognizes that there will be fluctuations in the direction and magnitude of the vorticity in heavy ion collisions. These fluctuations will drive the spins back towards equilibium, just as fluctuations around a constant magnetic field drive electron spins towards equilibrium. The second mechanism considers the scattering of massive strange quarks with massless up and down quarks and gluons in the plasma. Since helicity is conserved in QCD interactions when the quark is massless, helicity flip can only occur when the quark has a mass. At the scales of interest, the current quark masses of the up and down quarks are in the range from 4 to 7 MeV and may be considered massless. If their helicities are out of equilibrium they cannot be brought into equilibrium by scattering. They evolve without change. The current quark mass of the strange quark is in the range of 100 to 120 MeV. Their equilibration times will be nonzero due to scattering. The outline of the paper is as follows. In Sec. II we review the use of tetrads for spin-1/2 fermions in an accelerated system, in particular for a rotating system. In Sec. III we find the eigenvalues and eigenvectors for spin-1/2 fermions in a model Hamiltonian where the orbital angular momentum is small enough that it may be neglected. In Sec. IV we review fluctuations at the level of second order response theory as linear response theory is insufficient. We apply this to a massive Pauli particle in Sec. V, to a massless Dirac particle in Sec. VI, and to a massive Dirac particle in Sec. VII. In the latter two sections we restrict our attention to the case when the momentum is parallel or anti-parallel to the direction of vorticity, both for reasons of simplicity and because we neglect orbitatal angular momentum in these sections. In Sec. VIII we apply kinetic theory to the rate of helicity flip of strange quarks and anti-quarks using lowest order QCD perturbation theory. In Sec. IX we present numerical results, and in Sec. X we give our conclusions. Some useful commutation and anti-commutation relations are recalled in the appendix. The appendix also contains many of the tedious mathematical details. Readers mainly interested in the results and not the mathematical details may wish to read Secs. IX and X first. II. TETRADS AND SPIN Consider a fluid element undergoing linear acceleration, expansion, and rotation. Although we are mostly interested in rotation the tetrad formalism is able to handle all types. See Ref. [13] for a clear review of the topic. 1 The idea is to set up an inertial coordinate system at rest with respect to a fluid element at every space-time point. Let x µ represent the space-time coordinates of an observer at rest in the fluid element and ξ a the coordinates of an inertial frame. Then g µν (x)dx µ dx ν = η ab dξ a dξ b .(1) When there is no cause for confusion we use Greek indices for the x-coordinates, Latin indices a, b, ... for the ξ-coordinates, and Latin indices i, j, ... for spatial indices. The Minkowski metric is η ab = diag(1, −1, −1, −1). The tetrad is defined as e a µ (x) = ∂ξ a ∂x µ(2) while the inverse tetrad is e µ a (x) = g µν (x)η ab e b ν (x) . (3) Note that Greek indices are raised and lowered with g µν (x) and its inverse, while Latin indices are raised and lowered by η ab and its inverse. The tetrads obey the orthogonality properties e a µ (x)e µ b (x) = δ a b e µ a (x)e a ν (x) = δ µ ν .(4) In analogy to the affine connection Γ λ µν there is a spin connection ω a µ b which is used to take covariant derivatives of spinors. It satisfies the equation ω a µ b = e a ν e λ b Γ ν µλ − e λ b ∂ µ e a λ .(5) The Dirac matricesγ µ (x) become space-time dependent. They are obtained from the usual Dirac matrices γ a byγ µ (x) = e µ a (x)γ a . They satisfyγ µγν +γ νγµ = 2g µν compared to γ a γ b + γ b γ a = 2η ab .(8) One finds that the gradient of a spinor is replaced by a covariant derivative. ∂ µ ψ → D µ ψ = (∂ µ + Γ µ − ieA µ ) ψ(9) Here an electromagnetic vector potential is included for reference. The symbol Γ µ is also called the spin connection, which is confusing in several ways. The Dirac equation is [14] iγ µ (x)D µ ψ − mψ = 0 .(10) The spin connection is Γ µ = − 1 2 ω µab S ab(11) where S ab = i 2 σ ab and σ ab = i 2 [γ a , γ b ] .(12) Consider a region of space where a fluid element is rotating in an anti-clockwise sense around the z axis with angular speed ω which may be considered constant within that region. Here we follow Ref. [15] and choose the tetrad as the 4 × 4 matrix e a µ (x) =     1 v x v y 0 0 1 0 0 0 0 1 0 0 0 0 1    (13) where v x ≡ −ωy and v y ≡ ωx. From this is it straightforward to find the metric g µν (x) =     1 − v 2 −v x −v y 0 −v x −1 0 0 −v y 0 −1 0 0 0 0 −1     ,(14) the inverse metric g µν (x) =     1 −v x −v y 0 −v x −1 + v 2 x v x v y 0 −v y v x v y −1 + v 2 y 0 0 0 0 −1     ,(15) and the inverse tetrad e µ a (x) =     1 0 0 0 −v x 1 0 0 −v y 0 1 0 0 0 0 1     .(16) To get the spin connection it is useful to have e νa (x) =     1 −v x −v y 0 0 −1 0 0 0 0 −1 0 0 0 0 −1     .(17) The nonzero components of the affine connection are Γ 1 00 = ωv y Γ 2 00 = −ωv x Γ 2 01 = ω Γ 1 02 = −ω .(18) The nonzero components of ω µab are ω 012 = −ω 021 = ω .(19) Hence the only nonzero component of Γ µ is Γ 0 = − i 2 ωΣ 3(20) where Σ j = σ j 0 0 σ j .(21) Finally the Dirac matrices areγ 0 = γ 0 γ 1 = γ 1 − v x γ 0 γ 2 = γ 2 − v y γ 0 γ 3 = γ 3(22) The single particle Hamiltonian can be found by writing the Dirac equation in the form i∂ 0 ψ = Hψ with the result H = βm − iα j ∂ j + iω(x∂ 2 − y∂ 1 ) − 1 2 ωΣ 3(23) Defining the vorticity 1 2 ∇ × v = ω(24) we can express the Hamiltonian in terms of the orbital and spin angular momentum as H = βm + α · p − ω · (L + S) .(25) It can also be written as H = βm + α · p − v · p − ω · S .(26) This Hamiltonian is consistent with the literature. When taking the nonrelativistic limit via the Foldy-Wouthuysen procedure, it is known that the orbital angular momentum term gives rise to the usual Coriolis and centrifugal forces [16,17]. The last term is the spin-rotation coupling. The conserved current density is j µ =ψγ µ ψ (27) which, as a 4-vector, should satisfy ∂ µ j µ + Γ ν να j α = 0 .(28) Since Γ ν να = 1 √ −g ∂ α √ −g(29) where g = det (g µν ), and g = −1 in the present case, it follows that the current is ordinarily conserved. One also finds by direct calculation from the Dirac equation that ∂ µ j µ = 0 .(30) III. EIGENVALUES AND EIGENVECTORS As mentioned earlier, in this paper we are interested in the spin-rotation coupling. The vorticity couples to the total angular momentum J = L + S, and it is J which commutes with the Haniltonian of Eq. (25) or (26). Nevertheless we shall henceforth drop the term ω · L = v · p. Because the vorticity in energy units is so small in high energy heavy ion collisions, typically on the order of several MeV, this appears justifiable. Alternatively, one may restrict attention to the region near the origin where the orbital angular momentum is small and \|v\| 1. Keeping the coupling of vorticity to orbital angular momentum complicates the problem significantly, and one should perhaps use an angular momentum basis rather than a momentum basis. Consider the Hamiltonian H = mβ + α · p − 1 2 ω 0 Σ 3 . Due to rotational symmetry around the z axis we take the transverse momentum to be in the x direction and set p 2 = 0. Define E = p 2 + m 2 and E 3 = p 2 3 + m 2 . The two positive energy states have eigenvalues E ± = E 2 + 1 4 ω 2 0 ± ω 0 E 3 . The unnormalized eigenvector for the upper sign is u + =                −p 1 p 3 (E 3 + m) E + + E 3 + 1 2 ω 0 1 p 1 E + + E 3 + 1 2 ω 0 −(E 3 − m) p 3                .(31) and the unnormalized eigenvector for the lower sign is u − =                1 p 3 (E − − E 3 + 1 2 ω 0 ) p 1 (E 3 + m) p 3 E 3 + m E − − E 3 + 1 2 ω 0 p 1                .(32) These eigenvectors are orthogonal. When p 1 → 0, u + is an eigenstate of Σ 3 with eigenvalue −1 while u − is an eigenstate of Σ 3 with eigenvalue +1. The two negative energy states have eigenvalues −E ± . The unnormalized eigenvector for the upper sign is v + =                − (E 3 − m) p 3 −p 1 E + + E 3 + 1 2 ω 0 1 p 1 (E 3 − m) p 3 E + + E 3 + 1 2 ω 0                .(33) and the unnormalized eigenvector for the lower sign is v − =                − E − − E 3 + 1 2 ω 0 p 1 p 3 E 3 + m −p 3 (E − − E 3 + 1 2 ω 0 ) p 1 (E 3 + m) 1                .(34) When p 1 → 0, v + is an eigenstate of Σ 3 with eigenvalue +1 while v − is an eigenstate of Σ 3 with eigenvalue −1. IV. FLUCTUATION THEORY The expressions given in Sec. II remain valid no matter what direction in space the angular velocity is pointing in. They also remain true if the vorticity is allowed to depend on t, although it cannot depend on space. We write it as ω(t) = (ω 1 (t), ω 2 (t), ω 0 + ω 3 (t))(35) where ω 0 is the constant, average angular velocity and the ω i (t) are small, fluctuating quantities whose averages are zero. We want to calculate the equivalent of the Bloch equations for this situation, which entails using second order perturbation theory (second order response theory). We mostly follow the notation of Ref. [18]. See also Ref. [19]. Note that the latter reference uses the density matrix formalism whereas we use the commutator formalism. Consider a time independent Hamiltonian H 0 . The eigenvalues and eigenstates of H 0 are such that H 0 \|n = E n \|n . This is in the Heisenberg picture where \|n ≡ \|n H = e iH 0 t \|n, t S with the subscripts H and S referring to the Heisenberg and Schrödinger pictures. Consider a time independent operator A S in the Schrödinger picture. In the Heisenberg picture it is A H (t) = e iH 0 t A S e −iH 0 t . The thermal average of this operator is A H (t) 0 = 1 Z 0 Tr e −β(H 0 −µN ) A H (t) = 1 Z 0 Tr e −β(H 0 −µN ) A S = 1 Z 0 n e −β(En−µNn) n\|A S \|n = A S 0 .(36) The subscript 0 on the right angular bracket indicates that the average is taken with respect to the Hamiltonian H 0 together with any conserved charge N . The average is clearly time independent. Next consider the Hamiltonian H(t) = H 0 +V (t), where V (t) is a time dependent perturbation that vanishes when t < 0. The time evolution operator for the full H(t) is denoted by U (t). It satisfies the equation of motion d dt U (t) = −iH(t)U (t) .(37) It can be factorized as U (t) = e −iH 0 t U I (t) .(38) This leads to the equation of motion for U I (t) d dt U I (t) = −iV I (t)U I (t)(39) where V I (t) = e iH 0 t V (t) e −iH 0 t is the perturbation in the interaction picture. This can be solved iteratively to yield U I (t) = 1 + 1 i t 0 dt V I (t ) + 1 i 2 t 0 dt t 0 dt V I (t )V I (t ) + · · · .(40) The density matrix now becomes time dependent, ρ(t) = U (t)ρ 0 U † (t) = e −iH 0 t U I (t)ρ 0 U † I (t)e iH 0 t ,(41) and ρ I (t) = e iH 0 t ρ(t)e −iH 0 t = U I (t)ρ 0 U † I (t) ,(42) where ρ 0 = e −β(H 0 −µN ) Z 0 .(43) The ρ I satisfies dρ I dt = −i [V I , ρ I ] .(44) Averages are taken with ρ(t) instead of ρ 0 . This is denoted by replacing the subscript 0 with t on the right angular bracket. Thus A t = Tr (ρ(t)A S ) = Tr e −iH 0 t ρ I (t) e iH 0 t A S = Tr (ρ I (t)A I ) .(45) The average is representation independent, as it must be. We are interested in the equation of motion for A t . To this end we calculate d dt ρ 0 U † (t)A S U (t) = d dt ρ 0 U † I (t)A I (t) U I (t) = ρ 0 U † I dA I dt U I + dU † I dt A I U I + U † I A I dU I dt(46) and take the trace. Using the equation of motion dA I /dt = i[H 0 , A I ], the first term on the far RHS contributes to d A t /dt the term iTr ρ 0 U † I (t)[H 0 , A I (t)]U I (t) = iTr (ρ I (t)[H 0 , A I (t)]) = iTr (ρ(t)[H 0 , A S ]) = i [H 0 , A S ] t . (47) The second and third terms on the far RHS are iρ 0 [V I (t), A I (t)] − ρ 0 t 0 dt [V I (t ), [V I (t), A I (t)]] up to and including terms of second order in the perturbation. Taking the trace yields d dt A t = i [H 0 , A S ] t + i [V I (t), A I (t)] 0 − t 0 dt [V I (t ), [V I (t), A I (t)]] 0 .(48) Next we shall perform an ensemble or time average over the fluctuating fields. We assume that V (t) = V I (t) = 0. The average of the product V (t)V (t ) = V I (t)V I (t ) is not zero but is assumed to be correlated on a time scale of τ c . It is also assumed that fluctuations induced in A I (t) are small enough that we may ignore V I (t )A I (t). This is coarse graining, also sometimes called the Born approximation. Therefore, up to second order in the fluctuations we have d dt A t = i [H 0 , A S ] t − t 0 dt [V I (t ), [V I (t), A I (t)]] 0 .(49) Here the double angular bracket means that averaging over the fluctuating fields is performed in addition to the thermal ensemble average of Eq. (36). It is a more convenient notation than the overline. V. MASSIVE PAULI PARTICLE The Hamiltonian for a massive, nonrelativistic particle with spin one-half is H 0 = p 2 2m − 1 2 ω 0 σ 3 .(50) Since the kinetic energy commutes with the spin operator, this is basically the simple spin model presented in Section IV of Ref. [19]. Let us apply the results of Sec. IV with H 0 = − 1 2 ω 0 σ 3 . Then e iH 0 t = cos 1 2 ω 0 t − i sin 1 2 ω 0 t σ 3 .(51) Recall the well known similarity transformations σ 1 (t) ≡ e iH 0 t σ 1 e −iH 0 t = cos(ω 0 t)σ 1 + sin(ω 0 t)σ 2 σ 2 (t) ≡ e iH 0 t σ 2 e −iH 0 t = cos(ω 0 t)σ 2 − sin(ω 0 t)σ 1 σ 3 (t) ≡ e iH 0 t σ 3 e −iH 0 t = σ 3 .(52) From here on in, whenever a Pauli or Dirac matrix appears without a time argument it is understood to remain unaffected by the time evolution. Equivalently, it is evaluated at t = 0. We are most interested in the operator A S = σ 3 . It quickly follows that [V I (t), σ 3 ] = iω 1 (t)σ 2 (t) − iω 2 (t)σ 1 (t) .(53) The model assumes that fluctuations in different directions in Cartesian coordinates are uncorrelated, namely ω i (t)ω j (t ) = ω 2 i e −\|t−t \|/τc δ ij (54) where τ c is a correlation time. We can write the double commutator as [[V I (t ), [V I (t), σ 3 ]] = W 11 + W 22 + cross terms (55) with W 11 = ω 1 (t)ω 1 (t ) cos[ω 0 (t − t )]σ 3 W 22 = ω 2 (t)ω 2 (t ) cos[ω 0 (t − t )]σ 3 .(56) Cross terms involve ω i (t)ω j (t ) with i = j; these average to zero. Averaging involves the integral t 0 dt e −\|t−t \|/τc cos[ω 0 (t − t )] .(57) For t τ c this integral becomes T 0 = τ c 1 + ω 2 0 τ 2 c .(58) Recognizing that we are interested in the small departure from the equilibrium value of the z component of the spin s eq 3 = 1 2 tanh(βω 0 /2)(59) we find that ds 3 dt = − s 3 − s eq 3 τ (60) where the relaxation time is given by 1 τ = ω 2 1 + ω 2 2 T 0 .(61) This agrees with the Eq. (IV.33) of Ref. [19] which based its calculations on the density matrix. Analogous calculations can be done for the components of spin perpendicular to the vorticity. The double communators needed for the x component of spin are [σ 2 (t ), [σ 2 (t), σ 1 (t)]] = 4σ 1 (t ) [σ 3 , [σ 3 , σ 1 (t)]] = 4σ 1 (t)(62) and for the y component [σ 1 (t ), [σ 1 (t), σ 2 (t)]] = 4σ 2 (t ) [σ 3 , [σ 3 , σ 2 (t)]] = 4σ 2 (t) .(63) One encounters integrals like t 0 dt e −(t−t )/τc σ 2 (t ) = T 0 [σ 2 (t) + ω 0 τ c σ 1 (t)] .(64) Putting them all together results in the remaining two spin equations. ds 1 dt = ω 0 1 + ω 2 2 τ c T 0 s 2 − ω 2 3 τ c + ω 2 2 T 0 s 1 ds 2 dt = −ω 0 1 + ω 2 1 τ c T 0 s 1 − ω 2 3 τ c + ω 2 1 T 0 s 2(65) VI. MASSLESS DIRAC PARTICLE In this section we apply the general formulas to the case of a massless Dirac particle. We focus on the situation where the momentum is parallel to the vorticity. The general case is much more involved and does not provide significantly more useful information. The Hamiltonian is H 0 = pα 3 − 1 2 ω 0 Σ 3 .(66) Note that Σ 3 commutes with H 0 . The time evolution operator is (for details see the appendix) e iH 0 t = cos 1 2 ω 0 t cos(pt) I + sin 1 2 ω 0 t sin(pt) γ 5 + i cos 1 2 ω 0 t sin(pt) α 3 − i sin 1 2 ω 0 t cos(pt) Σ 3 .(67) The similarity transformations of the Σ matrices are Σ 1 (t) = e iH 0 t Σ 1 e −iH 0 t = cos(ω 0 t) cos(2pt)Σ 1 + sin(ω 0 t) cos(2pt)Σ 2 + sin(ω 0 t) sin(2pt)α 1 − cos(ω 0 t) sin(2pt)α 2 Σ 2 (t) = e iH 0 t Σ 2 e −iH 0 t = cos(ω 0 t) cos(2pt)Σ 2 − sin(ω 0 t) cos(2pt)Σ 1 + sin(ω 0 t) sin(2pt)α 2 + cos(ω 0 t) sin(2pt)α 1 Σ 3 (t) = e iH 0 t Σ 3 e −iH 0 t = Σ 3 .(68) The single and double commutators needed for the fluctuations of ω i (t) in the i = 1 direction are [Σ 1 (t), Σ 3 ] = −2iΣ 2 (t)(69) and [Σ 1 (t ), [Σ 1 (t), Σ 3 ]] = 2 {cos[(ω 0 + 2p)(t − t )] + cos[(ω 0 − 2p)(t − t )]} Σ 3 + 2 {cos[(ω 0 − 2p)(t − t )] − cos[(ω 0 + 2p)(t − t )]} α 3 .(70) The latter shows that Σ 3 and α 3 are coupled. Therefore we also need [Σ 1 (t), α 3 ] = 2i sin(ω 0 t) cos(2pt)α 1 − 2i cos(ω 0 t) cos(2pt)α 2 − 2i cos(ω 0 t) sin(2pt)Σ 1 − 2i sin(ω 0 t) sin(2pt)Σ 2 = −2iα 2 (t)(71) and [Σ 1 (t ), [Σ 1 (t), α 3 ]] = 2 {cos[(ω 0 + 2p)(t − t )] + cos[(ω 0 − 2p)(t − t )]} α 3 + 2 {cos[(ω 0 − 2p)(t − t )] − cos[(ω 0 + 2p)(t − t )]} Σ 3 .(72) Averaging over the fluctuating fields ω i (t) can be performed using Eqs. (57) and (58). This results in d dt Σ 3 t = − Σ 3 t τ 1 − α 3 t τ 2 d dt α 3 t = − α 3 t τ 1 − Σ 3 t τ 2(73) where 1 τ 1 = 1 2 ω 2 1 + ω 2 2 (T − + T + ) 1 τ 2 = 1 2 ω 2 1 + ω 2 2 (T − − T + )(74) and T ± = τ c 1 + (2p ± ω 0 ) 2 τ 2 c .(75) This makes use of the rotational symmetry around the vorticity axis. The normal modes are d dt Σ 3 ± α 3 t = − Σ 3 ± α 3 t τ ±(76) where 1 τ ± = ω 2 1 + ω 2 2 T ∓ .(77) In Eq. (76) it is to be understood that these represent departures from the equilibrium values, otherwise the equilibrium values should be inserted by hand on the right hand side, as in Eq. (60). Of phenomenological interest is the situation where ω 0 \|p\|. In that limit 1 τ ± ≈ ω 2 1 + ω 2 2 τ c 1 + 4p 2 τ 2 c .(78) Now add a mass term but keep the momentum parallel to the vorticity. The Hamiltonian is H 0 = mβ + pα 3 − 1 2 ω 0 Σ 3 .(79) The evolution operator has the form e iH 0 t = C 1 I + C 2 γ 5 + C 3 βΣ 3 + iC 4 β + iC 5 α 3 + iC 6 Σ 3 .(80) The coefficients are calculated in the appendix with the result C 1 = cos(Et) cos 1 2 ω 0 t C 2 = p E sin(Et) sin 1 2 ω 0 t C 3 = m E sin(Et) sin 1 2 ω 0 t C 4 = m E sin(Et) cos 1 2 ω 0 t C 5 = p E sin(Et) cos 1 2 ω 0 t C 6 = − cos(Et) sin 1 2 ω 0 t .(81) Operators in the interaction picture can be obtained from those in the Schrödinger picture with tedious algebra. Of particular interest are Σ 1 (t) = B 1 (t)Σ 1 + B 2 (t)Σ 2 + B 3 (t)α 1 − B 4 (t)α 2 + iB 5 (t)βα 1 + iB 6 (t)βα 2 ,(82)Σ 2 (t) = B 1 (t)Σ 2 − B 2 (t)Σ 1 + B 3 (t)α 2 + B 4 (t)α 1 + iB 5 (t)βα 2 − iB 6 (t)βα 1 ,(83) while Σ 3 (t) = Σ 3 on account of the fact that it commutes with H 0 . The B i are given in the appendix. We shall also need α 3 (t) = p 2 E 2 + m 2 E 2 cos(2Et) α 3 + mp E 2 [1 − cos(2Et)] β + i m E sin(2Et)βα 3 ,(84) and β(t) = m 2 E 2 + p 2 E 2 cos(2Et) β + mp E 2 [1 − cos(2Et)] α 3 − i p E sin(2Et)βα 3 .(85) One can check that mβ(t) + pα 3 (t) = mβ + pα 3 , as it should be since this operator commutes with H 0 . From these the final relevant one is easily found to be β(t)α 3 (t) = cos(2Et)βα 3 − i p E sin(2Et)β + i m E sin(2Et)α 3 .(86) What is needed for fluctuations for Σ 3 are the double commutators [Σ 1 (t ), [Σ 1 (t), Σ 3 ]] = [Σ 2 (t ), [Σ 2 (t), Σ 3 ]] [Σ 1 (t ), [Σ 1 (t), α 3 ]] = [Σ 2 (t ), [Σ 2 (t), α 3 ]] [Σ 1 (t ), [Σ 1 (t), β]] = [Σ 2 (t ), [Σ 2 (t), β]] [Σ 1 (t ), [Σ 1 (t), βα 3 ]] = [Σ 2 (t ), [Σ 2 (t), βα 3 ]] . (87) E 2 cos θ − + cos θ + cos(2Et) + p 2 E 2 sin θ − + sin θ + sin(2Et) βα 3(91) The following integrals are useful for averaging over the fluctuations. The arrows represent the steady state where τ c t. t 0 dt e −(t−t )/τc cos[ω 0 (t − t )] → T 0 .(92)t 0 dt e −(t−t )/τc sin[ω 0 (t − t )] → ω 0 τ c T 0 .(93)t 0 dt e −(t−t )/τc cos[(2E ± ω 0 )(t − t )] → T ± (94) t 0 dt e −(t−t )/τc sin[(2E ± ω 0 )(t − t )] → (2E ± ω 0 )τ c T ±(95) Here T ± = τ c 1 + (2E ± ω 0 ) 2 τ 2 c(96) and T 0 = τ c 1 + ω 2 0 τ 2 c(97) as defined earlier. The results of performing the integration over t in the steady state are t 0 dt e −(t−t )/τc [Σ 1 (t ), [Σ 1 (t), Σ 3 ]] = 2 p 2 E 2 T − + T + + 2 m 2 E 2 T 0 Σ 3 + 2 p E T − − T + p 2 E 2 + m 2 E 2 cos(2Et) + τ c m 2 E 2 2ω 0 T 0 + (2E − ω 0 )T − − (2E + ω 0 )T + sin(2Et) α 3 + 2 mp 2 E 3 T − − T + 1 − cos(2Et) − τ c 2ω 0 T 0 + (2E − ω 0 )T − − (2E + ω 0 )T + sin(2Et) β + 2i mp E 2 T − − T + sin(2Et) − τ c 2ω 0 T 0 + (2E − ω 0 )T − − (2E + ω 0 )T + cos(2Et) βα 3 (98) t 0 dt e −(t−t )/τc [Σ 1 (t ), [Σ 1 (t), α 3 ]] = 2 p E T − − T + Σ 3 + 2 p 2 E 2 T − + T + + 2 m 2 E 2 T 0 cos(2Et) α 3 + 2 mp E 2 T − + T + − 2T 0 cos(2Et) β + 4i m E T 0 sin(2Et) βα 3 (99) t 0 dt e −(t−t )/τc [Σ 1 (t ), [Σ 1 (t), βα 3 ]] = 2iτ c mp E 2 2ω 0 T 0 + (2E − ω 0 )T − − (2E + ω 0 )T + Σ 3 + 2i m E 2 m 2 E 2 T 0 + p 2 E 2 T − + T + sin(2Et) + τ c p 2 E 2 (2E − ω 0 )T − + (2E + ω 0 )T + 1 − cos(2Et) α 3 + 2i p E τ c (2E − ω 0 )T − + (2E + ω 0 )T + m 2 E 2 + p 2 E 2 cos(2Et) − 2 m 2 E 2 T 0 + p 2 E 2 T − + T + sin(2Et) β + 2 2 m 2 E 2 T 0 + p 2 E 2 T − + T + cos(2Et) + τ c p 2 E 2 (2E − ω 0 )T − + (2E + ω 0 )T + sin(2Et) βα 3(100) These still need to be expressed in terms of the time-dependent operators in the Dirac basis. The relationships are given in the appendix. Finally, we need the commutators of the operators in the interaction picture with the unperturbed Hamiltonian, which are [H 0 , Σ 3 (t)] = 0 [H 0 , α 3 (t)] = 2mβα 3 (t) [H 0 , β(t)] = −2pβα 3 (t) [H 0 , β(t)α 3 (t)] = 2mα 3 (t) − 2pβ(t) .(101) Then the equations of motion can be written in matrix form as d dt     Σ 3 (t) α 3 (t) iβ(t)α 3 (t) β(t)     =     −h 0 −h 2 h 1 0 −h 2 −h 0 2m −h 3 h 1 −2m −h 0 (2p + h 4 ) 0 0 −2p 0         Σ 3 (t) α 3 (t) iβ(t)α 3 (t) β(t)    (102) where h 0 = p 2 2E 2 T − + T + + m 2 E 2 T 0 ω 2 ⊥ h 1 = τ c mp 2E 2 2ω 0 T 0 + (2E − ω 0 )T − − (2E + ω 0 )T + ω 2 ⊥ h 2 = p 2E T − − T + ω 2 ⊥ h 3 = mp 2E 2 T − + T + − 2T 0 ω 2 ⊥ h 4 = τ c p 2E (2E − ω 0 )T − + (2E + ω 0 )T + ω 2 ⊥(103) with ω 2 ⊥ = ω 2 1 + ω 2 2 .(104) As in the case of massless Dirac particles, it is understood that we are solving for the departures from the equilibrium values. The eigenvalues λ are found from a fourth order polynomial. Defining λ = λ + h 0 for convenience this polynomial is P = λ 4 − h 0 λ 3 + 4E 2 + 2ph 4 − h 2 1 − h 2 2 λ 2 + 4mph 3 + h 0 h 2 1 + h 2 2 − 4m 2 λ − 2ph 2 (2ph 2 + h 2 h 4 − h 1 h 3 ) .(105) Consider some limiting cases. When p = 0 then h 1 = h 2 = h 3 = h 4 = 0. There is one zero eigenvalue belonging to β(t) . The spin Σ 3 (t) has eigenvalue −ω 2 ⊥ T 0 . The quantities α 3 (t) and iβ(t)α 3 (t) are coupled with complex eigenvalues −ω 2 ⊥ T 0 ± 2mi. There is only one relaxation time and it is the same as found earlier. When m = 0 then h 1 = h 3 = 0. Defining λ ± = ω 2 ⊥ T ∓ , with λ + ≥ λ − , the remaining h's are h 0 = 1 2 (λ + + λ − ) h 2 = 1 2 (λ + − λ − ) h 4 = pτ c (λ + + λ − ) − 1 2 ω 0 τ c (λ + − λ − ) .(106) As we saw earlier, Σ 3 (t) and α 3 (t) are coupled with eigenvalues −λ + and −λ − . The quantities β(t) and iβ(t)α 3 (t) are coupled with eigenvalues − 1 4 (λ + + λ − ) ± 1 4 (λ + + λ − ) 2 − 32p (2p + h 4 ) . These are real for small momentum and become complex at larger momentum. This momentum scale is essentially (λ + + λ − )/8. Then the pair of conjugate eigenvalues at larger momenta are to good approximation just − 1 4 (λ + + λ − ) ± 2pi . VIII. STRANGE QUARK HELICITY FLIP IN QUARK GLUON PLASMA In this section we explore another mechanism for the relaxation rate for strange quark spin, which is spin/helicity flip in collisions of strange quarks with up or down quarks, antiquarks, or gluons. As is well-known, the helicity of a massless quark is conserved in such collisions due to the vector coupling to gluons. For a quark whose mass is small compared to its energy the cross section for helicity flip is proportional to m 2 . The current quark mass of the strange quark is around 100 MeV, whereas the temperature of the plasma might range from 500 MeV down to 200 MeV. Therefore we may use this as an approximation to estimate the rate of helicity flip in the plasma. A common approximation is the energy-dependent relaxation time approximation. Consider the reaction a + b → c + d. The relaxation time τ a (E a ) for species a with energy E a as measured in the rest frame of the plasma is given by [20,21] 1 + d a f eq a τ a (E a ) = bcd N 1 + δ ab dΓ b dΓ c dΓ d W (a, b\|c, d)f eq b (1 + d c f eq c ) (1 + d d f eq d ) .(107) Here the f eq i are Fermi-Dirac or Bose-Einstein distribution functions, and the d i are −1 for fermions and +1 for bosons. The integration over phase space involves dΓ i = d 3 p i (2π) 3 .(108) The W is related to the dimensionless amplitude M by W (a, b\|c, d) = (2π) 4 δ 4 (p a + p b − p c − p d ) 2E a 2E b 2E c 2E d \|M(a, b\|c, d)\| 2 .(109) The \|M(a, b\|c, d)\| 2 is averaged over spin in both the initial and final states; this compensates the spin factor 2s i + 1 in the phase space integration. It is further related to the differential cross section by dσ dt = 1 64πs 1 p 2 * \|M\| 2 (110) where p * is the initial state momentum in the center-of-momentum frame. Finally, the N is a degeneracy factor for spin, color, and any other internal degrees of freedom. Its value depends on how these variables are summed or averaged over in \|M\| 2 . In order to obtain analytical results we shall drop the Pauli suppression and Bose enhancement factors in the final state. For the reactions s + q → s + q and s +q → s +q, where q = u or d, this will only slightly enhance the rate. For the reaction s + g → s + g these two final state effects should approximately cancel. Therefore we expect this approximation to result in an accurate or slight overestimate of the rate. In Eq. (107) let a be the incoming strange quark with momentum p µ , b be the incoming light quark, anti-quark, or gluon with momentum p µ 2 , c be the outgoing strange quark with momentum p µ 3 , and d be the outgoing light quark, anti-quark, or gluon with momentum p µ 4 . With no loss of generality we will work in the rest frame of the plasma and takê p = (1, 0, 0) p 3 = (cos φ 3 , sin φ 3 , 0) p 4 = (cos φ 4 sin θ 4 , sin φ 4 sin θ 4 , cos θ 4 ) . The Mandelstam variables are s = (p + p 2 ) 2 = (p 3 + p 4 ) 2 and t = (p 3 − p) 2 = (p 4 − p 2 ) 2 . Following Ref. [22] we insert integrations over s and t with a Dirac δ-function for each. This is a natural thing to do since \|M\| 2 depends only on those two variables. We can use the 3-dimensional δ-function in Eq. (107) to eliminate the integration over p 2 . Then consider the integral J = dΩ 3 dΩ 4 δ E + E 2 − E 3 − E 4 δ s − 2E 3 E 4 (1 −p 3 ·p 4 ) δ t + 2EE 3 (1 −p ·p 3 ) ,(112) where E 2 = \|p 3 + p 4 − p\|. In these coordinates dΩ 3 dΩ 4 = The result is J = 2πE 2 EE 3 E 4 θ(y − E) √ Ax 2 + Bx + C (113) where x = E 3 − E 4 y = E 3 + E 4 A = − 1 4 s 2 B = 1 2 s(s + 2t)(2E − y) C = st(s + t) + s 2 E(y − E) − 1 4 (s + 2t) 2 y 2 .(114) Part of the remaining integration is dE 3 dE 4 = 1 2 dxdy. Integration over x can be done immediately x − x + dx √ Ax 2 + Bx + C = 2π s (115) where x ± = −B ± √ B 2 − 4AC 2A since B 2 − 4AC ≥ 0 and A < 0. The integral over y is simply ∞ E+s/4E dy f eq 2 (y − E) = T ln 1 ± e −s/4ET ±1 (116) where the upper sign is for fermions and the lower sign for bosons. The expression (107) reduces to 1 − f eq (E) τ (E) = N T 32(2π) 3 E 2 ds s ln 1 ± e −s/4ET ±1 dt \|M(s, t)\| 2 .(117) First consider the reaction s + q → s + q shown in Fig. 1. The strange quark is allowed to flip for any one of these four combinations. Here g is the QCD coupling constant. In this expression initial colors are averaged over while final colors are summed over. If we want the rate for helicity flip of the strange quark we should use the above expression along with N = 6 because of scattering from a light quark of either helicity and of any of three colors. The integral over t is essentially the cross section. Since a massless particle, the gluon, is being exchanged the cross section would be logarithmically divergent. Many-body effects are necessary to screen this divergence. Here we follow Ref. [22]. We remove the region of phase space causing the divergence. We integrate over − (s − m 2 ) 2 s + k 2 c ≤ t ≤ −k 2 c ,(119) and similarly for u, where k 2 c is an infrared cutoff. This way of regulating the divergence treats t and u symmetrically. It maintains s + t + u = 2m 2 . In order that the restricted range of t be sensible means that a small region of s must also be removed. We should take s ≥ s 0 where (s 0 − m 2 ) 2 s 0 = 2k 2 c ,(120) or s 0 = m 2 + k 2 c + k c 2m 2 + k 2 c . For the photon production processes addressed in Ref. [22] all quarks were massless so then s 0 = 2k 2 c . The contribution from the region of phase space removed must be added back in using the method of hard thermal loops, as Ref. [22] did. Now the t integration gives −k 2 c −(s−m 2 ) 2 /s+k 2 c dt \|M\| 2 helicity flip = 8 9 g 4 m 2 ln (s − m 2 ) 2 sk 2 c − 1 − 1 + 2s k 2 c (s − m 2 ) 2 . (121) The remaining integral over s is I q = 4 ∞ s 0 ds s ln (s − m 2 ) 2 sk 2 c − 1 − 1 + 2s k 2 c (s − m 2 ) 2 ln 1 + e −s/4ET .(122) In Ref. [22] it was shown that by adding the hard thermal loop contribution (the region of phase space removed) essentially replaced k 2 c with the effective mass of the exchanged quark in the plasma, as defined not at zero momentum but at high momentum, such that k 2 c ∝ g 2 T 2 . It is worth noting that the hard thermal loop contribution came from a branch cut, not a pole. In the present case we would expect k 2 c to be replaced by the effective mass of the exchanged gluon, k 2 c = m 2 P , with m 2 P = 1 6 g 2 N c + 1 2 N f T 2(123) where N c is the number of colors and N f is the number of light flavors. Performing the hard thermal loop calculation to validate this is outside the scope of this paper. Allowing for scattering from u,ū, d andd provides another factor of 4. Putting it all together yields 1 τ q (E) = α 2 s T 3π m 2 E 2 1 + e −E/T I q .(124) Unfortunately the expression for I q cannot be evaluated analytically. There is another way to approach this problem, which is to insert a static color electric screening mass in the gluon propagator. Replace Eq. (118) with \|M\| 2 helicity flip = − 2 9 g 4 m 2 t + st 2 (s − m 2 ) 2 1 (t − m 2 el ) 2(125) where m 2 el = 2m 2 P . This is the approach used in the parton cascade model ZPC [23] as implemented in the AMPT model which simulates high energy heavy ion collisions [24]. Although appealing, this approach is not manifestly gauge invariant so we do not pursue it here. Next consider the reaction s+g → s+g shown in Fig. 2. A lengthy calculation (see Appendix D) yields \|M\| 2 helicity flip = 4g 4 m 2 (−t) 3s 4 m 4s 2 3 + 3t u m + 4t 2 3u 3 m + 8(m 4 − su) 2 3 t 2 − 3 u m t + 4 3u 2 m + 7m 4 t 2 3u 2 m + 3 (m 4 − su − s m u m ) u m − 2(m 4 − su) t 2 .(126) Here s m = s − m 2 , u m = u − m 2 , and r 2 = m 4 − su. Then s m + t + u m = 0. This is obtained by summing over all colors in the initial and final states, and dividing by 3 to obtain the rate for scattering of a strange quark with a specified (average) color. When integrating over t or u it is helpful to express this as − 1 − c ln s(s m − k 2 c ) s(m 2 + k 2 c ) − m 4 + b(s 2 m − 2k 2 c s) (s m − k 2 c ) (s(m 2 + k 2 c ) − m 4 ) + ds m 2 s 3 m s 2 − 2(s m + k 2 c )s m s + 4k 2 c − e s 2 m s − 2k 2 c .(130) The remaining integral over s is I g = ∞ s 0 ds s 1 s 4 m b(s 2 m − 2k 2 c s) (s m − k 2 c ) (s(m 2 + k 2 c ) − m 4 ) + ds m 2 s 3 m s 2 − 2(s m + k 2 c )s m s + 4k 2 c − e s 2 m s − 2k 2 c + a ln s 2 m sk 2 c − 1 − c ln s(s m − k 2 c ) s(m 2 + k 2 c ) − m 4 × ln 1 − e −s/4ET −1 .(131) Finally we obtain the rate as 1 τ g (E) = α 2 s T 3π m 2 E 2 1 + e −E/T I g .(132) In principle one should use a density matrix to determine the spin or helicity relaxation time along the vorticity axis. We shall be content to use this formula as a proxy, recognizing that it should be a very close estimate. IX. NUMERICAL RESULTS In this section we provide numerical results for the relaxation times using the formulas derived in previous sections. The strange quark mass is set at m = 110 MeV. We begin with vorticity fluctuations and move on to helicity flip in quark-gluon plasma. Apart from the mass and momentum of the strange quark, the vorticity fluctutations require knowledge of the average vorticity ω 0 , the magnitude of the fluctutations ω 2 ⊥ , and the correlation time τ c for these fluctuations. These cannot be known a priori but must be found by a combination of experimental measurements and numerical simulations of high energy heavy ion collisions. They clearly depend on the beam energy, size of the colliding nuclei, and centrality (impact parameter). Measurements by the STAR Collaboration [6] of the hyperon polarization indicate that ω 0 = (9 ± 1) × 10 21 s −1 , with a systematic error of a factor of two, when averaging over the entire RHIC energy range. This converts to an energy of ω 0 = 6 MeV. For illustrative purposes we take ω 2 ⊥ = 8 MeV 2 and τ c = 4 fm/c. Figure 3 shows the imaginary part of the four eigenvalues coming from vorticity fluctuations. Two of the eigenvalues are purely real, while the other two are complex conjugates of each other. With the chosen parameters, and within numerical accuracy, one of the purely real eigenvalues has the eigenvector (m/E) β(t) + (p/E) α 3 (t) . The other one has the eigenvector Σ 3 (t) with an admixture of (m/E) β(t) + (p/E) α 3 (t) at the level of 10 −4 or less. These are associated with the two operators which commute with the Hamiltonian. The complex eigenvalues have eigenvectors which are linear combinations of (m/E) β(t) − (p/E) α 3 (t) and iβ(t)α 3 (t) , whose associated operators do not commute with the Hamiltonian. The imaginary parts are nearly equal to ±2Ei, following the discussion at the end of Sec. VII in the limits m = 0 and p = 0. the real parts of the eigenvalues. The largest equilibration time diverges like 1/p 2 as p → 0 and is associated with the zero eigenvalue of β(t) , as discussed at the end of Sec. VII. This mode has a minimum when p ≈ m, which arises from the transition from β(t) to α 3 (t) as described above. When p m this eigenvalue has the limiting form λ → − N D p 2 (133) where N = 2 T − + T + T 0 ω 2 ⊥ τ c − T − + T + 2T 0 − 1 ω 2 ⊥ τ c 3 m 2 + τ c 2 T − + T + T 0 − ω 0 m T − − T + T 0 ω 2 ⊥ τ c 2 .(134) In this expression and this expression only the T ± are evaluated at E = m. Also D = 4m 2 + ω 2 ⊥ τ c 2 .(135) When p m it has the limiting form λ → − ω 2 ⊥ τ c 1 + 4E 2 τ 2 c .(136) A good representation of the dashed curve is τ ≈ D N 1 p 2 + 1 + 4E 2 τ 2 c ω 2 ⊥ τ c .(137) The other three modes have smaller equilibration times and appear to all be equal, but that is only because of the logarithmic scale used in the figure. As we saw at the end of Sec. VII, these three modes become degenerate when p → 0. At that point the equilibration time is 1 + ω 2 0 τ 2 c ω 2 ⊥ τ c . Within the thickness of the solid curve in the figure they can all be approximated by the formula τ ≈ 1 + 4E 2 τ 2 c ω 2 ⊥ τ c 1 + ω 2 0 τ 2 c 1 + 4m 2 τ 2 c(138) as a function of p. Since the time scale for quark-gluon plasma expansion, cooling and entering the hadronization stage is in the range from several to at most ten fm/c, it is clear that these equilibration times are far too large to influence the evolution of strange quark spin. For numerical estimates of the helicity flip rate we take g = 2 corresponding to α s = 1/π. This may not seem like a small number, and it is not, but it is a realistic number for plasma temperatures on the order of 200 and 400 MeV. For comparison the fine structure constant 1/137 implies that the electromagnetic coupling is e = 0.30. Figure 5 shows the dimensionless integrals I q and I g appearing in Eqs. (122) and (131). Both integrals increase with momentum and decrease with temperature. The integral coming from scattering with massless quarks is comparable to the integral coming from scattering with gluons. Figure 6 shows the equilibration times for strange quark helicity separately for the reactions s + q and s + g. At a given temperature the equilibration time for scattering with gluons is slightly smaller than scattering with massless quarks. Figure 7 shows the net equilibration time τ −1 net = τ −1 q + τ −1 g . These equilibration times are also far too large to influence the evolution of strange quark spin in high energy heavy ion collisions. X. CONCLUSION Measurements of the net polarization of Λ andΛ hyperons in heavy ion collisions at RHIC have been interpreted as being due to a coupling between their spin and the vorticity of the fluid created in these collisions. This motivated the present study of the equilibration time for strange quark spin in rotating quark-gluon plasma, as the spin in these hyperons is generally attributed to the strange quark. We considered two mechanisms: vorticity fluctuations and helicity flip in scatterings between strange quarks and light quarks and gluons. Our calculations lead to equilibration times orders of magnitude too large to be relevant to heavy ion collisions. Certainly our calculations and parameters can be improved upon. Regarding vorticity fluctuations, we made some rough estimates of the parameters involved, but ultimately they should come from numerical simulations of high energy heavy ion collisions. In order to make the calculations relatively tractable for a strange quark moving with relativistic speeds we studied only the case where its momentum was either parallel or anti-parallel to the vorticity. Otherwise we would have to include orbital angular momentum, which would probably entail using an angular momentum basis rather than a linear momentum basis. Relaxing that restriction cannot change the equilibration time by very much, as the nonrelativistic limit yields a result independent of the direction of the momentum. In fact, the nonrelativistic limit can be applied to the hyperons in the later hadronic phase. Our results strongly suggest that vorticity fluctuations cannot alter the polarization of a hyperon either. Regarding helicity flip, the contribution from hard thermal loops [25] should be calculated. In this paper we assumed that the infrared cutoff should be associated with the thermal mass of the exchanged gluon in the t-channel. Given how small the helicity flip rate is makes it questionable whether such a calculation has more than theoretical interest. How then do the Λ andΛ hyperons acquire their polarization in heavy ion collisions? One possibility is that the strange quarks are created with a net polarization. It is generally acknowledged that gluons are produced first, and they create the majority of the quarks and anti-quarks. Phase space alone would favor quarks and anti-quarks being created with a preference for their spin to be aligned with the vorticity. Vorticity fluctuations and helicity flip scattering would not change the polarization as the quark-gluon plasma evolves. The strange quarks and anti-quarks could then pass along their spin to the hyperons. A specific mechanism for polarizing quarks during the initial stage of heavy ion collisions was proposed in Ref. [3] and developed further in Refs. [26,27]. They considered parton-parton scattering in a longitudinal shear flow. By Fourier transforming the differential cross section from transverse momentum to transverse distance, and taking into account the asymmetry in coordinate space due to the shear, they find that the partons can become polarized. Recently this idea has been extended to include the scattering of wave packets [28]. Whether this is in fact the origin of the polaization observed remains to be seen. If it is, then the relationship between hyperon polarization and vorticity is much more complicated than equipartition of energy, as is usually assumed. Recently, a general analysis of the spin current in relativistic viscous fluids has been proposed [29]. It points out that the spin current is not conserved, because angular momentum can be transferred between orbital and spin degrees of freedom. As with other transport coefficients, the corresponding relaxation times are not determined by thermodynaics but must be calculated from the microscopic dynamics. Another possibility is that strange quarks and anti-quarks in the quark-gluon plasma phase in heavy ion collisions are not polarized at all. Rather, the hadronization from quarks and gluons to hadrons favors hyperons polarized parallel to the vorticity simply due to the available phase space. Finally, we point out the potential application of our work to the chiral magnetic effect (CME) and the chiral vortical effect (CVE) in high energy heavy ion collisions [30]. it is straightforward to calculate the first few terms in the expansion of e iH 0 t . H 2 0 = p 2 + 1 4 ω 2 0 I − pω 0 γ 5 H 3 0 = p p 2 + 3 4 ω 2 0 α 3 − 1 2 ω 0 3p 2 + 1 4 ω 2 0 Σ 3 H 4 0 = p 4 + 3 2 p 2 ω 2 0 + 1 16 ω 4 0 I − 2pω p 2 + 1 4 ω 2 0 γ 5 (B2) We observe that only four matrices enter the expansion. Therefore we can express the evolution operator in the general form e iH 0 t = C 1 I + C 2 γ 5 + iC 3 α 3 + iC 4 Σ 3 (B3) where all C i are real. Since the evolution operator is unitary e iH 0 t e −iH 0 t = 1 we find the two conditions C 2 1 + C 2 2 + C 2 3 + C 2 4 = 1 (B4) and C 1 C 2 + C 3 C 4 = 0 .(B5) The first condition suggests that we express the C i in terms of the Hopf angles for a sphere in four dimensions. C 1 = cos ξ 2 cos η C 2 = sin ξ 1 sin η C 3 = cos ξ 1 sin η C 4 = sin ξ 2 cos η . The second condition says that ξ 1 = −ξ 2 ≡ ξ. Hence the evolution operator can be expressed of those two angles by e iH 0 t = cos ξ cos η I + sin ξ sin η γ 5 + i cos ξ sin η α 3 − i sin ξ cos η Σ 3 . The angles must both vanish when t = 0. Doing a Taylor series expansion in ξ and η, and comparing to Eq. (B2), results in the identification η = pt and ξ = 1 2 ω 0 t. This is a clear generalization of Eq. (51). These are the same as Eqs. (IV.31) and (IV.32) of Ref.[19] apart from two points. First, the sign of our H is the opposite of theirs which flips the sign of the spin precession terms. Second, the spin precession terms in Eqs. (IV.31) and (IV.32) have a correction to the spin precession frequency which is reduced (minus sign), whereas our result indicates an enhancement (plus sign). This might be a misprint, or it might be traced to an incorrect reading of the sign of the imaginary part of Eq. (IV.22). FIG. 1 . 1Scattering of a massive strange quark (double line) which flips its helicity. The massless quark or anti-quark cannot change its helicity. its helicity because it has a small mass while the lighter quark is not. There are four combinations: a positive helicity strange quark can flip to negative helicity with a light quark of either helicity, and a negative helicity strange quark can flip to positive helicity with a light quark of either helicity. A straighforward calculation (see Appendix D) yields\|M\| 2 helicity flip = − 8 9 g 4 m 2 1 t + s (s − m 2 ) 2 FIG. 2 . 2Scattering of a massive strange quark (double line), which flips its helicity, with a gluon. FIG. 3 . 3Imaginary parts of the eigenvalues from Sec. VII on vorticity fluctuations. Two of them are zero. The other two are closely approximated by ±2E. Figure 4 FIG. 4 . 44shows the equilibration time for the four modes. These come from the inverse of Equilibration times for the four modes from Sec. VII on vorticity fluctuations. Three of them are nearly equal and shown as the solid curve. online) The dimensionless integrals I q (blue) and I g (red) appearing in the kinetic theory. The dashed curves correspond to a temperature of 200 MeV and the solid curves to 400 MeV. . 6. (Color online) The equilibration times from kinetic theory for the reactions s + q (blue) and s + g (red). The dashed curves correspond to a temperature of 200 MeV and the solid curves to 400 MeV. FIG. 7 . 7The equilibration time from kinetic theory including both the reactions s + q and s + g. The dashed curve corresponds to a temperature of 200 MeV and the solid curve to 400 MeV. Equation (188) in this reference is consistent with the literature. The more compact expression given in Eq. (194d)is wrong since it gives a different result for the Γ µ derived below. Appendix A: Dirac AlgebraFollowing is a compendium of commutators and anticommutators useful for the matrices that frequently enter the calculations.the first few terms in the expansion of e iH 0 t arewhere E 2 = p 2 + m 2 and E 2 \|\| = E 2 + 1 4 ω 2 0 . We observe that six matrices enter the expansion. Hence the evolution operator has the formAppendix D: Quark Helicity Flip AmplitudeConsider a massless quark scattering from a massive strange quark. To lowest order there is only a single Feynman diagram which involves the exchange of a gluon in the t channel. We are interested in the situation where the strange quark changes its helicity. The helicity of the massless quark cannot change. The amplitude is denoted by M(σσ , −σσ ) where σ is the helicity of the incoming strange quark and σ is the helicity of the massless quark. Using the method of Ref.[31]one readily finds thatirrespective of the sign of σ and σ . Here p * is the momentum and θ * the scattering angle in the center of momentum frame, and i, j, k, l are the quark colors. Squaring this, averaging over initial colors and summing over all colors usingfor an SU (N ) gauge theory, givesfor any choice of σ and σ . This result is also true for scattering of a massless anti-quark. To our knowledge the quark helicity flip amplitude for a massive quark scattering with a gluon has not been published before. Here we outline the major steps of the calculation at the tree level. The amplitude for scattering of a quark of helicity σ with a gluon of helicity λ to a quark of helicity σ and a gluon of helicity λ is denoted by M(σλ, σ λ ). It is useful to define s m = s−m 2 , u m = u − m 2 , and r 2 = m 4 − su. 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[ "Quantum fluctuations in FRLW space-time", "Quantum fluctuations in FRLW space-time" ]	[ "Yevgeniya Rabochaya *e-mail:[email protected] \nDipartimento di Fisica\nUniversità di Trento\nINFN-TIFPA\nTrento Via Sommarive 1438123PovoCentroItalia\n" ]	[ "Dipartimento di Fisica\nUniversità di Trento\nINFN-TIFPA\nTrento Via Sommarive 1438123PovoCentroItalia" ]	[]	In this paper we study a quantum field theoretical approach, where a quantum probe is used to investigate the properties of generic non-flat FRLW space time. The fluctuations related to a massless conformal coupled scalar field defined on a space-time with horizon is identified with a probe and the procedure to measure the local temperature is presented.	null	[ "https://arxiv.org/pdf/1511.07040v1.pdf" ]	119,214,144	1511.07040	01ebe4201625c275c768cfe1164e719fa274436e	Quantum fluctuations in FRLW space-time 22 Nov 2015 November 24, 2015 Yevgeniya Rabochaya e-mail:[email protected] Dipartimento di Fisica Università di Trento INFN-TIFPA Trento Via Sommarive 1438123PovoCentroItalia Quantum fluctuations in FRLW space-time 22 Nov 2015 November 24, 20151:39 WSPC Proceedings -9.75in x 6.5in main page 1 1Quantum fluctuationTemperatureUnruh effect In this paper we study a quantum field theoretical approach, where a quantum probe is used to investigate the properties of generic non-flat FRLW space time. The fluctuations related to a massless conformal coupled scalar field defined on a space-time with horizon is identified with a probe and the procedure to measure the local temperature is presented. Introduction The Hawking radiation 1 is one of the most robust and important predictions of quantum field theory in curved space-time. Here we would like to study some (local) properties of a generic Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time with non-flat topology. Let us remind some basic facts about the formalism. Any spherically symmetric four dimensional metric can be expressed in the form: ds 2 = γ ij (x i )dx i dx j + R 2 (x i )dΩ 2 2 , i, j ∈ {0, 1} ,(1) with γ ij (x i ) a tensor describing a two-dimensional space-time with coordinates x i , R(x i ) being the "areal radius" and dΩ 2 encoding the metric of a two-dimensional sphere orthogonal respect to the first one. The dynamical trapping horizon -if exists-is located in the correspondence of χ(x i ) H = 0 , ∂ i χ(x i )\| H 0 , χ(x i ) = γ ij (x i )∂ i R(x i )∂ j R(x i ) .(2) Thus, one may define the quasi-local Misner-Sharp gravitational energy as E MS (x i ) := 1 2G N R(x i ) 1 − χ(x i ) .(3) For example, the mass of a black hole described by this formalism results to be E = R H /(2G N ). The Killing vector fields ξ µ (x ν ) are the generators of the isometries with ∇ µ ξ ν (x ν ) + ∇ ν ξ µ (x ν ) = 0: in the static case, with the time-like Killing vector field K µ = (1, 0, 0, 0), the Killing surface gravity κ K is given by κ K K µ (x ν ) = K ν ∇ ν K µ (x ν ) .(4) In the dynamical case, the real geometric object which generalizes the Killing vector field is the Kodama vector field 2 , K i (x i ) := 1 √ −γ ε ij ∂ j R(x i ) , i = 0, 1 ; K i := 0 , i = 0, 1 .(5) Thus, the Hayward surface gravity associated with dynamical horizon is 3 κ H := 1 2 γ R(x i ) H .(6) The Hawking radiation is a thermal radiation of the black holes due to quantum effects. In the static case, all derivations of Hawking radiation lead to a semiclassical expression for the radiation rate Γ in terms of the exchange ∆E K of the Killing energy E K and the Killing/Hawking temperature T K , Γ ≡ e − 2π∆E K κ K , T K := κ K 2π .(7) In the dynamical case one may suggest the Kodama/Hayward temperature: T H := κ H 2π .(8) An important example that demonstrates the covariance of the formalism is given by the de Sitter space-time. The static patch reads ds 2 = −dt 2 (1 − H 2 0 r 2 ) + dr 2 (1 − H 2 0 r 2 ) + r 2 dΩ 2 ,(9) where R = r and the horizon is located at r H = 1/H 0 with surface gravity κ H = H 0 . The second patch is given by the expanding coordinates of the flat FLRW metric, ds 2 = −dt 2 + e 2H0t dr 2 + r 2 dΩ 2 ,(10) where R = e H0t r and the dynamical (cosmological) horizon is r H = 1/H 0 with κ H = H 0 . Finally, the global patch in non-flat FLRW metric is given by ds 2 = −dt 2 + cosh 2 (H 0 t) dr 2 (1 − H 2 0 r 2 ) + r 2 dΩ 2 ,(11) with R = r cosh(H 0 t), and r H = 1/H 0 and κ H = H 0 again. Now we will see how it is possible to associate a temperature to the dynamical horizon of flat and non-flat de Sitter space-time in (10)-(11). Quantization of massless field in FLRW metric We recall the quantization of a conformal coupled massless scalar field in the FRLW space-time. The metric reads ds 2 = a 2 (η)(−dη 2 + dΣ 2 3 ) , dΣ 2 3 = dr 2 1 − kh 2 0 r 2 + r 2 dS 2 2 .(12) where dη = dt/a(t) is the conformal time, h 0 is a mass scale and the topology of the spacial section can be flat, spherically or hyperbolic for k = 0, 1, −1, respectively. Given a massless scalar field, φ(x) = α f α (x)a α + f α (x)a + α ,(13) such that the modes are conformal invariant, namely ( − R/6)f α (x) = 0, the associated Wightman function W (x, x ′ ) =< φ(x)φ(x ′ ) > results to be W (x, x ′ ) = α f α (x)f * α (x ′ ) , − R 6 W (x, x ′ ) = 0 .(14) The Wightman function satisfies the following rule for the conformal transformations of the metric: ds 2 = Ω(x) 2 ds 2 0 , φ = 1 Ω φ 0 , W (x, x ′ ) = 1 Ω(x)Ω(x ′ ) W 0 (x, x ′ ) .(15) We may also take W (x, x ′ ) = W (η − η ′ , r − r ′ ) due to the homogeneity and isotropy of FLRW space-times. Let us consider the spherical case (k = 1) in (12), ds 2 = a 2 (η) −dη 2 + dχ 2 + 1 h 2 0 sin 2 h 0 χdS 2 2 , h 0 χ = arcsin h 0 r .(16) This metric is conformally related to the Minkowski space-time, ds 2 = a 2 (η)4 cos 2 h 0 η + χ 2 4 cos 2 h 0 η − χ 2 −dt 2 + dr 2 + r 2 dS 2 2 ,(17) with t ± r = 1 h 0 tan h 0 η ± χ 2 .(18) Thus, by starting from the well-known Wightman function in Minkowski space-time, one can use (15) and derive for the spherical FLRW metric W (x, x ′ ) = h 2 0 8π 2 a(η)a(η ′ ) 1 cos(h 0 (η − η ′ )) − cos(h 0 (χ − χ ′ )) . The hyperbolic case k = −1 is obtained with the substitution h 0 → ih 0 , while the flat case k = 0 corresponds to the limit h 0 → 0. Quantum fluctuations in flat space-time Let us consider a free massless quantum scalar field φ(x) in thermal equilibrium at the temperature T in flat space-time. We know that finite temperature field theory effects of this kind can be investigated by given that the scalar field defined in the Euclidean manifold S 1 × R 3 , where the imaginary time is τ = −it, compactified in the circle S 1 with period β = 1/T . We briefly review the local quantity < φ(x) 2 >, which is a divergent quantity due to the product of valued operator distribution in the same point x. By making use of the zeta-function regularization procedure, the quantum fluctuations read 4,5 : < φ(x) 2 >= ζ(1\|L β )(x) , L β = −∂ 2 τ − ∇ 2 ,(20) where ζ(z\|L β )(x) is the local zeta-function associated with the operator L β . It is easy to see that the analytic continuation of the local zeta-function is regular at z = 1 and finally one gets < φ(x) 2 >= 1 12β 2 = T 2 12 .(21) In this way we obtain the temperature of the quantum field in thermal equilibrium from the zeta-function renormalized vacuum expectation value, namely we have a quantum thermometer. Quantum fluctuations in FLRW space-time Now we extend the argument to generic FLRW metric. The off-diagonal Wightman function (19) leads to W (x, x ′ ) =< φ(x)φ(x ′ ) >= 1 4π 2 1 Σ 2 (x, x ′ ) ,(22) with Σ 2 (τ, τ − s) = a(τ )a(τ − s) 2 h 2 0 (cos h 0 (∆χ(s)) − cos h 0 (∆η(s))) ,(23) where a(τ ) is the conformal factor. Thus, in the limit s → 0, one has < φ(x) 2 >= W (τ, τ ) .(24) It is possible to show that W (τ, τ − s) = − 1 4πs 2 + B 48π 2 + O(s 2 ) ,(25) where B = H 2 + A 2 + 2Ḣṫ + h 2 0 a 2 (1 − 2ṫ 2 ) , A 2 = 1 t 2 − 1 ẗ + H(ṫ 2 − 1) ,(26) the dot being the derivative respect to the proper time, H = (da(t)/dt)/a(t) being the usual Hubble parameter, and A 2 the radial acceleration. Therefore, after the regularization for the divergent part at s → 0, < φ 2 > \| R = 1 48π 2 H 2 + A 2 + 2Ḣṫ ± h 2 0 a 2 (1 − 2ṫ 2 ) .(27) This result is quite general and it is valid also for spatial curvature k = 0. Quantum fluctuations in non flat de Sitter space-time In the case of de Sitter space-time with k = 1, we may put H 0 = h 0 and the expression for quantum fluctuations reads < φ 2 > R = 1 48 H 2 0 + A 2 = 1 48π 2 H 2 0 1 − R 2 0 H 2 0 ,(28) where R 0 = const is the areal radius of the Kodama observer and the acceleration has been computed as A 2 = R 2 0 H 4 0 1 − R 2 0 H 2 0 .(29) For a Kodama observer with R 0 = 0 we recover the Gibbons-Hawking temperature associated with de Sitter space-time, T = H 0 2π .(30) This is an important check of our approach, since it shows the coordinate independence of the result for the important case of de Sitter space-time. Quantum fluctuations in FRLW form of Minkowski space-time The Minkowski space-time may be written in a FRLW form with hyperbolic section k = −1 (Milne universe), ds 2 M = −dt 2 + t 2 dr 2 1 + r 2 + r 2 dΩ 2 2 , h 0 = 1 .(31) Making use of the Hayward formalism, it is easy to verify that there is no dynamical horizon and the surface gravity is vanishing. In this case we obtain < φ 2 > R = A 2 48π 2 ,(32) namely only the radial acceleration A 2 is present and the temperature is defined as T U = A 2π ,(33) recovering the well known Unruh effect. AcknowledgmentsThis proceeding is based on the paper 6 . I thank Sergio Zerbini for helpful discussions. . S W Hawking, Nature. 24830S. W. Hawking, Nature 248, 30 (1974); . Commun. Math. Phys. 43206Erratum-ibid.Commun. Math. Phys.43, 199 (1975) [Erratum-ibid. 46, 206 (1976)] . H Kodama, Prog. Theor. Phys. 631217H. Kodama, Prog. Theor. Phys. 63, 1217 (1980). . S A Hayward, Class. Quant. Grav. 153147S. A. Hayward, Class. Quant. Grav. 15, 3147 (1998). . A A Bytsenko, G Cognola, L Vanzo, S Zerbini, arXiv:hep-th/9505061Phys. Rept. 266hep-thA. A. Bytsenko, G. Cognola, L. Vanzo, S. Zerbini, Phys. Rept. 266, 1-126 (1996). [arXiv:hep-th/9505061 [hep-th]]. . S W Hawking, Commun. Math. Phys. 55133S. W. Hawking, Commun. Math. Phys. 55, 133 (1977). Quantum detectors in generic non flat FLRW space-times. Y Rabochaya, S Zerbini, arXiv:1505.00998gr-qcY. Rabochaya and S. Zerbini, "Quantum detectors in generic non flat FLRW space-times," arXiv:1505.00998 [gr-qc].	[]
[ "Delayed stability switches in singularly perturbed predator-prey models", "Delayed stability switches in singularly perturbed predator-prey models" ]	[ "J Banasiak [email protected] \nDepartment of Mathematics and Applied Mathematics\nUniversity of Pretoria\nPretoriaSouth Africa\n\nInstitute of Mathematics\nTechnical University of Lódź\nLódźPoland\n", "M S Seuneu Tchamga \nSchool of Mathematical Sciences\nUniversity of KwaZulu-Natal\n4041DurbanSouth Africa\n" ]	[ "Department of Mathematics and Applied Mathematics\nUniversity of Pretoria\nPretoriaSouth Africa", "Institute of Mathematics\nTechnical University of Lódź\nLódźPoland", "School of Mathematical Sciences\nUniversity of KwaZulu-Natal\n4041DurbanSouth Africa" ]	[]	In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an appropriate monotonicity assumption on the vector field to extend it to the two-dimensional case. The result is applied to identify all possible predator-prey models with quadratic vector fields allowing for the existence of canard solutions.	10.1016/j.nonrwa.2016.10.013	[ "https://arxiv.org/pdf/1605.07519v1.pdf" ]	119,316,241	1605.07519	74b8689524cdc75d9baf57f306fda26f2c701863	Delayed stability switches in singularly perturbed predator-prey models 24 May 2016 J Banasiak [email protected] Department of Mathematics and Applied Mathematics University of Pretoria PretoriaSouth Africa Institute of Mathematics Technical University of Lódź LódźPoland M S Seuneu Tchamga School of Mathematical Sciences University of KwaZulu-Natal 4041DurbanSouth Africa Delayed stability switches in singularly perturbed predator-prey models 24 May 2016Singularly perturbed dynamical systemsmultiple time scalesTikhonov theoremdelayed stability switchnon-isolated quasi steady statespredator-prey modelscanard solutions 2010 MSC : 34E1534E1792D40 In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an appropriate monotonicity assumption on the vector field to extend it to the two-dimensional case. The result is applied to identify all possible predator-prey models with quadratic vector fields allowing for the existence of canard solutions. Introduction In many multiple scale problems; that is, the problems in which processes occurring at vastly different rates coexist, the presence of such rates is manifested by the presence of a small (or large) parameter which expresses the ratio of the intrinsic time units of these processes. Mathematical modelling of such processes often leads to singularly perturbed systems of the form x ′ = f (t, x, y, ǫ), x(0) =x, ǫy ′ = g(t, x, y, ǫ), y(0) =ẙ, (1. 1) where f and g are sufficiently regular functions from open subsets of R × R n × R m × R + to, respectively, R n and R m , for some n, m ∈ N. It is of interest to determine the behaviour of solutions to (1. 1) as ǫ → 0 and, in particular, to show that they converge to solutions of the reduced system obtained from (1. 1) by letting ǫ = 0. There are several reasons for this. First, taking such a limit in some sense 'incorporates' fast processes into the slow dynamics and hence links models acting at different time scales, often leading to new descriptions of nature, see e.g. [2]. Second, letting formally ǫ = 0 in (1. 1) lowers the order of the system and hence reduces its computational complexity by offering an approximation that retains the main dynamical features of the original system. In other words, often the qualitative properties of the reduced system with ǫ > 0 can be 'lifted' to ǫ > 0 to provide a good description of dynamics of (1. 1). The first systematic analysis of problems of the form (1. 1) was presented by A.N. Tikhonov in the 40' and this theory, with corrections due to F. Hoppenstead, can be found in e.g. [2,15,31]. Later, a parallel theory based on the center manifold theory was given by F. Fenichel [11] and a reconciliation of these two theories can be found in [26]. To introduce the main topic of this paper one should understand the main features of either theory and, since our work is more related to the Tikhonov approach, we shall focus on presenting the basics of it. Letȳ(t, x) be the solution to the equation 0 = g(t, x, y, 0), (1. 2) often called the quasi steady state, andx(t) be the solution to x ′ = f (t, x,ȳ(t, x), 0), x(0) =x. (1. 3) We assume thatȳ is an isolated solution to (1. 2) in some set [0, T ] ×Ū and that it is a uniformly, in (t, x) ∈ [0, T ] ×Ū, asymptotically stable equilibrium of dỹ d τ = g(t, x,ỹ, 0) , (1. 4) where here (t, x) are treated as parameters. Further, assume thatx(t) ∈ U for t ∈ [0, T ] providedx ∈Ū and thatẙ is in the basin of attraction ofȳ. Theorem 1.1. Let the above assumptions be satisfied. Then there exists ε 0 > 0 such that for any ε ∈ ] 0, ε 0 ] there exists a unique solution (x ε (t), y ε (t)) of (1. 1) on [0, T ] and lim ε→0 x ε (t) =x(t), t ∈ [0, T ] , lim ε→0 y ε (t) =ȳ(t), t ∈ ] 0, T ] , (1. 5) wherex(t) is the solution of (1. 3) andȳ(t) =ȳ(t,x(t)) is the solution of (1. 2). We emphasize that the main condition for the validity of the Tikhonov theorem are that the quasi steady state be isolated and attractive; the latter in the language of dynamical systems is referred to as hyperbolicity. In applications, however, we often encounter the situation when either the quasi steady state ceases to be hyperbolic along some submanifold (a fold singularity), or two (or more) quasi steady states intersect. The latter typically involves the so called 'exchange of stabilities' as in the transcritical bifurcation theory: the branches of the quasi steady states change from being attractive to being repelling (or conversely) across the intersection. The assumptions of the Tikhonov theorem fail to hold in the neighbourhood of the intersection, but it is natural to expect that any solution that passes close to it follows the attractive branches of the quasi steady states on either side of the intersection. Such a behaviour is, indeed, often observed, see e.g. [8,20,21]. However, in many cases an unexpected behaviour of the solution is observed -it follows the attracting part of one of the quasi steady states and, having passed the intersection, it continues along the now repelling branch of it for some prescribed time and only then jumps to the attracting part of the other quasi steady state. Such a behaviour, called the delayed switch of stability, was first observed in [30] (and explained in [23]) in the case of a pitchfork bifurcation, in which an attracting quasi steady state produces two new attracting branches while itself continues as a repelling one. The delayed switch of stabilities for a fold singularity was observed in the van der Pol equation and have received explanations based on methods ranging from nonstandard analysis [5] to classical asymptotic analysis [10]; solutions displaying such a behaviour were named canard solutions. In this paper we shall focus on the so called transcritical bifurcation, in which two quasi steady states intersect and exchange stabilities at the intersection; here the delayed switch was possibly first observed in [12] and analysed in [27]. The interest in problems of this type partly stems from the applications to determine slow-fast oscillations [10,14,16,22,25], where the intersecting quasi steady states are used to prove the existence of cycles in the original problem and to approximate them. Another application is in the bifurcation theory, where the bifurcation parameter is driven by another, slowly varying, equation coupled to the original system [6,7]. In both cases using the naive approximation of the true solutions by a solution lying on the quasi steady states, without taking into account the possibility of the delay in the stability switch, results in a serious under-, or overestimate of the real dynamics of the system, see e.g. [6,7,25]. As we mentioned earlier, there is a rich literature concerning these topics and we do not claim that our paper offers significantly new theoretical results. However, by employing a monotonic structure of the equations and combining it with the method of upper and lower solution of [8] we have managed to give a constructive and rather elementary proof of the existence of the delayed stability switch for a large class of planar systems including, in particular, predator-prey models with quadratic vector field. As a by-product of the method, we also provided results on immediate stability switch. Here, our results pertain to a different class of problems than that considered in e.g. [20,21] but, when applied to the predator-prey system, they give the same outcome. As an added benefit of our approach we mention that, in contrast to the papers based on the orbit analysis, e.g. [25], we are able to give the precise value of time at which the stability switch occurs. Finally we note that, for completeness, we only proved the results for planar systems. Some of them, however, can be extended to multidimensional systems, [4]. The paper is structured as follows. In Section 2 we recall the one-dimensional delayed stability switch theorem of [8] and we formulate and prove its counterpart when the stability of quasi steady states is reversed. Section 3 contains the main results of the paper. In Theorem 3.1 we prove the existence of the delayed switch for a general predator-prey type model. Theorem 3.2 shows the convergence of the solution to the second quasi steady state after the switch. Finally, in Theorem 3.3 we give conditions ensuring an immediate stability switch. In Section 4 we apply these theorems to identify the cases of the delayed and immediate stability switches in classical predator-prey models. Finally, in Appendix we provide a sketch of the proof of the Butuzov et. al result with some amendments necessary for our considerations. Acknowledgement. The research of J.B. has been supported by the National Research Foundation CPRR13090934663. The results are part of Ph.D. research of M.S.S.T., supported by DAAD. Preliminary results The one-dimensional result In this section we shall recall a result on the delayed stability switch in a one dimensional case, given by V. F. Butuzov et.al., [8]. Let us consider a singularly perturbed scalar differential equation. ǫ dy dt = g(t, y, ǫ), y(t 0 , ǫ) =ẙ (2. 6) in D = I N × I T × I ǫ0 , where I N =] − N, N [, I T =]t 0 , T [, I ǫ0 = {ǫ : 0 < ǫ < ǫ 0 << 1}, with T > t 0 , N > 0 and g ∈ C 2 (D, R). Further, define G(t, ǫ) = t t0 g y (s, 0, ǫ)ds. (2. 7) Then we adopt the following assumptions. (α 1 ) g(t, y, 0) = 0 has two roots y ≡ 0 and y = φ(t) ∈ C 2 (Ī T ) in I N ×Ī T , which intersect at t = t c ∈ (t 0 , T ) and φ(t) < 0 for t 0 ≤ t ≤ t c , φ(t) > 0 for t c ≤ t ≤ T. (α 2 ) g y (t, 0, 0) < 0, g y (t, φ(t), 0) > 0 for t ∈ [t 0 , t c ), g y (t, 0, 0) > 0, g y (t, φ(t), 0) < 0 for t ∈ (t c , T ]. (α 3 ) g(t, 0, ǫ) ≡ 0 for (t, ǫ) ∈Ī T ×Ī ǫ0 . (α 4 ) The equation G(t, 0) = 0 has a root t * ∈]t 0 , T [. (α 5 ) There is a positive number c 0 such that ±c 0 ∈ I N and g(t, y, ǫ) ≤ g y (t, 0, ǫ)y for t ∈ [t 0 , t * ], ǫ ∈Ī ǫ0 , \|y\| ≤ c 0 . Theorem 2.1. Let us assume that all the assumptions (α 1 )-(α 5 ) hold. If y 0 ∈ (0, a), then for sufficiently small ǫ there exists a unique solution y(t, ǫ) of (2. 6) with 9) and the convergence is almost uniform on the respective intervals. lim ǫ→0 y(t, ǫ) = 0 for t ∈]t 0 , t * [, (2. 8) lim ǫ→0 y(t, ǫ) = φ(t) for t ∈]t * , T ],(2. Some ideas of the proof of the above theorem play a key role in the considerations of this paper and thus we give a sketch of it in Appendix A. Here we introduce essential notation and definitions which are necessary to formulate and prove the main results. By, respectively, lower and upper solutions to (2. 6) we understand continuous and piecewise differentiable (with respect to t) functions Y and Y that satisfy, for t ∈Ī T , Y (t, ǫ) ≤ Y (t, ǫ), Y (t 0 , ǫ) ≤ẙ ≤ Y (t 0 , ǫ), (2. 10) ǫ dY dt − g(t, Y , ǫ) ≥ 0, ǫ dY dt − g(t, Y , ǫ) ≤ 0. (2. 11) It follows that if there are upper Y and lower Y solutions to (2. 6), then there is a unique solution y to (2. 6) satisfying Y (t, ǫ) ≤ y(t, ǫ) ≤ Y (t, ǫ), t ∈Ī T , ǫ ∈ I ǫ0 . (2. 12) The proof of Theorem 2.1 uses an upper solution given by Y (t, ǫ) =ůe G(t,ǫ) ǫ . (2. 13) If we considerẙ > 0 then, by assumption (α 3 ), Y = 0 is an obvious lower solution to (2. 6). It is, however, too crude to analyze the behaviour of the solution close to t * and the modification of (2. 13), given by Y (t, ǫ) = ηe G(t,ǫ)−δ(t−t 0 ) ǫ , (2. 14) is used, where η, δ are appropriately chosen constants. As explained in detail in Appendix A, conditions on g can be substantially relaxed. Namely, we may assume that g is a Lipschitz function onD with respect to all variables such that g is twice continuously differentiable with respect to y uniformly in (t, y, ǫ) ∈D and that there is a neighbourhood of (t * , 0), V (t * ,0) := ]t * − α, t * + α[ × ] − ǫ 1 , ǫ 1 [ in which g u (t, 0, ǫ) is differentiable with respect to ǫ uniformly in t. The case of reversed stabilities of quasi steady states It is interesting to observe that the phenomenon of delayed exchange of stability, described in Theorem 2.1, does not occur if the role of the quasi steady states is reversed. Precisely, we have Theorem 2.2. Let us consider problem (2. 6) and assume (α ′ 1 ) g(t, y, 0) = 0 has two roots y ≡ 0 and y = φ(t) ∈ C 2 (Ī T ) in I N ×Ī T , which intersect at t = t c ∈ (t 0 , T ) and φ(t) > 0 for t 0 ≤ t ≤ t c , φ(t) < 0 for t c ≤ t ≤ T. Further, we assume that (α 2 ) and (α 3 ) are satisfied. Let y 0 ∈ (0, a). Then Proof. We see that y = φ(t) is an isolated attracting quasi steady state in the domain [0,t] × [a 0 , a], wherē t < t c is an arbitrary number close to t c and a 0 > 0 is an arbitrary number that satisfies a 0 < inf t∈[0,t] φ(t). lim ǫ→0 y(t, ǫ) = φ(t) for t ∈]t 0 , t c [, Then y 0 > 0 is in the domain of attraction of y = φ(t). Hence, the first equation of (2. 15) is satisfied. Let us take any t ′ > t c . Then y(t ′ , ǫ) > 0 and thus it is in the domain of attraction of the quasi-steady state y = 0. We cannot use directly the version of Tikhonov theorem, [20, Theorem 1B], as we do not know a priori whether y(t ′ , ǫ) converges. In the one dimensional case, however, we can argue as in Appendix A to see that the second equation of (2. 15) is satisfied on ]t c , T ]. Finally, denoting byφ the composite attracting quasi steady state,φ(t) = φ(t) for t 0 ≤ t < t c andφ(t) = 0 for t c ≤ t ≤ T , we see that g(t, y, 0) < 0 for y >φ and thus, for y > 0, g(t, y, ǫ) < 0 for y > φ + ω ǫ with ω ǫ → 0 as ǫ → 0. Remark 2.1. It is interesting to note that in this case the root t * of G(t, 0), see (2. 7), can satisfy t * > t c , but this does not have any impact on the switch of stabilities. Also, in general, the assumptions of theorem on an immediate switch of stabilities, e.g. [ Two-dimensional case We consider the following singularly perturbed system of equations x ′ (t) = f (t, x, y, ǫ), ǫy ′ (t) = g(t, x, y, ǫ) x(t 0 ) =x, y(t 0 ) =ẙ. (a1) Functions f, g are C 2 (V ) for some t 0 < T ≤ ∞, 0 < M, N ≤ ∞, ǫ 0 > 0. (a2) g(t, x, 0, ǫ) = 0 for (t, x, ǫ) ∈ I T × I M × I ǫ0 . (a3) f (t, x, y 1 , ǫ) ≤ f (t, x, y 2 , ǫ) for any (t, x, y 1 , ǫ), (t, x, y 2 , ǫ) ∈ V, y 1 ≥ y 2 . (a4) g(t, x 1 , y, ǫ) ≤ g(t, x 2 , y, ǫ) for any (t, x 1 , y, ǫ), (t, x 2 , y, ǫ) ∈ V, x 1 ≤ x 2 . Further, we need assumptions related to the structure of quasi steady states of (3. 1). (a5) The set of solutions of the equation 0 = g(t, x, y, 0) (3. 2) inĪ T ×Ī N ×Ī M consists of y = 0 (see assumption (a2)) and y = φ(t, x), with φ ∈ C 2 (Ī T ×Ī M ). The equation 0 = φ(t, x) (3. 3) for each t ∈Ī T has a unique simple solution ]0, M [∋ x = ψ(t) ∈ C 2 (Ī T ). To fix attention, we assume that φ(t, x) < 0 for x − ψ(t) < 0 and φ(t, x) > 0 for x − ψ(t) > 0. (a6) g y (t, x, 0, 0) < 0 and g y (t, x, φ(t, x), 0) > 0 for x − ψ(t) < 0, g y (t, x, 0, 0) > 0 and g y (t, x, φ(t, x), 0) < 0 for x − ψ(t) > 0. Since we are concerned with the behaviour of solutions close to the intersection of quasi steady state, we must assume that they actually pass close to it. Denote byx(t, ǫ) the solution of x ′ = f (t, x, 0, ǫ), x(t 0 , ǫ) =x. (3. 4) Then we assume that (a7) the solutionx =x(t) to the problem (3. 4) with ǫ = 0, called the reduced problem, x ′ = f (t, x, 0, 0), x(t 0 ) =x (3. 5) with −M <x < ψ(t 0 ) satisfiesx(T ) > ψ(T ) and there is exactly onet c ∈]t 0 , T [ such thatx(t c ) = ψ(t c ). Further, we defineḠ (t, ǫ) = t t0 g y (s,x(s, ǫ), 0, ǫ)ds (3. 6) and assume that (a8) the equationḠ (t, 0) = t t0 g y (s,x(s), 0, 0)ds = 0 has a roott * ∈]t 0 , T [. As in the one dimensional case, by assumption (a6),Ḡ attains a unique negative minimum att c and is strictly increasing for t >t c and thus assumption (a8) ensures thatt * is the only positive root in ]0, T [. Finally, (a9) There is 0 < c 0 ∈ I N and g(t,x(t, ǫ), y, ǫ) ≤ g y (t,x(t, ǫ), 0, ǫ)y for t ∈ [t 0 ,t * ], ǫ ∈Ī ǫ0 , \|y\| ≤ c 0 . We noted earlier, though the list of assumptions is quite long, they are quite natural. Apart from usual regularity assumptions, assumptions (a5) and (a6) ensure that we have two quasi steady states with interchange of stabilities. Crucial for the proof are assumptions (a3) and (a4) that allow to control solutions of (3. 1) by upper and lower solutions of appropriately constructed one dimensional problems, while (a7)-(a9) make sure that the latter satisfy the assumptions of Theorem 2.1. Remark 3.1. In what follows repeatedly we will use the following argument which uses monotonicity of f and g in (3. 1) and is based on e.g. [28, Appendix C]. Consider a system of differential equations x ′ = F (t, x, y), x(t 0 ) =x, y ′ = G(t, x, y), y(t 0 ) =ẙ, (3. 7) with F and G satisfying Lipschitz conditions with respect to x, y in some domain of R 2 , uniformly in t ∈ [t 0 , T ]. Assume that F satisfies F (t, x, y 1 ) ≤ F (t, x, y 2 ) for y 1 ≥ y 2 . If we know that a unique solution (x(t), y(t)) of (3. 7) satisfies φ 1 (t, x(t)) ≤ y(t) ≤ φ 2 (t, x(t)) on [t 0 , T ] for some Lipschitz functions φ 1 and φ 2 , then z 2 (t) ≤ x(t) ≤ z 1 (t), where z i satisfies z ′ i = F (t, z i , φ i (t, z i )), z i (t 0 ) =x, (3. 8) i = 1, 2. Indeed, consider z 1 satisfying z ′ 1 (t) ≡ F (t, z 1 (t), φ(t, z 1 (t))), z 1 (t 0 ) =x. Then we have x ′ (t) ≡ F (t, x(t), y(t)) ≤ F (t, x(t), φ 1 (t, x(t)) and we can invoke [28, Theorem B.1] to claim that x(t) ≤ z 1 (t) on [t 0 , T ] (note that in the one dimensional case the so-called type K assumption that is to be satisfied by F is always fulfilled). The other case follows similarly from the same result. We also note that if F satisfies F (t, x, y 1 ) ≤ F (t, x, y 2 ) for y 1 ≤ y 2 and we know that a unique solution (x(t), y(t)) of (3. 7) satisfies φ 1 (t, x(t)) ≤ y(t) ≤ φ 2 (t, x(t)) on [t 0 , T ] for some Lipschitz functions φ 1 and φ 2 , then z 1 (t) ≤ x(t) ≤ z 2 (t) where, as before, z i is a solution to (3. 8). Theorem 3.1. Let assumptions (a1)-(a9) be satisfied and −M < • x< ψ(t 0 ), 0 <ẙ < N . Then the solution (x(t, ǫ), y(t, ǫ)) of (3. 1) satisfies lim ǫ→0 x(t, ǫ) =x(t) on [t 0 ,t * [, (3. 9) lim ǫ→0 y(t, ǫ) = 0 on ]t 0 ,t * [, (3. 10) wherex(t) satisfies (3. 5) withx(t 0 ) =x and the convergence is almost uniform on respective intervals. Furthermore, ]t 0 ,t * [ is the largest interval on which the convergence in (3. 10) is almost uniform. Proof. First we shall prove that there ist * such that y(t, ǫ) → 0 almost uniformly on ]0,t * [. Let us fix initial conditions (x,ẙ) as in the assumptions and consider the solution (x(t, ǫ), y(t, ǫ)) originating from this initial condition. Since y(t, ǫ) ≥ 0 on [t 0 , T ], assumption (a3) gives x(t, ǫ) ≤x(t, ǫ),(3. 11) see (3. 4). Then assumptions (a2) and (a4) give 0 ≤ y(t, ǫ) ≤ȳ(t, ǫ),(3. 12) whereȳ(t, ǫ) is the solution to ǫy ′ =ḡ(t, y, ǫ),ȳ(t 0 , ǫ) =ẙ, (3. 13) and we denotedḡ(t, y, ǫ) := g(t,x(t, ǫ), y, ǫ). Since (3. 4) is a regularly perturbed equation, by e.g. [31], (t, ǫ) →x(t, ǫ) is also twice differentiable with respect to both variables and thusḡ retains the regularity of g. Furthermore,ḡ(t, y, 0) = g(t,x(t), y, 0). By (3. 2), the only solutions toḡ(t, y, 0) = 0 are y = 0 and y = φ(t,x(t)). Denote ϕ(t) = φ(t,x(t)). From (3. 3), φ(t, x) = 0 if and only if x = ψ(t) and thus ϕ(t) = 0 if and only ifx(t) = ψ(t); that is, by (a7), for t =t c . Indeed, we have ϕ(t c ) = φ(t c ,x(t c )) = φ(t c , ψ(t c )) = 0, with ϕ(t) < 0 for t <t c and ϕ(t) > 0 for t >t c . Hence, assumption (α 1 ) is satisfied for (3. 13). Further, sinceḡ y (t, y, ǫ) = g y (t,x(t, ǫ), y, ǫ), we see that assumption (a6) implies (α 2 ). Then assumptions (a8) and (a9) show that assumptions (α 4 ) and (α 5 ) are satisfied for (3. 13) and thusȳ(t, ǫ) satisfies (2. 9); in particular lim ǫ→0ȳ (t, ǫ) = 0 for t ∈]t 0 ,t * [. This result, combined with (3. 12), shows that lim ǫ→0 y(t, ǫ) = 0 for t ∈]t 0 ,t * [. Now, for anyx satisfying (a7), there is a neighbourhood U ∋x andt > t 0 such that y = 0 is an isolated quasi steady state on [t 0 ,t] ×Ū so that (3. 1) satisfies the assumptions of the Tikhonov theorem, see [2]. Thus, lim ǫ→0 x(t, ǫ) =x(t) on [t 0 ,t] and hence the problem x ′ = f (t, x, y(t, ǫ), ǫ), with initial condition x(t, ǫ) is regularly perturbed on [t,t * [. Therefore, lim ǫ→0 x(t, ǫ) =x(t) on [t,t * [. Combining the above observations, we have lim ǫ→0 x(t, ǫ) =x(t) almost uniformly on [t 0 ,t * [. In the next step we shall show that this is the largest interval on which y(t, ǫ) converges to zero almost uniformly. Assume to the contrary that lim ǫ→0 y(t, ǫ) = 0 almost uniformly on ]t 0 , t 1 ] for some t 1 >t * ; that is, for any ρ > 0 and any θ > 0 there is ǫ 1 = ǫ 1 (ρ, θ) such that for any t ∈ [t 0 + θ, t 1 ] and ǫ < ǫ 1 we have 0 ≤ y(t, ǫ) ≤ ρ. (3. 14) Then, by assumption (a3), on [t 0 + θ, t 1 ] we have f (t, x, ρ, ǫ) ≤ f (t, x, y(t, ǫ), ǫ). At the same time, y(t, ǫ) ≤ C for some constant C > 0, see e.g. [2, Proposition 3.4.1]. In fact, in our case we see that g < 0 for y > 0, sufficiently small ǫ and t close to t 0 , hence y(t, ǫ) ≤ẙ on [t 0 , t 0 + θ] if θ is sufficiently small. Then the function x 1 (t) = x 1 (t, ǫ) for t ∈ [t 0 , t 0 + θ[, x 2 (t, ǫ) for t ∈ [t 0 + θ, t 1 ], (3. 15) where x ′ 1 = f (t, x 1 , C, ǫ), x 1 (t 0 ) =x and x ′ 2 = f (t, x 2 , ρ, ǫ), x 2 (t 0 +θ) = x 1 (t 0 +θ, ǫ) satisfies x 1 (t, ǫ) ≤ x(t, ǫ) . However, this function is not differentiable and cannot be used to construct the lower solution for y(t, ǫ). Hence, we consider the solution x 3 to x 3 ′ = f (t, x 3 , ρ, 0), x 3 (t 0 ) =x on [t 0 , t 1 ] . By Gronwall's lemma, using the regularity of f with respect to all variables, we get \|x 1 (t, ǫ) − x 3 (t)\| ≤ Lθ (3. 16) for some constant L (note that L can be made independent of ǫ as f is C 2 in all variables). Thus, summarizing, for a given ρ, there is θ 0 such that for any θ < θ 0 and sufficiently small ǫ, − M < x(t, ρ, θ) := x 3 (t, ρ) − Lθ ≤ x 1 (t, ǫ) ≤ x(t, ǫ), t ∈ [t 0 , t 1 ]. (3. 17) Then, using assumption (a4), we find that the solution y = y(t, ρ, θ, ǫ) to ǫy ′ = g(t, y, ρ, θ, ǫ), y(0, ρ, θ, ǫ) =ẙ, (3. 18) where g(t, y, ρ, θ, ǫ) := g(t, x(t, ρ, θ), y, ǫ), satisfies y(t, ρ, θ, ǫ) ≤ y(t, ǫ), t ∈ [t 0 , t 1 ]. By construction, equation (3. 18) is in the form allowing for the application of Theorem 2.1. We will not need, however, the full theorem but only the considerations for the lower solution. As withḡ, we note that g is a C 2 function with respect to all variables. We consider the function G(t, ρ, θ, ǫ) = t t0 g y (s, 0, ρ, θ, ǫ)ds (3. 19) and observe that g(t, 0, 0, 0, ǫ) =ḡ(t, ǫ) = g(t,x(t, ǫ), 0, ǫ) and also g y (t, 0, 0, 0, ǫ) =ḡ y (t, ǫ) = g y (t,x(t, ǫ), 0, ǫ). Then G(t 0 , ρ, θ, 0) = 0. Further, since G(t * , 0, 0, 0) =Ḡ(t * , 0) = 0 and G t (t * , 0, 0, 0) = g y (t * , 0, 0) > 0, the Implicit Function Theorem shows that for sufficiently small ρ, θ there is a C 2 function t * = t * (ρ, θ) such that G(t * (ρ, θ), ρ, θ, 0) ≡ 0 with t * (ρ, θ) →t * as ρ, θ → 0. Furthermore, since by (a4) and (a2) we have g(t, x 1 , y, 0) ≤ g(t, x 2 , y, 0) for x 1 ≤ x 2 and g(t, x, 0, 0) = 0, we easily obtain g y (t, x 1 , 0, 0) ≤ g y (t, x 2 , 0, 0), x 1 ≤ x 2 . (3. 20) Since x(t, ρ, θ) ≤ x(t, ǫ) ≤x(t), t ∈ [t 0 , t 1 ] we find that G(t, ρ, θ, 0) ≤Ḡ(t, 0) and thus t * (ρ, θ) ≥t * . Denote by Y (t, ρ, θ, δ, η, ǫ) the solution defined by (2. 14) with G replaced by G. We observe that the parameter δ is defined independently of ρ, θ and η, hence G(t(ρ, θ, δ, ǫ), ρ, θ, ǫ) − δ(t − t 0 ) ≡ 0 and Y (t(ρ, θ, δ, ǫ), ρ, θ, δ, η, ǫ) = η. This function Y is a lower solution to (3. 18) provided η ≤ δ/k, see (A.3), where k can be also made independent of any of the parameters. So, we can find ρ 0 , θ 0 such that sup 0≤ρ≤ρ0,0≤θ≤θ0 t * (ρ, θ) ≤t < t 1 . Then, for a given ρ, θ satisfying the above, we have t(ρ, θ, δ, ǫ) = t * (ρ, θ) + ω(δ, ǫ) and we can take δ, ǫ 1 such that ω(δ, ǫ) +t < t 1 for all ǫ < ǫ 1 . For such a δ, we fix η < δ/k and then ρ < η. Then, for sufficiently small ǫ, y(t(ρ, θ, δ, ǫ), ǫ) < ρ and, on the other hand, y(t(ρ, θ, δ, ǫ), ǫ) ≥ Y (t(ρ, θ, δ, ǫ), ρ, θ, δ, η, ǫ) = η > ρ. Thus, the assumption that there is t 1 >t * such that y(t, ǫ) converges almost uniformly to zero on ]t 0 , t 1 [ is false. In the next step, we will investigate the behaviour of the solution beyondt * . Clearly, we cannot use y defined by (3. 18) as a lower solution there since it is a lower solution only as long as x(t, ǫ) ≤ ρ which, as we know, is only ensured for t <t * . Thus, we have to find another a priori upper bound for x(t, ǫ) that takes into account the behaviour of x(t, ǫ) beyondt * . For this we need to adopt an additional assumption which ensures that x(t, ǫ) does not return to the region of attraction of y = 0. Let g t g x + f (t,x,y,ǫ)=(t,ψ(t),0,0) > 0, t ∈ [0, T ].(3. 21) Remark 3.2. Condition (3. 21) has a clear geometric interpretation, see Fig.1. The normal to the curve x = ψ(t) pointing towards the region {(t, x); x > ψ(t)} is given by (−ψ ′ (t), 1). However, we have 0 ≡ φ(t, ψ(t)), hence ψ ′ = −φ t /φ x \| (t,x)=(t,ψ(t)) which, in turn, is given by −g t /g x on (t, x, y, ǫ) = (t, ψ(t), φ(t, ψ(t)), 0) = (t, ψ(t), 0, 0) on account of 0 ≡ g(t, x, φ(t, x), 0). Thus (3. 21) is equivalent to (−ψ ′ , 1) · (1, x ′ ) = (−ψ ′ , 1) · (1, f ), (t, x, y, ǫ) = (t, ψ(t), 0, 0), so that it expresses the fact that the solution x of (3. 5) cannot cross x = ψ(t) from above. If the problem is autonomous, then (3. 21) turns into f \| (x,y,ǫ)=(c,0,0) > 0, t ∈ [0, T ], where x = ψ(t c ) ≡ c, which means thatx(t) is strictly increasing crossing the line x = c. Theorem 3.2. Assume that, in addition to (a1)-(a9), inequality (3. 21) is satisfied. Then lim ǫ→0 x(t, ǫ) = x φ (t), ]t * , T ], (3. 22) lim ǫ→0 y(t, ǫ) = φ(t, x φ (t)), ]t * , T ], (3. 23) wherex φ (t) satisfies x ′ φ = f (t, x φ , φ(t, x φ ), 0), x φ (t * ) =x(t * ) (3. 24) and the convergence is almost uniform on ]t * , T ]. Proof. Since the proof is quite long, we shall begin with its brief description. Note that in the notation here we suppress the dependance of the construction on all auxiliary parameters. The idea is to use the one dimensional argument, as in Theorem 3.1; that is, to construct an appropriate lower solution but this time on [t 0 , T ]. As mentioned above, for t <t * we can use x and y, but beyondt * we must provide a new construction. First, using the classical Tikhonov approach, we show that if y(t, ǫ), with sufficiently small ǫ, enters the layer φ − ω < y < φ + ω at some t >t c , then it stays there. Hence, in particular, we obtain an upper bound for y(t, ǫ) for t > t c . Combining it with the upper bound obtained in the proof of Theorem 3.1, we obtain an upper bound for y on [t 0 , T ] which is, however, discontinuous. Using (a3), this gives a lower solution X for x(t, ǫ) on [t 0 , T ], that can be modified to be a differentiable function. It is possible to prove that X stays uniformly bounded away from ψ but only up to somet >t * . This fact is essential as otherwise the equation for Y , constructed using X as in (3. 18), would have quasi steady states intersecting in more than one point (whenever X(t) = ψ(t), see the considerations following (3. 13)). Hence, we only can continue considerations on [t 0 ,t ]. Now, as in the one dimensional case, the constructed Y converges on ]t 0 ,t ] to some quasi steady state, which is close to φ(t, X(t)) but, since we only have y(t, ǫ) ≥ Y (t, ǫ), this is not sufficient for the convergence of y(t, ǫ). However, this estimate allows for constructing an upper solution for x(t, ǫ) and hence an upper solution for y(t, ǫ). By careful application of the regular perturbation theory for x(t, ǫ) we prove that y(t, ǫ) is sandwiched between two functions which are small perturbations of φ(t, x φ (t)), where x φ satisfies (3. 24). Thus y(t, ǫ) converges to φ(t, x φ (t)) on ]t 0 ,t ]. This shows, in particular, that the solution enters the layer φ − δ < y < φ + δ for arbitrarily small δ provided ǫ is small enough, and the application of the Tikhonov approach with a Lyapunov function allows for extending the convergence up to T . Step 1. An upper bound for y(t, ǫ) aftert c . Let us take arbitrary t 1 ∈]t c ,t * [. By (3. 21), there is ̺ 0 > 0 such thatx(t 1 ) > ψ(t 1 ) + ̺ 0 . Since x(t 1 , ǫ) →x(t 1 ) and y(t 1 , ǫ) → 0, there is ǫ 0 such that for any 0 < ǫ < ǫ 0 we have x(t 1 , ǫ) > ψ(t 1 ) + ρ 0 /2 and 0 < y(t 1 , ǫ) < ρ, as established in the proof of the previous theorem. Let Ψ(t, x, y, ǫ) := g t (t, x, y, ǫ) g x (t, x, y, ǫ) + f (t, x, y, ǫ). By (3. 21), we have Ψ(t, ψ(t), 0, 0) > 0 for t ∈ [0, T ] and thus there is α 1 , r 1 , r 2 , ǫ 0 such that Ψ(t, ψ(t) + ̺, y, ǫ) ≥ α 1 (3. 25) for all \|y\| ≤ r 1 , \|̺\| < r 2 , \|ǫ\| < ǫ 0 . Consider now the surface S = {(t, x, y); t ∈ [0, T ], x = ψ(t)+̺, 0 ≤ y ≤ r 1 }. By continuity, there is 0 < ̺ < min{̺, r 2 } such that max t∈[0,T ] φ(t, ψ(t) + ̺) < r 1 . Σ ω = {(t, x, y); t ∈ [0, T ], ψ(t) + ̺ ≤ x ≤ M, φ(t, x) − ω ≤ y ≤ φ(t, x) + ω}. (3. 26) and the domain V ω = {(t, x, y); t ∈ [0, T ], ψ(t) + ̺ ≤ x ≤ M, 0 ≤ y ≤ φ(t, x) + ω}. Note that 'left' wall of V ω , V ω,l := V ω ∩ S is contained in the set {(t, x, y); Ψ(t, x, y, ǫ) > 0} and thus, by Remark 3.2, no trajectory can leave V ω across V ω,l . Using a standard argument with the Lyapunov type function V (t) = (y(t, ǫ) − φ(t, x(t, ǫ))) 2 , see e.g. [2, pp. 86-90] or [31, p. 203], if the solution is in Σ ω , it cannot leave this domain through the surfaces y = φ(x, t) ± ω. Hence, in particular, we have {x(t, ǫ), y(t, ǫ)} t1≤t≤T ∈ V ω . Step 2. Construction of the lower solution for x(t, ǫ) on [t 0 , T ]. By Step 1, for an arbitrary fixed t 1 ∈]t c ,t * [, there is ω such that y(t, ǫ); 0 < y(t, ǫ) < φ(t, x(t, ǫ)) + ω for t ∈ [t 1 , T ]. On the other hand, for any ρ > 0 and sufficiently small θ > 0 we have 0 < y(t, ǫ) < ρ on [t 0 + θ,t * − θ] for all ǫ < ǫ 1 = ǫ(ρ, θ). Then, by (3. 17), we have in particular x(t, θ, ρ) ≤ x(t, ǫ) for t ∈ [t 0 ,t * − θ]. Consider now the solution to x ′ 4 = f (t, x 4 , φ(t, x 4 ) + ω, ǫ), x 4 (t) = x(t, θ, ρ), t ∈ [t, T ], for some somet ∈]t 1 ,t * − θ[. Using Remark 3.1, we see that x 4 (t, θ, ρ, ǫ) ≤ x(t, ǫ) for all sufficiently small ǫ. At the same time, using the regular perturbation theory, for any ϑ > 0 there is, possibly smaller, ǫ 5 such that for all ǫ < ǫ 5 and t ∈ [t, T ] the solution x 5 (t) = x 5 (t,t, θ, ρ) to x ′ 5 = f (t, x 5 , φ(t, x 5 ), 0), x 5 (t) = x(t, θ, ρ), t ∈ [t, T ],(3. 27) satisfies \|x 5 (t, θ, ρ) − x 4 (t, θ, ρ, ǫ)\| < Cϑ on [t, T ], with C independent of θ, ρ, ǫ, ϑ,t. Then we construct the function X(t, θ, ρ, ϑ) = −Cϑ + x(t, θ, ρ) for t ∈ [t 0 ,t], x 5 (t, θ, ρ) for t ∈]t, T ], which clearly satisfies X(t, θ, ρ, ϑ) ≤ x(t, ǫ), t ∈ [t 0 , T ]. (3. 28) Next we prove that X stays uniformly away from ψ(t) in some neighbourhood oft * . For this, we note that bothx and x are defined on [t 0 , T ] and close to each other, by the definition of x 3 and (3. 17) (for small ρ). Thus, by (a7), there are Ω ′′ ≤ Ω ′ and t # <t * such thatx ≥ ψ + Ω ′ and x ≥ ψ + Ω ′′ on [t # ,t * ]. Let 0 < Ω < Ω ′′ . Then, by (a1), we see that infV f ≥ K for some K > −∞ (which follows, in particular, since 0 ≤ y(t, ǫ) ≤ φ(t, x(t, ǫ)) for t ≥t) and hence x 5 (t) ≥ x 5 (t) + K(t −t). Then, for anyt ∈]t # ,t * [, we have X(t, θ, ρ, ϑ) = x 5 (t) − Cϑ ≥ x 5 (t) + K(t −t) = x(t) + K(t −t) ≥ ψ(t) + Ω ′′ + K(t −t) = ψ(t) + Ω + (ψ(t) − ψ(t) + K(t −t) − Cϑ + Ω ′′ − Ω). Since the constants C, Ω, Ω ′′ can be made independent oft ∈ [t # ,t * ], and by the regularity of ψ, we see that there ist >t * ,t sufficiently close tot * , and ϑ > 0 such that X(t, , θ, ρ, ϑ) ≥ ψ(t) + Ω, t ∈ [t,t ]. (3. 29) Step 3. Construction of the lower solution for y(t, ǫ) on [t 0 , T ] and its behaviour for t ∈]t * ,t ]. Let us now consider the solution Y (t, θ, ρ, ϑ, ǫ) of the Cauchy problem ǫY ′ = g(t, X(t, θ, ρ, ϑ), Y , ǫ), Y (t 0 , θ, ρ, ϑ, ǫ) =ẙ. (3. 30) We observe that the above equation has two quasi-steady states, y ≡ 0 and y = φ(t, X(t, θ, ρ, ϑ)), that only intersect at t c , which is close tot c , at least on [t 0 ,t ]. Moreover, for t <t the lower solution x can be made as close as one wishes tox. Though X is not a C 2 function, as required by Theorem 2.1, we can use the comment at the end of Appendix A and only consider t ≥t. Here, instead of only a Lipschitz function X, we have the function x 5 (t, θ, ρ) − Cϑ that is smooth with respect to all parameters -note that ρ and θ enter into the formula through a regular perturbation of the equation and the initial condition. We define the function G for (3. 30) by G(t, ρ, θ, ϑ, ǫ) = t t0 g y (s, X(s, θ, ρ, ϑ), 0, ǫ)ds. (3. 31) We observe that for t <t we have, by (3. 20), G(t, ρ, θ, ϑ, 0) = t t0 g y (s, x(s, θ, ρ) − Cϑ, 0, 0)ds ≤ G(t, ρ, θ, 0). and also, since X(t) ≤ x(t, ǫ) ≤x(t) for any t ∈ [t 0 , T ], G(t, ρ, θ, ϑ, 0) ≤Ḡ(t, 0). (3. 32) This means that G < 0 on ]0,t] and G → 0 witht →t * and θ, ρ, ϑ → 0. Now, writing G(t, ρ, θ, ϑ, 0) = t t0 g y (s, x(s, θ, ρ) − Cϑ, 0, 0)ds + t t g y (s, x 5 (t, θ, ρ) − Cϑ, 0, 0)ds and, using (a6) and (3. 29) to the effect that g y (t, x 5 (t, θ, ρ) − Cϑ, 0, 0) ≥ L on [t,t ] for some L > 0, we see that for sufficiently smallt * −t, θ, ρ and ϑ we have t t * g y (s, x 5 (s, θ, ρ) − Cϑ, 0, 0)ds ≥ L(t −t * ) > t t0 g y (s, x(s, θ, ρ) − Cϑ, 0, 0)ds, since the last term is negative. Therefore there is a solution t * = t * (t, ρ, θ, ϑ) <t to G(t, ρ, θ, ϑ, 0) = 0. Moreover, this solution is unique as G is strictly monotonic for t ≥t, by (3. 32) it satisfies t * >t * and t * →t * ift * −t, θ, ρ, ϑ → 0. Now, for a fixedt, ρ, θ, ϑ, G is a C 2 -function of (t, ǫ) ∈]t,t * [× ]−ǭ,ǭ[ wherē ǫ is chosen so that (3. 28) is satisfied for all 0 < ǫ <ǭ. Thus, we can apply Theorem 2.1 with the weaker assumptions discussed at the end of Appendix A to claim that lim ǫ→0 Y (t, θ, ρ, ϑ, ǫ) = φ(t, x 5 (t) − Cϑ) (3. 33) almost uniformly on ]t * ,t ]. Because of this, for any τ ∈]t * ,t[ and any δ ′ > 0 we can findǫ > 0,θ > 0 such that for any ǫ <ǫ, ϑ <θ and t ∈ [τ,t] we have y(t, ǫ) ≥ φ(t, x 5 (t)) − δ ′ . (3. 34) Step 4. Upper solutions for x(t, ǫ) and y(t, ǫ) on [t 0 ,t ]. Thanks to these estimates, we see that the solution x 6 = x 6 (t, ǫ) of the problem x ′ 6 = f (t, x 6 , φ(t, x 5 ) − δ ′ , ǫ), x 6 (τ, ǫ) =x(τ, ǫ) (3. 35) satisfies, for sufficiently small ǫ, x 6 (t, ǫ) ≥ x(t, ǫ) on t ∈ [τ,t ]. Thus, we can construct a composite upper bound for x(t, ǫ) on [t 0 ,t ] as X(t, ǫ) = x(t, ǫ) for t ∈ [t 0 , τ ], x 6 (t, ǫ) for t ∈]τ,t ] and hence a new upper bound for y(t, ǫ), defined to be the solution to ǫȲ ′ = g(t,X(t, ǫ),Ȳ , ǫ),Ȳ (t 0 , ǫ) =ẙ. (3. 36) We observe that for t ∈ [t 0 , τ ] we have g(t,X(t), 0, 0) = g(t,x(t), 0, 0). Step 5. Convergence of (x(t, ǫ), y(t, ǫ)) on ]t * ,t ]. Now, x 6 (t, 0) is the solution to HenceḠ x ′ 6 = f (t, x 6 , φ(t, x 5 ) − δ ′ , 0), x 6 (τ, ǫ) =x(τ, 0), (3. 39) which is a regular perturbation of x ′ = f (t, x, φ(t, x 5 ), 0), x(t) = x(t, θ, ρ), t ∈ [t, T ]. But, by the uniqueness, the solution of the latter is x 5 and thus, for any δ ′′ > 0 we can findt, τ, θ, ρ, ϑ, δ ′ , ǫ ′′ such that for all ǫ < ǫ ′′ we have \|x 6 (t, 0) − x 5 (t)\| < δ ′′ on [τ,t ]. We need some reference solution independent of the auxiliary parameters so we denote by x φ the function satisfying x ′ φ = f (t, x φ , φ(x φ ), 0), x φ (t * ) =x(t * ) Clearly, this equation is a regular perturbation of both (3. 39) and (3. 27) and thus for any δ ′′′ > 0, after possibly further adjusting ǫ, we find φ(t, x φ (t)) − δ ′′′ ≤ y(t, ǫ) ≤ φ(t, x φ (t)) + δ ′′′ , t ∈ [τ,t ] (3. 40) which shows that lim ǫ→0 y(t, ǫ) = φ(t, x φ (t)) (3. 41) uniformly on t ∈ [τ,t ]. This in turn shows that lim ǫ→0 x(t, ǫ) = x φ (t) (3. 42) uniformly on t ∈ [τ,t ]. Step 6. Convergence of (x(t, ǫ), y(t, ǫ)) on ]t * , T ]. Eq. (3. 42) allows us to re-write (3. 40) as φ(t, x(t, ǫ)) −δ ≤ y(t, ǫ) ≤ φ(t, x(t, ǫ)) +δ, t ∈ [τ,t ], for some, arbitrarily small,δ > 0. Using the argument with the Lyapunov function and the notation from Step 1, the trajectory will not leave the layer Σδ. But then, by the standard argument as in e.g. [ y(t, ǫ) = φ(t, x φ (t)) uniformly on t ∈ [τ, T ]. Since we could take τ >t * arbitrarily close tot * , we obtain the thesis. Next, we provide a two-dimensional counterpart of Theorem 2.2, in which the stability of the quasi steady states is reversed. It provides conditions for an immediate switch of stabilities but, due to the structure of the problem, covers a different class of problems than e.g. [20,Theorem 2] for each t ∈Ī T has a unique simple solution ]0, M [∋ x = ψ(t) ∈ C 2 (Ī T ). We assume that φ(t, x) > 0 for x − ψ(t) < 0 and φ(t, x) < 0 for x − ψ(t) > 0. (a6') g y (t, x, 0, 0) > 0 and g y (t, x, φ(t, x), 0) < 0 for x − ψ(t) < 0, g y (t, x, 0, 0) < 0 and g y (t, x, φ(t, x), 0) > 0 for x − ψ(t) > 0. (a7') The solution x φ to the problem x ′ = f (t, x, φ(t, x), 0), x(t 0 ) =x, (3. 46) with −M <x < ψ(t 0 ) satisfies x φ (T ) > ψ(T ) and there is exactly one t c ∈]t 0 , T [ such that x φ (t c ) = ψ(t c ). Then the solution (x(t, ǫ), y(t, ǫ)) of (3. 1) satisfies (a) wherex(t) satisfies (3. 5) withx(t c ) = x φ (t c ) and the convergence is uniform. lim ǫ→0 x(t, ǫ) = x φ (t) on [t 0 , t c [, lim ǫ→0 y(t, ǫ) = φ(t, x φ (t)) on ]t 0 , t c [,(3. Proof. Some technical steps of the proof are analogous to those in the proofs of Theorems 3.1 and 3.2 and thus here we shall give only a sketch of them. From (a7') we see that for any t c < t c there is δ t c such that inf t0≤t≤t c (ψ(t)−x φ (t)) ≥ δ t c . For any 0 < η < δ t c define U η = {(t, x); t 0 ≤ t ≤ t c , 0 ≤ x ≤ ψ(t) − η}. By (a5'), we have ξ η = inf (t,x)∈Uη φ(t, x) > 0 and thus φ is an isolated quasi steady state on U η . Note that in the original formulation of the Tikhonov theorem, [2,31], U η should be a cartesian product of t and x intervals, but the current situation can be easily reduced to that by the change of variables z(t) = x(t) − ψ(t). Hence, (3. 47) follows from the Tikhonov theorem. We observe that for any η > 0 we can find t c so that y(t c , ǫ) < η and ψ(t c )−η < x(t c , ǫ) < ψ(t c )+η. Now, as in (3. 25), there are α 1 > 0, ζ 0 , ǫ 0 such that Ψ(t, ψ(t) + ζ, y, ǫ) ≥ α 1 (3. 49) for all \|y\| ≤ ζ 0 , \|ζ\| < ζ 0 , \|ǫ\| < ǫ 0 . Further, denote byφ the composite stable quasi steady state: φ(t, x) = φ(t, x) for t 0 ≤ t < T, 0 < x ≤ ψ(t) andφ(t, x) = 0 for t 0 ≤ t ≤ T, ψ(t) < x ≤ M . Then, by (a6'), we see that g(t, x, y, 0) < 0 for t 0 ≤ t ≤ T, 0 ≤ x ≤ M,φ(t, x) < y ≤ N. Therefore, for any ω > 0 there is β > 0 such g(t, x, y, 0) < −β for y ≥φ + ω and thus also g(t, x, y, ǫ) ≤ 0 for y ≥φ + ω for sufficiently small ǫ. Now, let us take arbitrary ζ < ζ 0 , ω < ζ and η such that φ(t, ψ(t) − η) + ω < ζ. Then we take t c such that x(t c , ǫ) > ψ(t c ) − η. It is clear that y(t, ǫ) ≤ ζ for t ≥ t c . Indeed, by (3. 49), the trajectory cannot cross back through {(t, x, y); t 0 ≤ t ≤ T, x = ψ(t) − η, 0 ≤ y ≤ φ(t, ψ(t) − η) + ω}, hence the only possibility would be to go throughφ + ω < η for x > ψ(t) − η but then, by the selection of constants, the trajectory would enter the region where y ′ (t, ǫ) ≤ 0. Thus, a standard argument shows that lim ǫ→0 y(t, ǫ) = 0, uniformly on [t c , T ]. Then the problem x ′ = f (t, x, y(t, ǫ), ǫ), x(t c , ǫ) = x(t c , ǫ) on [t c , T ] is a regular perturbation of x ′ = f (t, x, 0, 0), x(t c ) = x φ (t c ), whose solution isx. Therefore (3. 48) is satisfied. Using (3. 49) we can get a more detailed picture of the solution. Indeed, we see that x(t, ǫ) > ψ(t) + η for t <t c := t c + 2η/α 1 and sufficiently small ǫ and (x(t, ǫ), y(t, ǫ)) cannot cross back through {(t, x, y); t 0 ≤ t ≤ T, x = ψ(t) + η, y ≥ 0}, by 0 ≤ y(t, ǫ) ≤ ζ for t ≥ t c . Thus the solution stays in the domain of attraction of the quasi steady state y = 0 after x(t, ǫ) crosses the line x = ψ(t). An application to predator-prey models Let us consider a general mass action law model of two species interactions, x ′ = x(A + Bx + Cy), x(0) =x, ǫy ′ = y(D + Ey + F x), y(0) =ẙ,(4. 1) where none of the coefficients equals zero. It is natural to consider this system in the first quadrant Q = {(x, y); x ≥ 0, y ≥ 0}. It is clear that y = 0 is one quasi steady state, while the other is given by the formula y = φ(t, x) = − F E x − D E , with ψ(t) = −D/F . This quasi steady state lies in Q only if −D/F > 0. Under this assumption, the geometry of Theorem 3.1 is realized if −F/E > 0, while that of Theorem 3.3 if −F/E < 0. At the same time, g y (x, y) = D + 2Ey + F x. Hence, g y (x, 0) < 0 if and only if D + F x < 0, while g y (x, φ(t, x)) < 0 if and only if D + F x > 0. Summarizing, for the switch to occur in the biologically relevant region, D and F must be of opposite sign. In what follows we use positive parameters a, b, c, d, e, f do denote absolute values of capital case ones. Then we have the following cases. Case 1. D < 0, F > 0. Case 1a. E > 0. Then the right hand side of the second equation in (4. 1) is of the form y(−d + ey + f x) with y describing a predatory type population but with a very specific vital dynamics. It may describe a population of sexually reproducing generalist predator, see e.g. [9, Section 1.5, Exercise 12], but its dynamics is not very interesting -without the prey it either dies out or suffers a blow up. Also, in the coupled case of (4. 1), the only attractive quasi steady state in Q is y = 0 for x < d/f as the attracting part of φ is negative. We shall not study this case. Case 1b. E < 0. In this case the right hand side of the second equation of (4. 1) is of the form y(−d−ey+f x) which may describe a specialist predator (one that dies out in the absence of a particular prey). In this case the second quasi steady state is given by y = φ(x) = f e x − d e , and the quasi steady state y = 0 is attractive for x < d/e and repelling for x > d/e, where φ becomes attractive. Hence we are in the geometric setting of Theorem 3.1. For its applicability, f (x, y) = x(A + Bx + Cy) must be decreasing with respect to y, which requires C < 0 (for x > 0). Then the assumptions of Theorem 3.1 require either A, B > 0, or A > 0, B < 0 with a/b > d/f with 0 <x < d/f , or A < 0, B > 0 with a/b < d/f and a/b <x < d/f , as in each case the solutionx to x ′ = x(A + Bx), x(0) =x (4. 2) crosses d/f at some finite time t c . We observe thatx is increasing in all three cases. Thus, the function G, defined by (2. 7), satisfies G ′′ (t) = fx ′ (t) > 0 and thus there is a unique t * > t c for which G(t * ) = 0. Finally, we see that g yy (x, y) = −e < 0 and thus (a9) is satisfied. Case 2. D > 0 and F < 0. Case 2a. E > 0. Then the right hand side of the second equation in (4. 1) is of the form y(d + ey − f x), thus y describes a prey type population but with a specific vital dynamics: if not preyed upon, y blows up in finite time. Also, in the coupled case of (4. 1), the only attractive quasi steady state in Q is y = 0 for x > d/f as the attracting part of φ for x < d/f is negative. As before, we shall not study this case. Case 2b. E < 0. Here, the right hand side of the second equation of (4. 1) is y(d − ey − f x), which describes a prey with logistic vital dynamics. The second quasi steady state is given by y = φ(x) = − f e x + d e ,(4. 3) and the quasi steady state y = 0 is repelling for x < d/e and attractive for x > d/e, while φ is attractive for x < d/e. Thus the geometry of the problem is that of Theorem 3.3 and we have to identify conditions on A, B and C that ensure that the solution x φ , see (3. 46), originating fromx < d/f, crosses the line x = d/f in finite time. In this case (3. 46) is given by The assumptions of Theorem 3.3 will be satisfied if and only ifx < d/f and x eq > d/f and it is attracting, x ′ = x Ae + Cd e + Be − Cf e x .or Be − Cf = 0, Ae + Cd > 0, or x eq ∈ [0, d/f [ is repelling with x eq <x. To express these conditions in algebraic terms, we see that if Be − Cf = 0, then we must have It is interesting that Cases 1b and 2b have, in some sense, their duals. Consider, in the geometry of Case 2 b,x > d/f and assume that the coefficients are such that the solutionx(t) to (4. 2) decreases and crosses d/f . Then the solutions (x ǫ (t), y ǫ (t)) are first attracted by (x(t), 0) as long as they are above x > d/f and later they enter the region of attraction of (4. 3). So, under some technical assumptions, one can expect again a delay in the exchange of stabilities. We prove this by transforming this case to Case 1b. Hence, consider (4. 1) in the geometric configuration of Case 2b, − A < B d f ,(4.x ′ = x(A + Bx + Cy), x(0) =x ǫy ′ = y(d − ey − f x), y(0) =ẙ,(4. 6) and assume thatx > 0. Then the solutionx to x < a/b in the latter case. Let us change the variable according to x = −z + 2d/f. Then the system (4. 6) becomes z ′ = z − 2d f Af + 2Bd f − Bz + Cy , z(0) = 2d f −x < d f , ǫy ′ = y(−d − ey + f z), y(0) =ẙ. (4. 7) We observe that the second equation is the same as in Case 1b, so the assumptions of Theorem 3.1 concerning the function g are satisfied. We only have to ascertain that the assumptions concerning the function f of Theorem 3.1 also hold. We note that we consider the problem for z < 2d/f where the multiplier (z − 2d/f ) < 0. Thus, to have (a3) we need C = c > 0. For (a7), we observe that the equlibria of z are z 1 = 2d/f and z 2 = A B + 2d f . As before, (a7) will be satisfied if z 2 < d/f is repelling withz > z 2 , or z 2 > d/f and is attracting, or z 2 > 2d/f and z 1 is attracting. It is easy to see that the first case occurs when A/B < −d/f and B > 0, the second when A/B > −d/f and B < 0, and the last when both A > 0, B > 0. Thus, we obtain −A > B d f . Since the case when z 2 < d/f and it is repelling is possible if and only if B = b > and A = −a < 0, we see that d/f >z > z 2 is equivalent to d/f <x < a/b. We observe that if we consider the geometry of Case 1 b, but assume thatx > d/f and the solution to x ′ = x Ae − Cd e + Be + Cf e x , x(0) =x (4. 8) is decreasing and passes through x = d/f , then, by the same change of variables as above, we can transform this problem to the one discussed in Case 2b and obtain that there is an immediate switch of stabilities as in Theorem 3.3. To summarize, we obtain the delayed switch of stabilities in the following six cases: Fast predator a) x ′ = x(a + bx − cy), x(0) =x ∈]0, d/f [, ǫy ′ = y(−d − ey + f x), y(0) =ẙ > 0, b) x ′ = x(a − bx − cy), x(0) =x ∈]0, d/f [, ǫy ′ = y(−d − ey + f x), y(0) =ẙ > 0, with a/b > d/f , c) x ′ = x(−a + bx − cy), x(0) =x ∈]a/b, d/f [, ǫy ′ = y(−d − ey + f x), y(0) =ẙ > 0, with a/b < d/f . Fast prey a) x ′ = x(−a − bx + cy), x(0) =x > d/f, ǫy ′ = y(d − ey − f x), y(0) =ẙ > 0, b) x ′ = x(a − bx + cy), x(0) =x > d/f, ǫy ′ = y(d − ey − f x), y(0) =ẙ > 0, with a/b < d/f , c) x ′ = x(−a + bx + cy), x(0) =x ∈]d/f, a/b[ ǫy ′ = y(d − ey − f x), y(0) =ẙ > 0, with a/b > d/f . Appendix A. Sketch of the proof of Theorem 2.1. To explain the construction of the upper solution (2. 13), first we observe that, by the Tikhonov theorem, for any c 0 > 0 (see assumption (α 5 )) and δ > 0 (such that t 0 + δ < t c ), there is an ǫ(δ) > such that 0 < y(t 0 + δ, ǫ) ≤ c 0 . Thus, using (α 3 ), all solutions y(t, ǫ) are nonnegative and bounded from above by the solution of (2. 6) witht = t 0 + δ andv b = c 0 . Since in the first identity of (2. 9) we have to prove the convergence on the open interval ]t 0 , T ], it is enough to prove it for any δ with the initial condition at t 0 + δ being smaller than c 0 . Thus, without loosing generality, we can assume that y(t 0 , ǫ) =ẙ ≤ c 0 . Then assumption (α 5 ) asserts that the right hand side of (2. 6) is dominated by its linearization at y = 0 as long as the solution remains small (that is, at least on [t 0 ,t] for anyt < t c ). The author then considers the linearization ǫ dY dt = g y (t, 0, ǫ)Y , Y (t 0 , ǫ) =ů ∈]0, c 0 ], whose solution is (2. 13), Y (t, ǫ) =ů exp ǫ −1 G(t, ǫ). Crucial for the estimates are the properties of G. From the regularity of g and (α 2 ) we see that g y (t, 0, ǫ) is negative and separated from zero for sufficiently small ǫ and thus, by (2. 7), G(t, ǫ) ≤ 0 on [t 0 , t 0 + ν] for some small ν > 0. Similarly, from (α 4 ) and the regularity of G with respect to ǫ we find that there is a constant κ such that G(t, ǫ) ǫ ≤ G(t, 0) ǫ + κ (A.1) on [t 0 , t * ], ǫ ∈ I ǫ0 , so that G(t, ǫ)/ǫ < 0 on [t 0 + ν, t * − ν] for sufficiently small ǫ. Hence Y (t, ǫ) ≤ c 0 on [t 0 , t * − ν] and sufficiently small ǫ, and thus the inequality of assumption (α 5 ) can be extended on [t 0 , t * − ν]. But then, again by (α 5 ), we have ǫ dY dt − g(t, Y , ǫ) = g y (t, 0, ǫ)Y − g(t, Y , ǫ) ≥ 0 and Y is an upper solution of (2. 6). Hence 0 ≤ lim ǫ→0 + y(t, ǫ) ≤ lim ǫ→0 + Y (t, ǫ) = 0 uniformly on [t 0 + ν, t * − ν]. Since ν was arbitrary, we obtain the first identity of (2. 9). We can also derive an upper bound for y(t, ǫ) for t ∈ [t * − ν, T ]. From the above, there isǭ such that for ǫ <ǭ we have y(t * − ν, ǫ) < φ(t * − ν). Then, as in [31, p. 203] (see also 3. 26), we fix (sufficiently small) ω and select ǫ so that any solution y(t, ǫ) with ǫ <ǫ that enters the strip {(y, t); t ∈ [t * − ν, T ], φ(t) − ω < y < φ(t) + ω}, stays there. Hence, we have y(t, ǫ) ≤ φ(t) + ω, t ∈ [t * − ν, T ], for any ǫ ≤ min{ǭ,ǫ}. To prove the second identity of (2. 9) we first have to prove that y(t, ǫ) detaches from zero soon after t * . Clearly, Y (t, ǫ) has this property as G(t, ǫ) > 0 for t > t * . However, this is an upper solution so its behaviour does not give any indication about the properties of y(t, ǫ). Hence, we consider the function (2. 14), Y (t, ǫ) = η exp ǫ −1 (G(t, ǫ) − δ(t − t 0 )), with η ≤ min{ẙ, min t∈[t * ,T ] φ(t)}. Using assumptions (α 2 ) and (α 4 ) and the implicit function theorem (first for G(t, 0) − δ(t − t 0 ) and then for G(t, ǫ) − δ(t − t 0 )) we find that for any sufficiently small δ there exists ǫ(δ), such that for any 0 < ǫ < ǫ(δ) there is a simple root t(δ, ǫ) > t * of G(t, ǫ) − δ(t − t 0 ) = 0. Moreover, t(δ, ǫ) → t * as δ, ǫ → 0. Then we have Y (t, ǫ) ≤ η for t 0 ≤ t ≤ t(δ, ǫ) (A.2) with Y (t(δ, ǫ), ǫ) = η. On the other hand ǫ dY dt − g(t, Y , ǫ) = g y (t, 0, ǫ)Y − g(t, Y , ǫ) − δY . Since 0 ≤ η ≤ẙ ≤ c 0 (see the first part of the proof), for any y ∈ [0, c 0 ] we obtain, by assumption (α 3 ), g(t, y, ǫ) = g y (t, 0, ǫ)y + 1 2 g yy (t, y * , ǫ)y 2 with 0 ≤ y * ≤ c 0 . Then g y (t, 0, ǫ)y − g(t, y, ǫ) = − 1 2 g yy (t, y * , ǫ)y 2 ≤ ky 2 (A. 3) for k = supD \|g yy \| < ∞ and hence ǫ dY dt − g(t, Y , ǫ) = k 2 Y 2 − δY ≤ 0 on [t 0 , t(δ, η)], provided η ≤ δ/k. Observe, that the constants are correctly defined. Indeed, k depends on the properties of g that are independent of ǫ, and on c 0 , that is selected a priori as the constant for which assumption (α 5 ) is satisfied. Thus, it is independent of δ and η. Next, we can fix δ and ǫ(δ) which are related to solution of G(t, ǫ) − δ(t − t 0 ) = 0 and independent of η. Finally, we can select η to satisfy the above condition. Thus, Y is a subsolution of (2. 6) on [t 0 , t(δ, ǫ)]. Next we have to make these considerations independent of ǫ. Since the solution t(δ, ǫ) is a C 1 function, for a fixed δ we can consider t(δ) = sup 0<ǫ≤ǫ(δ) t(δ, ǫ). As before, t(δ) → t * as δ → 0. By the regularity of g and second part of assumption (α 2 ) we see that g(t, η, 0) > 0 on [t * , T ] for sufficiently small η > 0 and then g(t, η, ǫ) > 0 for sufficiently small ǫ on [t * , T ]. Thus, Y (t, ǫ) = η is a subsolution on [t(δ, ǫ), t(δ)]. Hence we see that η ≤ y(t(δ), ǫ) ≤ φ(t(δ)) + ω (A.4) for sufficiently small ω and for sufficiently small corresponding ǫ. Clearly, the points (t(δ), η) and (t(δ), φ(t(δ))+ ω) are in the basin of attraction of φ and hence solutions originating from these two points converge to φ for t > t(δ). Since solutions cannot intersect we have, by (A.4), lim ǫ→0 + y(t, ǫ) = φ(t),t > t(δ) (A.5) uniformly on [t, T ] and thus the convergence is almost uniform on ]t(δ), T ]. Since, however, t(δ) → t * as δ → 0, we obtain the second identity of (2. 9). A closer scrutiny of the proof shows that the assumption that g is a C 2 function with respect to all variables is too strong. Indeed, for (A.1) we need that g y (t, 0, ǫ) be Lipschitz continuous in ǫ ∈ I ǫ0 uniformly in t ∈ [t 0 , t * ]. Further, (A.3) together with earlier calculations require g to be twice continuously differentiable with respect to y. Finally, the construction of the root t(δ, ǫ) requires G to be a C 1 function in some neighborhood of (t * , ǫ) for which it is sufficient that g u (t, 0, ǫ) be a C 1 function in ǫ for sufficiently small ǫ, uniformly in t in a neighbourhood of t * . lim ǫ→0 y ǫ→0(t, ǫ) = 0 for t ∈ [t c , T ].(2. 15) ( 3 . 1 ) 31Let V := I T × I M × I N × I ǫ0 =]t 0 , T [ × ] − M, M [ × ] − N, N [ × ]0, ǫ 0 [. We introduce the following general assumptions concerning the structure of the system. Note that, apart the monotonicity assumptions (a3) and (a4), they are natural extensions of the assumptions of Theorem 2.1 to two dimensions. Figure 1 : 1Illustration of the assumptions for Theorem 3.1. Figure 2 : 2The cross-section of the construction for a given t. for arbitrary 0 < ω < min{α ̺ /2, r 1 − max t∈[0,T ] φ(t, ψ(t) + ̺), consider the layer (t, 0) on [t 0 , τ ] with τ >t * and thusḠ(t, 0) < 0 for t ∈]t 0 ,t * [,Ḡ(t * , 0) = 0 andḠ(t, 0) > 0 for t ∈]t * ,t[ since, by (3. 28) and (3.29), x(t, ǫ) > ψ(t) on [t * , τ ] and x 6 (t, ǫ) > ψ(t) on [τ,t ]. Thus the assumptions of Theorem 2.1 are satisfied and we see that lim ǫ→0Ȳ (t, ǫ) = φ(t, x 6 (t, 0)) (3. 38) uniformly on [τ,t ]. Theorem 3. 3 . 3Consider problem (3. 1) with assumptions (a1), (a2), (a8)-(a9), (3. 21) and (a5') The solution of the equation 0 = g(t, x, y, 0) (3. 44) inĪ T ×Ī N ×Ī M consists of y = 0 and y = φ(t, x), where φ ∈ C 2 (Ī T ×Ī M ). The equation 0 = φ(t, x) (3. 45) x (t, ǫ) =x(t) on [t c , T ], lim ǫ→0 y(t, ǫ) = 0 on [t c , T [, (3. 48) dynamics of this equation. If Be − Cf = 0, then there is only one equilibrium x = 0 and the solution grows or decays depending on whether Ae + Cd is positive or negative. If Be − Cf = 0, then there is another equilibrium, given byx eq = − Ae + Cd Be − Cf . Figure 3 : 3Delayed stability switch in the Case 1b. The orbits are traversed from left to right. Be − Cf = 0, then B and C must be of the same sign and for the solution to be increasing we must have Ae + Cd > 0 which again yields (4. 5). Summarizing, (4. 5) is equivalent to B = b > 0, A = a > 0, or B = b > 0, A = −a < 0 and a/b < d/f, or B = −b < 0, A = a > 0 and a/b > d/f. It is important to note that these conditions do not involve the position of x eq . Just to recall, we must have either x eq > d/f and it is attracting, or x eq < d/f and it is repelling (here we can think of the case Be − Cf = 0 with A, C > 0 as having x eq = −∞.) Thus, assumptions of Theorem 3.3 are satisfied if and only if the geometry is as in this point, (4. 5) is satisfied andx ∈ ]x eq , d/f [ if x eq < d/f . Then the x component of the solution (x(t, ǫ), y(t, ǫ)) to (4. 1) grows above d/f and an immediate change of stability occurs when the solution passes close to (d/f, 0). We note that Case 2b can be transformed to a problem that satisfies the assumptions of[20, Theorem 2]. On the other hand, not all assumptions of [8, Theorem 1.1] are satisfied. x ′ = x(A + Bx), x(0) =x, will decrease and pass through x = d/f if and only if −A > Bd/f (which is equivalent to either A = −a < 0, B = −b < 0, or A = a > 0, B = −b < 0 and a/b < d/f , or A = −a < 0, B = b > 0 and a/b > d/f ) and Figure 4 : 4Stability switch without delay in the geometry of the case Case 1b withx > d/f . The orbits are traversed from right to left. . H Amman, Escher, J. Analysis II. Birkhäuser. Amman, H.; Escher, J. Analysis II. Birkhäuser, Basel (2008). 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Introduction to System Order Reduction Methods with Applications, LNM 2114, Springer, Cham, (2014). Study of a system of differential equations with a small parameter at the highest derivatives. M A Shishkova, Dokl. Akad. Nauk SSSR. 209576179Shishkova, M. A. Study of a system of differential equations with a small parameter at the highest derivatives, Dokl. Akad. Nauk SSSR 209 (1973) 576179. Differential Equations. A N Tikhonov, A B Vasilyeva, A G Sveshnikov, Springer VerlagMoscow; Berlinin Russian; English translationTikhonov, A.N.; Vasilyeva, A.B.; Sveshnikov, A.G. Differential Equations. Nauka, Moscow (1985), in Russian; English translation: Springer Verlag, Berlin (1985).	[]
[ "Central exclusive production as a probe of the gluonic component of the η", "Central exclusive production as a probe of the gluonic component of the η" ]	[ "L A Harland-Lang \nDepartment of Physics\nInstitute for Particle Physics Phenomenology\nUniversity of Durham\nDH1 3LEUK\n", "V A Khoze \nDepartment of Physics\nInstitute for Particle Physics Phenomenology\nUniversity of Durham\nDH1 3LEUK\n\nPetersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia\n", "M G Ryskin \nPetersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia\n", "W J Stirling \nCavendish Laboratory\nUniversity of Cambridge\nJ.J. Thomson AvenueCB3 0HECambridgeUK\n" ]	[ "Department of Physics\nInstitute for Particle Physics Phenomenology\nUniversity of Durham\nDH1 3LEUK", "Department of Physics\nInstitute for Particle Physics Phenomenology\nUniversity of Durham\nDH1 3LEUK", "Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia", "Petersburg Nuclear Physics Institute\nNRC Kurchatov Institute\n188300Gatchina, St. PetersburgRussia", "Cavendish Laboratory\nUniversity of Cambridge\nJ.J. Thomson AvenueCB3 0HECambridgeUK" ]	[]	Currently, the long-standing issue concerning the size of the gluonic content of the η ′ and η mesons remains unsettled. With this in mind we consider the central exclusive production (CEP) of η ′ , η meson pairs in the perturbative regime, applying the Durham pQCD-based model of CEP and the 'hard exclusive' formalism to evaluate the meson production subprocess. We calculate for the first time the relevant leading order parton-level processes gg → qqgg and gg → gggg, where the final-state gg and qq pairs form a pseudoscalar flavour-singlet state. We observe that these amplitudes display some non-trivial and interesting theoretical properties, and we comment on their origin. Finally, we present a phenomenological study, and show that the cross sections for the CEP of η ′ , η meson pairs are strongly sensitive to the size of the gluon content of these mesons. The observation of these processes could therefore provide important and novel insight into this problem.	10.1140/epjc/s10052-013-2429-3	[ "https://arxiv.org/pdf/1302.2004v2.pdf" ]	119,103,074	1302.2004	a3ebe3689c748e79cbb6565208a51e61050c08b6	Central exclusive production as a probe of the gluonic component of the η May 2013 L A Harland-Lang Department of Physics Institute for Particle Physics Phenomenology University of Durham DH1 3LEUK V A Khoze Department of Physics Institute for Particle Physics Phenomenology University of Durham DH1 3LEUK Petersburg Nuclear Physics Institute NRC Kurchatov Institute 188300Gatchina, St. PetersburgRussia M G Ryskin Petersburg Nuclear Physics Institute NRC Kurchatov Institute 188300Gatchina, St. PetersburgRussia W J Stirling Cavendish Laboratory University of Cambridge J.J. Thomson AvenueCB3 0HECambridgeUK Central exclusive production as a probe of the gluonic component of the η May 2013Preprint typeset in JHEP style -PAPER VERSION IPPP/13/05 DCPT/13/10Central exclusive productionDiffractionMesonMHVgluon * KRYSTHAL collaboration Currently, the long-standing issue concerning the size of the gluonic content of the η ′ and η mesons remains unsettled. With this in mind we consider the central exclusive production (CEP) of η ′ , η meson pairs in the perturbative regime, applying the Durham pQCD-based model of CEP and the 'hard exclusive' formalism to evaluate the meson production subprocess. We calculate for the first time the relevant leading order parton-level processes gg → qqgg and gg → gggg, where the final-state gg and qq pairs form a pseudoscalar flavour-singlet state. We observe that these amplitudes display some non-trivial and interesting theoretical properties, and we comment on their origin. Finally, we present a phenomenological study, and show that the cross sections for the CEP of η ′ , η meson pairs are strongly sensitive to the size of the gluon content of these mesons. The observation of these processes could therefore provide important and novel insight into this problem. Introduction Central exclusive production (CEP) processes of the type pp(p) → p + X + p(p) , (1.1) can significantly extend the physics programme at high energy hadron colliders. Here X represents a system of invariant mass M X , and the '+' signs denote the presence of large rapidity gaps. Such reactions provide a very promising way to investigate both QCD dynamics and new physics in hadron collisions, and consequently they have been widely discussed in the literature, with recently there being a renewal of interest in the CEP process, see for example [1][2][3] for reviews and further references. It is well known that the η and η ′ mesons, the isoscalar members of the nonet of the lightest pseudoscalar mesons, play an important role in the understanding of low energy QCD. Knowledge of the quark and gluon components of their wave functions would provide important information about various aspects of non-perturbative QCD, see for instance [4] and references therein. A clear observation of the presence of a purely gluonic, gg, component in the η ′ (and η) mesons would also confirm that the gluons play an independent important role in hadronic spectroscopy. The presence of such a component of the η ′ meson, due to the so called gluon anomaly [5][6][7][8], is also related to the old question of why the η ′ mass is much larger than that of the η (the well-known 'U (1) problem', see [9]). Currently, while different determinations of the η-η ′ mixing parameters are generally consistent, the long-standing issue concerning the extraction of the gluon content of the η ′ (and η) remains uncertain, in particular due to non-trivial theory assumptions and approximations that must be made, as well as the current experimental uncertainties and limitations, see for example [10] for a discussion of the theoretical uncertainties (in e.g. the decay form factors) present in such extractions and [4] for a review of the experimental situation. The results of a detailed analysis of various radiative processes and heavy particle decays, see for example [4,11,12], therefore do not currently allow a conclusive confirmation or otherwise of a non-qq component in the the η ′ and η mesons, within the experimental uncertainties. Moreover, as discussed in [4], it is unlikely that lattice simulations of QCD will provide a determination of the gluonic contribution of the η ′ , η wave functions in the near future. On the other hand, in [13] a study of the ηγ and η ′ γ transition form factors, F (η,η ′ ),γ (Q 2 ), appears to indicate that the two-gluon Fock component of the η ′ meson may be quite large. In this paper we will show that the CEP of pseudoscalar meson pairs (η ′ η ′ , ηη ′ , ηη) at sufficiently high transverse momenta p ⊥ can provide a potentially powerful tool to probe the structure of the η ′ , η mesons, and is especially well suited to addressing the old problem of the value of the gluonic flavour-singlet contribution, as well as clarifying the issue of η-η ′ mixing. We show that any sizeable gg component of the η ′ (and η) can have a strong effect on the CEP cross section, and therefore such an exclusive process represents a sensitive probe of this. Moreover, we may expect that in the near future, after analysing the existing 4 photon candidates with E T > 2.5 GeV and forward rapidity gaps, CDF will collect a large number of η ′ , η events and also significantly improve the current limits on π 0 π 0 CEP [14]. η and η ′ meson CEP may also be studied within the CMS/TOTEM special low-pileup runs with sufficient luminosity [14,15]. In [16] the observation of 43 γγ events with \|η(γ)\| < 1.0 and E T (γ) > 2.5 GeV, with no other particles detected in −7.4 < η < 7.4 was reported, which corresponds to a cross section of σ γγ = 2.48 +0. 40 −0.35 (stat) +0.40 −0.51 (syst) pb. In [17] the π 0 π 0 → 4γ CEP background was calculated for the first time and found to be small (with σ(π 0 π 0 )/σ(γγ) ∼ 1%), a prediction that is in agreement with the CDF measurement, which finds that the contamination caused by π 0 π 0 CEP is very small (< 15 events, corresponding to a ratio N (π 0 π 0 )/N (γγ) < 0.35, at 95% C.L.). As well as representing a potential observable, we note that η( ′ )η( ′ ) CEP, via the η( ′ ) → γγ decay may also in principle represent a background to γγ production, if this cannot be suppressed experimentally. It is therefore important to calculate the predicted η( ′ )η( ′ ) CEP cross sections, in particular in the presence of a potentially large gg flavour-singlet component, which may enhance the corresponding rates. We will apply the 'hard exclusive' formalism described in [18] (see also [19]) to the production of a meson, M anti-meson, M , pair via the gg → M M subprocess, which may then be used to calculate the corresponding CEP cross section, for X = M M in (1.1). In [17] the gg → qqqq parton-level amplitudes were calculated, where each qq pair forms a meson state, and from this the flavour singlet and non-singlet meson pair CEP cross section was calculated. However, any non-zero gg component of the flavour singlet η ′ and, through mixing, η mesons will also be accessed via the gg → ggqq and gg → gggg subprocesses. We will show that these amplitudes, which on the face of it are quite unrelated, in fact display a striking similarity, being identical with each other and with the 'ladder-type' gg → qqqq amplitude calculated in [17], that only contribute for flavour-singlet mesons, up to overall normalization factors. We show how this remarkable result may be explained in the MHV framework by the fact that the same external parton orderings contribute in all three cases. Theoretical studies of meson pair CEP in fact have a long history, which predates the perturbative Durham approach depicted in Fig. 1. Exclusive ππ production, for example, mediated by Pomeron-Pomeron fusion, has been a subject of theoretical studies within a Regge-pole framework since the 1970s (see, for instance [20][21][22] for early references and [23][24][25] for more recent ones). However, as discussed in [24,25], at comparatively large meson transverse momenta, k ⊥ , CEP should be dominated by the perturbative 2gluon exchange mechanism discussed above and shown in Fig. 1. At lower k ⊥ a study of the transition region between these 'non-perturbative' and 'perturbative' regimes may be necessary, as was performed in [25] for the case of ππ CEP. For the (E ⊥ < 2.5 GeV, \|η\| < 1) meson pair event selection we will consider in this paper, the perturbative contribution was found to be dominant, and this is certainly expected to be true for the η( ′ )η( ′ ) perturbative cross sections, which we will show to be enhanced relative to π 0 π 0 production. We will therefore neglect such a Regge-based non-perturbative contribution throughout this paper. The outline of this paper proceeds as follows. In Section 2 we introduce the CEP formalism for the process (1.1). In Section 3 we introduce the 'hard exclusive' formalism used to model the gg → M M subprocess amplitudes, and calculate for the first time the gg → ggqq and gg → gggg amplitudes through which the gg component of the η ′ and η is accessed in the CEP process. In Appendix B we show how these amplitudes may also be calculated using the MHV formalism, in a way which sheds some light on the interesting theoretical features which these amplitudes display. This aims to provide some theoretical insight into these amplitudes, but can be skipped by the reader who is only interested in the phenomenological implications of our analysis. In Section 4 we present numerical results for the CEP of η ′ η ′ , ηη ′ and ηη meson pairs, for a range of different sizes of the gg component of the flavour-singlet distribution amplitude. Finally, we conclude in Section 5. Central exclusive production The formalism used to calculate the perturbative CEP cross section is explained in detail elsewhere [26] and so we will only review the relevant aspects here (for recent reviews and references see [1][2][3]). The amplitude is described by the diagram shown in Fig. 1, where the hard subprocess gg → X is initiated by gluon-gluon fusion and the second t-channel gluon is needed to screen the colour flow across the rapidity gap intervals. We can write the 'bare' amplitude in the factorized form [25,27] T = π 2 d 2 Q ⊥ M Q 2 ⊥ (Q ⊥ − p 1 ⊥ ) 2 (Q ⊥ + p 2 ⊥ ) 2 f g (x 1 , x ′ 1 , Q 2 1 , µ 2 ; t 1 )f g (x 2 , x ′ 2 , Q 2 2 , µ 2 ; t 2 ) , (2.1) where the f g 's in (2.1) are the skewed unintegrated gluon densities of the proton: in the kinematic region relevant to CEP, they are given in terms of the conventional (integrated) densities g(x, Q 2 i ). t i is the 4-momentum transfer squared to proton i and µ is the hard scale of the process, taken typically to be of the order of the mass of the produced state: as in [27], we use µ = M X /2 in what follows. The t-dependence of the f g 's is isolated in a proton form factor, which we take to have the phenomenological form F N (t) = exp(bt/2), with b = 4 GeV −2 . The M is the colour-averaged, normalised sub-amplitude for the gg → X process Figure 1: The perturbative mechanism for the exclusive process pp → p + X + p, with the eikonal and enhanced survival factors shown symbolically. M ≡ 2 M 2 X 1 N 2 C − 1 a,b δ ab q µ 1 ⊥ q ν 2 ⊥ V ab µν . (2.2) X Q ⊥ x 2 x 1 S eik S enh p 2 p 1 f g (x 2 , · · · ) f g (x 1 , · · · ) Here a and b are colour indices, M X is the central object mass, V ab µν represents the gg → X vertex and q i ⊥ are the transverse momenta of the incoming gluons, given by q 1 ⊥ = Q ⊥ − p 1 ⊥ , q 2 ⊥ = −Q ⊥ − p 2 ⊥ , (2.3) where Q ⊥ is the momentum transferred round the gluon loop and p i ⊥ are the transverse momenta of the outgoing protons. Only one transverse momentum scale is taken into account in (2.1) by the prescription Q 1 = min{Q ⊥ , \|(Q ⊥ − p 1 ⊥ )\|} , Q 2 = min{Q ⊥ , \|(Q ⊥ + p 2 ⊥ )\|} . (2.4) The longitudinal momentum fractions carried by the gluons satisfy x ′ ∼ Q ⊥ √ s ≪ x ∼ M X √ s , (2.5) where x ′ is the momentum fraction of the second t-channel gluon. The differential cross section at X rapidity y X is then given by dσ dy X = S 2 enh d 2 p 1 ⊥ d 2 p 2 ⊥ \|T (p 1 ⊥ , p 2 ⊥ )\| 2 16 2 π 5 S 2 eik (p 1 ⊥ , p 2 ⊥ ) , (2.6) where T is given by (2.1) and S 2 eik is the 'eikonal' survival factor, the probability of producing no additional particles due to soft proton-proton rescattering. This is calculated using a generalisation of the 'two-channel eikonal' model for the elastic pp amplitude (see [28] and references therein for details). Besides the effect of eikonal screening, S eik , there is an additional suppression caused by the rescatterings of the intermediate partons (inside the unintegrated gluon distribution, f g ). This effect is described by the so-called enhanced Reggeon diagrams and usually denoted as S 2 enh , see Fig. 1. The value of S 2 enh depends mainly on the transverse momentum of the corresponding partons, that is on the argument Q 2 i of f g (x, x ′ , Q 2 i , µ 2 ; t) in (2.1), and depends only weakly on the p ⊥ of the outgoing protons (which formally enters only at NLO). While in [26,29] the S 2 enh -factor was calculated using the formalism of [30], here, following [17,25], we use a newer version of the multi-Pomeron model [31] which incorporates the continuous dependence on Q 2 i and not only three 'Pomeron components' with different 'mean' Q i . We therefore include the S enh factor inside the integral (2.1), with S 2 enh being its average value integrated over Q ⊥ . If we consider the exact limit of forward outgoing protons, p i ⊥ = 0, then we find that after the Q ⊥ integration (2.2) reduces to M ∝ q i 1 ⊥ q j 2 ⊥ V ij → 1 2 Q 2 ⊥ (V ++ + V −− ) ∼ λ 1 ,λ 2 δ λ 1 λ 2 V λ 1 λ 2 ,(2.7) where λ (1,2) are the gluon helicities in the gg rest frame. The only contributing helicity amplitudes are therefore those for which the gg system is in a J z = 0 state, where the z-axis is defined by the direction of motion of the gluons in the gg rest frame, which, up to corrections of order ∼ q 2 ⊥ /M 2 X , is aligned with the beam axis. In general, the outgoing protons can pick up a small p ⊥ , but large values are strongly suppressed by the proton form factor, and so the production of states with non-J z = 0 quantum numbers is correspondingly suppressed (see [26,29] for examples of this in the case of χ (c,b) and η (c,b) CEP). In particular, we find roughly that \|T (\|J z \| = 2)\| 2 \|T (J z = 0)\| 2 ∼ p 2 ⊥ 2 Q 2 ⊥ 2 , (2.8) which is typically of order ∼ 1/50 − 1/100, depending on the central object mass, cms energy √ s and choice of PDF set. As discussed in [17], this 'J z = 0 selection rule' [32] will have important consequences for the case of meson pair CEP. Finally, we note that in (2.7) the incoming gluon helicities are averaged over at the amplitude level: this result is in complete contrast to a standard inclusive production process where the amplitude squared is averaged over all gluon helicities. Eq. (2.7) can be readily generalised to the case of non-J z = 0 gluons which occurs away from the forward proton limit, see in particular Section 4.1 (Eq. (41)) of [26], which we make use of throughout to calculate the M M CEP amplitude from the corresponding gg → M M helicity amplitude. 3. gg → η( ′ )η( ′ ) amplitudes: Feynman diagram calculation The hard exclusive formalism The leading order contributions to the γγ → M M process were first calculated in [18] (see also [19,33,34]), where M (M ) is in this case a flavour non-singlet meson(anti-meson). The cross section has been calculated at NLO in [35]. For the case of mesons with flavorsinglet Fock states there is also a contribution coming from the LO two-gluon component of the meson, and this was calculated in [36] (see also [37,38]). In particular, in [36] they considered the process γγ → η 1 M (where M = η 1 ), where η 1 is a flavour-singlet meson state that can has both a qq and a gg component. The amplitude can be written as M λλ ′ (ŝ, θ) = 1 0 dx dy φ 1 (x)φ M (y) T q λλ ′ (x, y;ŝ, θ) + 1 0 dx dy φ G (x)φ M (y) T g λλ ′ (x, y;ŝ, θ) ,(3. 1) whereŝ is the η 1 M invariant mass, x, y are the meson momentum fractions carried by the quarks or gluons in the meson, λ, λ ′ are the photon helicities and θ is the scattering angle in the γγ cms frame. T q(g) λλ ′ is the hard scattering amplitude for the parton level process γγ → qq qq(gg), see Fig. 2, where each qq and gg pair is collinear and has the appropriate colour, spin, and flavour content projected out to form the parent meson. In the meson rest frame, the relative motion of the partons is small: thus for a meson produced with large momentum, \| k\|, we can neglect the transverse component of the parton momentum, q, with respect to k, and simply write q = xk in the calculation of T λλ ′ . φ(x) is the leading twist meson distribution amplitude, representing the probability amplitude of finding a γ 2 (λ 2 ) Figure 2: Representative Feynman diagrams for the γγ → qqqq(gg) processes. There are 20 Feynman diagrams of type (a), and the corresponding helicity amplitudes are given in [17] (see also [18,19]). There are 24 diagrams of type (b), and the corresponding helicity amplitudes are given by (3.19, 3.20). There are 31 diagrams of type (a), but with the photons replaced by gluons, and the corresponding helicity amplitudes are given in [17]. γ 1 (λ 1 ) k 3 k 4 (a) γ 2 (λ 2 ) γ 1 (λ 1 ) k 3 k 4 (b) valence parton in the meson carrying a longitudinal momentum fraction x of the meson's momentum, integrated up to the scale Q over the quark transverse momentum q t (with respect to meson momentum k). While φ M (x) and φ 1 (y) represent the qq distribution amplitudes of the mesons M and η 1 , respectively, φ G corresponds to the gg distribution amplitude of the η 1 . We recall (see [17] and references therein for further details) that the meson distribution amplitude depends on the (non-perturbative) details of hadronic binding and cannot be predicted in perturbation theory. However, the overall normalization of the qq distribution amplitude can be set by the meson decay constant f M via [18] 1 1 0 dx φ M (x) = f M 2 √ 3 . (3.2) It was also shown in [40] that for very large Q 2 the meson distribution amplitude evolves towards the asymptotic form φ M (x, Q) → Q 2 →∞ √ 3f M x(1 − x) . (3.3) However this logarithmic evolution is very slow and at realistic Q 2 values the form of φ M can in general be quite different. Indeed, the recent BABAR measurement of the pion transition form factor F πγ (Q 2 ) [41,42], for example, strongly suggests that φ π (x, Q) does not have the asymptotic form out to Q 2 40 GeV 2 , although new Belle data [43] are in conflict with this. Another possible choice is the 'Chernyak-Zhitnisky' (CZ) form, which 1 We note that this normalization, which we use throughout this paper, corresponds to the definition of fM given in [18] for the case of the pion, that is it with fπ ≈ 93 MeV. In the literature the value fπ → √ 2fπ ≈ 133 MeV, is often used, in particular in [13,39], which we will refer to later. In Section 5 of [17], the more conventional value fπ ≈ 133 MeV is confusingly quoted, but in all numerical calculations the lower value was correctly taken. we will make use of later on [44] φ CZ M (x, Q 2 = µ 2 0 ) = 5 √ 3f M x(1 − x)(2x − 1) 2 , (3.4) where the starting scale is roughly µ 0 ≈ 1 GeV. For the two-gluon distribution amplitude, φ G (x), the normalization cannot be set as in (3.2), as we have φ G (x) = −φ G (1 − x) , as required by the antisymmetry of the pseudoscalar spin projection (3.18) of the gluons under this interchange, but an analogous formula can be written down [36,38,45 ] 1 0 dy φ G (x, Q 2 )(2x − 1) ∝ f G (Q 2 ) . (3.5) Such an expression serves to define f G , the value of which is to be determined. While as Q 2 → ∞, it can be shown that the gg distribution amplitude vanishes due to QCD evolution [45,46] lim Q 2 →∞ φ G (x) = 0 , (3.6) there is no reason to assume this will be the case at experimentally relevant energies. More precisely, the qq flavour-singlet and gg distribution amplitudes can be expanded in terms of the Gegenbauer polynomials C n [40,45,46] φ (1,8) ,M (x, µ 2 F ) = 6f M (1,8) 2 √ N C x(1 − x)[1 + n=2,4,··· a (1,8) n (µ 2 F )C 3/2 n (2x − 1)] , φ G,M (x, µ 2 F ) = f M 1 2 √ N C C F 2n f x(1 − x) n=2,4,··· a G n (µ 2 F )C 5/2 n−1 (2x − 1) , (3.7) where µ F is the factorization scale, taken as usual to be of the order of the hard scale of the process being considered, and n f = 3 for η( ′ ) mesons. The f M 1,8 (with M = η, η ′ in the present case) are given by (3.11), with the M dependence expressing the difference due to the mixing of the η, η ′ states and decay constants. We assume that apart from this the distribution amplitudes, i.e. the a n , are independent of the meson being considered. We choose to include the decay constants f M 1 , f M 8 explicitly in our definition of the distribution amplitudes, in contrast to, e.g. [13,39], where they are introduced in the hard amplitude T λλ ′ in (3.1) 2 . The evolution of the distribution amplitude is then dictated by the µ 2 F dependence of 2 Our choice of normalization of φ1,G differs further by a factor of 1/2 √ NC , which in [13] is included in the hard amplitude, and an additional factor of CF /2n f x(1 − x) in φG due to the different normalization of the gg spin projection (3.18). Of course, the final physical result will not depend on the choice of convention, which simply corresponds to a choice of which overall factors to include in the distribution amplitude φG,1, and which to include in the hard amplitude T λλ ′ in (3.1). the coefficients a n via a 1 n (µ 2 F ) = a (+) n (µ 2 0 ) α s (µ 2 0 ) α s (µ 2 F ) γ (+) n /β 0 + ρ (−) n a (−) n (µ 2 0 ) α s (µ 2 0 ) α s (µ 2 F ) γ (−) n /β 0 , a G n (µ 2 F ) = ρ (+) n a (+) n (µ 2 0 ) α s (µ 2 0 ) α s (µ 2 F ) γ (+) n /β 0 + a (−) n (µ 2 0 ) α s (µ 2 0 ) α s (µ 2 F ) γ (−) n /β 0 . (3.8) That is, the quark and gluon components mix under evolution. Here β 0 = 11 − 2n f /3, the a (±) n (µ 2 0 ) at the starting scale µ 0 are inputs which must be, e.g., extracted from data, and the remaining factors γ ± , ρ (±) are defined in Appendix A. We note that despite the different normalization convention we take in (3.7) for the quark and gluon distribution amplitudes, the coefficients a n are defined so that they obey the same evolution equation as in for example [13]. While the size of the a G n (µ 2 F = Q 2 ) will decrease with increasing scale, their initial value can in principle be quite large, and the evolution of for example the leading coefficient a G 2 → 0 with Q 2 is slow 3 . Finally, to make contact with the physical η, η ′ states we will be considering in this paper, we introduce the flavour-singlet and non-singlet quark basis states \|qq 1 = 1 √ 3 \|uu + dd + ss , \|qq 8 = 1 √ 6 \|uu + dd − 2ss ,(3.9) and the two-gluon state \|gg , (3.10) with corresponding distribution amplitudes given by (3.7). Here, we follow [47] (see also [48,49]) in taking a general two-angle mixing scheme for the η and η ′ mesons. That is, the mixing of the η, η ′ decay constants is not assumed to follow the usual (one-angle) mixing of the states. This is most easily expressed in terms of the η and η ′ decay constants f η 8 = f 8 cos θ 8 , f η 1 = −f 1 sin θ 1 , f η ′ 8 = f 8 sin θ 8 , f η ′ 1 = f 1 cos θ 1 ,(3.11) with the fit of [50] giving f 8 = 1.26f π , θ 8 = −21.2 • , f 1 = 1.17f π , θ 1 = −9.2 • . (3.12) We then take the distribution amplitudes (3.7) with the decay constants given as in (3.11), for the corresponding Fock components (3.9) and (3.10). That is, the η and η ′ states are given schematically by (see for example [13,39] for more details and discussion) \|η = f 8 cos θ 8 φ 8,η (x, µ 2 F )\|qq 8 − f 1 sin θ 1 φ 1,η (x, µ 2 F )\|qq 1 +φ G,η (x, µ 2 F )\|gg , \|η ′ = f 8 sin θ 8 φ 8,η ′ (x, µ 2 F )\|qq 8 + f 1 cos θ 1 φ 1,η ′ (x, µ 2 F )\|qq 1 +φ G,η ′ (x, µ 2 F )\|gg ,(3.13) where to make things explicit the distribution amplitudesφ are defined as in (3.7), but with the decay constants divided out (i.e.φ 8,η (x, µ 2 F ) = φ 8,η (x, µ 2 F )/f η 8 ...) , and these do not represent the conventional, normalized expressions for the η ′ , η Fock states, but simply indicate the decay constants and distribution amplitudes that should be associated with each qq and gg state in this two-angle mixing scheme. Preliminary consideration: γγ → M η 1 We begin this section by recalculating the γγ → η 1 M helicity amplitudes, as in [36]. For simplicity, we consider the case of a pure flavour-singlet state η 1 , but the following result can readilty be applied to the more realistic case of an η ′ or η meson. The appropriate qq and gg spin and colour quantum numbers are projected onto the relevant pseudoscalar meson states using [39,40] P qq = δ ij √ N C γ 5 p M √ 2 , (3.14) P gg = − δ ab N 2 C − 1 iǫ µν ⊥ √ 2 , (3.15) where ǫ 12 ⊥ = −ǫ 21 ⊥ = 1 and all other components are zero. It can be expressed as ǫ µν ⊥ = 2 s e µναβ p M α p η 1 β ,(3.16) where ǫ 0123 = −1. Explicitly, these correspond to the following combinations of spin states 1 √ 2 u + (y) √ y v − (1 − y) √ 1 − y + u − (y) √ y v + (1 − y) √ 1 − y = γ 5 p M √ 2 , (3.17) 1 √ 2 ǫ * µ + (x)ǫ * ν − (1 − x) − ǫ * µ − (x)ǫ * ν + (1 − x) = −i ǫ µν ⊥ √ 2 ,(3.18) where u(z), v(z) are the usual Dirac spinors for the quarks and ǫ(z) is the polarization vector of the gluon, carrying momentum fraction z of the parent mesons, while the ± signs indicate the particle helicities. The normalizations are conventional (other choices are possible, provided the distribution amplitude is suitably re-defined to compensate), and we can see that these projections are odd under the parity transformation + ↔ −, as required for a P = −1 meson state. We make use of (3.14, 3.15) throughout this paper. Explicitly calculating the amplitudes corresponding to the 24 contributing Feynman diagrams (see e.g. Fig. 1 (c) of [36]) 4 , we find T g(γγ) ++ = T g(γγ) −− = 0 , (3.19) T g(γγ) +− = T g(γγ) −+ ! = N 2 C − 1 N C 64π 2 αe 2 q α ŝ sy(1 − y) b cos 2 θ − (2x − 1)a a 2 − b 2 cos 2 θ , (3.20) where the (γγ) is to distinguish these from the amplitudes with initial-state gluons that we will consider shortly. e q is the quark charge in the meson M and a, b are given by a = (1 − x)(1 − y) + xy , (3.21) b = (1 − x)(1 − y) − xy . (3.22) The '!' in (3.20) indicates that this is not an exact equality, but rather the result of the explicit calculation and (3.20) are equivalent after the amplitude T g(γγ) has been integrated over the (antisymmetric) η 1 distribution amplitude φ G,η 1 (x). That is, they are equivalent up to terms which are even under the interchange x → 1 − x. We note that (3.19) is consistent with the results of [36,37], while (3.20) is consistent up to overall numerical factors (in particular our result is a factor of 2 larger than that of [37]). As no definition of the gluon spin projector (3.15) is given in [36,37], while the results also differ between these two papers by a factor of 2x(1 − x), a precise comparison is quite difficult to make, and we can reasonably assume that the difference between (3.20) and the results [36,37] is due to differing normalization conventions for the spin projectors and gluon wavefunctions. gg → ggqq(gg) amplitudes Representative diagrams for the three processes we will consider are shown in Fig. 3: in the case of the 4g and 6g amplitudes, we note that all diagrams allowed by colour conservation and the antisymmetry of the gluon spin projection contribute to the total amplitude, and not just diagrams of this ladder type. After a quite lengthy calculation, we find that the gg → ggqq amplitude for J z = 0 incoming gluons is given by T gq ++ = T gq −− = −2 δ ab N C N 2 C − 1 64π 2 α 2 ŝ sxy(1 − x)(1 − y) (1 + cos 2 θ) (1 − cos 2 θ) 2 (2x − 1) ,(3.23) The gg → gggg amplitude for J z = 0 incoming gluons is given by T gg ++ = T gg −− = −4 δ ab N 2 C N 2 C − 1 64π 2 α 2 ŝ sxy(1 − x)(1 − y) (1 + cos 2 θ) (1 − cos 2 θ) 2 (2x − 1)(2y − 1) ,(3.24) We also reproduce for completeness the gg → qqqq amplitudes for flavour-singlet mesons 4 We have made use of the FORM symbolic manipulation programme [51] throughout this paper. corresponding to the 'ladder diagrams' as in Fig. 3 (a), see [17]. In the case of the gg → ggqq and gg → gggg amplitudes for \|J z \| = 2 incoming gluons we can find no simple closed form. However, by numerical evaluation we can see that they exhibit a similar angular behaviour to the J z = 0 counterparts. We show this in Fig. 4, where we plot the differential cross sections dσ λ 1 λ 2 /d\| cos θ\| corresponding to the amplitudes T gq +− and T gg +− (or equivalently, T gq −+ and T gg −+ ). We recall that, due to the selection rule which operates for CEP [32] the contribution of the \|J z \| = 2 amplitudes is strongly suppressed. As observed in [17], this is particularly important in the case of flavour-non-singlet meson production. These mesons (which only have a qq component), cannot be produced in the 'ladder-type' gg → qqqq or the gg → ggqq subprocesses discussed above, where the qq pairs forming the mesons come from the same quark line. The contributing diagrams are instead of the type shown in Fig. 2 (left), with the photon pair replaced by gluons, but in this case it was found in [17] that the corresponding amplitude vanishes for J z = 0 incoming gluons, and so the CEP cross section for flavour-non-singlet meson production is expected to be strongly suppressed. g(λ1) g(λ2) k 3 k 4 (a) g(λ1) g(λ2) k 3 k 4 (b) g(λ1) g(λ2) k 3 k 4 (c)T qq. ++ = T qq. −− = − δ ab N C 64π 2 α 2 Ŝ sxy(1 − x)(1 − y) (1 + cos 2 θ) (1 − cos 2 θ) 2 , (3.25) T qq. +− = T qq. −+ = − δ ab N C 64π 2 α 2 Ŝ sxy(1 − x)(1 − y) (1 + 3 cos 2 θ) 2(1 − cos 2 θ) 2 . We note that it can readily be shown from the Feynman rules for fermion fields that we should associate an additional factor of (−1) with the flavour-non-singlet amplitude of That the amplitudes corresponding to these two types of diagrams have a relative minus sign is not in fact surprising: if we consider the case that both qq pairs are of the same flavour, the diagram of the type shown in Fig. 2 (a) is completely equivalent (replacing the photon pair with gluons) to that of Fig. 3 (a) under the interchange of the identical fermion quarks (or anti-quarks). Finally, it can be shown that the remarkable fact that (3.23), (3.24) and (3.25) are identical up to overall colour and normalization factors is not accidental, but can in fact be understood in a 'MHV' framework. We refer the reader to Appendix B for references and a detailed discussion of this. Results From Section 3.3 we can see that to first approximation (ignoring the flavour non-singlet component and the \|J z \| = 2 amplitudes), the effect of including a gg component to the η ′ η ′ CEP amplitude will simply be to multiply it by an overall normalization factor N gg , given by (omitting the µ 2 F dependence for simplicity) N gg = dx dy φ 1,η ′ (x)φ 1,η ′ (y)T qq (x, y) + 2φ 1,η ′ (x)φ G (y)T qg (x, y) + φ G (x)φ G (y)T gg (x, y) dx dy φ 1,η ′ (x)φ 1,η ′ (y)T qq (x, y) ,(4. 1) with all dependence on the final-state kinematic variables (i.e. cos(θ) andŝ) dropping out, due to the identity of the corresponding amplitudes, up to overall colour and normalization factors. For example for the qq term we may take dx dy φ 1,η ′ (x)φ 1,η ′ (y)T qq (x, y) → 1 N C n f dx dy φ 1,η ′ (x)φ 1,η ′ (y) xy(1 − x)(1 − y) , (4.2) where the factor of n f (= 3) in the numerator comes from summing over the three quark flavours contributing to each qq state, divided by the wavefunction normalizations (3.9). To give an initial estimate of the size of this effect, we recall from (3.7) that the gluon distribution amplitude can be written in the form φ G,η ′ (x, µ 2 F ) = f η ′ 1 2 √ N C C F 2n f x(1 − x) n=2,4··· a G n (µ 2 F )C 5/2 n−1 (2x − 1) . (4.3) Neglecting higher order terms in n, we therefore have explicitly φ G,η ′ (x, µ 2 F ) ≈ 5f η ′ 1 3 √ 6 a G 2 (µ 2 F )x(1 − x)(2x − 1) . (4.4) For the quark distribution, we will consider both the asymptotic (3.3) and CZ (3.4) distribution amplitudes, i.e. taking a 1 2 (µ 2 0 ) = 0 and a 1 2 (µ 2 0 ) = 2/3 in (3.7), respectively, with all higher n terms being zero. We have dx φ asym. 1,η ′ (x) x(1 − x) = √ 3f η ′ 1 , (4.5) dx φ CZ 1,η ′ (x) x(1 − x) = 5 √ 3 f η ′ 1 , (4.6) dx φ G,η ′ (x)(2x − 1) x(1 − x) = 5a G 2 9 √ 6 f η ′ 1 . (4.7) (4.8) Using these we then have, as in (4.2) dx dy φ asym. 1,η ′ (x)φ asym. 1,η ′ (y)T qq (x, y) → 3(f η ′ 1 ) 2 , 2 dx dy φ asym. 1,η ′ (x)φ G,η ′ (y)T qg (x, y) → 5a G 2 3 (f η ′ 1 ) 2 , dx dy φ CZ 1,η ′ (x)φ CZ 1,η ′ (y)T qq (x, y) → 25 3 (f η ′ 1 ) 2 , 2 dx dy φ CZ 1,η ′ (x)φ G,η ′ (y)T qg (x, y) → 25a G 2 9 (f η ′ 1 ) 2 , dx dy φ G,η ′ (x)φ G,η ′ (y)T gg (x, y) → 25(a G 2 ) 2 108 (f η ′ 1 ) 2 ,(4.9) where again the µ 2 F dependence is omitted for simplicity, and it is understood that φ CZ 1,η ′ (x) is evaluated at the scale µ 0 , so that it is given by (3.4). F . We assume that the flavour-singlet qq distribution amplitude with which this mixes under evolution is given by the CZ form (3.4). The µ 2 F dependence exhibited in the plot is largely insensitive to the form of the qq distribution amplitude. For a G 2 < 0 the µ 2 F dependence is identical up to the overall factor of (−1). a G 2 (µ 2 0 ) = 2 a G 2 (µ 2 0 ) = 5 a G 2 (µ 2 0 ) = 9.5 a G 2 (µ 2 0 ) = 19 a G 2 (µ 2 F ) - µ 2 F [GeV 2 ] To give an initial numerical estimate we may then for illustration take the value from [13] a G 2,fit (µ 2 0 ) = 19 ± 5 , (4.10) which is extracted from the transition form factors F η( ′ )γ (Q 2 ), and where all higher (n = 4, 6...) order terms are neglected. However, this corresponds to the gg distribution amplitude at the scale µ 2 0 = 1 GeV 2 . In fact (see also [13]), the leading coefficient a G 2 (µ 2 F ) of φ G (x, µ 2 F ) evolves rapidly away from its value at µ 0 , decreasing by roughly a factor of ∼ 3 for µ 2 F ≈ 10 GeV 2 , and decreasing at a much slower rate after that, see Fig. 5. Thus, to give an initial numerical estimate we can take a G n (µ 2 F = 10 GeV 2 ) ≈ 7, corresponding to an experimentally realistic value of M X ≈ 5 GeV. In this case, we find from (4.9) that N asym gg ≈ 9 , (4.11) N CZ gg ≈ 5 ,(4.12) and therefore we will expect a potentially large increase (∼ N 2 gg ) in the η( ′ )η( ′ ) CEP cross section for this value (4.10) of a G 2 (µ 2 0 ). While (4.10) may give some rough guidance for the expected size of the gg component of the η( ′ ), we note that the fit of [13] contains important uncertainties, see the discussion at the end of this section, and so the question of the size of a G 2 (µ 2 0 ) is not settled. Therefore, to give a conservative evaluation of the sensitivity of the CEP process to the size of this gg component, we will consider in what follows a band of cross section predictions, corresponding to the range a G 2 (µ 2 0 ) ∈ (−a G 2,fit /2, +a G 2,fit /2) = (−9.5, 9.5) . (4.13) As we will see, even within this quite narrow and conservative range of values, the predicted CEP cross section changes considerably. The results we present below are based on an exact numerical calculation, using a modification of the SuperCHIC MC [52], and including the precise effect of the evolution of the gg and qq wavefunctions, in contrast to the rough estimate described above. We take µ 0 = 1 GeV and µ 2 F = M 2 X /2 (being of order the squared momentum transfer ∼ \|t\|, \|û\|) throughout, although other choices of factorization scale are of course possible. We show in Figs. 6, 7 the cross sections for η ′ η ′ , ηη and ηη ′ production using the CZ form (3.4) for the quark distribution amplitude, for three different choices a G 2 (µ 2 0 ) = (−9.5, 0, 9.5) of the gg distribution amplitude normalization (4.13), and for the mesons required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. All cross section predictions in this paper are calculated using MSTW08LO PDFs [53]: while there is in general a reasonably large variation in the predicted cross sections (which are sensitive to the low x and Q 2 region of the gluon density) on the choice of PDF, we use this set as it gives a prediction which is consistent with the existing CDF exclusive γγ data [16], see for example [25] for further discussion. We show the cross sections for √ s = 0.9, 1.96 GeV, as these are the energies at which the existing CDF data on exclusive final-states have been taken, and from a further analysis of which we may hope for an observation of these meson pair CEP processes to come [14]. At the LHC, we expect the cross section (for the same event selection) to be roughly a factor of ∼ 3-5 larger for √ s = 7-14 TeV, with the particle distributions almost unchanged. For a G 2 (µ 2 0 ) = 9.5, we can see that the expected cross section increases significantly, while for negative a G 2 the gq interferes destructively with the gg and qq contributions, leading to a suppression in the predicted cross section. The predicted ηη cross section is less suppressed, due to the \|J z \| = 2 flavour-non-singlet contribution (see the discussion in section 3.3), which for the lower flavour-singlet cross sections (corresponding to a G 2 = 0, −9.5) becomes important. We note that the extracted value of a G 2 (µ 2 0 ) in general depends on the form of the flavour-singlet quark distribution amplitude, i.e. the size of the a 1 n (µ 2 0 ) in (3.7). With this in mind, in Fig. 8 we show the same cross sections at √ s = 1.96 TeV, but for the asymptotic form of the quark distribution amplitude (3.3). The a G 2 (µ 2 0 ) = 0 cross section is predicted to be somewhat smaller than for the CZ distribution amplitude choice, while a similar, somewhat larger, relative enhancement is seen for the case that a G 2 (µ 2 0 ) = 9.5. For a G 2 (µ 2 0 ) = −9.5 the destructive interference is in fact almost exact for this M X region, and so the cross sections are predicted to be very strongly affected. However, we would not necessarily expect to see such a large destructive interference effect in the data, as such an exact cancellation may not occur at higher orders. Again, we stress that these only correspond to specific cases in a band of possible a G 2 (µ 2 0 ), and that the real size of this may be smaller. Neglecting contributions with \|J z \| = 2 incoming gluons, the ratio of η ′ η ′ to ηη (ηη ′ ) cross sections are determined by the mixing in (3.11-3.13) to be σ(η ′ η ′ ) : σ(ηη ′ ) : σ(ηη) = 1 : 2 tan 2 (θ 1 ) : tan 4 (θ 1 ) , where we have taken the value of θ 1 from (3.12) 6 and the factor of '2' in the ηη ′ case accounts for the non-identity of the final-state particles. A measurement of the cross section ratios (in particular, σ(η ′ η ′ )/σ(ηη ′ )), would therefore serve as a probe of the mixing parameter, θ 1 , see (3.12). This ratio is predicted to be unchanged by the inclusion of a non-zero gluon component a G 2 = 0, as for J z = 0 incoming gluons only the flavour-singlet component η 1 of the η and η ′ mesons contributes, while we have seen in Section 3 that the gg → ggqq and gg → gggg amplitudes are identical in form to the purely quark case gg → qqqq, and will therefore only effect the overall normalization of this flavour-singlet contribution. In the ratio of cross sections, this overall factor cancels and we are left with the scaling of (4.14) irrespective of the size of the gg component. In Table 1 we show numerical results for the ratios (4.14): due to the relative importance of the \|J z \| = 2 flavour non-singlet contribution, in the ηη case, this scaling is only expected to be approximate, see below : Differential cross section dσ/dM X for X = η ′ η ′ , ηη, ηη ′ production at √ s = 0.9 TeV with MSTW08LO PDFs [53], taking the CZ form (3.4) for the quark distribution amplitude, and for a band of a G 2 (µ 2 0 ) values for the gg distribution amplitude. The mesons are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. ≈ 1 : 1 19 : 1 1450 , (4.14) a G 2 (µ 2 0 ) = 9.5 a G 2 (µ 2 0 ) = 0 a G 2 (µ 2 0 ) = −9.5 dσ(η ′ η ′ )/dM X [pb/GeV], √ s = 1.96 TeV, φ CZ . - M X [GeV]a G 2 (µ 2 0 ) = 9.5 a G 2 (µ 2 0 ) = 0 a G 2 (µ 2 0 ) = −9.5 dσ(η ′ η ′ )/dM X [pb/GeV], √ s = 0.9 TeV, φ CZ . - M X [GeV] (there is also in all cases a small effect due to the differing η and η ′ masses in the MC simulation). Thus, to first approximation we can only look at absolute value of the various η( ′ )η( ′ ) CEP cross sections to determine the size of the gg component, a G 2 (µ 2 0 ). This is potentially problematic because of the other uncertainties in the CEP calculation, due primarily to the value of the survival factors S 2 eik , S 2 enh , which are not known precisely, and potential higher-order corrections in the hard process, which combined are expected to a give a factor of ∼ × ÷ 2 − 3 uncertainty, as well as a sizeable PDF uncertainty in the low-x, Q 2 regime relevant to such processes (see [25] for a detailed discussion of this and further references). Nevertheless, given the sensitivity of the CEP cross section to the gg → gggg and gg → ggqq subprocess, if the gg component of the η( ′ ) is sizeable enough (as the result of [13], for example, suggests), such a measurement may still provide useful information. However, it is more reliable to look at the ratio of the η( ′ )η( ′ ) cross section to other processes, in which case many of the uncertainties due to PDFs and survival factors largely cancel out and a potentially much cleaner measurement of the gg component of the η( ′ ) becomes possible. With this in mind we show for illustration in Table 2 to π 0 π 0 and η ′ η ′ to γγ cross sections 7 , for the same three choices a G 2 (µ 2 0 ) = (−9.5, 0, 9.5) of the gg distribution amplitude normalization (4.13). As mentioned above, the scaling of (4.14) is only approximate, as it ignores the possibility of a flavour-non-singlet contribution for the case that the incoming gluons in the gg → η( ′ )η( ′ ) subprocess are in a \|J z \| = 2 state. Such a contribution may in particular be relevant for ηη production, when the suppression in the flavour-singlet cross section due to the small mixing ∼ sin 4 θ 1 can be comparable to the suppression of the flavour-non-singlet contribution due to the J z = 0 selection rule that operates for CEP [32]. This can be seen in Table 1, where the ratio of η ′ η ′ to ηη (and ηη ′ ) cross sections differs from (4.14) due to this 7 The π + π − CEP cross section is predicted to be identical to the neutral pion case, up to a factor of 2 due to the non-identity of the final-state particles. The observation of such a process has recently been reported in [15], although it is not clear that this measurement probes sufficiently high pion k ⊥ for the perturbative approach discussed here to be applicable. In order to better clarify the situation, a measurement of the meson k ⊥ -distributions corresponding to the same data would be very useful, while a comparison between the k ⊥ (and Mππ) distributions at 1.96 TeV and 900 GeV would probe the size of any possible contamination due to proton dissociation [3]. We also note that we expect π + π − (and K + K − ) data to come soon from CMS [56], with a veto applied on any additional particles within their rapidity coverage, which should contain a sizeable exclusive component. additional contribution. For larger values of a G 2 (µ 2 0 ), the flavour non-singlet contribution becomes relatively less important and we approach the expected values; a measurement of these ratios may therefore serve as an additional probe of the gg contribution. Recalling that the π 0 π 0 CEP cross section is also predicted to be strongly suppressed, due to the vanishing at LO of the gg → π 0 π 0 amplitude for J z = 0 incoming gluons, a measurement of the ratios σ(η( ′ )η( ′ ))/σ(π 0 π 0 ) would also represent as an important probe of the J z = 0 selection rule 8 . We can also see from Table 2 (see also [17]) that the η ′ η ′ cross section is expected to be somewhat larger than for γγ CEP. However, multiplying by the corresponding branching ratios, Br(η ′ → γγ) ≈ 2% and Br(η → γγ) ≈ 40% [57], squared, we can see in Table 2 that the cross sections for η ′ η ′ (and also for ηη) CEP after branching to the 4γ final state are predicted to be a small fraction of the direct γγ CEP cross section for the relevant event selection, with a similar result holding for the ηη ′ final state. We therefore do not expect these to represent an important background to the existing CDF [16] and any forthcoming CMS [58] γγ data 9 . 20 20 20 σ(ηη ′ )/σ(ηη) 11 66 78 Table 1: Ratios of η( ′ )η( ′ ) CEP cross sections at √ s = 1.96 TeV with MSTW08LO PDFs [53], for a gg distribution amplitude with different choices of a G 2 (µ 2 0 ) and with the qq distribution amplitude given by the CZ form (3.4). The meson are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. a G 2 (µ 2 0 ) -9.5 0 9.5 σ(ηη)/σ(π 0 π 0 ) 2.7 12 66 σ(η ′ η ′ )/σ(π 0 π 0 ) 570 16000 100000 σ(η ′ η ′ )/σ(γγ) 3.5 100 660 σ(η ′ η ′ → 4γ)/σ(γγ) 0.0017 0.049 0.33 σ(ηη → 4γ)/σ(γγ) 0.0025 0.012 0.066 Table 2: Ratios of η( ′ )η( ′ ) to π 0 π 0 and γγ CEP cross sections at √ s = 1.96 TeV with MSTW08LO PDFs [53], for a gg distribution amplitude with different choices of a G 2 (µ 2 0 ) and with the qq distribution amplitude given by the CZ form (3.4). The meson/photons are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. Also show are the ratios σ(η ′ η ′ )/σ(γγ) and σ(ηη)/σ(γγ) cross sections multiplied by the η( ′ ) → γγ branching ratios squared. 8 It should be noted that any NNLO corrections or higher twist effects which allow a Jz = 0 contribution may cause the precise value of the flavour-non-singlet ηη (and π 0 π 0 ) cross section to be somewhat larger than the leading-order, leading-twist \|Jz\| = 2 estimate. 9 Moreover there should be a further reduction, in any contamination from η, η ′ → 4γ due to the experimental exclusivity and isolation cuts and event selection. On the other hand it is worth mentioning that the multi-photon decay modes of the η and η ′ are quite sizeable (about 72% and 18% respectively [57]) and, in principle, these may also contribute to the background. a G 2 (µ 2 0 ) -9.5 0 9.5 σ(η ′ η ′ )/σ(ηη) 210 1300 1600 σ(η ′ η ′ )/σ(ηη ′ ) We recall that while various estimates are available in the literature (see for example [4,10,[59][60][61][62]), no firm consensus exists about the precise size of the gg component 10 of the η( ′ ), which is indicative of the uncertainty present in these 'standard' approaches, see for example [10] for a discussion of the theoretical uncertainties ( in e.g. decay form factors) present in such extractions. In [13], for example, the gluonic contribution to the η( ′ ) transition form factor F η( ′ )γ (Q 2 ) only enters at NLO, and so has a relatively small effect, so that the extracted value relies on a precision fit to the data 11 . In the case of CEP, on the other hand, the gluonic amplitudes (3.23), (3.24) enter at LO and so, as we have seen above, it is potentially strongly sensitive to such a gg contribution. An observation of ηη, η ′ η ′ and/or ηη ′ CEP could therefore shed important light on this uncertain issue. Conclusion and Outlook In this paper we have performed a detailed analysis of the central exclusive pair production of pseudoscalar η and η ′ mesons, pp → p + X + p, with X = ηη, η ′ η ′ and ηη ′ . We have concentrated on the case that the mesons are produced at sufficiently high transverse momentum k ⊥ , that a perturbative approach which combines the Durham model of the general CEP process (see e.g. [26] and references therein) with the 'hard exclusive' formalism (see e.g. [18,19,33] and references therein) used to model the specific gg → η( ′ )η( ′ ) subprocess, can be applied. We have in particular extended the previous work of [17] to include the possibility of a non-zero flavour-singlet gg valence component of the η ′ (and η) mesons. We recall that knowledge of the quark and gluon components of these states would provide important information about various aspects of non-perturbative QCD, see the Introduction for more details. With this in mind we have calculated for the first time the amplitudes gg → ggqq and gg → gggg, where the gg, qq pairs form collinear pseudoscalar meson states with the right spin and colour quantum numbers, see Fig. 3. It is worth emphasizing that the parton-level amplitudes relevant to the CEP of flavour-singlet (and non-singlet) meson pairs in the perturbative regime exhibit some remarkable features. In [17] it was shown that, for the case that the incoming gluons are in a J z = 0 state, the parton-level amplitude for the production of flavour-non-singlet states, gg → qqqq, vanishes completely at LO, while the \|J z \| = 2 amplitude exhibit a 'radiation zero' at a particular value of the c.m.s. scattering angle. On the other hand, the production of flavour-singlet qq states can take place via a separate set of 'ladder diagrams', see Fig. 3 (a), and in this case the J z = 0 amplitude does not vanish, and can be written in a very simple form. In this paper, we have shown that if one or both of these qq pairs forming the flavour-singlet meson states is replaced by a gluon pair gg, as in e.g. Fig. 3 (b,c), then these apparently unrelated amplitudes, gg → ggqq and gg → gggg, are identical to each other and to this 'ladder-type' amplitude, up to overall normalization factors. That is, they are predicted to have exactly the same angular dependence in the incoming gg rest-frame. In [17] and in Section B we have shown how these remarkable and non-trivial results may be explained in the MHV framework by the fact that the same external parton orderings contribute in all three cases. In CEP, this contribution from a gg valence component of the η ′ , η mesons enters at the same (leading) order to the purely qq contribution. We have seen in Section 4 that as a result the predicted η( ′ )η( ′ ) CEP cross sections display quite a strong sensitivity to any potential gg component of the η ′ , η mesons. An observation of ηη, η ′ η ′ and/or ηη ′ CEP could therefore provide a potentially powerful handle on the structure of the η ′ , η states. More generally, the observation of such meson pair (η( ′ )η( ′ ), ππ, KK...) CEP processes would serve as a test of the unique properties of the gg → M M parton-level helicity amplitudes, in particular the vanishing of the J z = 0 amplitudes for flavour nonsinglet mesons and the identical form of the gg and qq mediated flavour-singlet amplitudes. We emphasize in particular that the η ′ η ′ CEP cross section is predicted to be very large, see Section 4. These results, in conjunction with the J z = 0 CEP selection rule, lead to highly non-trivial predictions for the meson pair CEP cross sections, which it would be very interesting to compare to data. Finally, we note that we may expect new results on η( ′ )η( ′ ) and meson pair CEP more generally to come from further analysis of the existing CDF data (in particular the existing 4γ candidates with E T > 2.5 GeV and forward rapidity gaps) as well as from the CMS/Totem (ATLAS) special low-pileup runs with sufficient luminosity [14,15]. At CDF, the observation of γγ CEP has already been reported [16], and it was determined experimentally that the contamination caused by π 0 π 0 → 4γ CEP, with the photons in the π 0 decay merging or one photon being undetected, is very small (< 15 events, corresponding to a ratio N (π 0 π 0 )/N (γγ) < 0.35, at 95% C.L.), in agreement with the prediction of [17], for which σ(π 0 π 0 )/σ(γγ) ∼ 1%. This prediction is a non-trivial result of the perturbative CEP framework and the hard exclusive formalism, and an observation of π 0 π 0 CEP, which may hopefully come with the increased statistics that a further analysis of the existing data can bring [14], would certainly represent an interesting further test of the theoretical formalism. In Section 4 we observed that the η ′ η ′ and ηη to 4γ cross sections are expected to be of about the same size or maybe even be larger than the π 0 π 0 cross section, but that they should not be an important background to γγ CEP. Nonetheless, this raises the possibility of a future observation of these processes via the γγ decay chain. We also note that the branching ratios for the ρ 0 γ and ηπ + π − decays of the η ′ are sizeable, and may be viable channels for the observation of η ′ η ′ (and ηη ′ ) CEP, a possibility which would hopefully be confirmed after further analysis and simulations. To conclude, the CEP of η ′ and η meson pairs, in the perturbative regime, represents a novel (and complementary) probe of the size of a flavour-singlet gg component of these mesons. It is our hope that future η ′ , η pair CEP data and analysis will be forthcoming from the Tevatron and the LHC and that through this we can shed some light on this interesting and currently uncertain question. 2 fermion-anti-fermion pairs (recalling that the helicities of a connected fermion-antifermion pair must be opposite) no MHV amplitudes exist. With this in mind, we will show that the fact that (3.23), (3.24) and (3.25) are identical up to overall colour and normalization factors is not accidental, but follows from the observation that the J z = 0 helicity amplitudes considered in Section 3.3 are MHV, with n − 2 = 4 partons (the two incoming gluons, and two outgoing partons) having the same helicity. We will in particular show that the same set of MHV partial amplitudes, i.e. with the same orderings of the parton momenta, contributes in all cases. This section explores purely theoretical aspects of the previously calculated amplitudes, and so a reader who is only interested in the CEP cross section predictions may skip forward to Section 4. We recall that in general it is well known that a full n-parton amplitude M n can be written in the form of a 'dual expansion', as a sum of products of colour factors T n and purely kinematic partial amplitudes A n M n ({p i , h i , c i }) = ig n−2 σ T n (σ{c i })A n (σ{1 λ 1 , · · · , n λn ) , (B.1) where c i are colour labels, i λ i corresponds to the ith particle (i = 1 · · · n), with momentum p i and helicity λ i , and the sum is over appropriate simultaneous non-cyclic permutations σ of colour labels and kinematics variables. The purely kinematic part of the amplitude A n encodes all the non-trivial information about the full amplitude, M n , while the factors T n are given by known colour traces, see for instance [67] for more details. Adjusting the notation from (B.1) slightly for clarity, the n-gluon and qq ((n − 2)-gluon) MHV partial amplitudes are given by A(g + 1 , g + 2 , ..., g − i , ..., g − j , ..., g + n ) = i j 4 n k=1 k k + 1 , (B.2) A(g + 1 , g + 2 , ..., g − i , ..., q − j , q + j+1 , ..., g + n ) = i j 3 i j + 1 n k=1 k k + 1 , (B.3) where k i k j ≡ k − i \|k + j = u − (k i )u + (k j ) = v + (k i )v − (k j ) is the standard spinor contraction, and all momenta are defined as incoming. For the case that the quark (anti-quark) has positive (negative) helicity, it is sufficient to simply interchange j and j +1 in the numerator of the right hand side. The form of (B.3) expresses the important requirement that gluons are always emitted from the 'same side' of the connected quark-antiquark line (see for example Fig. 1 of [68] and the discussion in the text); that is, the qq pair must appear consecutively and in the same order. We can then apply these general formulae to the specific n = 6 amplitudes in Section 3.3, where the outgoing qq and gg pairs must form collinear meson states of the correct colour and spin. Firstly, we can see that in the case of both of the n-parton amplitudes (B.2) and (B.3), the numerators are independent of the particular (non-cyclic) permutation of the partons being considered, and will therefore factorize. The form this takes is given by the corresponding spin projections, as in (3.17) and (3.18). In particular, if we define the following momenta l 3 = xp 3 l 4 = (1 − x)p 3 l 5 = yp 4 l 6 = (1 − y)p 4 , (B.4) where p 3,4 are the 4-momenta of the outgoing mesons, then the numerator in the 6-gluon case corresponds to (recalling that the helicity here is defined with respect to the incoming gluon momenta) (g − (l 3 )g + (l 4 ) − g + (l 3 )g − (l 4 ))(g − (l 5 )g + (l 6 ) − g + (l 5 )g − (l 6 )) → 4 6 4 + 3 5 4 − 3 6 4 − 4 5 4 =ŝ 2 (2y − 1)(2x − 1) . (B.5) In the qq case, the final expression is the same, but with the factor of '(2y − 1)' removed. We can see from (B.2) and (B.3) that the amplitudes for a given ordering of partons are identical between the two cases, up to these overall numerator factors. To justify the statement that it is indeed the same set of orderings which contribute in the case of both amplitudes (3.23) and (3.24), we must consider the colour factors, T n , in (B.1). In the qq case this is given by T n ((n − 2)g + qq) = (λ 1 · · · λ n−2 ) i 1 j 1 , (B.6) where i 1 (j 1 ) is the colour index the quark (anti-quark). In the n-gluon case, T n is given by T n (ng) = Tr (λ 1 · · · λ n ) . (B.7) For a given T n , the corresponding diagram has the same cyclic ordering of the quark and gluons as their colour labels in T n . The λ matrices are normalized (unconventionally) so that Tr(λ a λ b ) = δ ab , as required by the form of the expansion given in (B.1). Now, an explicit calculation in Section 3.3 has shown that in both cases it is only the leading terms in N C which give a non-zero contribution. In the gg → qqgg case the terms of order 1/N C which come from certain 'abelian diagrams' (i.e. those which are identical to the γγ → qqgg diagrams, with the photons replaced by gluons) do not contribute due to the vanishing of the J z = 0 two-photon amplitudes (3.19), while in the gg → 4g case, all colour factors coming from the colour-singlet final and initial states are of order N 2 C . We therefore know that all the partial amplitudes with colour factors T n which are sub-leading in N C must sum to give zero contribution, and can be neglected. By considering the various possible values of T n for different non-cyclic orderings of the quarks and gluons, it is easy to show that only those diagrams of the type shown in Fig. 9 (d-f) and Fig. 9 (g-i) give such a leading N C colour factor, with in particular T 6 (4g + qq) = N C + · · · , (B.8) T 6 (6g) = N 2 C + · · · , (B.9) in all cases. That is, the same type of diagrams (corresponding to a given ordering of the external parton momenta) contribute in both cases. Recalling that the denominator term (1 − y)p 4 (i) Figure 9: Representative contributing MHV diagrams for flavour singlet meson pair production, via (a-c) gg → qqqq (d-f) gg → qqgg (g-i) gg → gggg in (B.2) and (B.3) is identical for a given particle ordering, while the numerator and colour factors factorize, we can see that the resulting amplitudes will be equivalent up to these overall factors, as expected. Explicitly, calculating the partial amplitudes corresponding to the 6g diagrams shown in Fig. 9 (g-i), including those coming from interchanging p 1 ↔ p 2 , x → 1−x and y → 1−y, we find where these correspond to the kinematic amplitudes with the numerator factors, as in (B.5), as well as an additional factor of xy(1 − x)(1 − y) in the denominator, omitted for simplicity, and the subscript ('g,h,i') indicates the corresponding diagram in Fig. 9. The amplitudes in the 4g case are all a factor of 2 smaller due to the requirement that only those diagrams with the gluons emitted on one side of the quark line are included, while there is no such requirement for the purely gluonic case. In fact, the individual MHV amplitudes corresponding to these parton orderings are divergent, due to factors of l 3 l 4 ∼ p 2 3 = 0 and l 5 l 6 ∼ p 2 4 = 0 present in the denominators, but the sum over all particle interchanges (in particular, of the two gluons forming the meson states, i.e. l 3 (l 5 ) ↔ l 4 (l 6 ), where there is only one pair to be interchanged in the ggqq case) is not. To arrive at these (finite) results requires a careful grouping of the contributing amplitudes and application of the Schouten identity Combining (B.12) with the relevant numerator, colour and normalization factors, we can readily reproduce the expressions of (3.24) and (3.23), up to an overall phase, which we take to be due the differing convention for the overall phase of the amplitude which the normalization of (B.1) implies. Finally we turn to the gg → qqqq amplitude. We first consider the colour factor, T n , in (B.1) for the qqqq case. This is given by T n ((n − 4)g + 2qq) = (−1) p N p C (λ a 1 · · · λ a l ) i 1 α 1 (λ b 1 · · · λ b l ′ ) i 2 α 2 , (B.14) where i 1 , i 2 are the colour indices of the quarks, α 1 , α 2 are the colour indices of the antiquarks and the labels a i , b i refer to the gluons. The pair (α 1 α 2 ) is a permutation of the pair (j 1 j 2 ), where quark i k is connected by a fermion line to antiquark j k . p is then the number of times α k = j k , with p = 1 if (α 1 α 2 ) ≡ (j 1 j 2 ). The sum is over all the partitions of the gluons (l + l ′ = n − 4, l = 0, · · · , n − 4) and over the permutations of the gluon indices, with the product of zero λ matrices given by a Kronecker delta. In the case of (3.25), we can again see that it is only the leading terms in N C (O(1)), which contribute to the hard amplitude. Again, it can readily be shown that the relevant partial amplitudes for this have the same parton ordering as those in the 6g and 4g cases, as shown in Fig. 9 (a-c). Diagrams a and b correspond to the p = 0 and c to p = 1 in (B.14), with the additional factor of 1/N C in the case of diagram c canceled by the factor of δ ii in the numerator; in both cases, the colour factor is then O(1). In the case of diagram c both gluons must be emitted by one of the quark lines, otherwise the amplitude will vanish due a factor of Tr(λ) in the numerator. The kinematic amplitude for two non-identical quark anti-quark pairs, qq and QQ, is given by (see e.g. [69]) A(q h 1 1 , g + 2 , · · · , Q −h 2 m , Q h 2 m+1 , g + m+2 , · · · , q −h 1 n ) = We note that (B.15) and (B.16) are for the case that p = 0 in (B.14), i.e. for diagrams a and b of Fig. 9. For the (p = 1) case of diagram c, we must simply interchange n ↔ m in the above expressions. We can again readily show that summing over the contributing quark helicity states according to the spin projection (3.17), as was done in (B.5) for the 6g case, will simply give an overall factor ofŝ 2 for the numerator of (B.15), irrespective of the particle ordering. Thus, the denominator term in (B.15) is identical to that in (B.2) and (B.3) for a given particle ordering, while the numerator and colour factors factorize. As it is again the same type of diagrams (corresponding to a given ordering of the external parton momenta) which contributes, we find that the resulting amplitudes will be equivalent up to these overall factors, as expected 12 . Figure 3 : 3Representative Feynman diagrams for the gg → M M process, where the M are flavoursinglet mesons. There are 8 Feynman diagrams of type (a), and the corresponding helicity amplitudes are given by(3.25, 3.26). There are 76 Feynman diagrams of type (b), and the corresponding J z = 0 helicity amplitude is given by(3.23), while a numerical evaluation of the \|J z \| = 2 case is shown inFig. 4. There are 130 Feynman diagrams of type (c), and the corresponding J z = 0 helicity amplitude is given by(3.24), while a numerical evaluation of the \|J z \| = 2 case is shown inFig. 4. In the case of the amplitudes for (b) and (c), all diagrams allowed by colour conservation are included, and not just diagrams of this ladder type. three J z = 0 amplitudes are identical, up to overall colour and normalization factors (including the factors of '(1 − 2x), (1 − 2y)' which ensure that the amplitudes have the correct symmetry under the interchange x(y) ↔ 1 − x(y)). Figure 4 : 4Differential cross section dσ λ1λ2 /d\| cos θ\| for gg → η 1 η 1 via gg → ggqq and gg → gggg diagrams, shown for illustration, for √ŝ = 5 GeV. The qq distribution amplitude is given by(3.4), with n f = 3, while the gg distribution amplitude is given by(4.4), with f 1 = f π . λ 1,2 are the incoming gluon helicities. Fig. 2 2(a), for the case of incoming gluons, but not with the 'ladder' diagrams of Fig. 3 5 . Figure 5 : 5n = 2 coefficient of the expansion (4.3) of the gg distribution amplitude φ G,M (x, µ 2 F ), as a function of the scale µ 2 Figure 6 : 6Differential cross section dσ/dM X for X = η ′ η ′ , ηη, ηη ′ production at √ s = 1.96 TeV with MSTW08LO PDFs[53], taking the CZ form (3.4) for the quark distribution amplitude, and for a band of a G 2 (µ 2 0 ) values for the gg distribution amplitude. The mesons are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. Figure 7 7Figure 7: Differential cross section dσ/dM X for X = η ′ η ′ , ηη, ηη ′ production at √ s = 0.9 TeV with MSTW08LO PDFs [53], taking the CZ form (3.4) for the quark distribution amplitude, and for a band of a G 2 (µ 2 0 ) values for the gg distribution amplitude. The mesons are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. ηη ′ )/dM X [pb/GeV], √ s = 1.96 TeV, φasym. Figure 8 : 8Differential cross section dσ/dM X for X = η ′ η ′ , ηη, ηη ′ production at √ s = 1.96 TeV with MSTW08LO PDFs[53], taking the asymptotic form (3.3) for the quark distribution amplitude, and for a band of a G 2 (µ 2 0 ) values for the gg distribution amplitude. The mesons are required to have transverse energy E ⊥ > 2.5 GeV and pseudorapidity \|η\| < 1. i = 0 , (B.12) F (h 1 , h 2 12) The higher order n = 4, 6,... terms are found to evolve faster towards zero with increasing n. Combined with the fact that as n increases, the additional powers of C 5/2 n−1 (2x − 1) (or C 3/2 n (2x − 1) in the qq case) give a smaller numerical contribution to the distribution amplitude, this means that any fit can effectively truncate the series (3.7) after a limited number of terms. In[17], the opposite assignment was taken. However only the quark contribution was considered, in which only the relative sign between these two types of diagram was important. The quoted cross sections and results are therefore correct. We note that the predicted η ′ η and ηη cross sections are lower than in[17]. This is due to the different choice of mixing scheme (3.11-3.13) and in particular the lower value of θ1, which leads to a smaller flavoursinglet component of the η. It is found in for example[50,54,55] that this scheme(3.11-3.13) and choice of mixing parameters describe the available data well. A measurement of the cross section ratios in (4.14) would certainly shed further light on this. For example, the situation with regards to the χ c(0,2) decays into η and η ′ pairs appears somewhat puzzling. Experimentally, no enhancement in the production of the η, η ′ pairs as compared to pions is observed (after taking trivial phase space effects into account). This may indicate that there is some destructive interference between the qq and the gg components of the pseudoscalar bosons, or that the gg component is small. See[13] for other ideas and[63] for a review and further discussion.11 The potential importance of power corrections to F η( ′ )γ (Q 2 ) at lower values of Q 2 , where much of the existing data lies, may in fact cast doubt on the reliability of such a fit. We thank Viktor Chernyak for a useful discussion on this topic. AcknowledgementsWe thank Mike Albrow, Erik Brucken, Victor Chernyak, Risto Orava, Kornelija Passek-Kumericki and Antoni Szczurek for useful discussions. This work was supported by the grant RFBR 11-02-00120-a and by the Federal Program of the Russian State RSGSS-4801.2012.2. WJS is grateful to the IPPP for an Associateship. VAK thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work.A. Meson distribution amplitudes: anomalous dimensionsThe anomalous dimensions which control the evolution (3.8) are given by[45,46]In terms of these we can then write down the eigenvalues γ (±) n which diagonalise the anomalous dimensionsThe parameters ρB. MHV calculationIt is well known (see for example[64]) that the tree level n-gluon scattering amplitudes, in which the maximal number (n−2) of gluons have the same helicity, the so-called 'maximally helicity violating' (MHV), or 'Parke-Taylor', amplitudes, are given by remarkably simple formulae[65,66]. These results were extended using supersymmetric Ward identities to include amplitudes with one and two quark-antiquark pairs in[64], where 'MHV' refers to the case where (n − 2) partons have the same helicity. In these cases, simple analytic expressions can again be written down for the MHV amplitudes, while for greater than . A D Martin, M G Ryskin, V A Khoze, 0903.2980Acta Phys.Polon. 401841A. D. Martin, M. G. Ryskin, and V. A. Khoze, Acta Phys.Polon. B40, 1841 (2009), 0903.2980. . M G Albrow, T D Coughlin, J R Forshaw, 1006.1289Prog.Part.Nucl.Phys. 65149M. G. Albrow, T. D. Coughlin, and J. R. Forshaw, Prog.Part.Nucl.Phys. 65, 149 (2010), 1006.1289. . 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Passek-Kumericki, (2012), 1206.4870. . Mike Albrow, private communicationMike Albrow, private communication. In fact, in this gg → qqqq case, the situation is somewhat more complicated. In particular, as we do not include diagrams corresponding to permutations of the legs within each meson state, l3(l5) ↔ l4(l6), the singularities discussed above do not cancel and so the amplitudes A. 102g a,b,c (taking the notation of (B.In fact, in this gg → qqqq case, the situation is somewhat more complicated. In particular, as we do not include diagrams corresponding to permutations of the legs within each meson state, l3(l5) ↔ l4(l6), the singularities discussed above do not cancel and so the amplitudes A 2g a,b,c (taking the notation of (B.10)- However, when all three amplitudes are summed the result is finite. 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[ "An optimal series expansion of the multiparameter fractional Brownian motion ", "An optimal series expansion of the multiparameter fractional Brownian motion " ]	[ "Anatoliy Malyarenko " ]	[]	[]	We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.	10.1007/s10959-007-0122-x	[ "https://arxiv.org/pdf/math/0411539v4.pdf" ]	7,464,747	math/0411539	5f2a5c8a13617608fedc0bf4e4001572caa34249	An optimal series expansion of the multiparameter fractional Brownian motion * 5 Oct 2007 6th November 2018 Anatoliy Malyarenko An optimal series expansion of the multiparameter fractional Brownian motion * 5 Oct 2007 6th November 2018fractional Brownian motionseries expansionBessel functions We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal. Introduction The fractional Brownian motion with Hurst parameter H ∈ (0, 1) is defined as the centred Gaussian process ξ(t) with the autocorrelation function R(s, t) = Eξ(s)ξ(t) = 1 2 (\|s\| 2H + \|t\| 2H − \|s − t\| 2H ). This process was defined by Kolmogorov [8] and became a popular statistical model after the paper by Mandelbrot and van Ness [13]. There exist two multiparameter extensions of the fractional Brownian motion. Both extensions are centred Gaussian random fields on the space R N . The multiparameter fractional Brownian sheet has the autocorrelation function R(x, y) = 1 2 N N j=1 (\|x j \| 2H j + \|y j \| 2H j − \|x j − y j \| 2H j ), H j ∈ (0, 1) while the multiparameter fractional Brownian motion has the autocorrelation function R(x, y) = 1 2 ( x 2H + y 2H − x − y 2H ),(1) where · denotes the Euclidean norm in R N and where H ∈ (0, 1). Dzhaparidze and van Zanten [3] and Iglói [7] derived two different explicit series expansions of the fractional Brownian motion. In [4], Dzhaparidze and van Zanten extended their previous result to the case of the multiparameter fractional Brownian sheet. All the above mentioned expansions were proven to be rate optimal. We extend the results by Dzhaparidze and van Zanten to the case of the multiparameter fractional Brownian motion. To present and prove our result, we need to recall definitions of some special functions [6]. The Bessel function of the first kind of order ν is defined by the following series J ν (z) = ∞ m=0 (−1) m z 2m+ν 2 m m!Γ(ν + m + 1) .(2) Let j ν,1 < j ν,2 < · · · < j ν,n < . . . be the positive zeros of J ν (z). Let g m (u) = 2 (N −2)/2 Γ(N/2) J m+(N −2)/2 (u) u (N −2)/2 , m ≥ 0, and let δ n m denote the Kronecker's delta. The Gegenbauer polynomials of order m, C λ m , are given by the generating function 1 (1 − 2xt + t 2 ) λ = ∞ m=0 C λ m (x)t m . Let m be a nonnegative integer, and let m 0 , m 1 , . . . , m N −2 be integers satisfying the following condition m = m 0 ≥ m 1 ≥ · · · ≥ m N −2 ≥ 0. Let x = (x 1 , x 2 , . . . , x N ) be a point in the space R N . Let r k = x 2 k+1 + x 2 k+2 + · · · + x 2 N , where k = 0, 1, . . . , N − 2. Consider the following functions H(m k , ±, x) = x N −1 + ix N r N −2 ±m N−2 r m N−2 N −2 N −3 k=0 r m k −m k+1 k × C m k+1 +(N −k−2)/2 m k −m k+1 x k+1 r k , and denote Y (m k , ±, x) = r −m 0 H(m k , ±, x). The functions Y (m k , ±, x) are called the (complex-valued) spherical harmon- ics. For a fixed m, there exist h(m, N) = (2m + N − 2)(m + N − 3)! (N − 2)!m!(4) spherical harmonics. They are orthogonal in the Hilbert space L 2 (S N −1 ) of the square integrable functions on the unit sphere S N −1 , and the square of the length of the vector Y (m k , ±, x) is L(m k ) = 2π N −2 k=1 π2 k−2m k −N +2 Γ(m k−1 + m k + N − 1 − k) (m k−1 + (N − 1 − k)/2)(m k−1 − m k )![Γ(m k + (N − 1 − k)/2)] 2 . Let l = l(m k , ±) be the number of the symbol (m 0 , m 1 , . . . , m N −2 , ±) in the lexicographic ordering. The real-valued spherical harmonics, S l m (x), can be defined as S l m (x) =      Y (m k , +, x)/ L(m k ), m N −2 = 0, √ 2 Re Y (m k , +, x)/ L(m k ), m N −2 > 0, l = l(m k , +), − √ 2 Im Y (m k , −, x)/ L(m k ), m N −2 > 0, l = l(m k , −). The hypergeometric function is defined by the series 2 F 1 (a, b; c; z) = ∞ k=0 (a) k (b) k (c) k k! z k , where (u) k = u(u + 1) . . . (u + k − 1), (u) 0 = u. The incomplete beta function is defined as B z (α, β) = z 0 t α−1 (1 − t) β−1 dt.ξ(x) = ∞ m=0 ∞ n=1 h(m,N ) l=1 τ mn [g m (j \|m−1\|−H,n x ) − δ 0 m ]S l m x x ξ l mn ,(5) where ξ l mn are independent standard normal random variables, and τ mn = 2 H+1 π (N −2)/2 Γ(H + N/2)Γ(H + 1) sin(πH) Γ(N/2)J \|m−1\|−H+1 (j \|m−1\|−H,n )j H+1 \|m−1\|−H,n .(6) The series (5) converges with probability 1 in the space C(B) of continuous functions in B = { x ∈ R N : x ≤ 1 }. Ayache and Linde [1] derived another representation of the multiparameter fractional Brownian motion, using wavelets. Hence it is natural to compare their result with Theorem 1. Consider the multiparameter fractional Brownian motion ξ(x), x ∈ B as the centred Gaussian random variable ξ on the Banach space C(B) of all continuous functions on B. The pth l-approximation number of ξ [10] is defined by: l p (ξ) = inf        E ∞ j=p f j ξ j 2 C(B)   1/2 : ξ = ∞ j=1 f j ξ j , f j ∈ C(B)      , where ξ j are independent standard normal random variables and the infimum is taken over all possible series representations for ξ. Ayache and Linde [1] determined the convergence rate of l p (ξ) → 0 as p → ∞. To formulate their result, introduce the following notation. If a n , n ≥ 1, and b n , n ≥ 1 are sequences of positive real numbers, we write a n b n provided that a n ≤ cb n for a certain c > 0 and for any positive integer n. Then a n ≈ b n means that a n b n as well as b n a n . In the same way, we write f (u) g(u) provided that f (u) ≤ cg(u) for a certain c > 0 and uniformly for all u, and f (u) ≈ g(u) if f (u) g(u) as well as g(u) f (u). Ayache and Linde [1, Theorem 1.1] proved that l p (ξ) ≈ p −H/N (log p) 1/2 . They also proved that their wavelet series representation possesses the optimal approximation rate. We will prove that our representation (5) is optimal as well. Theorem 2. The representation (5) possesses the optimal approximation rate for ξ on B. Theorem 1 is proved in Section 2 while Theorem 2 is proved in Section 3. I am grateful to Professors K. Dzhaparidze, M. Lifshits, and H. van Zanten for useful discussions. I thank the two anonymous referees for their very careful reading of the manuscript and for their helpful remarks. Proof of Theorem 1 It is well known [1], that the multiparameter fractional Brownian motion can be represented as the stochastic integral ξ(x) = c HN R N e i(p,x) − 1 p N/2+H dŴ (p), where dŴ is the complex-valued white noise obtained by Fourier transformation of the real-valued white noise. It follows that the covariance function (1) of the multiparameter fractional Brownian motion can be represented as R(x, y) = c 2 HN R N [e i(p,x) − 1][e −i(p,y) − 1] p −N −2H dp.(7) The following Lemma gives the explicit value of the constant c 2 HN . This result was announced by Malyarenko [12]. Lemma 1. The constant c 2 HN has the following value: c 2 HN = 2 2H−1 Γ(H + N/2)Γ(H + 1) sin(πH) π (N +2)/2 . Proof. For N = 1, our formula has the form c 2 H1 = 2 2H−1 Γ(H + 1/2)Γ(H + 1) sin(πH) π 3/2 , or c 2 H1 = Γ(2H + 1) sin(πH) 2π . Here we used the doubling formula for gamma function. This result is known [17]. Therefore, in the rest of the proof we can and will suppose that N ≥ 2. Rewrite (7) as R(x, y) = c 2 HN R N [1 − e i(p,x) ] p −N −2H dp + c 2 HN R N [1 − e −i(p,y) ] p −N −2H dp − c 2 HN R N [1 − e i(p,x−y) ] p −N −2H dp.(8) Consider the first term in the right hand side of this formula. Using formula 3.3.2.3 from [14], we obtain c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = 2π (N −1)/2 c 2 HN Γ((N − 1)/2) ∞ 0 λ N −1 dλ π 0 [1 − e iλ x cos u ]λ −N −2H sin N −2 u du. It is clear that the integral of the imaginary part is equal to 0. The integral of the real part may be rewritten as c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = 2π (N −1)/2 c 2 HN Γ((N − 1)/2) ∞ 0 λ −1−2H π 0 [1 − cos(λ x cos u)] sin N −2 u du dλ. (9) To calculate the inner integral, we use formulas 2.5.3.1 and 2.5.55.7 from [14]: π 0 sin N −2 u du = √ πΓ((N − 1)/2) Γ(N/2) , π 0 cos(λ x cos u) sin N −2 u du = √ π2 (N −2)/2 Γ((N − 1)/2) J (N −2)/2 (λ x ) (λ x ) (N −2)/2 . It follows that π 0 [1 − cos(λ x cos u)] sin N −2 u du = √ πΓ((N − 1)/2) Γ(N/2) [1 − g 0 (λ x )].(10) Substituting (10) in (9), we obtain c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = 2π N/2 c 2 HN Γ(N/2) ∞ 0 λ −2H−1 [1 − g 0 (λ x )] dλ.(11) To calculate this integral, we use formula 2.2.3.1 from [14]: 1 0 (1 − v 2 ) (N −3)/2 dv = √ πΓ((N − 1)/2) 2Γ(N/2) . It follows that 1 = 2Γ(N/2) √ πΓ((N − 1)/2) 1 0 (1 − v 2 ) (N −3)/2 dv.(12) On the other hand, according to formula 2.5.6.1 from [14] we have 1 0 (1 − v 2 ) (N −3)/2 cos(λ x v) dv = √ π2 (N −4)/2 Γ((N − 1)/2) J (N −2)/2 (λ x ) (λ x ) (N −2)/2 . It follows that g 0 (λ x ) = 2Γ(N/2) √ πΓ((N − 1)/2) 1 0 (1 − v 2 ) (N −3)/2 cos(λ x v) dv.(13) Subtracting (13) from (12), we obtain 1 − g 0 (λ x ) = 4Γ(N/2) √ πΓ((N − 1)/2) 1 0 (1 − v 2 ) (N −3)/2 sin 2 λ x 2 v dv. Substitute this formula in (11). We have c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = 8π (N −1)/2 c 2 HN Γ((N − 1)/2) ∞ 0 1 0 λ −2H−1 (1 − v 2 ) (N −3)/2 sin 2 λ x 2 v dv dλ. (14) Consider two integrals: 1 0 λ −2H−1 sin 2 v x 2 λ dλ and ∞ 1 λ −2H−1 sin 2 v x 2 λ dλ. In the first integral, we bound the second multiplier by λ 2 /4. In the second integral, we bound it by 1. It follows that the integral ∞ 0 λ −2H−1 sin 2 v x 2 λ dλ converges uniformly, and we are allowed to change the order of integration in the right hand side of (14). After that the inner integral is calculated using formula 2.5.3.13 from [14]: ∞ 0 λ −2H−1 sin 2 v x 2 λ dλ = πv 2H 4Γ(2H + 1) sin(πH) x 2H . Now formula (11) may be rewritten as c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = 2π (N −1)/2 c 2 HN Γ((N − 1)/2)Γ(2H + 1) sin(πH) 1 0 v 2H (1 − v 2 ) (N −3)/2 dv · x 2H . The integral in the right hand side can be calculated by formula 2.2.4.8 from [14]: 1 0 v 2H (1 − v 2 ) (N −3)/2 dv = Γ((N − 1)/2)Γ(H + 1/2) 2Γ(H + N/2) and (11) is rewritten once more as c 2 HN R N [1 − e i(p,x) ] p −N −2H dp = π (N +2)/2 c 2 HN 2 2H Γ(H + 1)Γ(H + N/2) sin(πH) x 2H . (15) On the other hand, the left hand side of (15) is clearly equal to 1 2 x 2H . The statement of the Lemma follows. Taking into account (11), one can rewrite (8) as R(x, y) = 2π N/2 c 2 HN Γ(N/2) ∞ 0 λ −1−2H [1 − g 0 (λ x ) − g 0 (λ y ) + g 0 (λ x − y )] dλ.(16) Let x = 0, y = 0 and let ϕ denote the angle between the vectors x and y. Two addition theorems for Bessel functions (formulas 7.15(30) and 7.15(31) from [6, vol. 2]) may be written in our notation as g 0 (λ x − y ) = ∞ m=0 h(m, N)g m (λ x )g m (λ y ) C (N −2)/2 m (cos ϕ) C (N −2)/2 m (1) ,(17) where g m is as in (3). Substituting (17) in (16), we obtain R(x, y) = 2π N/2 Γ(N/2) ∞ m=0 C (N −2)/2 m (cos ϕ) C (N −2)/2 m (1) h(m, N) ∞ 0 κ m x (λ)κ m y (λ) dλ,(18) where κ m r (λ) = c HN λ −H−1/2 [g m (rλ) − δ 0 m ]. Recall that the Hankel transform of order ν > −1 of a function κ ∈ L 2 (0, ∞) (see [18,Section 8.4]) is defined aŝ κ(u) = ∞ 0 κ(λ)J ν (uλ) √ uλ dλ. Define the kernelκ m r (u) as the Hankel transform of order \|m − 1\| − H of the kernel κ m r (λ):κ m r (u) = ∞ 0 κ m r (λ)J \|m−1\|−H (uλ) √ uλ dλ.(19) Therefore, the Parceval identity for the Hankel transform [18, Section 8.5, Theorem 129] implies that (18) may be rewritten as R(x, y) = 2π N/2 Γ(N/2) ∞ m=0 C (N −2)/2 m (cos ϕ) C (N −2)/2 m (1) h(m, N) ∞ 0κ m x (u)κ m y (u) du.(20) To calculateκ m r (u) in the case of m ≥ 1, we use formula 2.12.31.1 from [15].κ where χ (0,r) (u) denote the indicator function of the interval (0, r). For m = 0, the integral in (19) can be rewritten aŝ κ 0 r (u) = 2 (N −2)/2 Γ(N/2)c HN r 1−N/2 √ u ∞ 0 λ 1−H−N/2 J (N −2)/2 (rλ)J 1−H (uλ) dλ − √ uc HN ∞ 0 λ −H J 1−H (uλ) dλ. To calculate the second integral, we use formula 2.12.2.2 from [15]. ∞ 0 λ −H J 1−H (uλ) dλ = Γ(1 − H) 2 H u 1−H . For the first integral, we use formula 2.12.31.1 from [15] once more. In the case of u > r we obtain ∞ 0 λ 1−H−N/2 J (N −2)/2 (rλ)J 1−H (uλ) dλ = Γ(1 − H)r (N −2)/2 2 H+(N −2)/2 Γ(N/2)u 1−H . In the case of u < r, we have ∞ 0 λ 1−H−N/2 J (N −2)/2 (rλ)J 1−H (uλ) dλ = Γ(1 − H)r 2H+N/2−3 2 H+(N −2)/2 Γ(H + (N − 2)/2)Γ(2 − H) × 2 F 1 (1 − H, 2 − H − N/2; 2 − H; u 2 /r 2 ). Using formula 7.3.1.28 from [16], we can rewrite the last expression as ∞ 0 λ 1−H−N/2 J (N −2)/2 (rλ)J 1−H (uλ) dλ = r (N −2)/2 2 H+(N −2)/2 Γ(H + (N − 2)/2)u 1−H × B u 2 /r 2 (1 − H, H + (N − 2)/2). Combining everything together, we obtain To calculate it, we use the inversion formula for Hankel transform: According to the addition theorem for spherical harmonics [6, Vol. 2, Chapter XI, section 4, Theorem 4], C (N −2)/2 m (cos ϕ) C (N −2)/2 m (1) = 2π N/2 Γ(N/2)h(m, N) h(m,N ) l=1 S l m x x S l m y y .(24) Substituting this equality in (23), we obtain R(x, y) = 8π N (Γ(N/2)) 2 ∞ m=0 ∞ n=1 h(m,N ) l=1 κ m x (j \|m−1\|−H,n )κ m y (j \|m−1\|−H,n ) J 2 \|m−1\|−H+1 (j \|m−1\|−H,n )j \|m−1\|−H,n × S l m x x S l m y y . It follows immediately that the random field ξ(x) itself has the form (5), (6), and the series (5) converges in mean square for any fixed x ∈ B. Since the functions in (5) are continuous and the random variables are symmetric and independent, the Itô-Nisio theorem [19] implies that for proving that the series (5) converges uniformly on B with probability 1, it is sufficient to show that the corresponding sequence of partial sums ξ M (x) = M m=0 M n=1 h(m,N ) l=1 τ mn [g m (j \|m−1\|−H,n x ) − δ 0 m ]S l m x x ξ l mn (25) is weakly relatively compact in the space C(B). To prove this, one can use the same method as in the proof of Theorem 4.1 in [5]. Proof of Theorem 1 is finished. Consider some particular cases of Theorem 1. In the case of N = 1, there exists only two spherical harmonics on the 0-dimensional sphere S 0 = {−1, 1}. They are S 1 0 (x) = 1 √ 2 , S 1 1 (x) = x √ 2 , where x ∈ S 0 . Moreover, we have g 0 (j 1−H,n \|x\|) = cos(j 1−H,n \|x\|), g 1 (j −H,n \|x\|) = sin(j −H,n \|x\|). Therefore formula (5) becomes ξ(x) = 2c H1 ∞ n=1 cos(j 1−H,n x) − 1 J 2−H (j 1−H,n )j 1+H 1−H,n ξ 1 0n + ∞ n=1 sin(j −H,n x) J 1−H (j −H,n )j 1+H −H,n ξ 1 1n . (26) This is the result of [3]. Consider the case of the multiparameter fractional Brownian motion on the plane (N = 2). Let (r, ϕ) be the polar coordinates. The spherical harmonics are: S 1 0 (ϕ) = 1 √ 2π , S 1 m (ϕ) = cos(mϕ) √ π , S 2 m (ϕ) = sin(mϕ) √ π . It follows that ξ(r, ϕ) = 2 H+1 Γ(H + 1) sin(πH) √ π 1 √ 2 ∞ n=1 J 0 (j 1−H,n r) − 1 J 2−H (j 1−H,n )j 1+H 1−H,n ξ 1 0n + ∞ m=1 ∞ n=1 J m (j m−1−H,n r) cos(mϕ) J m−H (j m−1−H,n )j 1+H m−1−H,n ξ 1 mn + ∞ m=1 ∞ n=1 J m (j m−1−H,n r) sin(mϕ) J m−H (j m−1−H,n )j 1+H m−1−H,n ξ 2 mn . Proof of Theorem 2 It is enough to prove that the rate of convergence in (5) is not more than the optimal rate p −H/N (log p) 1/2 , where p denote the number of terms in a suitable truncation of the series. Since for N > 1 we have a triple sum in our expansion (5), it is not clear a priori how we should truncate the series. We need a lemma. 1. j ν,n ≈ n + ν/2 − 1/4. J 2 ν+1 (j ν,n ) ≈ 1 jν,n . 3. \|g 0 (u)\| ≤ 1. 4. \|g m (u)\| 1 [m(m+N −2)] (N−1)/4 , m ≥ 1, N ≥ 2. 5. \|g ′ m (u)\| 1 [m(m+N −2)] (N−3)/4 , m ≥ 1, N ≥ 2. 6. h(m, N) m N −2 , n ≥ 2. 7. \|S l m (x/ x )\| m (N −2)/2 , N ≥ 2. Proof. Property 1 is proved in [20,Section 15.53]. It is shown in [20, Section 7.21] that J 2 ν (u) + J 2 ν+1 (u) ≈ 1 u ,(27) since Property 2. Property 3 follows from the fact that g 0 (u) is the element of the unitary matrix [21]. Let µ 1 < µ 2 < . . . be the sequence of positive stationary values of the Bessel function J m+(N −2)/2 (u). It is shown in [20,Section 15.31] that \|J m+(N −2)/2 (µ 1 )\| > \|J m+(N −2)/2 (µ 2 )\| > . . . . Let u 1 be the first positive maximum of the function g m (u). Let x 1 denote the maximal value of the function \|g m (u)\| in the interval (0, j m+(N −2)/2,1 ), let x 2 denote the maximal value of \|g m (u)\| in the interval (j m+(N −2)/2,1 , j m+(N −2)/2,2 ), and so on. Then we have x 1 \|J m+(N −2)/2 (µ 1 )\| u (N −2)/2 1 , x 2 \|J m+(N −2)/2 (µ 2 )\| j (N −2)/2 m+(N −2)/2,1 , x 3 \|J m+(N −2)/2 (µ 3 )\| j (N −2)/2 m+(N −2)/2,2 , and so on. The right hand sides form a decreasing sequence. Then \|g m (u)\| \|J m+(N −2)/2 (µ 1 )\| u (N −2)/2 1 . It follows from (27) that \|J ν (u)\| 1 √ u , and we obtain \|g m (u)\| 1 µ 1 u (N −2)/2 1 To estimate u 1 , consider the differential equation u 2 d 2 f du 2 + (N − 1)u df du + [u 2 − m(m + N − 2)]f = 0.(28) It follows from formulas (3) and (4) in [20,Section 4.31] that the function g m satisfies this equation. It follows from the series representation (2) of the Bessel function and the definition (3) of the function g m (u) that for m ≥ 1 we have g m (0) = 0 and g m increases in some right neighbourhood of zero. Therefore we have g m (u 1 ) > 0, g ′ m (u 1 ) = 0 and g ′′ m (u 1 ) ≤ 0. It follows from equation (28) that u 1 ≥ m(m + N − 2). For µ 1 , we have the estimate µ 1 > m + (N − 2)/2 ([20, Section 15.3]). Using the inequality m+(N −2)/2 ≥ m(m + N − 2), we obtain Property 4. In any extremum of g ′ m (u) we have g ′′ m (u) = 0. It follows from (28) that \|g ′ m (u)\| = [u 2 − m(m + N − 2)]g m (u) (N − 1)u \|ug m (u)\|. Property 5 follows from this formula and Property 4. Using Lemma 2, we can write the following estimate: E   h(m,N ) l=1 u l mn (x)ξ l mn   2 1 (m + 1)(m/2 + n) 2H+1 .(30) For any positive integer q, consider the truncation To prove this relation, consider the partial sum η k (x) defined by η k (x) = 2 k−1 <(m+1)(m/2+n) 2H+1 ≤2 k h(m,N ) l=1 u l mn (x)ξ l mn . For a given ε k > 0, let x 1 , . . . , x P k ∈ S N −1 be a maximal ε k -net in S N −1 , i.e., the angle ϕ(x j , x k ) between any two different vectors x j and x k is greater than ε k and the addition of any new point breaks this property. Then P k ≈ ε −N +1 k . A proof of this fact for the case of N = 3 by Baldi et al [2,Lemma 5], is easy generalised to higher dimensions. The Voronoi cell S(x j ) is defined as S(x j ) = { x ∈ S N −1 : ϕ(x, x j ) ≤ ϕ(x, x k ), k = j }. Divide the ball B onto [ε −1 k ] concentric spherical layers of thickness ε k . Voronoi cells determine the division of each layer onto P k segments. The angle ϕ(x, y) between any two vectors x and y in the same segment is ε k (we never choose the point 0). Call all segments in all layers B j , 1 ≤ j ≤ M k = P k [ε −1 k ] ε −N k . Choose a point x j inside each segment. Then we have E sup x∈B \|η k (x)\| ≤ E sup 1≤j≤M k \|η k (x j )\| + E sup 1≤j≤M k sup x,y∈B j \|η k (x) − η k (y)\|. By a maximal inequality for Gaussian sequences [22, Lemma 2.2.2], the first term is bounded by a positive constant times 1 + log M k sup 1≤j≤M k E(η k (x j )) 2 . Using the estimate (30), we obtain E(η k (x)) 2 2 k−1 <(u+1)(u/2+v) 2H+1 ≤2 k ,u≥0,v≥1 du dv (u + 1)(u/2 + v) 2H+1 . The integral in the right hand side is easily seen to be 2 −kH/(H+1) . It follows that E sup 1≤j≤M k \|η k (x j )\| ≤ 2 −kH/(2H+2) 1 + log M k .(31) To estimate the second term we write E sup 1≤j≤M k sup x,y∈B j \|η k (x) − η k (y)\| ≤ E sup 1≤j≤M k sup x,y∈B j 2 k−1 <(m+1)(m/2+n) 2H+1 ≤2 k h(m,N ) l=1 \|u l mn (x) − u l mn (y)\| · \|ξ l mn \|. The difference \|u l mn (x) − u l mn (y)\| can be estimated as \|u l mn (x) − u l mn (y)\| ≤ \|g m (j \|m−1\|−H,n x ) − δ 0 m \| · \|S l m (x/ x ) − S l m (y/ y )\| + \|S l m (y/ y )\| · \|g m (j \|m−1\|−H,n x ) − g m (j \|m−1\|−H,n y )\|. Let ∆ 0 denote the angular part of the Laplace operator in the space R N . Using the Mean Value Theorem for g m , Lemma 2 and formulas ∆ 0 S l m (x/ x ) = −m(m + N − 2)S l m (x/ x ), \|S l m (x/ x ) − S l m (y/ y )\| sup z \| −∆ 0 S l m (z/ z )\|ϕ(x, y), where the sup is taken over all points z belonging to the segment of the geodesic circle connecting the points x/ x and y/ y , we obtain \|u l mn (x) − u l mn (y)\| ε k τ mn √ m(m/2 + n). Hence, we have Using Lemma 2 once more, the last formula may be rewritten as E sup 1≤j≤M k sup x,y∈B j \|η k (x)−η k (y)\| ε k 2 k−1 <(m+1)(m/2+n) 2H+1 ≤2 k m N −3/2 (m/2+n) 1/2−H . The integral, which corresponds to the sum in the right hand side, is easily seen to be asymptotically equal to 2 kN/(2H+2) . Therefore, E sup 1≤j≤M k sup x,y∈B j \|η k (x) − η k (y)\| ε k · 2 kN/(2H+2) . Combining this with the estimate (31) for the first term, we obtain the asymptotic relation E sup x∈B \|η k (x)\| 2 −kH/(2H+2) 1 + log M k + ε k · 2 kN/(2H+2) . Now set ε k = 2 −k(H+N )/(2H+2) and recall that M k ε −N k = 2 kN (H+N )/(2H+2) . Then we see that the first term is bounded by a constant times 2 −kH/(2H+2) √ k and the second one is of lower order. We proved that E sup x∈B \|η k (x)\| 2 −kH/(2H+2) √ k.(32) To complete the proof, it suffices to show that Let r be the positive integer such that 2 r−1 < q ≤ 2 r . Then by the triangle inequality The arguments in the proof of (32) show that since 2 r−1 < q ≤ 2 r , we also have Relation (33) is proved. HN u −H+m−1/2 (r 2 − u 2 ) H+N/2−1 2 H+N/2−1 Γ(H + N/2)r m+N −2 χ (0,r) (u), x B u 2 /r 2 (1 − H, H + (N − 2)/2)]χ (0,r) (u).It follows from the last display and from(21) that the support of the kernelŝκ m x (u) andκ m y (u) lies in [0, 1] since x , y ≤ 1 for x, y ∈ B. x (u)k m y (u) du(22)for x, y ∈ B. For any ν > −1, the Fourier-Bessel functionsϕ ν,n (u) = √ 2u J ν+1 (j ν,n ) J ν (j ν,n u), n ≥ 1 form a complete, orthonormal system in L 2 [0, 1] [20, Section 18.24]. Put ν = \|m − 1\| − H. (u)ϕ \|m−1\|−H,n (u) du. \|m−1\|−H+1 (j \|m−1\|−H,n ) j \|m−1\|−H,n κ m x (j \|m−1\|−H,n ).Substituting the calculated values in(22)x (j \|m−1\|−H,n )κ m y (j \|m−1\|−H,n ) J 2 \|m−1\|−H+1 (j \|m−1\|−H,n )j \|m−1\|−H,n . Lemma 2 . 2The Bessel functions, the functions g m defined by (3), the numbers h(m, N), and the spherical harmonics S m l (x/ x ) have the following properties. mn (x) = τ mn [g m (j \|m−1\|−H,n x ) q expansion(5). The number of terms in this sum is asymptotically equal to the integral(u+1)(u/2+v)≤q,u≥0,v≥1 u N −2 du dvand therefore is bounded by a constant times q N/(2H+2) (N/(2H+2))(−H/N ) (log q N/(2H+2) ) 1/2 q −H/(2H+2) (log q) 1/2 . By equivalence of moments ([9, Proposition 2.1]), the last formula is equivalent to the following asymptotic relation. H/(2H+2) (log q) 1/2 . q −H/(2H+2) (log q) 1/2 . q −H/(2H+2) (log q) 1/2 . 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[ "Peierls bounds from Toom contours", "Peierls bounds from Toom contours" ]	[ "Jan M Swart ", "Réka Szabó ", "Cristina Toninelli " ]	[]	[]	We review and extend Toom's classical result about stability of trajectories of cellular automata, with the aim of deriving explicit bounds for monotone Markov processes, both in discrete and continuous time. This leads, among other things, to rigorous bounds for a two-dimensional interacting particle system with cooperative branching and deaths. Our results can be applied to derive bounds for other monotone systems as well.MSC 2010. Primary: 60K35; Secondary: 37B15, 82C22	null	[ "https://arxiv.org/pdf/2202.10999v1.pdf" ]	247,025,534	2202.10999	6e65dd1768b19fa5c81113f4bb17cdc1a28b0325	Peierls bounds from Toom contours February 23, 2022 Jan M Swart Réka Szabó Cristina Toninelli Peierls bounds from Toom contours February 23, 2022Random Cellular AutomataMonotone Interacting Particle SystemsToom's the- orem We review and extend Toom's classical result about stability of trajectories of cellular automata, with the aim of deriving explicit bounds for monotone Markov processes, both in discrete and continuous time. This leads, among other things, to rigorous bounds for a two-dimensional interacting particle system with cooperative branching and deaths. Our results can be applied to derive bounds for other monotone systems as well.MSC 2010. Primary: 60K35; Secondary: 37B15, 82C22 Introduction Monotone systems We will be interested in Markov processes, both in discrete and continuous time, that take values in the space {0, 1} Z d of configurations x = (x(i)) i∈Z d of zeros and ones on the ddimensional integer lattice Z d . By definition, a map ϕ : {0, 1} Z d → {0, 1} is local if ϕ depends only on finitely many coordinates, i.e., there exists a finite set ∆ ⊂ Z d and a function ϕ : {0, 1} ∆ → {0, 1} such that ϕ (x(i)) i∈Z d = ϕ (x(i)) i∈∆ for each x ∈ {0, 1} Z d . We say that ϕ is monotone if x ≤ y (coordinatewise) implies ϕ(x) ≤ ϕ(y). We say that ϕ is monotonic if it is both local and monotone. The discrete time Markov chains (X n ) n≥0 taking values in {0, 1} Z d that we will be interested in are uniquely characterised by a finite collection ϕ 1 , . . . , ϕ m of monotonic maps and a probability distribution p 1 , . . . , p m on {1, . . . , m}. They evolve in such a way that independently for each n ≥ 0 and i ∈ Z d , X n+1 (i) = ϕ k (θ i X n ) with probability p k (1 ≤ k ≤ m), (1.1) where for each j ∈ Z d , we define a translation operator θ i : {0, 1} Z d → {0, 1} Z d by (θ i x)(j) := x(i + j) (i, j ∈ Z d ) . We call such a Markov chain (X n ) n≥0 a monotone random cellular automaton. The continuous time Markov chains (X t ) t≥0 taking values in {0, 1} Z d that we will be interested in are similarly characterised by a finite collection ϕ 1 , . . . , ϕ m of monotonic maps and a collection of nonnegative rates r 1 , . . . , r m . They evolve in such a way that independently for each i ∈ Z d , X t (i) is replaced by ϕ k (θ i X t ) at the times of a Poisson process with rate r k (1.2) (1 ≤ k ≤ m). We call such a Markov process a monotone interacting particle system. Wellknown results [Lig85,Thm I.3.9] show that such processes are well-defined. They are usually constructed so that t → X t (i) is piecewise constant and right-continuous at its jump times. Let P x denote the law of the discrete time process started in X 0 = x and let 0 and 1 denote the configurations that are constantly zero or one, respectively. Well-known results imply that there exist invariant laws ν and ν, called the lower and upper invariant law, such that P 0 [X n ∈ · ] =⇒ n→∞ ν and P 1 [X n ∈ · ] =⇒ n→∞ ν, (1.3) where ⇒ denotes weak convergence of probability laws on {0, 1} Z d with respect to the product topology. Each invariant law ν of (X n ) n≥0 satisfies ν ≤ ν ≤ ν in the stochastic order, and one has ν = ν if and only if ρ = ρ, where ]. We will be interested in methods to derive lower bounds on ρ. It will be convenient to give names to some special monotonic functions. We start with the constant monotonic functions ϕ 0 (x) := 0 and ϕ 1 (x) := 1 (x ∈ Z d ). (1.5) Apart from these constant functions, all other monotonic functions have the property that ϕ(0) = 0 and ϕ(1) = 1, and therefore monotone systems that do not use the function ϕ 0 (resp. ϕ 1 ) have the constant configuration 1 (resp. 0) as a fixed point of their evolution. We will discuss whether this fixed point is stable when the original system is perturbed by applying ϕ 0 (resp. ϕ 1 ) with a small probability or rate. The next monotonic function of interest is the "identity map" ϕ id (x) := x(0) x ∈ {0, 1} Z d . (1.6) Monotone systems that only use ϕ id do not evolve at all, of course. We can think of the continuous-time interacting particle systems as limits of discrete-time cellular automata where time is measured in steps of some small size ε, the maps ϕ 1 , . . . , ϕ m are applied with probabilities εr 1 , . . . , εr m , and with the remaining probability, the identity map ϕ id is applied. where round denotes the function that rounds off a real number to the nearest integer. The function ϕ NEC is known as North-East-Center voting or NEC voting, for short, and also as Toom's rule. In analogy to ϕ NEC , we let ϕ NWC , ϕ SWC , ϕ SEC denote maps that describe North-West-Center voting, South-West-Center voting, and South-East-Center voting, respectively, defined in the obvious way. We will call the map ϕ NN from (1.7) Nearest Neigbour voting or NN voting, for short. Another name found in the literature is the symmetric majority rule. Figure 1 shows numerical data for random perturbations of the cellular automata defined by ϕ NEC and ϕ NN . Both ϕ NEC and ϕ NN have obvious generalisations to higher dimensions, but we will not need these. We call ϕ coop the cooperative branching rule. It is also known as the sexual reproduction rule because of the interpretation that when ϕ coop is applied at a site (i 1 , i 2 ), two parents at (i 1 + 1, i 2 ) and (i 1 , i 2 + 1) produce offspring at (i 1 , i 2 ), provided the parents' sites are both occupied and (i 1 , i 2 ) is vacant. Toom's stability theorem Recall the definition of the constant monotonic map ϕ 0 in (1.5). In what follows, we fix a monotonic map ϕ : {0, 1} Z d → {0, 1} that is not constantly zero or one. For each p ∈ [0, 1], we let (X p k ) k≥0 denote the monotone random cellular automaton defined by the monotonic functions ϕ 0 and ϕ that are applied with probabilities p and 1 − p, respectively. We let ρ(p) denote the density of the upper invariant law as a function of p. Since ϕ is not constant, 1 is a fixed point of the deterministic system (X 0 k ) k≥0 , and hence ρ(0) = 1. We say that (X k ) k≥0 = (X 0 k ) k≥0 is stable if ρ(p) → 1 as p → 0. Furthermore, we say that ϕ is an eroder if for each initial state X 0 0 that contains only finitely many zeros, one has X 0 n = 1 for some n ∈ N. We quote the following result from [Too80, Thm 5]. Toom's stability theorem (X k ) k≥0 is stable if and only if ϕ is an eroder. In words, this says that the all-one fixed point is stable under small random perturbations if and only if ϕ is an eroder. For general local maps that need not be monotone, it is known that there exists no algorithm to decide whether a given map is an eroder, even in one dimension [Pet87]. By contrast, for monotonic maps, there exists a simple criterion to check whether a given map is an eroder. Each monotonic map ϕ : {0, 1} Z d → {0, 1} can uniquely be written as ϕ(x) = A∈A(ϕ) i∈A x(i), (1.8) where A(ϕ) is a finite collection of finite subsets of Z d that have the interpretation that their indicator functions 1 A (A ∈ A(ϕ)) are the minimal configurations on which ϕ gives the outcome 1. In particular, A(ϕ 0 ) = ∅ and A(ϕ 1 ) = {∅}, where in (1.8) we use the convention that the supremum (resp. infimum) over an empty set is 0 (resp. 1). We let Conv(A) denote the convex hull of a set A, viewed as a subset of R d . Then [Too80,Thm 6], with simplifications due to [Pon13, Thm 1], says that a monotonic map ϕ that is not constantly zero or one is an eroder if and only if A∈A(ϕ) Conv(A) = ∅. (1.9) We note that by Helly's theorem [Roc70,Corollary 21.3.2], if (1.9) holds, then there exists a subset A ⊂ A(ϕ) of cardinality at most d + 1 such that A∈A Conv(A) = ∅. Using (1.9), it is straightforward to check that the maps ϕ NEC and ϕ coop , defined in (1.7), are eroders. On the other hand, one can easily check that ϕ NN is not an eroder. Indeed, if (X 0 n ) n≥0 is started in an initial state with a zero on the sites (0, 0), (0, 1), (1, 0), (1, 1) and ones everywhere else, then the deterministic system remains in this state forever. Toom's model that applies the maps ϕ 0 , ϕ 1 , and ϕ NEC with probabilities p, r, and 1 − p − r, respectively. On the right: the mononotone random cellular automaton that applies the maps ϕ 0 , ϕ 1 , and ϕ NN with probabilities p, r, and 1 − p − r, respectively. Contrary to ϕ NEC , the map ϕ NN is not an eroder. By the symmetry between the 0's and the 1's, in both models, the density ρ of the lower invariant law equals 1 − ρ. Due to metastability effects, the area where the upper invariant law differs from the lower invariant law is shown too large in these numerical data. For Toom's model with r = 0, the data shown above suggest a first order phase transition at p c ≈ 0.057 but based on numerical data for edge speeds we believe the true value is p c ≈ 0.053. We conjecture that the model on the right has a unique invariant law everywhere except on the diagonal p = r for p sufficiently small. Main results While Toom's stability theorem is an impressive result, it is important to realise its limitations. As Toom already remarked [Too80, Section V], his theorem does not apply to monotone cellular automata whose local state space is not {0, 1}, but {0, 1, 2}, for example. Also, his theorem only applies in discrete time and only to random perturbations of cellular automata defined by a single non-constant monotonic map ϕ. The most difficult part in the proof of Toom's stability theorem is showing that if ϕ is an eroder, then ρ(p) → 1 as p → 0. To give a lower bound on ρ(p) for small values of p, Toom uses a Peierls contour argument. The main result of our article is extending this Peierls argument to monotone cellular automata whose definition involves, apart from the constant monotonic map ϕ 0 , several non-constant monotonic maps ϕ 1 , . . . , ϕ m . We are especially interested in the case when one of these maps is the identity map ϕ id and in the closely related problem of giving lower bounds on ρ(p) for monotone interacting particle systems, which evolve in continuous time. Another result of our work is obtaining explicit lower bounds for ρ(p) for concrete models, which has not been attempted very much. In particular, we extend Toom's definition of a contour to monotone cellular automata that apply several non-constant monotonic maps and to monotone interacting particle systems. We show that X n (i) = 0 for some i ∈ Z d (or equivalently X t (i) = 0 in continuous time) implies the presence of a Toom contour "rooted at" (n, i) (or (t, i) respectively), which in turn can be used to obtain lower bounds for ρ(p) via a Peierls argument. Our main results are contained in Theorems 7, 9 and 41. At this point rather than formally stating these results, which would require dwelling into technical details, we state the explicit bounds we obtain as a result of our construction. Our extension of Toom's result allows us to establish or improve explicit lower bounds for ρ(p) for concrete models. First we consider Toom's set-up, that is monotone random cellular automata that apply the maps ϕ 0 and ϕ with probabilities p and 1 − p, respectively, where ϕ is an eroder. An easy coupling argument shows that the intensity ρ(p) of the upper invariant law is a nonincreasing function of p, so we can define a critical parameter p c := sup{p : ρ(p) > 0} ∈ [0, 1]. (1.10) Since ϕ is an eroder, Toom's stability theorem tells us that p c > 0. We show how to derive explicit lower bounds on p c for any choice of the eroder ϕ, and do this for two concrete examples. We first take for ϕ the map ϕ NEC and obtain the bound p c ≥ 3 −21 , which does not compare well to the estimated value p c ≈ 0.053 coming from numerical simulations. Nevertheless, this is probably the best rigorous bound currently available. Then we take for ϕ the map ϕ coop and, improving on Toom's method, we get the bound p c ≥ 1/64. This is also some way off the estimated value p c ≈ 0.105 coming from numerical simulations. Then we consider the monotone random cellular automaton on Z d that applies the maps ϕ 0 , ϕ id , and ϕ coop with probabilities p, q, r, respectively with q = 1 − p − r. For each p, r ≥ 0 such that p + r ≤ 1, let ρ(p, r) denote the intensity of the upper invariant law of the process with parameters p, 1 − p − r, r. Arguing as before, it is easy to see that for each 0 ≤ r < 1 we can define a critical parameter p c (r) := sup{p : ρ(p, r) > 0} ∈ [0, 1 − r]. (1.11) By carefully examining the structure of Toom contours for this model, we prove the bound p c (r) > 0.00624r. Finally, we consider the interacting particle system on Z 2 that applies the monotonic maps ϕ 0 and ϕ coop with rates 1 and λ, respectively. This model was introduced by Durrett [Dur86] as the sexual contact process, and we can think of it as the limit of the previous discrete-time cellular automata. For each λ > 0 we let ρ(λ) denote the intensity of the upper invariant law of the process with parameters 1, λ. Again, we define a critical parameter λ c := inf{λ ≥ 0 : ρ(λ) > 0} ∈ (0, ∞). (1.12) Numerical simulations suggest the value λ c ≈ 12.4, we show the upper bound λ c ≤ 161.1985. Durrett claimed a proof that λ c ≤ 110, which he describes as ridiculous, but for which he challenges the reader to do better. We have quite not managed to beat his bound, though we are not far off. The proofs of all results in [Dur86] are claimed to be contained in a forthcoming paper with Lawrence Gray [DG85] that has never appeared. In [Gra99], Gray refered to these proofs as "unpublished" and in [BD17], Durrett cites the paper as an "unpublished manuscript". Although for monotone cellular automata that apply several non-constant monotonic maps and for monotone interacting particle systems our methods do not seem to be enough to obtain bounds on the critical value in general, we believe that our examples are instructive of how one can try to do it for a concrete model. Discussion The cellular automaton defined by the NEC voting map ϕ NEC is nowadays known as Toom's model. In line with Stigler's law of eponymy, Toom's model was not invented by Toom, but by Vasilyev, Petrovskaya, and Pyatetski-Shapiro, who simulated random perturbations of this and other models on a computer [VPP69]. The function p → ρ(p) appears to be continuous except for a jump at p c (see Figure 1). Toom, having heard of [VPP69] during a seminar, proved in [Too74] that there exist random cellular automata on Z d with at least d different invariant laws. Although Toom's model is not explicitly mentioned in the paper, his proof method can be applied to prove that p c > 0 for his model. In [Too80], Toom improved his methods and proved his celebrated stability theorem. His paper is quite hard to read. One of the reasons is that Toom tries to be as general as possible. For example, he allows for cellular automata that look back more than one step in time, which severely complicates the statement of conditions like (1.9). He also allows for noise that is not i.i.d. and cellular automata that are not monotone, even though all his results in the general case can easily be obtained by comparison with the i.i.d. monotone case. Toom's Peierls argument in the original paper is quite hard to understand. A more accessible account of Toom's original argument (with pictures!) in the special case of Toom's model can be found in the appendix of [LMS90]. 1 Although in principle, Toom's Peierls argument can be used to derive explicit bounds on p c , Toom did not attempt to do so, no doubt in the belief that more powerful methods would be developed in due time. Bramson and Gray [BG91] have given another alternative proof of Toom's stability theorem that relies on comparison with continuum models (which describe unions of convex sets in R d evolving in continuous time) and renormalisation-style block arguments. They somewhat manage to relax Toom's conditions but the proof is very heavy and any explicit bounds derived using this method would presumably be very bad. Gray [Gra99] proved a stability theorem for monotone interacting particle systems. The proofs use ideas from [Too80] and [BG91] and do not lend themselves well to the derivation of explicit bounds. Gray also derived necessary and sufficient conditions for a monotonic map to be an eroder [Gra99, Thm 18.2.1], apparently overlooking the fact that Toom had already proved the much simpler condition (1.9). Motivated by abstract problems in computer science, a number of authors have given alternative proofs of Toom's stability theorem in a more restrictive setting [GR88,BS88,Gac95,Gac21]. Their main interest is in a three-dimensional system which evolves in two steps: letting e 1 , e 2 , e 3 denote the basis vectors in Z 3 , they first replace X n (i) by X n (i) := round (X n (i) + X n (i + e 1 ) + X n (i + e 2 ))/3 , and then set X n+1 (i) := round (X n (i) + X n (i + e 3 ) + X n (i − e 3 ))/3 . They prove explicit bounds for finite systems, although for values of p that are extremely close to zero. 2 The proofs of [GR88] do not use Toom's Peierls argument but rely on different methods. Their bounds were improved in [BS88]. Still better bounds can be found in the unpublished note [Gac95]. The proofs in the latter manuscript are very similar to Toom's argument, with some crucial improvements at the end that are hard to follow due to missing definitions. This version of the argument seems to have inspired the incomplete note by John Preskill [Pre07] who links it to the interesting idea of counting "minimal explanations". His definition of a "minimal explanation" is a bit stronger than the definition we will adopt in Subsection 7.1 below, but sometimes, such as in the picture in Figure 3 on the right, the two definitions coincide. Figure 3 shows that the relation between Toom contours and minimal explanations is not so straightforward as suggested in [Gac95,Pre07]. We have not found a good way to control the number of minimal explanations with a given number of defective sites and we do not know how to derive the lower bounds on the density of the upper invariant law stated in [Gac95,Pre07]. Hwa-Nien Chen [Che92,Che94], who was a PhD student of Lawrence Gray, studied the stability of various variations of Toom's model under perturbations of the initial state and the birth rate. The proofs of two of his four theorems depend on results that he cites from the as yet nonexisting paper [DG85]. Ponselet [Pon13] gave an excellent account of the existing literature and together with her supervisor proved exponential decay of correlations for the upper invariant law of a large class of randomly perturbed monotone cellular automata [MP11]. There exists duality theory for general monotone interacting particle systems [Gra86,SS18]. The basic idea is that the state in the origin at time zero is a monotone function of the state at time −t, and this monotone function evolves in a Markovian way as a function of t. Durrett [Dur86] mentions this dual process as an important ingredient of the proofs of the forthcoming paper [DG85] and it is also closely related to the minimal explanations of Preskill [Pre07]. A good understanding of this dual process could potentially help solve many open problems in the area, but its behaviour is already quite complicated in the mean-field case [MSS20]. Outline The paper is organized as follows. We define Toom contours and give an outline of the main idea of the Peierls argument in Subsection 2.1. In Subsection 2.2 we prove Toom's stability theorem. In Susbsection 2.3 we introduce a stronger notion of Toom contours, that allows us to improve bounds for certain models. We then present two explicit bounds in Toom's set-up in Subsection 2.4. In Subsection 2.5 we consider monotone random cellular automata that apply several non-constant monotonic maps and in Subsection 2.6 we discuss continuous time results and bounds. The rest of the paper is devoted for proofs and technical arguments. The results stated in Subsections 2.1 are proved in Section 3. Section 4 contains all the proofs of the results stated in Subsections 2.2, 2.3 and 2.4. The results of Subsection 2.5 are proved in Section 5. Section 6 gives the precise definitions and results together with their proofs in the continuoustime setting. Finally, the relation between Toom contours and minimal explanations in the sense of John Preskill [Pre07] is discussed in Section 7, where we also discuss the open problem of counting minimal explanations. 2 Setting and definitions Toom's Peierls argument In this subsection, we derive a lower bound on the intensity of the upper invariant law for a class of monotone random cellular automata. We use a Peierls argument based on a special type of contours that we will call Toom contours. In their essence, these are the contours used in [Too80], though on the face of it our definitions will look a bit different from those of [Too80]. This pertains especially to the "sources" and "sinks" defined below that are absent from Toom's formulation and that we think help elucidate the argument. We start by defining a special sort of directed graphs, which we will call Toom graphs (see Figure 2). After that we first give an outline of the main idea of the Peierls argument and then provide the details. Toom graphs Recall that a directed graph is a pair (V, E) where V is a set whose elements are called vertices and E is a subset of V × V whose elements are called directed edges. For each directed edge (v, w) ∈ E, we call v the starting vertex and w the endvertex of (v, w). We let E in (v) := (u, v ) ∈ E : v = v and E out (v) := (v , w) ∈ E : v = v (2.1) denote the sets or directed edges entering and leaving a given vertex v ∈ V , respectively. We will need to generalise the concept of a directed graph by allowing directed edges to have a type in some finite set {1, . . . , σ}, with the possibility that several edges of different types connect the same two vertices. To that aim, we define an directed graph with σ types of edges to be a pair (V, E), where E = ( E 1 , . . . , E σ ) is a sequence of subsets of V × V . We interpret E s as the set of directed edges of type s. Definition 1 A Toom graph with σ ≥ 2 charges is a directed graph with σ types of edges (V, E) = (V, E 1 , . . . , E σ ) such that each vertex v ∈ V satisfies one of the following four conditions: (i) \| E s,in (v)\| = 0 = \| E s,out (v)\| for all 1 ≤ s ≤ σ, (ii) \| E s,in (v)\| = 0 and \| E s,out (v)\| = 1 for all 1 ≤ s ≤ σ, (iii) \| E s,in (v)\| = 1 and \| E s,out (v)\| = 0 for all 1 ≤ s ≤ σ, (iv) there exists an s ∈ {1, . . . , σ} such that \| E s,in (v)\| = 1 = \| E s,out (v)\| and \| E l,in (v)\| = 0 = \| E l,out (v)\| for each l = s. See Figure 2 for a picture of a Toom graph with three charges. We set V • := v ∈ V : \| E s,in (v)\| = 0 ∀1 ≤ s ≤ σ , V * := v ∈ V : \| E s,out (v)\| = 0 ∀1 ≤ s ≤ σ , V s := v ∈ V : \| E s,in (v)\| = 1 = \| E s,out (v)\| (1 ≤ s ≤ σ). (2.2) Vertices in V • , V * , and V s are called sources, sinks, and internal vertices with charge s, respectively. Vertices in V • ∩ V * are called isolated vertices. Informally, we can imagine that at each source there emerge σ charges, one of each type, that then travel via internal vertices of the corresponding charge through the graph until they arrive at a sink, in such a way that at each sink there converge precisely σ charges, one of each type. It is clear from this description that \|V • \| = \|V * \|, i.e., the number of sources equals the number of sinks. We let E := σ s=1 E s denote the union of all directed edge sets and we let E := {v, w} : (v, w) ∈ E denote the corresponding set of undirected edges. We say that a Toom graph (V, E) is connected if the associated undirected graph (V, E) is connected. Toom contours Our next aim is to define Toom contours, which are connected Toom graphs that are embedded in space-time Z d+1 in a special way. Let (V, E) = (V, E 1 , . . . , E σ ) be a Toom graph with σ charges. Recall that E = σ s=1 E s . Definition 2 An embedding of (V, E) is a map V v → ψ(v) = ψ(v), ψ d+1 (v) ∈ Z d × Z (2.3) that has the following properties: (i) ψ d+1 (w) = ψ d+1 (v) − 1 for all (v, w) ∈ E, (ii) ψ(v 1 ) = ψ(v 2 ) for each v 1 ∈ V * and v 2 ∈ V with v 1 = v 2 , (iii) ψ(v 1 ) = ψ(v 2 ) for each v 1 , v 2 ∈ V s with v 1 = v 2 (1 ≤ s ≤ σ). We interpret ψ(v) and ψ d+1 (v) as the space and time coordinates of ψ(v) respectively. Condition (i) says that directed edges (v, w) of the Toom graph (V, E) point in the direction of decreasing time. Condition (ii) says that sinks do not overlap with other vertices and condition (iii) says that internal vertices do not overlap with other internal vertices of the same charge. See Figure 3 for an example of an embedding of a Toom graph. Not every Toom graph can be embedded. Indeed, it is easy to see that if (V, E) has an embedding in the sense defined above, then \| E 1 \| = · · · = \| E σ \|,(2.4) i.e., there is an equal number of charged edges of each charge. The Toom graph of Figure 2 can be embedded, but if we would change the number of internal vertices on one of the paths from a source to a sink, then the resulting graph would still be a Toom graph but it would not be possible to embed it. . On the right: a minimal explanation for a monotone cellular automaton Φ that applies the maps ϕ 0 and ϕ coop with probabilities p and 1 − p, respectively. The origin has the value zero because the sites marked with a star are defective. This explanation is minimal in the sense that removing any of the defective sites results in the origin having the value one. The Toom contour in the middle picture is present in Φ. In particular, the sinks of the Toom contour coincide with some, though not with all of the defective sites of the minimal explanation. Definition 3 A Toom contour is a quadruple (V, E, v • , ψ), where (V, E) is a connected Toom graph, v • ∈ V • is a specially designated source, and ψ is an embedding of (V, E) that has the additional properties that: (iv) ψ d+1 (v • ) > t for all (i, t) ∈ ψ(V )\{ψ(v • )}, where ψ(V ) := {ψ(v) : v ∈ V } denotes the image of V under ψ. We call v • the root of the Toom contour and we say that the Toom contour Figure 3 for an example of a Toom contour with two charges. (V, E, v • , ψ) is rooted at the space-time point ψ(v • ) ∈ Z d+1 . See For any Toom contour (V, E, v • , ψ), we write E * := σ s=1 E * s with E * s := (v, w) ∈ E s : v ∈ V s ∪ {v • } (1 ≤ s ≤ σ), E • := σ s=1 E • s with E • s := (v, w) ∈ E s : v ∈ V • \{v • } (1 ≤ s ≤ σ). (2.5) i.e., E * is the set of directed edges that have an internal vertex or the root as their starting vertex, and E • are all the other directed edges, that start at a source that is not the root. The special role played by the root will become important in the next subsection, when we define what it means for a Toom contour to be present in a collection of i.i.d. monotonic maps. If (V, E, v • , ψ) is a Toom contour, then we let ψ(V * ) := ψ(v) : v ∈ V * , ψ( E * s ) := ψ(v), ψ(w) : (v, w) ∈ E * s , ψ( E • s ) := ψ(v), ψ(w) : (v, w) ∈ E • s (1 ≤ s ≤ σ), (2.6) denote the images under ψ of the set of sinks V * and the sets of directed edges E * s and E • s , respectively. We call two Toom contours (V, E, v • , ψ) and (V , E , v • , ψ ) equivalent if ψ(v • ) = ψ (v • ), ψ(V * ) = ψ (V * ), ψ( E * s ) = ψ ( E * s ), ψ( E • s ) = ψ ( E • s ). (2.7) The main idea of the construction We will be interested in monotone random cellular automata that are defined by a probability distribution p 0 , . . . , p m and monotonic maps ϕ 0 , . . . , ϕ m , of which ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are non-constant. This generalises Toom's set-up, who only considered the case m = 1. We fix an i.i.d. collection Φ = (Φ (i,t) ) (i,t)∈Z d+1 of monotonic maps such that P[Φ (i,t) = ϕ k ] = p k (0 ≤ k ≤ m). A space-time point (i, t) with Φ (i,t) = ϕ 0 is called a defective site. In Lemmas 4 and 5 below, we show that Φ almost surely determines a stationary process (X t ) t∈Z that at each time t is distributed according to the upper invariant law ν. Our aim is to give an upper bound on the probability that X 0 (0) = 0, which then translates into a lower bound on the intensity ρ of the upper invariant law. To achieve this, we first describe a special way to draw a Toom graph inside space-time Z d+1 . Such an embedding of a Toom graph in space-time is then called a Toom contour. Since our argument requires looking backwards in time, it will be convenient to adopt the convention that in all our pictures (such as Figure 3), time runs downwards. Next, we define when a Toom contour is present in the random collection of maps Φ. Theorem 7 then states that the event X 0 (0) = 0 implies the presence of a Toom contour in Φ. This allows us to bound the probability that X 0 (0) = 0 from above by the expected number of Toom contours that are present in Φ. In later subsections, we will then discuss conditions under which this expectation can be controlled and derive explicit bounds. Before we state the remaining definitions, which are mildly complicated, we explain the main idea of the construction. We will define presence of Toom contours in such a way that the space-time point (0, 0) is a source and all the sinks correspond to defective sites where the map ϕ 0 is applied. Let M n denote the number of Toom contours that have (0, 0) as a source and that have n sinks. One would like to show that if the map ϕ 0 is applied with a sufficiently small probability p, then the expression ∞ n=1 M n p n is small. This will not be true, however, unless one imposes additional conditions on the contours. In fact, it is rather difficult to control the number of contours with a given number of sinks. It is much easier to count contours with a given number of edges. Letting N n denote the number of contours with n edges (rather than sinks), it is not hard to show that N n grows at most exponentially as a function of n. To complete the argument, therefore, it suffices to impose additional conditions on the contours that bound the number of edges in terms of the number of sinks. If at a certain space-time point (i, t), the stationary process satisfies X t (i) = 0, and the map Φ (i,t) that is applied there is ϕ k , then for each set A ∈ A(ϕ k ) (with A(ϕ k ) defined in (1.8)), at least one of the sites j ∈ A must have the property that X t−1 (j) = 0. We will use this to steer edges in a certain direction, in such a way that different charges tend to move away from each other, except for edges that originate in a source. Since in the end, edges of all charges must convene in each sink, this will allow us to bound the total number of edges in terms of the "bad" edges that originate in a source. Equivalently, this allows us to bound the total number of edges in terms of the number of sources, which is the same as the number of sinks. This is the main idea of the argument. We now continue to give the precise definitions. The contour argument Having defined the right sort of contours, we now come to the core of the argument: the fact that X 0 (0) = 0 implies the existence of a Toom contour with certain properties. We first need a special construction of the stationary process (X t ) t∈Z . We let {0, 1} Z d+1 denote the space of all space-time configurations x = (x t (i)) (i,t)∈Z d+1 . For x ∈ {0, 1} Z d+1 and t ∈ Z, we define x t ∈ {0, 1} Z d by x t := (x t (i)) i∈Z d . We will call a collection φ = (φ (i,t) ) (i,t)∈Z d+1 of monotonic maps from {0, 1} Z d to {0, 1} a monotonic flow. By definition, a trajectory of φ is a space-time configuration x such that x t (i) = φ (i,t) (θ i x t−1 ) (i, t) ∈ Z d+1 . (2.8) We need the following two simple lemmas. Lemma 4 (Minimal and maximal trajectories) Let φ be a monotonic flow. Then there exist trajectories x and x that are uniquely characterised by the property that each trajectory x of φ satisfies x ≤ x ≤ x (pointwise). Lemma 5 (The lower and upper invariant laws) Let ϕ 0 , . . . , ϕ m be monotonic functions, let p 0 , . . . , p m be a probability distribution, and let ν and ν denote the lower and upper invariant laws of the corresponding monotone random cellular automaton. Let Φ = Φ (i,t) (i,t)∈Z d+1 be an i.i.d. collection of monotonic maps such that P[Φ (i,t) = ϕ k ] = p k (0 ≤ k ≤ m) , and let X and X be the minimal and maximal trajectories of Φ. Then for each t ∈ Z, the random variables X t and X t are distributed according to the laws ν and ν, respectively. From now on, we fix a monotonic flow φ that takes values in {ϕ 0 , . . . , ϕ m }, of which ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are nonconstant. Recall that A(ϕ k ), defined in (1.8), corresponds to the set of minimal configurations on which ϕ k gives the outcome 1. We fix an integer σ ≥ 2 and for each 1 ≤ k ≤ m and 1 ≤ s ≤ σ, we choose a set A s (ϕ k ) ∈ A(ϕ k ). (2.9) Informally, the aim of these sets is to steer edges of different charges away from each other. In later subsections, when we derive bounds for concrete models, we will make an explicit choice for σ and sets A s (ϕ k ). For the moment, we allow these to be arbitrary. The integer σ corresponds to the number of charges. The definition of what it means for a contour to be present will depend on the choice of the sets in (2.9). As a concrete example, consider the case m = 1 and ϕ 1 = ϕ coop , the cooperative branching map defined in (1.7). The set A(ϕ coop ) from (1.8) is given by A(ϕ coop ) = {A 1 , A 2 } with A 1 := {(0, 0)} and A 2 := {(0, 1), (1, 0)}. Using (1.9) we see that ϕ coop is an eroder. In this concrete example, we will set σ := 2 and for the sets A s (ϕ 1 ) (s = 1, 2) of (2.9) we choose the sets A 1 , A 2 we have just defined. Definition 6 A Toom contour (V, E, v • , ψ) with σ charges is present in the monotonic flow φ if: (i) φ ψ(v) = ϕ 0 for all v ∈ V * , (ii) φ ψ(v) ∈ {ϕ 1 , . . . , ϕ m } for all v ∈ V \V * , (iii) ψ(w) − ψ(v) ∈ A s (φ ψ(v) ) for all (v, w) ∈ E * s (1 ≤ s ≤ σ), (iv) ψ(w) − ψ(v) ∈ σ s=1 A s (φ ψ(v) ) for all (v, w) ∈ E • , where E * s and E • are defined in (2.5) and ψ(v), defined in (2.3), denotes the spatial coordinates of the space-time point ψ(v). Note that the definition of what it means for a contour to be present depends on the choice of the sets A s (ϕ k ) in (2.9). Conditions (i) and (ii) say that sinks of (V, E) are mapped to defective space-time points, where the constant map ϕ 0 is applied, and all other vertices are mapped to space-time points where one of the non-constant maps ϕ 1 , . . . , ϕ m is applied. Together with our earlier definition of an embedding, condition (iii) says that if (v, w) is an edge with charge s that comes out of the root or an internal vertex, then (v, w) is mapped to a pair of space-time points of the form (i, t), (i + j, t − 1) with j ∈ A s (φ ψ(v) ). Condition (iv) is similar, except that if v is a source different from the root, then we only require that j ∈ σ s=1 A s (φ ψ(v) ). It is clear from this definition that if (V, E, v • , ψ) and (V , E , v • , ψ ) are equivalent Toom contours, then (V, E, v • , ψ) is present in φ if and only if the same is true for (V , E , v • , ψ ). For our example of the monotone cellular automaton with ϕ 1 = ϕ coop , Definition 6 is demonstrated in Figure 3. Directed edges of charge 1 and 2 are indicated in red and blue, respectively. Because of our choice A 2 (ϕ 1 ) := {(0, 1), (1, 0)}, blue edges that start at internal vertices or the root point in directions where one of the spatial coordinates increases by one. Likewise, since A 1 (ϕ 1 ) := {(0, 0)}, red edges that start at internal vertices or the root point straight up, i.e., in the direction of decreasing time. Sinks of the Toom contour correspond to defective sites, as indicated in Figure 3 on the right. In view of Lemma 5, the following crucial theorem links the upper invariant law to Toom contours. Theorem 7 (Presence of a Toom contour) Let φ be a monotonic flow on {0, 1} Z d that take values in {ϕ 0 , . . . , ϕ m }, where ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are non-constant. Let x denote the maximal trajectory of φ. Let σ ≥ 2 be an integer and for each 1 ≤ s ≤ σ and 1 ≤ k ≤ m, let A s (ϕ k ) ∈ A(ϕ k ) be fixed. If x 0 (0) = 0, then, with respect to the given choice of σ and the sets A s (ϕ k ), a Toom contour (V, E, v • , ψ) rooted at (0, 0) is present in φ. We note that the converse of Theorem 7 does not hold, i.e., the presence in φ of a Toom contour (V, E, v • , ψ) that is rooted at (0, 0) does not imply that X 0 (0) = 0. This can be seen from Figure 3. In this example, if there would be no other defective sites apart from the sinks of the Toom contour, then the origin would have the value one. This is a difference with the Peierls arguments used in percolation theory, where the presence of a contour is a necessary and sufficient condition for the absence of percolation. Let T 0 denote the set of Toom contours rooted at (0, 0) (up to equivalence). We formally denote a Toom contour by T = (V, E, v • , ψ). Let Φ = (Φ (i,t) ) (i,t)∈Z d+1 be an i.i.d. collection of monotonic maps taking values in {ϕ 0 , . . . , ϕ m }. Then Theorem 7 implies the Peierls bound: 1 − ρ = P[X 0 (0) = 0] ≤ T ∈T 0 P T is present in Φ . (2.10) In Section 2.2 below, we will show how (2.10) can be used to prove the most difficult part of Toom's stability theorem, namely, that the upper invariant law of eroders is stable under small random perturbations. Toom contours with two charges Although Theorem 7 is sufficient to prove stability of eroders, when deriving explicit bounds, it is often useful to have stronger versions of Theorem 7 at one's disposal that establish the presence of Toom contours with certain additional properties that restrict the sum on the right-hand side in (2.10) and hence lead to improved bounds. Here we formulate one such result that holds specifically for Toom contours with two charges. As before, we let φ be a monotonic flow taking values in {ϕ 0 , . . . , ϕ m }, of which ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are non-constant. We set σ = 2 and choose sets A s (ϕ k ) ∈ A(ϕ k ) (1 ≤ k ≤ m, 1 ≤ s ≤ 2) as in (2.9). Definition 8 A Toom contour (V, E, v • , ψ) with 2 charges is strongly present in the monotonic flow φ if in addition to conditions (i)-(iv) of Definition 6, for each v ∈ V • \{v • } and w 1 , w 2 ∈ V with (v, w s ) ∈ E s,out (v) (s = 1, 2), one has: (v) ψ(w 1 ) − ψ(v) ∈ A 2 (φ ψ(v) ) and ψ(w 2 ) − ψ(v) ∈ A 1 (φ ψ(v) ), (vi) ψ(w 1 ) = ψ(w 2 ). Condition (v) can informally be described by saying that charged edges pointing out of any source other than the root must always point in the "wrong" direction, compared to charged edges pointing out of an internal vertex or the root. Note that for the Toom contour in Figure 3, this is indeed the case. With this definition, we can strengthen Theorem 7 as follows. Theorem 9 (Strong presence of a Toom contour) If σ = 2, then the Toom contour (V, E, v • , ψ) from Theorem 7 can be chosen such that it is strongly present in φ. Our proof of Theorem 9 follows quite a different strategy from the proof of Theorem 7. We do not know to what extent Theorem 9 can be generalised to Toom contours with three or more charges. In the following subsections, we will show how the results of the present subsection can be applied in concrete situations. In Subsection 2.2, we show how Theorem 7 can be used to prove stability of eroders, which is the difficult implication in Toom's stability theorem. In Subsection 2.3, building on the results of Subsection 2.2, we show how for Toom contours with two charges, the bounds can be improved by applying Theorem 9 instead of Theorem 7. In Subsection 2.4, we derive explicit bounds for two concrete eroders. In Subsection 2.5, we leave the setting of Toom's stability theorem and discuss monotone random cellular automata whose definition involves more than one non-constant monotonic map. In Subsection 6.2 we derive bounds for monotone interacting particle systems in continuous time. Stability of eroders In this subsection, we restrict ourselves to the special set-up of Toom's stability theorem. We fix a non-constant monotonic map ϕ that is an eroder and let Φ p = (Φ p (i,t) ) (i,t)∈Z d be an i.i.d. collection of monotonic maps that assume the values ϕ 0 and ϕ with probabilities p and 1 − p, respectively. We let (X p t ) t∈Z denote the maximal trajectory of Φ p and let ρ(p) := P[X p 0 (0) = 1] denote the intensity of the upper invariant law. We will show how the Peierls bound (2.10) can be used to prove that ρ(p) → 1 as p → 0, which is the most difficult part of Toom's stability theorem. To do this, first we will need another equivalent formulation of the eroder property (1.9). By definition, a polar function is a linear function R d z → L(z) = (L 1 (z), . . . , L σ (z)) ∈ R σ such that σ s=1 L s (z) = 0 (z ∈ R d ). (2.11) We call σ ≥ 2 the dimension of L. The following lemma is adapted from [Pon13, Lemma 12], with the basic idea going back to [Too80]. Recall the definition of A(ϕ) in (1.8). Lemma 10 (Erosion criterion) A non-constant monotonic function ϕ : {0, 1} Z d → {0, 1} is an eroder if and only if there exists a polar function L of dimension σ ≥ 2 such that σ s=1 sup A∈A(ϕ) inf i∈A L s (i) > 0. (2.12) If ϕ is an eroder, then L can moreover be chosen so that its dimension σ is at most d + 1. To understand why the condition (2.12) implies that ϕ is an eroder, for 1 ≤ s ≤ σ, let δ s := sup A∈A(ϕ) inf i∈A L s (i) and r s (x) := sup L s (i) : i ∈ Z d , x(i) = 0 x ∈ {0, 1} Z d , (2.13) with r s (1) := −∞, and let (X 0 k ) k≥0 denote the deterministic cellular automaton that applies the map ϕ in each space-time point, started in an arbitrary initial state. In the proof of Lemma 33 below, we will show that r s (X 0 n ) ≤ r s (X 0 0 ) − δ s n (n ≥ 0). (2.14) This says that δ s has the interpretation of an edge speed in the direction defined by the linear function L s . If x is a configuration containing finitely many zeros, then we define the extent of x by ext(x) := σ s=1 r s (x). (2.15) Then ext(1) = −∞, while on the other hand, by the defining property (2.11) of a polar function, ext(x) ≥ 0 for each x that contains at least one zero. Now (2.14) implies that if X 0 0 contains finitely many zeros, then ext(X 0 n ) ≤ ext(X 0 0 ) − nδ with δ := σ s=1 δ s . (2.16) It follows that X 0 n = 1 for all n such that ext(X 0 0 ) − nδ < 0. Since δ > 0 by (2.12), we conclude that ϕ is an eroder. We use Lemma 10 and the polar functions to choose the number of charges σ and to make a choice for the sets A s (ϕ) ∈ A(ϕ) (1 ≤ s ≤ σ) as in (2.9) when defining Toom contours. For a given choice of a polar function L and sets A s (ϕ), let us set B(ϕ) := σ s=1 A s (ϕ), (2.17) and define ε := σ s=1 ε s with ε s := inf i∈As(ϕ) L s (i) (1 ≤ s ≤ σ), R := σ s=1 R s with R s := − inf i∈B(ϕ) L s (i) (1 ≤ s ≤ σ). (2.18) Then Lemma 10 tells us that since ϕ is an eroder, we can choose the polar function L and sets A s (ϕ) in such a way that ε > 0, which we assume from now on. Recall that in the example where ϕ = ϕ coop , we earlier made the choices σ := 2, A 1 (ϕ) := {(0, 0)}, and A 2 (ϕ) := {(0, 1), (1, 0)}. We will now also choose a polar function by setting L 1 (z 1 , z 2 ) := −z 1 − z 2 and L 2 := −L 1 (z 1 , z 2 ) ∈ R 2 , (2.19) One can check that for this choice of L the constants from (2.18) are given by ε = 1 and R = 1. (2.20) Returning to the setting where ϕ is a general eroder, we let T 0 denote the set of Toom contours rooted at (0, 0) (up to equivalence). Since we apply only one non-constant monotonic map, conditions (iii) and (iv) of Definition 6 of what it means for a contour to be present in Φ p do not involve any randomness, i.e., these conditions now simplify to the deterministic conditions: (iii)' ψ(w) − ψ(v) ∈ A s (ϕ) for all (v, w) ∈ E * s (1 ≤ s ≤ σ), (iv)' ψ(w) − ψ(v) ∈ B(ϕ) for all (v, w) ∈ E • . Definition 11 We let T 0 denote the set of Toom contours rooted at (0, 0) (up to equivalence) that satisfy conditions (iii)' and (iv)'. For each T = (V, E, v • , ψ) ∈ T 0 , let n * (T ) := \|V • \| = \|V * \| and n e (T ) := \| E 1 \| = · · · = \| E σ \| (2.21) denote its number of sinks and sources, each, and its number of directed edges of each charge. As already explained informally, the central idea of Toom contours is that differently charged edges move away from each other except for edges starting at a source, which allows us to bound the number n e (T ) of edges in terms of the number n * (T ) of sources (or equivalently sinks). We now make this informal idea precise. It is at this point of the argument that the eroder property is used in the form of Lemma 10 which allowed us to choose the sets A s (ϕ) and the polar function L such that the constant ε from (2.18) is positive. We also need the following simple lemma. 3 Lemma 12 (Zero sum property) Let (V, E) be a Toom graph with σ charges, let ψ : V → Z d+1 be an embedding of (V, E), and let L : R d → R σ be a polar function with dimension σ. Then σ s=1 (v,w)∈ Es L s ( ψ(w)) − L s ( ψ(v)) = 0. (2.22) Proof We can rewrite the sum in (2.22) as v∈V σ s=1 (u,v)∈ E s,in (v) L s ( ψ(v)) − σ s=1 (v,w)∈ Es,out(v) L s ( ψ(v)) . (2.23) At internal vertices, the term inside the brackets is zero because the number of incoming edges of each charge equals the number of outgoing edges of that charge. At the sources and sinks, the term inside the brackets is zero by the defining property (2.11) of a polar function, since there is precisely one outgoing (resp. incoming) edge of each charge. As a consequence of Lemma 12, we can estimate n e (T ) from above in terms of n * (T ). Lemma 13 (Upper bound on the number of edges) Let ε and R be defined in (2.18). Then each T ∈ T 0 satisfies n e (T ) ≤ (1 + R/ε) n * (T ) − 1 . Proof Since \| E • s \| = n * (T ) − 1 and \| E * s \| = n e (T ) − n * (T ) + 1 (1 ≤ s ≤ σ) , Lemma 12 and rules (iii)' and (iv)' imply that 0 = σ s=1 (v,w)∈ E * s L s ( ψ(w)) − L s ( ψ(v)) + (v,w)∈ E • s L s ( ψ(w)) − L s ( ψ(v)) ≥ σ s=1 n e (T ) − n * (T ) + 1 ε s − n * (T ) − 1 R s = εn e (T ) − (ε + R) n * (T ) − 1 , (2.24) where we have used that L s ( ψ(w)) − L s ( ψ(v)) = L s ψ(w) − ψ(v) by the linearity of L s . By condition (ii) of Definition 2 of an embedding, sinks of a Toom contour do not overlap. By condition (i) of Definition 6 of what it means for a Toom contour to be present, each sink corresponds to a space-time point (i, t) that is defective, meaning that Φ (i,t) = ϕ 0 , which happens with probability p, independently for all space-time points. By Lemma 13, we can then estimate the right-hand side of (2.10) from above by T ∈T 0 P T is present in Φ ≤ T ∈T 0 p n * (T ) = p T ∈T 0 p n * (T )−1 ≤ p T ∈T 0 p ne(T )/(1+R/ε) = p ∞ n=0 N n p n/(1+R/ε) , (2.25) where N n := {T ∈ T 0 : n e (T ) = n} (n ≥ 0) (2.26) denotes the number of (nonequivalent) contours in T 0 that have n edges of each charge. The following lemma gives a rough upper bound on N n . Recall the definition of B(ϕ) in (2.17). Lemma 14 (Exponential bound) Let M := B(ϕ) and let τ := 1 2 σ denote 1 2 σ rounded up to the next integer. Then N n ≤ n τ −1 (τ + 1) 2τ n M σn (n ≥ 1). (2.27) Combining (2.25) and Lemma 14, we see that the right-hand side of (2.10) is finite for p sufficiently small and hence (by dominated convergence) tends to zero as p → 0. This proves that ρ(p) → 1 as p → 0, which is the most difficult part of Toom's stability theorem. Contours with two charges For Toom contours with two charges, the bounds derived in the previous subsection can be improved by using Theorem 9 instead of Theorem 7. To make this precise, for Toom contours with two charges, we define a subset T 0 of the set of contours T 0 from Definition 11 as follows: Definition 15 For Toom contours with σ = 2 charges, we let T 0 denote the set of Toom contours rooted at (0, 0) (up to equivalence) that satisfy: (iii)' ψ(w) − ψ(v) ∈ A s (ϕ) for all (v, w) ∈ E * s (1 ≤ s ≤ 2), (iv)" ψ(w) − ψ(v) ∈ A 3−s (ϕ) for all (v, w) ∈ E • s (1 ≤ s ≤ 2), (v)" ψ(w 1 ) = ψ(w 2 ) for all v ∈ V • \{v • }, w 1 ∈ E 1,out , and w 2 ∈ E 2,out . Note that condition (iii)' above is the same condition as (iii)' of Definition 11. Condition (iv)" strengthens condition (iv)' of Definition 11. Conditions (iv)" and (v)" correspond to conditions (v) and (vi) of Definition 8, which in our present set-up do not involve any randomness. We will need analogues of Lemmas 13 and 14 with T 0 replaced by T 0 . We define R := σ s=1 R s with R 1 := − inf i∈A 2 (ϕ) L 1 (i) and R 2 := − inf i∈A 1 (ϕ) L 2 (i). (2.28) The following lemma is similar to Lemma 13. Lemma 16 (Upper bound on the number of edges for σ = 2) Let ε and R be defined in (2.18) and (2.28). Then each T ∈ T 0 satisfies n e (T ) ≤ (1 + R /ε) n * (T ) − 1 . Proof The proof is the same as that of Lemma 13, with the only difference that condition (iv)" of Definition 15 allows us to use R s instead of R s (s = 1, 2) as upper bounds. Similarly to (2.26), we let N n := {T ∈ T 0 : n e (T ) = n} (n ≥ 0) (2.29) denote the number of (nonequivalent) contours in T 0 that have n edges of each charge. Then Theorem 9 implies the Peierls bound: 1 − ρ(p) ≤ T ∈T 0 P T is strongly present in Φ ≤ T ∈T 0 p n * (T ) ≤ p ∞ n=0 N n p n/(1+R /ε) . (2.30) The following lemma is similar to Lemma 14. Lemma 17 (Exponential bound for σ = 2) Let M s := A s (ϕ) (s = 1, 2). Then N n ≤ 1 2 (4M 1 M 2 ) n (n ≥ 1). (2.31) Some explicit bounds We continue to work in the set-up of the previous subsections, i.e., we consider monotone random cellular automata that apply the maps ϕ 0 and ϕ with probabilities p and 1 − p, respectively, where ϕ is an eroder. An easy coupling argument shows that the intensity ρ(p) of the upper invariant law is a nonincreasing function of p, so there exists a unique p c ∈ [0, 1] such that ρ(p) > 0 for p < p c and ρ(p) = 0 for p > p c . Since ϕ is an eroder, Toom's stability theorem tells us that p c > 0. In this subsection, we derive explicit lower bounds on p c for two concrete choices of the eroder ϕ. If one wants to use (2.10) to show that ρ > 0, then one must show that the right-hand side of (2.10) is less than one. In practice, when deriving explicit bounds, it is often easier to show that a certain sum is finite than showing that it is less than one. We will prove a generalisation of Theorems 7 and 9 that can in many cases be used to show that if a certain sum is finite, then ρ > 0. In the set-up of Theorem 7, we choose j s ∈ A s (ϕ 1 ) (1 ≤ s ≤ σ). We fix an integer r ≥ 0 and we let φ (r) denote the modified monotonic flow defined by φ (r) (i,t) := ϕ 1 if − r < t ≤ 0, φ (i,t) otherwise. (2.32) Below, we let x (r) denote the maximal trajectory of the modified monotonic flow φ (r) . As before, we let Conv(A) denote the convex hull of a set A. Proposition 18 (Presence of a large contour) In the set-up of Theorem 7, on the event that x (r) −r (i) = 0 for all i ∈ Conv({rj 1 , . . . , rj σ }), there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) present in φ (r) such that ψ d+1 (v) ≤ −r for all v ∈ V * and ψ d+1 (v) ≤ 1 − r for all v ∈ V • \{v • }. If σ = 2, then such a Toom contour is strongly present in φ (r) . As a simple consequence of this proposition, we obtain the following lemma. Lemma 19 (Finiteness of the Peierls sum) If T ∈T 0 p n * (T ) < ∞, then ρ(p) > 0. If σ = 2, then similarly T ∈T 0 p n * (T ) < ∞ implies ρ(p) > 0. We prove Proposition 18 and Lemma 19 in Section 4.3. Cooperative branching Generalizing the definition in (1.7), for each dimension d ≥ 1, we define a monotonic map ϕ coop,d : {0, 1} Z d → {0, 1} by ϕ coop,d (x) := x(0) ∨ x(e 1 ) ∧ · · · ∧ x(e d ) , (2.33) where 0 is the origin and e i denotes the ith unit vector in Z d . In particular, in dimension d = 2, this is the cooperative branching rule ϕ coop defined in (1.7). We chose σ := 2, A 1 (ϕ) := {0}, and A 2 (ϕ 1 ) := {e 1 , . . . , e d }, and as our polar function L we chose L 1 (z 1 , . . . , z d ) := − d i=1 z i and L 2 (z 1 , . . . , z d ) := d i=1 z i , (2.34) which has the result that the constants from (2.18) and (2.28) are given by ε = 1, R = 1 and R = 1. Arguing as in (2.25), using Lemmas 13 and 14 with M = d + 1, σ = 2 and τ = 1, we obtain the Peierls bound: T ∈T 0 P T is present in Φ ≤ T ∈T 0 p n * (T ) ≤ p ∞ n=0 2 2n (d + 1) 2n p n/2 . (2.35) This is finite when 4(d + 1) 2 p 1/2 < 1, so using Lemma 19 we obtain the bound p c (d) ≥ 16 −1 (d + 1) −4 . This bound can be improved by using Theorem 9 and its consequences. Applying Lemmas 16 and 17 with M 1 = d, M 2 = 1, we obtain the Peierls bound: T ∈T 0 P T is strongly present in Φ ≤ T ∈T 0 p n * (T ) ≤ p 2 ∞ n=0 4 n d n p n/2 . (2.36) This is finite when 4dp 1/2 < 1, so using Lemma 19 we obtain the bound p c (d) ≥ 1 16d 2 . (2.37) In particular, in two dimensions this yields p c (2) ≥ 1/64. This is still some way off the estimated value p c (2) ≈ 0.105 coming from numerical simulations but considerably better than the bound obtained from Lemmas 13 and 14. Toom's model We take for ϕ the map ϕ NEC . Then the set A(ϕ) from (1.8) is given by Using (1.9) we see that ϕ NEC is an eroder. We set σ := 3 and for the sets A s (ϕ NEC ) s = 1, 2, 3 of (2.9) we choose the sets A 1 , A 2 , A 3 we have just defined. We define a polar function L with dimension σ = 3 by A(ϕ) = {A 1 , A 2 , A 3 } with A 1 := {(0,L 1 (z 1 , z 2 ) := −z 1 , L 2 (z 1 , z 2 ) := −z 2 , L 3 (z 1 , z 2 ) := z 1 + z 2 , (2.38) (z 1 , z 2 ) ∈ R 2 . One can check that for this choice of L and the sets A s (ϕ NEC ) (1 ≤ s ≤ 3), the constants from (2.18) are given by ε = 1 and R = 2. (2.39) Using Lemma 14 with M = 3, σ = 3, and τ = 2, we can estimate the Peierls sum in (2.25) from above by p ∞ n=0 n3 4n 3 3n p n/3 . (2.40) This is finite when 3 7 p 1/3 < 1, so using Lemma 19 we obtain the bound p c ≥ 3 −21 , (2.41) which does not compare well to the estimated value p c ≈ 0.053 coming from numerical simulations. Nevertheless, this is probably the best rigorous bound currently available. Cellular automata with intrinsic randomness In this subsection we will be interested in monotone random cellular automata whose definition involves more than one non-constant monotonic map. We fix a dimension d ≥ 1, a collection ϕ 1 , . . . , ϕ m of non-constant monotonic maps ϕ k : {0, 1} Z d → {0, 1} , and a probability distribution p 1 , . . . , p m . Let (X k ) k≥0 denote the monotone random cellular automaton that applies the maps ϕ 1 , . . . , ϕ m with probabilities p 1 , . . . , p m and let ϕ 0 := ϕ 0 be the constant map that always gives the outcome zero. By definition, an δ-perturbation of (X k ) k≥0 is a monotone random cellular automaton (X k ) k≥0 that applies the maps ϕ 0 , . . . , ϕ m with probabilities p 0 , . . . , p m that satisfy p 0 ≤ δ and p k ≤ p k for all k = 1, . . . , m. We say that (X k ) k≥0 is stable if for each ε > 0, there exists a δ > 0 such that the density ρ of the upper invariant law of any δ-perturbation of (X k ) k≥0 satisfies ρ ≥ 1 − ε. Note that in the special case that m = 1, which corresponds to the set-up of Toom's stability theorem, these definitions coincide with our earlier definition. For deterministic monotone cellular automata, which in our set-up corresponds to the case m = 1, we have seen in Lemma 10 and formula (2.14) that the eroder property can equivalently be formulated in terms of edge speeds. For a random monotone cellular automaton (X k ) k≥0 , the intuition is similar, but it is not entirely clear how to define edges speeds in the random setting and it can be more difficult to determine whether (X k ) k≥0 is an eroder. Fix a polar function L of dimension σ ≥ 2 and let ε k s := sup A∈A(ϕ k ) inf i∈A L s (i) (1 ≤ k ≤ m, 1 ≤ s ≤ σ) (2.42) denote the edge speed in the direction defined by the linear function L s of the deterministic automaton that only applies the map ϕ k . If σ s=1 ε s > 0 with ε s := inf 1≤k≤m ε k s , (2.43) then (2.14) remains valid almost surely. In such a situation, it is not very hard to adapt the arguments of Section 2.2 to see that (X k ) k≥0 is stable. The condition (2.43) is, however, very restrictive and excludes many interesting cases. In particular, it excludes the case when one of the maps ϕ 1 , . . . , ϕ m is the identity map ϕ id , which, as explained below (1.6) is relevant in view of treating continuous-time interacting particle systems. Indeed, observe that, if ϕ k = ϕ id , then ε k s = 0 for each polar function L of dimension σ and each 1 ≤ s ≤ σ, implying σ s=1 ε s ≤ 0. The following example, which is an adaptation of [Gra99, Example 18.3.5], shows that in such situations it can be much more subtle whether a random monotone cellular automaton is stable. Fix an integer n ≥ 1 and let ϕ 1 : {0, 1} Z 2 → {0, 1} be the monotonic map defined as in (1.8) by the set of minimal configurations A(ϕ 1 ) := {(−1, 0), (0, 0)}, {(−2, 0), (0, 0)}, {(m, k) : −3 ≤ m ≤ −2, \|k\| ≤ n} . (2.44) Using (1.9), it is straightforward to check that ϕ 1 is an eroder. Now consider the random monotone cellular automaton (X k ) k≥0 that applies the maps ϕ 1 and ϕ id with probabilities p and 1 − p, respectively, for some 0 ≤ p ≤ 1. We claim that if p < 1, then for n sufficiently large, (X k ) k≥0 is not stable. To see this, fix l ≥ 2 and consider an initial state such that X 0 (i) = 0 for i ∈ {0, . . . , l} × {0, . . . , n} and X 0 (i) = 1 otherwise. Set α k := inf 0≤i 2 ≤n inf{i 1 : X k (i 1 , i 2 ) = 0} and β j k := sup{i 1 : X k (i 1 , j) = 0} (0 ≤ j ≤ n). (2.45) As long as at each height 0 ≤ j ≤ n, there are at least two sites of type 0, the right edge processes (β j k ) k≥0 with 0 ≤ j ≤ n behave as independent random walks that make one step to the right with probability p. Therefore, the right edge of the zeros moves with speed p to the right. In each time step, all sites in {α k , α k + 1} × {0, . . . , n} that are of type 0 switch to type 1 with probability p. When p = 1, the effect of this is that the left edge of the zeros moves with speed two to the right and eventually catches up with the right edge, which explains why ϕ 1 is an eroder. However, when p < 1, the left edge can move to the right only once all sites in {α k } × {0, . . . , n} have switched to type 1. For n large enough, this slows down the speed of the left edge with the result that in (X k ) k≥0 the initial set of zeros will never disappear. It is not difficult to prove that this implies that (X k ) k≥0 is not stable. To see a second example that demonstrates the complications that can arise when we replace deterministic monotone cellular automata by random ones, recall the maps ϕ NEC , ϕ NWC , ϕ SWC , and ϕ SEC defined in and below (1.7). For the map ϕ NEC , the edge speeds in the directions defined by the linear functions L 1 and L 2 from (2.38) are zero but the edge speed corresponding to L 3 is not, which we used in Subsection 2.4 to prove that the deterministic monotone cellular automaton that always applies the map ϕ NEC is stable. By contrast, for the cellular automaton that applies the maps ϕ NEC , ϕ NWC , ϕ SWC , and ϕ SEC with equal probabilities, by symmetry in space and since these maps treat the types 0 and 1 symmetrically, the edge speed in each direction is zero. As a result, we conjecture that, although each map applied by this random monotone cellular automaton is an eroder, it is not stable. In spite of these complications, Toom contours can sometimes be used to prove stability of random monotone cellular automata, even in situations where the simplifying assumption (2.43) does not hold. In these cases we cannot rely on the use of polar functions, instead we have to carefully examine the structure of the contour to be able to bound the number of contours in terms of the number of defective sites. Furthermore, one can generally take σ := m k=1 \|A(ϕ k )\|. We will demonstrate this on a cellular automaton that combines the cooperative branching map defined in (2.33) with the identity map. Cooperative branching with identity map We consider the monotone random cellular automaton on Z d that applies the maps ϕ 0 , ϕ id , and ϕ coop,d with probabilities p, q, r, respectively with q = 1 − p − r. For each p, r ≥ 0 such that p + r ≤ 1, let ρ(p, r) denote the intensity of the upper invariant law of the process with parameters p, 1 − p − r, r. A simple coupling argument shows that for fixed 0 ≤ r < 1, the function p → ρ(p, r) is nonincreasing on [0, 1 − r], so for each 0 ≤ r < 1, there exists a p c (r) ∈ [0, 1 − r] such that ρ(p, r) > 0 for 0 ≤ p < p c (r) and ρ(p, r) = 0 for p c (r) < p ≤ 1 − r. We will derive a lower bound on p c (r). Recall that setting p := ε and r := λε, rescaling time by a factor ε, and sending ε → 0 corresponds to taking the continuous-time limit, where in the limiting interacting particle system the maps ϕ 0 and ϕ coop,d are applied with rates 1 and λ, respectively. For this reason, we are especially interested in the asymptotics of p c (r) when r is small. In line with notation introduced in Subsection 2.4, we define A 1 := {0} and A 2 := {e 1 , . . . , e d }. We have A(ϕ id ) = A 1 and A(ϕ coop,d ) = A 1 , A 2 , (2.46) thus we set σ := \|A(ϕ id )\| ∨ \|A(ϕ coop,d )\| = 2, and for the sets A s (ϕ k ) in (2.9) we make the choices A 1 (ϕ id ) := A 1 , A 2 (ϕ id ) := A 1 , A 1 (ϕ coop,d ) := A 1 , A 2 (ϕ coop,d ) := A 2 . (2.47) Let Φ = (Φ (i,t) ) (i,t)∈Z 3 be an i.i.d. collection of monotonic maps so that P[Φ (i,t) = ϕ 0 ] = p, P[Φ (i,t) = ϕ id ] = q, and P[Φ (i,t) = ϕ coop,d ] = r. We let T 0 denote the set of Toom contours (V, E, 0, ψ) rooted at the origin with respect to the given choice of σ and the sets A s (ϕ k ) in (2.47). Theorem 7 then implies the Peierls bound 1 − ρ ≤ T ∈T 0 P T is strongly present in Φ . (2.48) In Section 5, we give an upper bound on this expression by carefully examining the structure of Toom contours for this model. We will prove the following lower bound on p c (r) for each r ∈ [0, 1): p c (r) > (d + 0.5) 2 + 1/(16d) − d − 0.5 r. In particular for d = 2 we obtain the bound p c (r) > 0.00624r. Continuous time In this subsection, we consider monotone interacting particle systems of the type described in (1.2). We briefly recall the set-up described there. We are given a finite collection ϕ 1 , . . . , ϕ m of non-constant monotonic maps ϕ k : {0, 1} Z d → {0, 1} and a collection of nonnegative rates r 1 , . . . , r m , and we are interested in interacting particle systems (X t ) t≥0 taking values in {0, 1} Z d that evolve in such a way that independently for each i ∈ Z d , X t (i) is replaced by ϕ k (θ i X t ) at the times of a Poisson process with rate r k (2.49) (1 ≤ k ≤ m). Without loss of generality we can assume that ϕ k = ϕ id for all 0 ≤ k ≤ m. For each r ≥ 0, let (X r t ) t≥0 denote the perturbed monotone interacting particle system that apart from the non-constant monotonic maps ϕ 1 , . . . , ϕ m , that are applied with rates r 1 , . . . , r m , also applies the constant monotonic map ϕ 0 := ϕ 0 with rate r 0 := r. We let ρ(r) denote the density of its upper invariant law. We say that the unperturbed interacting particle system (X t ) t≥0 is stable if ρ(r) → 1 as r → 0. Gray [Gra99,Theorem 18.3.1] has given (mutually non-exclusive) sufficient conditions on the edge speeds for a monotone interacting particle system to be either stable or unstable. Furthermore, [Gra99, Examples 18.3.5 and 6] he has shown that (X t ) t≥0 may fail to be stable even when m = 1 and the map ϕ 1 is an eroder in the sense of (1.9), and conversely, in such a situation, (X t ) t≥0 be stable even ϕ 1 is not an eroder. The reason for this is that we can think of interacting particle systems as continuous-time limits of cellular automata that apply the identity map ϕ id most of the time, and, as we have seen in the previous subsection, combining an eroder ϕ 1 with the identity map ϕ id can change the stability of a cellular automaton in subtle ways. However, for a certain type of interacting particle system called generalized contact process Gray's conditions on the edge speed turn out to be sufficient and necessary for the stability of (X t ) t≥0 . We now briefly describe this argument, as it is not present in [Gra99]. Recall that A(ϕ k ) defined in (1.8) denotes the set of minimal configurations on which ϕ k gives the outcome 1. We say that a monotone interacting particle system that applies the non-constant monotonic maps ϕ 1 , . . . , ϕ m is a generalized contact process, if {0} ∈ A(ϕ k ) for each 1 ≤ k ≤ m. The perturbed system (X r t ) t≥0 then can be seen as a model for the spread of epidemics: vertices represent individuals that can be healthy (state 0) or infected (state 1). Each healthy vertex can get infected, if a certain set of vertices in its neighbourhood is entirely infected, and each infected vertex can recover at rate r independently of the state of the other vertices. For a monotone interacting particle system that applies the non-constant monotonic maps ϕ 1 , . . . , ϕ m Gray defines the Toom operator ϕ( x) : {0, 1} Z d → {0, 1} as the map ϕ(x) := 1 − x(0) m k=1 ϕ k (x) + x(0) m k=1 ϕ k (x) x ∈ {0, 1} Z d . (2.50) That is, ϕ flips the state of the origin if at least one of the maps ϕ 1 , . . . , ϕ m would flip its state in configuration x. As each ϕ k is monotonic, it is easy to see that ϕ is monotonic as well. Recall from (2.18) that for each fixed polar function L of dimension σ we defined ε := σ s=1 ε s , ε s := inf i∈As(ϕ) L s (i) (1 ≤ s ≤ σ). (2.51) For a Toom operator ϕ with {0} ∈ A(ϕ) we have ε s ≥ 0 for each s. In this case, Gray's condition for stability simplifies as follows. A monotone interacting particle system with Toom operator ϕ satisfying {0} ∈ A(ϕ) is stable if and only if there exists a polar function L for which ε > 0. It is easy to see, that finding such a polar function is equivalent to finding a set A ∈ A(ϕ) which is entirely contained in an open halfspace in Z d . As {0} ⊂ A(ϕ), this is further equivalent to A∈A(ϕ) Conv(A) = ∅, which is the eroder condition in (1.9). Let (X t ) t≥0 be a generalized contact process. As {0} ⊂ A(ϕ k ) for each 1 ≤ k ≤ m, we clearly have {0} ⊂ A(ϕ) for the corresponding Toom operator ϕ in (2.50). Thus in this case we can formulate Gray's theorem [Gra99, Theorem 18.3.1] as follows. The generalized contact process (X t ) t≥0 is stable if and only if the corresponding Toom operator ϕ is an eroder. While Gray's results can be used to show stability of certain models, his ideas do not lend themselves well to the derivation of explicit bounds. It is with this goal in mind that we have extended Toom's framework to continuous time. Toom contours in continuous time are defined similarly as in the discrete time setting and can be thought of as the limit of the latter. Since this is very simiar to what we have already seen in Subsection 2.1, we do not give the precise definitions in the continuous-time setting here but refer to Section 6 instead. We will demonstrate how Toom contours can be used to give bounds on the critical parameters of some monotone interacting particle systems. As mentioned in the previous subsection, in our methods we cannot rely on the use of polar functions. Again, one can generally take σ := m k=1 \|A(ϕ k )\|. Sexual contact process on Z d (d ≥ 1) We consider the interacting particle system on Z d that applies the monotonic maps ϕ 0 and ϕ coop,d defined in (1.5) and (2.33) with rates 1 and λ, respectively. We let ρ(λ) denote the intensity of the upper invariant law as a function of λ and we define the critical parameter as λ c := inf{λ ≥ 0 : ρ(λ) > 0}. In line with notation introduced in Subsection 2.4, we define A 1 := {0} and A 2 := {e 1 , . . . , e d }. We have A(ϕ coop,d ) = A 1 , A 2 , (2.52) thus we set σ := \|A(ϕ coop,d )\| = 2, and for the sets A s (ϕ k ) in (2.9) we make the choices A 1 (ϕ coop,d ) := A 1 , A 2 (ϕ coop,d ) := A 2 . (2.53) In Section 6 we will show that X t (i) = 0 implies the presence of a continuous Toom contour rooted at (i, t) with respect to the given choice of σ and sets A s (ϕ coop,d ), and use these contours to carry out a similar Peierls argument as in the discrete time case. In one dimension, this process is called the one-sided contact process, and our computation yields the bound λ c (1) ≤ 49.3242 . . . . (2.54) There are already better estimates in the literature: in [TIK97] the authors prove the bound λ c (1) ≤ 3.882 and give the numerical estimate λ c (1) ≈ 3.306. In two dimensions this is the sexual contact process defined in [Dur86], and we prove the bound λ c (2) ≤ 161.1985 . . . . (2.55) In [Dur86] Durrett claimed a proof that λ c (2) ≤ 110, while numerical simulations suggest the value λ c (2) ≈ 12.4. Toom contours Outline In this section, we develop the basic abstract theory of Toom contours. In particular, we prove all results stated in Subsection 2.1. In Subsection 3.1, we prove the preparatory Lemmas 4 and 5. Theorems 7 and 9 about the (strong) presence of Toom contours are proved in Subsections 3.4 and 3.5, respectively. In Section 3.6, we briefly discuss "forks" which played a prominent role in Toom's [Too80] original formulation of Toom contours and which can be used to prove a somewhat stronger version of Theorem 7. The maximal trajectory In this subsection we prove Lemmas 4 and 5. Proof of Lemma 4 By symmetry, it suffices to show that there exists a trajectory x that is uniquely characterised by the property that each trajectory x of φ satisfies x ≤ x. For each s ∈ Z, we inductively define a function x s : Z d × {s, s + 1, . . .} → {0, 1} by x s s (i) := 1 (i ∈ Z d ) and x s t (i) = φ (i,t) (θ i x s t−1 ) i ∈ Z d , s < t . (3.1) Then x s−1 s (i) ≤ 1 = x s s (i) and hence by induction x s−1 t (i) ≤ x s t (i) for all s ≤ t, which implies that the pointwise limit x t (i) := lim s→−∞ x s t (i) (i, t) ∈ Z d+1 (3.2) exists. It is easy to see that x is a trajectory. If x is any other trajectory, then x s (i) ≤ 1 = x s s (i) and hence by induction x t (i) ≤ x s t (i) for all s ≤ t, which implies that x ≤ x. Thus, x is the maximal trajectory, and such a trajectory is obviously unique. Proof of Lemma 5 By symmetry, it suffices to prove the claim for the upper invariant law. We recall that two probability measures ν 1 , ν 2 on {0, 1} Z d are stochastically ordered, which we denoted as ν 1 ≤ ν 2 , if and only if random variables X 1 , X 2 with laws ν 1 , ν 2 can be coupled such that X 1 ≤ X 2 . The law µ of X t clearly does not depend on t and hence is an invariant law. The proof of Lemma 4 shows that P 1 [X t ∈ · ] ⇒ µ as t → ∞ as claimed in (1.3). Alternatively, µ is uniquely characterised by the fact that it is maximal with respect to the stochastic order, i.e., if ν is an arbitrary invariant law, then ν ≤ µ. Indeed, if ν is an invariant law, then for each s ∈ Z, we can inductively define a stationary process (X s t ) t≥s by X s t (i) = φ (i,t) (θ i X s t−1 ) i ∈ Z d , s < t , (3.3) where X s s has the law ν and is independent of Φ. Since ν is an invariant law, the laws of the processes X s are consistent in the sense of Kolmogorov's extension theorem and therefore we can almost surely construct a trajectory X of Φ such that X t has the law ν and is independent of (Φ (i,s) ) i∈Z d , t<s for each t ∈ Z. By Lemma 4, X ≤ X a.s. and hence ν ≤ µ in the stochastic order. We conclude that as claimed, µ = ν, the upper invariant law. Explanation graphs In this subsection we start preparing for the proof of Theorem 7. We fix a monotonic flow φ on {0, 1} Z d that take values in {ϕ 0 , . . . , ϕ m }, where ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are non-constant. We also fix an integer σ ≥ 2 and for each 1 ≤ s ≤ σ and 1 ≤ k ≤ m, we fix A s (ϕ k ) ∈ A(ϕ k ). Letting x denote the maximal trajectory of φ, our aim is to prove that almost surely on the event that x 0 (0) = 0, there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) present in φ. As a first step towards this aim, in the present subsection, we will show that the event that x 0 (0) = 0 almost surely implies the presence of a simpler structure, which we will call an explanation graph. Recall from Subsection 2.1 that a directed graph with σ types of edges is a pair (U, H), where H = ( H 1 , . . . , H σ ) is a sequence of subsets of U × U . We interpret H s as the set of directed edges of type s. For such a directed graph with σ types of edges, we let H s,in (u) and H s,out (u) denote the set of vertices with type s that end and start in a vertex u ∈ U , respectively. We also use the notation H := σ s=1 H s . Then (U, H) is a directed graph in the usual sense of the word. The following two definitions introduce the concepts we will be interested in. Although they look a bit complicated at first sight, in the proof of Lemma 22 we will see that they arise naturally in the problem we are interested in. Further motivation for these definitions is provided in Section 7 below, where it is shown that explanation graphs naturally arise from an even more elementary concept, which we will call a minimal explanation. Definition 20 An explanation graph for (0, 0) is a directed graph with σ types of edges (U, H) with U ⊂ Z d+1 for which there exists a subset U * ⊂ U such that the following properties hold: (i) each element of H is of the form (j, t), (i, t − 1) for some i, j ∈ Z d and t ∈ Z, (ii) (0, 0) ∈ U ⊂ Z d+1 and t < 0 for all (i, t) ∈ U \{(0, 0)}, (iii) for each (i, t) ∈ U \{(0, 0)}, there exists a (j, t + 1) ∈ U such that (j, t + 1), (i, t) ∈ H, (iv) if u ∈ U * , then H s,out (u) = ∅ for all 1 ≤ s ≤ σ, (v) if u ∈ U \U * , then H s,out (u) = 1 for all 1 ≤ s ≤ σ. Note that U * is uniquely determined by (U, H). We call U * the set of sinks of the explanation graph (U, H). Definition 21 An explanation graph (U, H) is present in φ if: Proof By condition (i) of Definition 21, the presence of an explanation graph clearly implies x 0 (0) = 0. To prove the converse implication, let x r : Z d × {r, r + 1, . . .} → {0, 1} be defined as in (3.1). We have seen in the proof of Lemma 4 that x r t (i) decreases to x t (i) as r → −∞. Therefore, since x 0 (0) = 0, there must be an r < 0 such that x r 0 (0) = 0. We fix such an r from now on. (i) x t (i) = 0 for all (i, t) ∈ U , (ii) U * = u ∈ U : φ u = ϕ 0 , (iii) j − i ∈ A s (φ (i,t) ) for all (i, t), (j, t − 1) ∈ H s (1 ≤ s ≤ σ). We will inductively construct a finite explanation for (0, 0) with the desired properties. At each point in our construction, (U, H) will be a finite explanation for (0, 0) such that: (i) x r t (i) = 0 for all (i, t) ∈ U , (ii)' φ (i,t) = ϕ 0 for all (i, t) ∈ U \U * , (iii) j − i ∈ A s (φ (i,t) ) for all (i, t), (j, t − 1) ∈ H s (1 ≤ s ≤ σ). The induction stops as soon as: (ii) U * = u ∈ U : φ u = ϕ 0 . We start with U = {(0, 0)} and H s = ∅ for all 1 ≤ s ≤ σ. In each step of the construction, we select a vertex (i, t) ∈ U * such that φ (i,t) = ϕ 0 . Since x r t (i) = 0 and A s (φ (i,t) ) ∈ A(φ (i,t) ) as defined in (1.8), for each 1 ≤ s ≤ σ we can choose j s ∈ A s (φ (i,t) ) such that x r t−1 (j s ) = 0. We now replace U by U ∪ {(j s , t − 1) : 1 ≤ s ≤ σ} and we replace H s by H s ∪ { (i, t), (j s , t − 1))} (1 ≤ s ≤ σ), and the induction step is complete. At each step in our construction, r < t ≤ 0 for all (i, t) ∈ U , since at time r one has x r r (i) = 1 for all i ∈ Z d . Since U can contain at most σ −t elements with time coordinate t, we see that the inductive construction ends after a finite number of steps. It is straightforward to check that the resulting graph is an explanation graph in the sense of Definition 20. Toom matchings In this subsection, we continue our preparations for the proof of Theorem 7. Most of the proof of Theorem 7 will consist, informally speaking, of showing that to each explanation graph, it is possible to add a suitable set of sources, such that the sources and sinks together define a Toom contour. It follows from the definition of an explanation graph that for each w ∈ U and 1 ≤ s ≤ σ, there exist a unique n ≥ 0 and w 0 , . . . , w n such that (i) w 0 = w and (w i−1 , w i ) ∈ H s for all 0 < i ≤ n, (ii) w n ∈ U * and w i ∈ U \U * for all 0 ≤ i < n. In other words, this says that starting at each w ∈ U , there is a unique directed path that uses only directed edges from H s and that ends at some vertex w n ∈ U * . We will use the following notation: P s (w) := w 0 , . . . , w n , π s (w) := w n , (w ∈ U, 1 ≤ s ≤ σ). (3.4) Then P s (w) is the path we have just described and π s (w) ∈ U * is its endpoint. By definition, we will use the word polar to describe any sequence (a 1 , . . . , a σ ) such that a s ∈ U for all 1 ≤ s ≤ σ and the points a 1 = (i 1 , t), . . . , a σ = (i σ , t) all have the same time coordinate. We call t the time of the polar. Definition 23 A Toom matching for an explanation graph (U, H) with N := \|U * \| sinks is an N × σ matrix a i,s 1≤i≤N, 1≤s≤σ (3.5) such that (i) (a i,1 , . . . , a i,σ ) is a polar for each 1 ≤ i ≤ N , (ii) π s : {a 1,s , . . . , a N,s } → U * is a bijection for each 1 ≤ s ≤ σ. We will be interested in polars that have the additional property that all their elements lie "close together" in a certain sense. By definition, a point polar is a polar (a 1 , . . . , a σ ) such that a 1 = · · · = a σ . We say that a polar (a 1 , . . . , a σ ) is tight if it is either a point polar, or there exists a v ∈ U such that (v, a s ) ∈ H for all 1 ≤ s ≤ σ, where we recall that H := σ s=1 H s . The following proposition is the main result of this subsection. Proposition 24 (Toom matchings) Let (U, H) be an explanation graph for (0, 0) with N := \|U * \| sinks. Then there exists a Toom matching for (U, H) such that in addition to the properties (i) and (ii) above, (iii) a 1,1 = · · · = a 1,σ = (0, 0), (iv) (a i,1 , . . . , a i,σ ) is a tight polar for each 1 ≤ i ≤ N . In the next subsection, we will derive Theorem 7 from Proposition 24. It is instructive to jump a bit ahead and already explain the main idea of the construction. Let (a i,s ) 1≤i≤N, 1≤s≤σ be the Toom matching from Proposition 24. For each i and s, we connect the vertices of the path P s (a i,s ) defined in (3.4) with directed edges of type s. By property (ii) of a Toom matching, this has the consequence that each sink u ∈ U * of the explanation graph is the endvertex of precisely σ edges, one of each type. Each point polar gives rise to a source where σ charges emerge, one of each type, that then travel through the explanation graph until they arrive at a sink. For each polar (a i,1 , . . . , a i,σ ) that is not a point polar, we choose v i ∈ U such that (v i , a i,s ) ∈ H for all 1 ≤ s ≤ σ, and for each 1 ≤ s ≤ σ we connect v i and a i,s with a directed edge of type s. These extra points v i then act as additional sources and, as will be proved in detail in the next subsection, our collection of directed edges now forms a Toom graph that is embedded in Z d+1 , and the connected component of this Toom graph containing the origin forms a Toom contour that is present in φ. This is illustrated in Figure 3. The picture on the right shows an explanation graph (U, H), or rather the associated directed graph (U, H), with sinks indicated with a star. The embedded Toom graph in the middle picture of Figure 3 originates from a Toom matching of this explanation graph. The proof of Proposition 24 takes up the remainder of this subsection. The proof is quite complicated and will be split over several lemmas. We fix an explanation graph (U, H) for (0, 0) with N := \|U * \| sinks. Because of our habit of drawing time downwards in pictures, it will be convenient to define a function h : U → N by h(i, t) := −t (i, t) ∈ U . (3.6) We call h(w) the height of a vertex w ∈ U . For u, v ∈ U , we write u H v when there exist u 0 , . . . , u n ∈ U with n ≥ 0, u 0 = u, u n = v, and (u k−1 , u k ) ∈ H for all 0 < k ≤ n. By definition, for w 1 , w 2 ∈ U , we write w 1 ≈ w 2 if h(w 1 ) = h(w 2 ) and there exists a w 3 ∈ U such that w i H w 3 for i = 1, 2. Moreover, for v, w ∈ U , we write v ∼ w if there exist m ≥ 0 and v = v 0 , . . . , v m = w such that v i−1 ≈ v i for 1 ≤ i ≤ m. Then ∼ is an equivalence relation. In fact, if we view U as a graph in which two vertices v, w are adjacent if v ≈ w, then the equivalence classes of ∼ are just the connected components of this graph. We let C denote the set of all (nonempty) equivalence classes. It is easy to see that the origin (0, 0) and the sinks form equivalence classes of their own. With this in mind, we set C * := {w} : w ∈ U * . Each C ∈ C has a height h(C) such that = 1, 2). Note that this implies that h(C 2 ) = h(C 1 ) + 1. The following lemma says that C has the structure of a directed tree with the sinks as its leaves. h(v) = h(C) for all v ∈ C. For C 1 , C 2 ∈ C, we write C 1 → C 2 if there exists a (v 1 , v 2 ) ∈ H such that v i ∈ C i (i Lemma 25 (Tree of equivalence classes) For each C ∈ C with C = {(0, 0)}, there exists a unique C ∈ C such that C → C. Moreover, for each C ∈ C\C * , there exists at least one C ∈ C such that C → C . Also, C ∈ C\C * implies C ∩ U * = ∅. Proof Since the sinks form equivalence classes of their own, C ∈ C\C * implies C ∩ U * = ∅. If C ∈ C\C * , then condition (v) in Definition 20 of an explanation graph implies the existence of a C ∈ C such that C → C . Similarly, if C ∈ C and C = {(0, 0)}, then the existence of a C ∈ C such that C → C follows from condition (iii) in Definition 20. It remains to show that C is unique. Assume that, to the contrary, there exist w, w ∈ C and (v, w), (v , w ) ∈ H so that v and v do not belong to the same equivalence class. Since w and w lie in the same equivalence class C, there exist w 0 , . . . , w m ∈ C with w = w 0 , w m = w , and w i−1 ≈ w i for all 0 < i ≤ m. Using condition (iii) in Definition 20, we can find v 0 , . . . , v m ∈ U such that (v i , w i ) ∈ H (0 ≤ i ≤ m). In particular we can choose v 0 = v and v m = v . Since v and v do not belong to the same equivalence class, there must exist an 0 < i ≤ m such that v i−1 and v i do not belong to the same equivalence class. Since w i−1 ≈ w i , there exists a u ∈ U such that w i−1 H u and w i H u. But then also v i−1 H u and v i H u, which contradicts the fact that v i−1 and v i do not belong to the same equivalence class. For C, C ∈ C, we describe the relation C → C in words by saying that C is a direct descendant of C. We let D C := {C ∈ C : C → C } denote the set of all direct descendants of C. We will view D C as an undirected graph with set of edges E C := {C 1 , C 2 } : ∃v ∈ C, w 1 ∈ C 1 , w 2 ∈ C 2 s.t. (v, w i ) ∈ H ∀i = 1, 2 . (3.7) The fact that this definition is reminiscent of the definition of a tight polar is no coincidence and will become important in Lemma 27 below. We first prove the following lemma. Lemma 26 (Structure of the set of direct descendants) For each C ∈ C\C * , the graph (D C , E C ) is connected. Proof Let D 1 , D 2 be nonempty disjoint subsets of D C such that D 1 ∪ D 2 = D C and let D i := v ∈ C : ∃C ∈ D i and w ∈ C s.t. (v, w) ∈ H (i = 1, 2). (3.8) To show that (D C , E C ) is connected, we need to show that D 1 ∩ D 2 = ∅ for all choices of D 1 , D 2 . By Lemma 25, C ∩ U * = ∅ and hence for each v ∈ C there exists a w ∈ U such that (v, w) ∈ H. Therefore, since D C contains all direct descendants of C, we have D 1 ∪ D 2 = C. Since D 1 and D 2 are nonempty, so are D 1 and D 2 . Assume that D 1 ∩ D 2 = ∅. Then, since C is an equivalence class, there must exist v i ∈ D i (i = 1, 2) such that v 1 ≈ v 2 , i.e., {w ∈ U : v 1 H w} ∩ {w ∈ U : v 2 H w} = ∅. (3.9) However, for i = 1, 2, the set {w ∈ U : v i H w} is entirely contained in the equivalence classes in D i and their descendants. Since by Lemma 25, C has the structure of a tree, this contradicts (3.9). We can now make the connection to the definition of tight polars. We say that a polar (a 1 , . . . , a σ ) lies inside a set D ⊂ U if a s ∈ D for all 1 ≤ s ≤ σ. Lemma 27 (Tight polars) Let C ∈ C\C * , let M := \|D C \| be the number of its direct descendants, and let D C := C ∈D C C be the union of all C ∈ D C . Let (a 1,1 , . . . , a 1,σ ) be a polar inside D C . Then, given that M ≥ 2, it is possible to choose tight polars (a i,1 , . . . , a i,σ ) (2 ≤ i ≤ M ) inside D C such that: For each C ∈ D C and 1 ≤ s ≤ σ, there is a unique 1 ≤ i ≤ M such that a i,s ∈ C . (3.10) Proof By Lemma 26, the graph D C is connected in the sense defined there. To prove the claim of Lemma 27 will prove a slightly more general claim. Let D C be a connected subgraph of D C with M elements, let D C := C ∈D C C , and let (a 1,1 , . . . , a 1,σ ) be a polar inside D C . Then we claim that it is possible to choose tight polars (a i,1 , . . . , a i,σ ) (2 ≤ i ≤ M ) inside D C such that (3.10) holds with D C and M replaced by D C and M respectively. We will prove the claim by induction on M . The claim is trivial for M = 1. We will now prove the claim for general M ≥ 2 assuming it proved for M − 1. Since D C is connected, we can find some C ∈ D C so that D C \{C } is still connected. If none of the vertices a 1,1 , . . . , a 1,σ lies inside C , then we can add a point polar inside C , use the induction hypothesis, and we are done. Likewise, if all of the vertices a 1,1 , . . . , a 1,σ lie inside C , then we can add a point polar inside D C \C , use the induction hypothesis, and we are done. We are left with the case that some, but not all of the vertices a 1,1 , . . . , a 1,σ lie inside C . Without loss of generality, we assume that a 1,1 , . . . , a 1,m ∈ C and a 1,m+1 , . . . , a 1,σ ∈ D C \C . Since D C is connected in the sense of Lemma 26, we can find a v ∈ C and w 1 ∈ C , w 2 ∈ D C \C such that (v, w i ) ∈ H (i = 1, 2). Setting a 2,1 = · · · = a 2,m := w 2 and a 2,m+1 = · · · = a 2,σ := w 1 then defines a tight polar such that: • For each 1 ≤ s ≤ σ, there is a unique i ∈ {1, 2} such that a i,s ∈ C . • For each 1 ≤ s ≤ σ, there is a unique i ∈ {1, 2} such that a i,s ∈ D C \C . In particular, the elements of (a i,s ) i∈{1,2}, 1≤s≤σ with a i,s ∈ D C \C form a polar in D C \C , so we can again use the induction hypothesis to complete the argument. Proof of Proposition 24 We will use an inductive construction. Let L := max{h(w) : w ∈ U }. For each 0 ≤ l ≤ L, we set U ≤l := {w ∈ U : h(w) ≤ l} and C l := {C ∈ C : h(C) = l}. We will inductively construct an increasing sequence of integers 1 = N 0 ≤ N 1 ≤ · · · ≤ N L and for each 0 ≤ l ≤ L, we will construct an N l × σ matrix a i,s (l) 1≤i≤N l , 1≤s≤σ such that a i,s (l) ∈ U ≤l for all 1 ≤ i ≤ N l and 1 ≤ s ≤ σ. Our construction will be consistent in the sense that a i,s (l + 1) = a i,s (l) ∀1 ≤ i ≤ N l , 1 ≤ s ≤ σ, 0 ≤ l < L, (3.11) that is at each step of the induction we add rows to the matrix we have constructed so far. In view of this, we can unambiguously drop the dependence on l from our notation. We will choose the matrices a i,s 1≤i≤N l , 1≤s≤σ (3.12) in such a way that for each 0 ≤ l ≤ L: (i) a 1,1 = · · · = a 1,σ = (0, 0), (ii) (a i,1 , . . . , a i,σ ) is a tight polar for each 2 ≤ i ≤ N l , (iii) For all C ∈ C l and 1 ≤ s ≤ σ, there is a unique 1 ≤ i ≤ N l such that P s (a i,s ) ∩ C = ∅, where P s (a i,s ) is defined as in (3.4). We claim that setting N := N L then yields a Toom matching with the additional properties described in the proposition. Property (i) of Definition 23 of a Toom matching and the additional properties (iii) and (iv) from Proposition 24 follow trivially from conditions (i) and (ii) of our inductive construction, so it remains to check property (ii) of Definition 23, which can be reformulated by saying that for each w ∈ U * and 1 ≤ s ≤ σ, there exists a unique 1 ≤ i ≤ N such that w ∈ P s (a i,s ). Since {w} ∈ C for each w ∈ U * (vertices in U * form an equivalence class of their own), this follows from condition (iii) of our inductive construction. We start the induction with N 0 = 1 and a 1,1 = · · · = a 1,σ = (0, 0). Since (0, 0) is the only vertex in U with height zero, this obviously satisfies the induction hypotheses (i)-(iii). Now assume that (i)-(iii) are satisfied for some 0 ≤ l < L. We need to define N l+1 and choose polars (a i,1 , . . . , a i,σ ) with N l < i ≤ N l+1 so that (i)-(iii) are satisfied for l + 1. We note that by Lemma 25, each C ∈ C l+1 is the direct descendent of a unique C ∈ C l \C * . By the induction hypothesis (iii), for each C ∈ C l \C * and 1 ≤ s ≤ σ, there exists a unique 1 ≤ i s ≤ N l such that P s (a is,s ) ∩ C = ∅. Let D C := {C ∈ C : C → C } denote the set of all direct descendants of C and let D C := D C denote the union of its elements. Then setting {b s } := P s (a is,s ) ∩ D C (1 ≤ s ≤ σ) defines a polar (b 1 , . . . , b σ ) inside D C . Applying Lemma 27 to this polar, we can add tight polars to our matrix in (3.12) so that condition (iii) becomes satisfied for all C ∈ D C . Doing this for all C ∈ C l \C * , using the tree structure of C (Lemma 25), we see that we can satisfy the induction hypotheses (i)-(iii) for l + 1. Construction of Toom contours In this subsection, we prove Theorem 7. With Proposition 24 proved, most of the work is already done. We will prove a slightly more precise statement. Below ψ(V ) and ψ(V * ) denote the images of V and V * under ψ and ψ( E s ) := ψ(v), ψ(w) : (v, w) ∈ E s . Theorem 7 is an immediate consequence of Lemma 22 and the following theorem. Theorem 28 (Presence of a Toom contour) Under the assumptions of Theorem 7, whenever there is an explanation graph (U, H) for (0, 0) present in φ, there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) present in φ with the additional properties that ψ(V ) ⊂ U , ψ(V * ) ⊂ U * , and ψ( E s ) ⊂ H s for all 1 ≤ s ≤ σ. Proof The main idea of the proof has already been explained below Proposition 24. We now fill in the details. Let (U, H) be an explanation graph for (0, 0) that is present in φ. Let N := \|U * \| be the number of sinks. By Proposition 24 there exists a Toom matching a i,s 1≤i≤N, 1≤s≤σ for (U, H) such that a 1,1 = · · · = a 1,σ = (0, 0), and (a i,1 , . . . , a i,σ ) is a tight polar for each 1 ≤ i ≤ N . Recall from (3.4) that P s (w) denotes the unique directed path starting at w that uses only directed edges from H s and that ends at some vertex in U * . For each 1 ≤ i ≤ N such that (a i,1 , . . . , a i,σ ) is a point polar, and for each 1 ≤ s ≤ σ, we will use the notation with (a l−1 i,s , a l i,s ) ∈ H s for all 0 < l ≤ m(i, s). For each 1 ≤ i ≤ N such that (a i,1 , . . . , a i,σ ) is not a point polar, by the definition of a tight polar, we can choose v i ∈ U such that (v i , a i,s ) ∈ H for all 1 ≤ s ≤ σ. In this case, we will use the notation where (a 0 i,s , a 1 i,s ) ∈ H and (a l−1 i,s , a l i,s ) ∈ H s for all 1 < l ≤ m(i, s). We can now construct a Toom graph (V, E) with a specially designated source v • as follows. We set w(i, s, l) :=        i if l = 0 < m(i, s), (i, s, l) if 0 < l < m(i, s), a m(i,s) i,s if l = m(i, s). (1 ≤ i ≤ N, 1 ≤ s ≤ σ),(3.15) and V := w(i, s, l) : 1 ≤ i ≤ N, 1 ≤ s ≤ σ, 0 ≤ l ≤ m(i, s) , E s := w(i, s, l − 1), w(i, s, l) : 1 ≤ i ≤ N, 0 < l ≤ m(i, s) (1 ≤ s ≤ σ), v • := w(1, 1, 0) = · · · = w(1, σ, 0). (3.16) It is straightforward to check that (V, E) is a Toom graph with sets of sources, internal vertices, and sinks given by V • = i : 1 ≤ i ≤ N, m(i, s) > 0 ∪ {a 0 i,s : m(i, s) = 0 , V s = (i, s, l) : 1 ≤ i ≤ N, 0 < l < m(i, s) (1 ≤ s ≤ σ), V * = a m(i,s) i,s : 1 ≤ i ≤ N, 1 ≤ s ≤ σ = U * . (3.17) Note that the vertices of the form a 0 i,s with m(i, s) = 0 are the isolated vertices, that are both a source and a sink. We now claim that setting ψ w(i, s, l) := a l i,s (1 ≤ i ≤ N, 1 ≤ s ≤ σ, 0 ≤ l ≤ m(i, s)) (3.18) defines an embedding of (V, E). We first need to check that this is a good definition in the sense that the right-hand side is really a function of w(i, s, l) only. Indeed, when l = 0 < m(i, s), we have w(i, s, l) = i and a 0 i,1 = · · · = a 0 i,σ by the way a 0 i,s has been defined in (3.13) and (3.14). For 0 < l < m(i, s), we have w(i, s, l) = (i, s, l), and finally, for l = m(i, s), we have w(i, s, l) = a l i,s . We next check that ψ is an embedding, i.e., (i) ψ d+1 (w) = ψ d+1 (v) − 1 for all (v, w) ∈ E, (ii) ψ(v 1 ) = ψ(v 2 ) for each v 1 ∈ V * and v 2 ∈ V with v 1 = v 2 , (iii) ψ(v 1 ) = ψ(v 2 ) for each v 1 , v 2 ∈ V s with v 1 = v 2 (1 ≤ s ≤ σ). Property (i) is clear from the fact that E ⊂ H and Definition 20 of an explanation graph. Property (ii) follows from the fact that ψ(V * ) = U * and ψ(V \V * ) ⊂ U \U * . Property (iii), finally, follows from the observation that P s (a i,s ) ∩ P s (a j,s ) = ∅ ∀1 ≤ s ≤ σ, 1 ≤ i, j ≤ N, i = j. (3.19) Indeed, P s (a i,s )∩P s (a j,s ) = ∅ would imply that π s (a i,s ) = π s (a j,s ), as in the explanation graph there is a unique directed path of each type from every vertex that ends at some w ∈ U * , which contradicts the definition of a Toom matching. Since moreover ψ(v • ) = (0, 0) and property (ii) of Definition 20 implies that t < 0 for all (i, t) ∈ ψ(V )\{(0, 0)}, we see that the quadruple (V, E, v • , ψ) satisfies all the defining properties of a Toom contour (see Definition 3), except that the Toom graph (V, E) may fail to be connected. To fix this, we restrict ourselves to the connected component of (V, E) that contains the root v • . To complete the proof, we must show that (V, E, v • , ψ) is present in φ, i.e., (i) φ ψ(v) = ϕ 0 for all v ∈ V * , (ii) φ ψ(v) ∈ {ϕ 1 , . . . , ϕ m } for all v ∈ V \V * , (iii) ψ(w) − ψ(v) ∈ A s (φ ψ(v) ) for all (v, w) ∈ E * s (1 ≤ s ≤ σ), (iv) ψ(w) − ψ(v) ∈ σ s=1 A s (φ ψ(v) ) for all (v, w) ∈ E • . We will show that these properties already hold for the original quadruple (V, E, v • , ψ), without the need to restrict to the connected component of (V, E) that contains the root. Since the explanation graph (U, H) is present in φ, we have U * = {u ∈ U : φ u = ϕ 0 }. Since ψ(V * ) = U * , this implies properties (i) and (ii). The fact that the explanation graph (U, H) is present in φ moreover means that j − i ∈ A s (φ (i,t) ) for all (i, t), (j, t − 1) ∈ H s (1 ≤ s ≤ σ). Since (a 0 i,s , a 1 i,s ) ∈ H and (a l−1 i,s , a l i,s ) ∈ H s for all 1 < l ≤ m(i, s) (1 ≤ i ≤ N, 1 ≤ s ≤ σ) , this implies properties (iii) and (iv). Construction of Toom contours with two charges In this subsection we prove Theorem 9. As in the previous subsection, we will construct the Toom contour "inside" an explanation graph. Theorem 9 is an immediate consequence of Lemma 22 and the following theorem. Theorem 29 (Strong presence of a Toom contour) If σ = 2, then Theorem 28 can be strengthened in the sense that the Toom contour (V, E, v • , ψ) is strongly present in φ. Although it is a strengthening of Theorem 28, our proof of Theorem 29 will be completely different. In particular, we will not make use of the Toom matchings of Subsection 3.3. Instead, we will exploit the fact that if we reverse the direction of edges of one of the charges, then a Toom contour with two charges becomes a directed cycle. This allows us to give a proof of Theorem 29 based on the method of "loop erasion" (as explained below) that seems difficult to generalise to Toom contours with three or more charges. Let n ≥ 0 be an even integer and let V := {0, . . . , n − 1}, equipped with addition modulo n. Let ψ : V → Z d+1 be a function such that ψ d+1 (k) − ψ d+1 (k − 1) = 1 (1 ≤ k ≤ n). (3.20) We write ψ(k) = ψ(k), ψ d+1 (k) (k ∈ V ) and for n ≥ 2 we define: V 1 := k ∈ V : ψ d+1 (k − 1) > ψ d+1 (k) > ψ d+1 (k + 1) , V 2 := k ∈ V : ψ d+1 (k − 1) < ψ d+1 (k) < ψ d+1 (k + 1) , V * := k ∈ V : ψ d+1 (k − 1) > ψ d+1 (k) < ψ d+1 (k + 1) , V • := k ∈ V : ψ d+1 (k − 1) < ψ d+1 (k) > ψ d+1 (k + 1) . (3.21) In the trivial case that n = 0, we set V 1 = V 2 := ∅ and V • = V * := {0}. Definition 30 Let V be as above. A Toom cycle is a function ψ : V → Z d+1 such that: (i) ψ satisfies (3.20), (ii) ψ(k 1 ) = ψ(k 2 ) for each k 1 ∈ V * and k 2 ∈ V with k 1 = k 2 , (iii) ψ(k 1 ) = ψ(k 2 ) for each k 1 , k 2 ∈ V s with k 1 = k 2 (1 ≤ s ≤ σ), (iv) t < ψ d+1 (0) for all (i, t) ∈ ψ(V )\{ψ(0)}, where V 1 , V 2 , V * , and V • are defined as in (3.21). If ψ : V → Z d+1 is a Toom cycle of length n ≥ 2, then we set: E 1 := (k, k + 1) : ψ d+1 (k) > ψ d+1 (k + 1), k ∈ V , ← E 2 := (k, k + 1) : ψ d+1 (k) < ψ d+1 (k + 1), k ∈ V , E 2 := (k, l) : (l, k) ∈ ← E 2 ,(3.22) where as before we calculate modulo n. If n = 0, then E 1 = E 2 := ∅. We let (V, E) := (V, E 1 , E 2 ) denote the corresponding directed graph with two types of directed edges. The following simple observation makes precise our earlier claim that if we reverse the direction of edges of one of the charges, then a Toom contour with two charges becomes a directed cycle. Lemma 31 (Toom cycles) If ψ : V → Z d+1 is a Toom cycle, then (V, E, 0, ψ) is a Toom contour with root 0, set of sources V • , set of sinks V * , and sets of internal vertices of charge s given by V s (s = 1, 2). Moreover, every Toom contour with two charges is equivalent to a Toom contour of this form. Proof Immediate from the definitions. Proof of Theorem 29 We will first show that Theorem 28 can be strengthened in the sense that the Toom contour (V, E, v • , ψ) also satisfies condition (v) of Definition 8. As in Theorem 28, let (U, H) be an explanation graph for (0, 0) that is present in φ. We let ← H s := {(k, l) : (l, k) ∈ H s } denote the directed edges we get by reversing the direction of all edges in H s (s = 1, 2). We will use an inductive construction. At each point in our construction, (V, E, 0, ψ) will be a Toom contour rooted at (0, 0) that is obtained from a Toom cycle ψ : V → Z d+1 as in Lemma 31, and T := inf{ψ d+1 (k) : k ∈ V } is the earliest time coordinate visited by the contour. At each point in our construction, it will be true that: (i)' φ ψ(k) = ϕ 0 for all k ∈ V * with T + 1 < ψ d+1 (k), (ii) φ ψ(v) ∈ {ϕ 1 , . . . , ϕ m } for all v ∈ V \V * , (iiia) ψ(k), ψ(k + 1) ∈ H 1 for each (k, k + 1) ∈ E 1 with k ∈ V 1 ∪ {0}, (iiib) ψ(k − 1), ψ(k) ∈ ← H 2 for each (k − 1, k) ∈ ← E 2 with k ∈ V 2 ∪ {0}, (iva) ψ(k), ψ(k + 1) ∈ H 2 for each (k, k + 1) ∈ E 1 with k ∈ V • \{0}, (ivb) ψ(k − 1), ψ(k) ∈ ← H 1 for each (k − 1, k) ∈ ← E 2 with k ∈ V • \{0}, (vi) ψ(k − 1) = ψ(k + 1) for each k ∈ V • \{0}. We observe that condition (i)' is a weaker version of condition (i) of Definition 6. Conditions (ii), (iiia), and (iiib) corresponds to conditions (ii) and (iii) of Definition 6. Conditions (iva) and (ivb) are a stronger version of condition (iv) of Definition 6, that implies also condition (v) of Definition 8. Finally, condition (vi) corresponds to condition (vi) of Definition 8. Our inductive construction will end as soon as condition (i) of Definition 6 is fully satisfied, i.e., when: We start the induction with the trivial Toom cycle defined by V := {0} and ψ(0) = (0, 0). We identify a Toom cycle ψ : {0, . . . , n − 1} → Z d+1 with the word ψ(0) · · · ψ(n − 1). In each step of the induction, as long as (i) is not yet satisfied, we modify our Toom cycle according to the following two steps, which are illustrated in Figure 4. I. Exploration. We pick k ∈ V * such that φ ψ(k) = ϕ 0 and ψ d+1 (k) = T + 1, or if such a k does not exist, with ψ d+1 (k) = T . We define w s by H s,out (ψ(k)) := (ψ(k), w s ) (s = 1, 2). In the word ψ(0) · · · ψ(n − 1), on the place of ψ(k), we insert the word ψ(k)w 1 ψ(k)w 2 ψ(k). (i) φ ψ(k) = ϕ 0 for all k ∈ V * . II. Loop erasion. If as a result of the exploration, there are k 1 , k 2 ∈ V * with k 1 < k 2 such that ψ(k 1 ) = ψ(k 2 ), then we remove the subword ψ(k 1 ) · · · ψ(k 2 ) from the word ψ(0) · · · ψ(n − 1) and on its place insert ψ(k 1 ). We repeat this step until ψ(k 1 ) = ψ(k 2 ) for all k 1 , k 2 ∈ V * with k 1 = k 2 . The effect of the exploration step is that one sink is replaced by a source and two internal vertices, one of each charge, and than two new sinks are created (see Figure 4). These new sinks are created at height −T or −T + 1 and hence can overlap with each other or with other preexisting sinks, but not with sources or internal vertices. If the exploration step has created overlapping sinks or the two new internal vertices overlap, then these are removed in the loop erasion step. After the removal of a loop, all remaining vertices are of the same type (sink, source, or internal vertex of a given charge) as before. Using these observations, it is easy to check that: (C) After exploration and loop erasion, the modified word ψ is again a Toom cycle rooted at (0, 0) (see Definition 30) and the induction hypotheses (i)', (ii), (iiia), (iiib), (iva), (ivb) and (vi) remain true. Let ∆ := {ψ(k) : k ∈ V * , φ ψ(k) = ϕ 0 }. In each step of the induction, we remove one element from ∆ with a given time coordinate, say t, and possibly add one or two new elements to ∆ with time coordinates t − 1. Since the explanation graph is finite, this cannot go on forever so the induction terminates after a finite number of steps. This completes the proof that Theorem 28 can be strengthened in the sense that the Toom contour (V, E, v • , ψ) also satisfies condition (v) of Definition 8. Forks We recall that for Toom contours with two charges, Theorem 9 strengthened Theorem 7 by showing the presence of a Toom contour with certain additional properties. As we have seen in Subsection 2.4, such additional properties reduce the number of Toom contours one has to consider and hence lead to sharper Peierls bounds. In the present subsection, we will prove similar (but weaker) strengthened version of Theorem 7 that holds for an arbitrary number of charges. Let (V, E, v • , ψ) be a Toom contour. By definition, a fork is a source v ∈ V • such that: {ψ(w) : (v, w) ∈ E} = 2. (3.23) As we will show in a moment, the proof of Theorem 28 actually yields the following somewhat stronger statement. In the original formulation of Toom [Too80], his contours contain no sources but they contain objects that Toom calls forks and that effectively coincide with our usage of this term. For Toom, the fact that the number of sinks equals the number of forks plus one was part of his definition of a contour. In our formulation, this is a consequence of the fact that the number of sources equals the number of sinks. Theorem 32 (Toom contour with forks only) Theorem 28 can be strengthened in the sense that all sources v ∈ V \{v • } are forks. Proof Let us say that v ∈ V • is a point source if \|{ψ(w) : (v, w) ∈ E}\| = 1. We first show that Theorem 28 can be strengthened in the sense that all sources v ∈ V \{v • } are forks or point sources. Indeed, this is a direct consequence of the fact that the tight polars (a i,1 , . . . , a i,σ ) (2 ≤ i ≤ M ) constructed in the proof of Lemma 27 are either point polars or have the property that the set {a i,s : 1 ≤ s ≤ σ} has precisely two elements. The latter give rise to forks while the former give rise to point sources or isolated vertices. Since a Toom countour is connected, sources other than the root can never be isolated vertices. This shows that Theorem 28 can be strengthened in the sense that all sources v ∈ V \{v • } are forks or point sources. Now if some v ∈ V • \{v • } is a point source, then we can simplify the Toom contour by removing this source from the contour and joining all elements of {w : (v, w) ∈ E} into a new source, that is embedded at the space-time point z ∈ Z d+1 defined by {z} := {ψ(w) : (v, w) ∈ E}. Repeating this process until it is no longer possible to do so we arrive at Toom contour (V, E, v • , ψ) with the additional property that all sources v ∈ V \{v • } are forks. Bounds for eroders Outline In this section, we apply the abstract theory developed in the previous section to concrete models. In Subsection 4.1, we discuss the erosion criteria (1.9) and (2.12). In particular, we prove Lemma 10 and show that (2.12) implies that ϕ is an eroder. In Subsection 4.2, we prove Lemmas 14 and 17 which give an exponential upper bound on the number of Toom contours and Toom cycles with a given number of edges. In Subsection 4.3, we prove Lemma 19 which shows that for eroders, finiteness of the Peierls sum is sufficient to conclude that ρ(p) > 0. At this point, we have proved all ingredients needed for the proof of Toom's stability theorem described in Subsection 2.2 and also for the explicit bounds for concrete eroders stated in Subsection 2.4. Eroders In this subsection we prove Lemma 10. Our proof depends on the equivalence of (1.9) and the eroder property, which is proved in [Pon13, Thm 1]. In Lemma 33, we give an alternative direct proof that (2.12) implies that ϕ is an eroder. Although we do not really need this alternative proof, we have included it since it is short and instructive. In particular, it links the eroder property to edge speeds, which we otherwise do not discuss but which are an important motivating idea behind the definition of Toom contours. Proof of Lemma 10 In [Pon13, Lemma 12] it is shown 4 that (1.9) is equivalent to the existence of a polar function L of dimension 2 ≤ σ ≤ d + 1 and constants ε 1 , . . . , ε σ such that σ s=1 ε s > 0 and for each 1 ≤ s ≤ σ, there exists an A s ∈ A(ϕ) such that ε s − L s (i) ≤ 0 for all i ∈ A s . It follows that σ s=1 sup A∈A(ϕ) inf i∈A L s (i) ≥ σ s=1 inf i∈As L s (i) ≥ σ s=1 ε s > 0, (4.1) which shows that (2.12) holds. Assume, conversely, that (2.12) holds. Since A(ϕ) is finite, for each 1 ≤ s ≤ σ we can choose A s (ϕ) ∈ A(ϕ) such that ε s := inf i∈As(ϕ) L s (i) = sup A∈A(ϕ) inf i∈A L s (i). (4.2) Then (2.12) says that σ s=1 ε s > 0. Let H s := {z ∈ R d : L s (z) ≥ ε s }. By the definition of a polar function, σ s=1 L s (z) = 0 for each z ∈ R d , and hence the condition σ s=1 ε s > 0 implies that for each z ∈ R d , there exists an 1 ≤ s ≤ σ such that L s (z) < ε s . In other words, this says that σ s=1 H s = ∅. For each 1 ≤ s ≤ σ, the set A s (ϕ) is contained in the half-space H s and hence the same is true for Conv(A s (ϕ)), so we conclude that Proof Most of the argument has already been given below Lemma 10. It only remains to prove (2.14). It suffices to prove the claim for n = 1; the general claim then follows by induction. Assume that i ∈ Z d satisfies L s (i) > r s (X 0 0 ) − δ s . We need to show that X 0 1 (i) = 1 for all such i. By the definition of δ s , we can choose A ∈ A(ϕ) such that inf j∈A L s (j) = δ s . It follows that L s (i + j) > r s (X 0 0 ) for all j ∈ A and hence X 0 0 (i + j) = 1 for all j ∈ A, which implies X 0 1 (i) = 1 by (1.8). Exponential bounds on the number of contours In this subsection, we prove Lemmas 14 and 17. Proof of Lemma 14 We first consider the case that the number of charges σ is even. Let E) is a directed graph with σ types of edges, that are called charges. In (V, E), all edges point in the direction from the sources to the sinks. We modify (V, E) by reversing the direction of edges of the charges 1 2 σ + 1, . . . , σ. Let (V, E ) denote the modified graph. In (V, E ), the number of incoming edges at each vertex equals the number of outgoing edges. Since moreover the undirected graph (V, E) is connected, it is not hard to see 5 that it is possible to walk through the directed graph (V, E ) starting from the root using an edge of charge 1, in such a way that each directed edge of E is traversed exactly once. T = (V, E, v • , ψ) ∈ T 0 . Recall that (V, Let m := σn e (T ) denote the total number of edges of (V, E ) and for 0 < k ≤ m, let (v k−1 , v k ) ∈ E s k denote the k-th step of the walk, which has charge s k . Let δ k := ψ(v k ) − ψ(v k−1 ) denote the spatial increment of the k-th step. Note that the temporal increment is determined by the charge s k of the k-th step. Let k 0 , . . . , k σ/2 denote the times when the walk visits the root v • . We claim that in order to specify (V, E, v • , ψ) uniquely up to equivalence, in the sense defined in (2.7), it suffices to know the sequences (s 1 , . . . , s m ), (δ 1 , . . . , δ m ), and (k 0 , . . . , k σ/2 ). (4.4) Indeed, the sinks and sources correspond to changes in the temporal direction of the walk which can be read off from the charges. Although the images under ψ of sources may overlap, we can identify which edges connect to the root, and since we also know the increment of ψ(v k ) in each step, all objects in (2.7) can be identified. The first charge s 1 is 1 and after that, in each step, we have the choice to either continue with the same charge or choose one of the other 1 2 σ available charges. This means that there are no more than ( 1 2 σ + 1) m−1 possible ways to specify the charges (s 1 , . . . , s m ). Setting M := σ s=1 A s (ϕ) , we see that there are no more than M m possible ways to specify the spatial increments (δ 1 , . . . , δ m ). Since k 0 = 0, k σ/2 = m, we can roughly estimate the number of ways to specify the visits to the root from above by n σ/2−1 . Recalling that m = σn e (T ), this yields the bound N n ≤ n σ/2−1 ( 1 2 σ + 1) σn−1 M σn . (4.5) This completes the proof when σ is even. When σ is odd, we modify (V, E) by doubling all edges of charge σ, i.e., we define (V, F) with F = ( F 1 , . . . , F σ+1 ) := ( E 1 , . . . , E σ , E σ ), (4.6) and next we modify (V, F) by reversing the direction of all edges of the charges 1 2 σ +1, . . . , σ+ 1. We can define a walk in the resulting graph (V, F ) as before and record the charges and spatial increments for each step, as well as the visits to the root. In fact, in order to specify (V, E, v • , ψ) uniquely up to equivalence, we do not have to distinguish the charges σ and σ + 1. Recall that edges of the charges σ and σ + 1 result from doubling the edges of charge σ and hence always come in pairs, connecting the same vertices. Since sinks do not overlap and since internal vertices of a given charge do not overlap, and since we traverse edges of the charges σ and σ + 1 in the direction from the sinks towards the sources, whenever we are about to traverse an edge that belongs to a pair of edges of the charges σ and σ + 1, we know whether we have already traversed the other edge of the pair. In view of this, for each pair, we only have to specify the spatial displacement at the first time that we traverse an edge of the pair. Using these considerations, we arrive at the bound Proof of Lemma 17 The proof goes along the same lines as that of Lemma 14 for the case σ is even. Observe that for σ = 2, the walk visits the root 0 twice: k 0 = 0, k 1 = m. Thus (k 0 , k 1 ) is deterministic, and we only need to specify the sequences (s 1 , . . . , s m ), (δ 1 , . . . , δ m ). (4.8) The first charge s 1 is 1 and after that, in each step, we have the choice to either continue with the same charge or choose charge 2. This means that there are no more than 2 m−1 possible ways to specify the charges (s 1 , . . . , s m ). Once we have done that, by condition (v) of Definition 8 of what it means for a cycle to be strongly present, we know for each 0 < k ≤ m whether the spatial increment δ k is in A 1 (ϕ) or A 2 (ϕ). Setting M s := \|A s (ϕ) (s = 1, 2), using the fact that \| E 1 \| = \| E 2 \| = 2n e (T ) = m/2, we see that there are no more than M m/2 1 · M m/2 2 possible ways to specify (δ 1 , . . . , δ m ). This yields the bound N n ≤ 2 2n−1 M n 1 · M n 2 . (4.9) Finiteness of the Peierls sum In this subsection, we prove Proposition 18 about the presence of a large contour. As a direct consequece of this proposition, we obtain Lemma 19 which says that for an eroder, finiteness of the Peierls sum in (2.25) suffices to conclude that the intensity of the upper invariant law is positive. We also prove a stronger version of Proposition 18, where we show the strong presence of a Toom contour in which all sources are forks. Proof of Proposition 18 Recall the definition of the modified collection of monotonic maps φ (r) in (2.32). Let x (r) denote the maximal trajectory of φ (r) . For each integer q ≥ 0, let C q := Conv({qj 1 , . . . , qj σ }). Then C q+1 = i + j s : i ∈ C q , 1 ≤ s ≤ σ (q ≥ 0). (4.10) Using this, it is easy to see by induction that our assumption that x (r) −r (i) = 0 for all i ∈ C r implies that x (r) −q (i) = 0 for all i ∈ C q and 0 ≤ q ≤ r. In particular, this holds for q = 0, so x (r) 0 (0) = 0. Using this, it is straightforward to adapt the proof of Lemma 22 and show that there is an explanation graph (U, H) for (0, 0) present in φ (r) which has the additional properties: • i ∈ Z d : (i, −q) ∈ U = C q (0 ≤ q ≤ r), • (i, −q), (i + j s , −q − 1) ∈ H s (0 ≤ q < r, i ∈ C q ). In particular, these properties imply that • t ≤ −r for all (i, t) ∈ U * . Theorem 28 tells us that there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) present in φ (r) with the additional properties that ψ(V ) ⊂ U , ψ(V * ) ⊂ U * , and ψ( E s ) ⊂ ψ( H s ) for all 1 ≤ s ≤ σ. This immediately implies that ψ d+1 (v) ≤ −r for all v ∈ V * . To see that the Toom contour can be chosen such that moreover ψ d+1 (v) ≤ 1 − r for all v ∈ V • \{v • }, we have to look into the proof of Theorem 28. In Subsection 3.3 we defined an equivalence relation ∼ on the set of vertices U of an explanation graph (U, H). In Lemma 25, we showed that the set of all equivalence classes has the structure of a directed tree. If we draw time downwards, then the root of this tree lies below. In the proof of Proposition 24, we constructed a Toom matching for (U, H) with the property that except for the root, all other polars lie at a level above the last level where the tree still consisted of a single equivalence class. Finally, in the proof of Theorem 28, we used these polars to construct sources that lie at most one level below the corresponding polar. The upshot of all of this is that in order to show that ψ d+1 (v) ≤ 1 − r for all v ∈ V • \{v • }, it suffices to show that the set of vertices {(i, t) ∈ U : t = 1 − r} forms a single equivalence class as defined in Subsection 3.3. To see that this indeed is the case, call two points i = (i 1 , . . . , i σ ), j = (j 1 , . . . , j σ ) ∈ C r−1 neighbours if there exist 1 ≤ s 1 , s 2 ≤ σ with s 1 = s 2 such that i s 1 = j s 1 − 1, i s 2 = j s 2 + 1, and i s = j s for all s ∈ {1, . . . , σ}\{s 1 , s 2 }. Define k ∈ C r by k s 1 = j s 1 , k s 2 = j s 2 + 1, and k s = j s for all other s. Then (i, 1 − r), (k, −r) ∈ H and (j, 1 − r), (k, −r) ∈ H which proves that (i, 1 − r) ≈ (j, 1 − r). Since any two points in C r−1 are connected by a path that in each step moves from a point to a neighbouring point, this shows that {(i, t) ∈ U : t = 1 − r} forms a single equivalence class. To complete the proof, we need to show that if σ = 2, then we can construct the Toom contour so that in addition it is strongly present in φ (r) . We use the same explanation graph (U, H) for (0, 0) with properties (i)-(iii) as above. Theorem 29 now tells us that there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) strongly present in φ (r) with the additional properties that ψ(V ) ⊂ U , ψ(V * ) ⊂ U * , and ψ( E s ) ⊂ ψ( H s ) for all 1 ≤ s ≤ σ. This again immediately implies that ψ d+1 (v) ≤ −r for all v ∈ V * , so again it remains to show that the Toom contour can be chosen such that moreover To see that this is the case, we have to look into the proof of Theorem 29. Instead of starting the inductive construction with the trivial Toom cycle of length zero, we claim that it is possible to start with a Toom cycle ψ of length 4r for which all sources except the root have the time coordinate 1 − r and all sinks have the time coordinate −r. Since the process of exploration and loop erasion will then only create new sources with time coordinate −r or lower, the claim then follows. A Toom cycle ψ with the described properties is drawn in Figure 5. More formally, this cycle has the following description. Starting from (0, 0), it first visits the points (−k, kj 1 ) with k = 1, . . . , r. Next, it alternatively visits the points (1 − r, (r − k)j 1 + (k − 1)j 2 ) and (−r, (r − k)j 1 + kj 2 ) with k = 1, . . . , r. Finally, it visits the points (k − r, (r − k)j 2 ) with k = 1, . . . , r, ending in (0, 0), where it started. ψ d+1 (v) ≤ 1 − r for all v ∈ V • \{v • }. (0, 0) (j 1 , −1) (rj 1 , −r) (j 2 , −1) (rj 2 , −r) Proposition 34 (Large contours with forks only) Proposition 18 can be strengthened in the sense that all sources v ∈ V \{v • } are forks. Proof A Toom contours with two charges that is strongly present in Φ (r) automatically has the property that all sources v ∈ V \{v • } are forks, because of condition (vi) of Definition 8. Thus, it suffices to prove the claim for Toom contours with three or more charges. In this case, as pointed out in the proof of Proposition 18, the fact that all sources v ∈ V \{v • } are forks is an automatic result of the construction used in the proof of Theorem 28. Since we used this same construction in the proof of Proposition 18, the contour constructed there also has this property. Proof of Lemma 19 Let T 0,r := (V, E, v • , ψ) ∈ T 0 : ψ d+1 (v) ≤ −r for all v ∈ V * . (4.11) By assumption, Fix j s ∈ A s (ϕ) (1 ≤ s ≤ σ) and set ∆ r := Z d ∩ Conv({rj 1 , . . . , rj σ }). Then Proposition 18 allows us to estimate T ∈T 0 p n * (T ) < ∞,P X −r (i) = 0 ∀i ∈ ∆ r ≤ T ∈T 0,r P T is present in Φ (r) ≤ ε, (4.13) where in the last step we have used that ψ d+1 (v) ≤ −r for all v ∈ V * and hence all sinks of V must be mapped to space-time points (i, t) where Φ (r) (i,t) = Φ (i,t) . By translation invariance, P X −r (i) = 1 for some i ∈ ∆ r ≤ i∈∆r P X −r (i) = 1 = \|∆ r \|P X 0 (0) = 1 . (4.14) Combining this with our previous formulas, we see that ρ(p) = P X 0 (0) = 1 ≥ \|∆ r \| −1 (1 − ε) > 0. (4.15) For Toom contours with two charges, Proposition 18 guarantees the strong presence of a large Toom contour, so we can argue similarly, replacing T 0 by T 0 . Remark In Peierls arguments, it is frequently extremely helpful to be able to draw conclusions based only on the fact that that the Peierls sum is finite (but not necessarily less than one). These sorts of arguments played an important role in [KSS14], where we took inspiration for Lemma 19, and can be traced back at least to [Dur88, Section 6a]. Cooperative branching and the identity map In this subsection, we study the monotone random cellular automaton that applies the maps ϕ 0 , ϕ id , and ϕ coop,d with probabilities p, q, r, respectively. For each p, r ≥ 0 such that p+r ≤ 1, let ρ(p, r) denote the intensity of the upper invariant law of the process with parameters p, 1 − p − r, r. For each 0 ≤ r < 1, there exists a p c (r) ∈ [0, 1 − r] such that ρ(p, r) > 0 for 0 ≤ p < p c (r) and ρ(p, r) = 0 for p c (r) < p ≤ 1 − r. We give lower bounds on p c (r). Recall from Subsection 2.5 that we set σ = 2 and for the sets A s (ϕ k ) in (2.9) we make the choices A 1 (ϕ id ) := A 1 , A 2 (ϕ id ) := A 1 , A 1 (ϕ coop,d ) := A 1 , A 2 (ϕ coop,d ) := A 2 , (5.1) with A 1 := {0} and A 2 := {e 1 , . . . , e d }. Let Φ = (Φ (i,t) ) (i,t)∈Z 3 be an i.i.d. collection of monotonic maps so that P[Φ (i,t) = ϕ 0 ] = p, P[Φ (i,t) = ϕ id ] = q, and P[Φ (i,t) = ϕ coop,d ] = r. We let T 0 denote the set of Toom contours (V, E, 0, ψ) rooted at the origin with respect to the given choice of σ and the sets A s (ϕ k ) in (2.47). Theorem 7 then implies the Peierls bound 1 − ρ ≤ T ∈T 0 P T is strongly present in Φ . (5.2) In the remainder of this section, we give an upper bound on this expression. Recall from Subsection 3.5 that if we reverse the direction of edges of charge 2, then the Toom graph becomes a directed cycle with edge set E 1 ∪ E 2 . For any set A ⊂ Z d , let us write −A := {−i : i ∈ A}. For any (v, w) ∈ E 1 ∪ E 2 we say that ψ (v, w) is (i) outward, if ψ 3 (w) = ψ 3 (v) − 1 and ψ(w) − ψ(v) ∈ A 2 , (ii) upward, if ψ 3 (w) = ψ 3 (v) − 1 and ψ(w) − ψ(v) ∈ A 1 , (iii) inward, if ψ 3 (w) = ψ 3 (v) + 1 and ψ(w) − ψ(v) ∈ −A 2 , (iv) downward, if ψ 3 (w) = ψ 3 (v) + 1 and ψ(w) − ψ(v) ∈ −A 1 . The use of the words "upward" and "downward" are inspired by our habit of drawing negative time upwards in pictures. As \|A 2 \| = d, we distinguish d types of outward and inward edges: we say that ψ (v, w) is type i, if \| ψ(w) − ψ(v)\| = e i . Our definitions in (5.1) together with Definitions 6 and 8 imply that a Toom contour is strongly present in Φ if and only if the following conditions are satisfied: (i) Φ ψ(v) = ϕ 0 for all v ∈ V * , (iia) Φ ψ(v) ∈ {ϕ id , ϕ coop,d } for all v ∈ V 1 ∪ V 2 ∪ {v • }, (iib) Φ ψ(v) = ϕ coop,d for all v ∈ V • \{v • }, (iiia) If (v, w) ∈ E * 1 , then ψ (v, w) is upward, (iiib) If (v, w) ∈ E * 2 , then ψ (v, w) is downward if Φ ψ(w) = ϕ id , ψ (v, w) is inward if Φ ψ(w) = ϕ coop,d , (iva)' If (v, w) ∈ E • 1 , then ψ (v, w) is outward, (ivb)' If (v, w) ∈ E • 2 , then ψ (v, w) is downward, where E • i and E * i are defined in (2.5). If (V, E, v • , ψ) is a Toom contour rooted at 0 that is strongly present in Φ, then we can fully specify ψ by saying for each (v, w) ∈ E 1 ∪ E 2 whether ψ (v, w) is upward, downward, inward or outward, and its type in the latter two cases. In other words, we can represent the contour by a word of length n consisting of the letters from the alphabet {o 1 , . . . , o d , u, d, i 1 , . . . , i d }, which represents the different kinds of steps the cycle can take. Then we obtain a word consisting of the letters o 1 , . . . , o d , u, d, i 1 , . . . , i d that must satisfy the following rules: • Each outward step must be immediately preceded by a downward step. • Between two occurrences of the string do · , and also before the first occurrence of do · and after the last occurrence, we first see a string consisting of the letter u of length ≥ 0, followed by a string consisting of the letters d, i 1 , . . . , i d , again of length ≥ 0. So, for example the contour in the middle of Figure 3 is described by the following word: uuuu do 1 do 1 uu do 2 do 2 di 1 di 1 do 1 do 1 di 1 di 1 di 2 di 2 . (5.3) We call a sequence of length ≥ 0 of consecutive downward/upward steps a downward/upward segment. We can alternatively represent ψ by a word of length n consisting of the letters from {o 1 , . . . , o d , U, D, i 1 , . . . , i d , i • 1 , . . . , i • d }, where U and D represent upward and downward segments. Let us for the moment ignore the • superscripts. Then we can obtain a word consisting of these letters that must satisfy the following rules: • Each outward step must be immediately preceded by a downward segment of length ≥ 1 and followed by an upward segment of length ≥ 0. • The first step is an upward segment. • Between two occurrences of the string Do · U , and also before the first and after the last occurrence, we see a sequence of the string Di · of length ≥ 0. • The last step is a downward segment. We add the superscript • to each inward step whose endpoint overlaps with the image of a source other than the root already visited by the cycle in one of the previous steps. For any Toom contour T denote by W (T ) the corresponding word satisfying these rules. The structure of such a representation of a contour becomes more clear if we indicate the vertices in V 1 , V 2 , V * , and V • with the symbols 1, 2, * , •, respectively. Then the contour in the middle of Figure 3 is described by the following word: • \|U * \| D • \|o 1 1 \|U * \| D • \|o 1 1 \|U * \| D • \|o 2 1 \|U * \| D • \|o 2 1 \|U * \| D 2 \|i 1 2 \| D 2 \|i 1 2 \| D • \|o 1 1 \|U * \| D • \|o 1 1 \|U * \| D 2 \|i • 1 2 \| D 2 \|i 1 2 \| D 2 \|i 2 2 \| D 2 \|i 2 2 \|D • \|. (5.4) Finally, let l + (T ), l − (T ) and l −,• (T ) denote the vectors containing the lengths of the upward segments, downward segments followed by o · or i · and downward segments followed by i • · respectively in the order we encounter them along the cycle. For the example above we have: l + (T ) =(4, 0, 2, 0, 0, 0, 0), (1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0 Proof Knowing the word describing T together with the lengths of all upward and downward segments uniquely determines the contour, so it is enough to show that W (T ), l + (T ) and l − (T ) determines l −,• (T ) = (l 1 , . . . , l j ) (j ≥ 0). l − (T ) = Assume we know l 1 , . . . , l i for some 0 ≤ i < j. We then know the length and type of each step along the cycle up to the downward segment corresponding to l i+1 , that is we know the coordinates of its starting point. This downward segment ends at a charge 2 internal vertex, and the consecutive step is inward ending at a source other than the root already visited by the cycle. The cycle enters each such source by a downward step and leaves it by an outward step, hence by the structure of the explanation graph the endpoints of this outward step must coincide with the endpoints of the inward step following the downward segment with length l i+1 . As each outward step is followed by an upward segment, the starting point of the consecutive upward segment must be the endpoint of our downward segment. The endpoint of every upward segment is a defective site, and each site along a downward segment (except maybe its endpoints) is an identity site, so this upward segment must contain every site of our downward segment. Furthermore, by (iii) of Definition 2 of an embedding there cannot be any other upward segment that overlaps with this downward segment. Therefore, given the starting coordinates of our downward segment, we check which upward segment visited these coordinates previously, and we let l i+1 be the distance between the starting points of this upward segment and our downward segment. By a small abuse of notation, let us also use the letters o, i to indicate the number of times the symbols o, i occur in our representation of the contour (regardless of the sub-and superscripts). As our contour is a cycle starting and ending at 0, we must have the same number of inward and outward steps, furthermore, the total lengths of upward and downward segments must be equal as well: o = i and l + (T ) 1 = l − (T ) 1 + l −,• (T ) 1 . (5.6) We observe that each source (other than the root) is followed by an outward step, thus \|V • \| = \|V * \| = i + 1. (5.7) Finally, in the representation W (T ) of a contour the first and last step is U and D respectively, and in between i strings of DoU alternate with i strings of Di. Thus, letting 0 ≤ j ≤ i denote the number of inward steps with the superscript • and using (5.6) we have Claim 2 For all 0 ≤ i, 0 ≤ j ≤ i we have l + (T ) ∈ Z + ∪ {0} i+1 , l − (T ) ∈ Z + ∪ {0} 2i−j+1 , l −,• (T ) ∈ Z + ∪ {0} j . (5.8) Let W (i,W (i, j) ≤ 2i i i j d 2i−j . (5.9) Proof In any W ∈ W (i, j) the first and last step is U and D respectively, and in between i strings of DoU alternate with i strings of Di. Thus (ignoring the super-and subscripts) we can arrange these strings in 2i i possible ways. We then choose j inward steps to which we add the superscript •, this can be done in i j ways. Finally, we can assign the o's and i's subscripts 1, . . . , d one by one. As we have seen in the proof of Claim 1, an inward step with the superscript • overlaps with an outward step previously visited by the cycle, so the type of this inward step is the same as the type of that outward step. Hence we can assign the types of o's and i's in d 2i−j different ways. Claim 3 Let W ∈ W (i, j) for some 0 ≤ i, 0 ≤ j ≤ i. Then T ∈T 0 :W (T )=W P T is strongly present in Φ ≤ 3i − j i p i+1 r 2i−j 1 1 − q 3i−j+1 . Using q = 1 − p − r and Claim 2 we can estimate the Peierls sum in (5.2) from above by ∞ i=0 i j=0 W (i, j) 3i − j i p i+1 r 2i−j 1 1 − q 3i−j+1 < p p + r ∞ i=0 16dpr (2d + 1)r + p (p + r) 3 i . (5.10) For any fixed r this sum is finite as soon as p < (d + 0.5) 2 + 1/(16d) − d − 0.5 r. In particular for d = 2 we obtain the following bound on the critical parameter p c (r) > 0.00624r. Proof Proof of Claim 3 The idea of the proof is similar to that of Lemma 9 in [GG82]. As the Toom cycle T is strongly present in Φ, each sink is mapped to a defective site, and each inward step ends and each outward step starts at a site where the cooperative branching map is applied. The definition of an embedding entails that sinks do not overlap, so using 5.7 they contribute to a factor p i+1 . To estimate the contribution of the in-and outward steps, we need to recall the construction of the Toom cycle in Section 3.5. We inductively add edges to the cycle by exploring its previously unexplored sites one by one. At an exploartion step, starting at the site we are exploring, an upward, a downward, an outward and an inward step is added in this order. Although during the loop erasion some of these steps might be erased, their relative order in the cycle does not change and the site is not visited again in later iterations. Therefore, each site is the starting point of at most one outward step and the endpoint of at most one inward step, and if both steps are present, the outward step is always visited first by the cycle. As outward steps start at a source, there are i inward and outward steps and j inward steps with the superscript •, we have that these steps contribute to a factor r 2i−j . Finally, the strong presence of T implies that every downward step, except for the ones ending at a source other than the root, ends at a site where the identity map is applied. 5.6 then yields that downward segments contribute to a factor q l + (T ) 1 −i . Let It remains to show that q −i (l + ,l − )∈L(W) q l + 1 ≤ 3i − j i 1 1 − q 3i−j+1 . (5.13) From now on, we will omit the last coordinate of l − . As we have seen in the proof of Claim 1, to determine the lengths in l −,• it is enough to know the type and length of each step along the cycle up to the corresponding downward step. Therefore, when the cycle visits the last downward segment, the length of every other down-and upward segment is already known. By (5.6) we then have l − 2i−j+1 = l + 1 − l −,• 1 − l − 1 − · · · − l − 2i−j . By a small abuse of notation we will denote l − = (l − 1 , . . . , l − 2i−j ) and l + = (l + 1 , . . . , l + i+1 ). Given l − and l + we merge all the lengths into a single vector in a certain order, that is we inductively construct two vectors k ∈ Z + ∪ {0} 3i−j+1 and k ± ∈ {1, −1} 3i−j+1 in the following way. We let K 0 = k + 0 = k − 0 = 0 and for each 1 ≤ s < 3i − j + 1 • if l − s−1 − l + s−1 > K s−1 or l − s−1 − l + s−1 = K s−1 < 0, then k s := l + k + s−1 +1 , k ± s := 1, k + s := k + s−1 + 1, k − s := k − s−1 , • otherwise k s := l − k − s−1 +1 , k ± s := −1, k + s := k + s−1 , k − s := k − s−1 + 1, and we let K s := K s−1 + k s k ± s . Finally we let k := (k 1 , . . . , k 3i−j+1 ) and k ± := (k ± 1 , . . . , k ± 3i−j+1 ). Note that each element k ± s is 1 or -1, depending on whether k s was chosen from l + or l − respectively, furthermore, the vectors k and k ± satisfy the property K s ≥ 0 iff k ± s = 1 ∀s. (5.14) Informally, this means that we rearrange the lengths such that every upward step ends at a non-negative height and every downward step ends at a negative height. As K 3i−j+1 = l + 1 − l − 1 ≥ 0, this implies that k ± 3i−j+1 = 1, that is the last element of k is an upward length. Let us further denote the sum of upward and downward lengths in k up to coordinate s by K + s := k 1 1{K ± 1 = 1} + · · · + k s 1{K ± s = 1}, K − s := k 1 1{K ± 1 = −1} + · · · + k s 1{K ± s = −1}. (5.15) Clearly, K + s ≥ K + s−1 and K − s ≥ K − s−1 for each s. Furthermore, (5.14) implies K − s−1 < K − s−1 ≤ K + s , if k ± s−1 = −1, k ± s = 1, K − s−1 ≤ K + s−1 < K − s , if k ± s−1 = 1, k ± s = −1. (5.16) Let K denote the set of all pairs of vectors (k, k ± ) such that k ∈ Z + ∪ {0} 3i−j+1 , k ± ∈ {1, −1} 3i−j+1 and that satisfy poperty (5.14), and let K ± denote the set of all vectors k ± that contain 2i − j (-1)'s and i + 1 1's such that k ± 3i−j+1 = 1. We then can further bound (l + ,l − )∈L(W) q l + 1 ≤ k ± ∈K ± k:(k,k ± )∈K q K + 3i−j+1 . (5.17) Let us fix for the moment the vector k ± and consider the sum k:(k,k ± )∈K q K + 3i−j+1 = k 1 ∈K 1 · · · k 3i−j+1 ∈K 3i−j+1 q K + 3i−j+1 , (5.18) where K s (k 1 , . . . , k s−1 ) denotes the set of all the possible k s 's given the first s − 1 coordinate of k. For any k ± s = 1, we can estimate k s−1 ∈K s−1 ks∈Ks q K + s ≤                k s−1 ∈K s−1 ∞ K + s =K + s−1 q K + s = 1 1 − q k s−1 ∈K s−1 q K + s−1 if k ± s−1 = 1, k s+1 ∈K s+1 ∞ K + s =K − s−1 q K + s = 1 1 − q k s−1 ∈K s+1 q K − s−1 if k ± s−1 = −1,(5.q K − s ≤                k s−1 ∈K s−1 ∞ K − s =K + s−1 q K − s = 1 1 − q k s−1 ∈K s−1 q K + s−1 if k ± s−1 = 1, k s+1 ∈K s+1 ∞ K − s =K − s−1 q K − s = 1 1 − q k s−1 ∈K s+1 q K − s−1 if k ± s−1 = −1. (5.20) Finally, if a length k s with k ± s = −1 corresponds to a downward segment ending at a source (of which we have i in total), we have k s ≥ 1. Then we can bound K − s ≥ K − s−1 + 1 if k ± s−1 = −1, and K − s ≥ K + s−1 + 1 if k ± s−1 = 1, as we have a strict inequality in (5.16) in this case. Thus these downward segments will each contribute to an additional factor of q. As k ± 3i−j+1 = 1, we can repeatedly apply these formulas in (5.18) for all s to obtain the upper bound q i 1 1−q 3i−j+1 . Observing that \|K ± \| = 3i−j i and using (5.17) we can conclude (5.13). Continuous time Outline In this section, we consider monotone interacting particle systems with a finite collection ϕ 0 , ϕ 1 , . . . , ϕ m of monotonic maps such that ϕ 0 = ϕ 0 , ϕ k = ϕ id for any 1 ≤ k ≤ m, and a collection of nonnegative rates r 0 , r 1 , . . . , r m , evolving according to (1.2). We extend the definition of Toom contours to continuous time, and show how to use them to obtain explicit bounds for certain models. Toom contours in continuous time Recall Definition 1 of a Toom graph (V, E) = (V, E 1 , . . . , E σ ) with σ charges and the definition of sources, sinks and internal vertices in (2.2). Continuous Toom contours are Toom graphs embedded in space-time Z d × R. Definition 35 A continuous embedding of (V, E) is a map V v → ψ(v) = ψ(v), ψ d+1 (v) ∈ Z d × R (6.1) that has the following properties: (i) either ψ d+1 (w) < ψ d+1 (v) and ψ(w) = ψ(v), or ψ d+1 (w) = ψ d+1 (v) and ψ(w) = ψ(v) for all (v, w) ∈ E, (ii) ψ(v 1 ) = ψ(v 2 ) for each v 1 ∈ V * and v 2 ∈ V with v 1 = v 2 , (iii) ψ(v 1 ) = ψ(v 2 ) for each v 1 , v 2 ∈ V s with v 1 = v 2 (1 ≤ s ≤ σ), (iv) ψ d+1 (v 3 ) / ∈ ψ d+1 (v 2 ), ψ d+1 (v 1 ) for each (v 1 , v 2 ) ∈ E s , v 3 ∈ V s ∪ V * with ψ(v 1 ) = ψ(v 2 ) = ψ(v 3 ) (1 ≤ s ≤ σ). We call ψ((v, w)) = (ψ(v), ψ(w)) a vertical segment, if ψ d+1 (w) < ψ d+1 (v), and a horizontal segment, if ψ d+1 (w) = ψ d+1 (v). Then (i) implies that ψ( E) is the union of vertical and horizontal segments. Property (iv) says that an internal vertex of charge s or a sink is not mapped into a point of a vertical segment in ψ( E s ) (1 ≤ s ≤ σ). Note that, unlike in the discrete time case, this definition of an embedding does not imply \| E 1 \| = · · · = \| E σ \|. Definition 36 A continuous Toom contour is a quadruple (V, E, v • , ψ), where (V, E) is a connected Toom graph, v • ∈ V • is a specially designated source, and ψ is a continuous embedding of (V, E) that has the additional property that: (v) ψ d+1 (v • ) > t for each (i, t) ∈ ψ(V )\ψ({v • }). We set V vert := v ∈ V : ψ((w, v)) is a vertical segment for some (w, v) ∈ E , V hor := v ∈ V : ψ((v, w)) is a horizontal segment for some (v, w) ∈ E , (6.2) that is V vert is the set of vertices in V whose images under ψ are the endpoints of a vertical segment, and V hor is the set of vertices in V whose images under ψ are the starting points of a horizontal segment. We let P r with r = (r 0 , . . . , r m ) be a probability measure under which we define a family of independent Poisson processes on R: P i,k for i ∈ Z d , 0 ≤ k ≤ m, each with rate r k . (6.3) We regard each P i,k as a random discrete subset of R. Note that P r -a.s. these sets are pairwise disjoint. P = P (i,k) i∈Z d ,0≤k≤m almost surely determines a stationary process (X t ) t∈R that at each time t is distributed according to the upper invariant law ν. As in the discrete time case, we need a special construction of this process. Let P = P i,k i∈Z d ,0≤k≤m denote a realization of the Poisson processes. We will call a point in P i,k (i ∈ Z d ) a type k arrival point, and call type 0 arrival points defective points. Furthermore, let {0, 1} Z d ×R denote the space of all space- time configurations x = (x t (i)) i∈Z d ,t∈R . For x ∈ {0, 1} Z d and t ∈ R, we define x t ∈ {0, 1} Z d by x t := (x t (i)) i∈Z d . By definition, a trajectory of P is a space-time configuration x such that x t (i) = ϕ k (θ i x t− ) ∀ 0 ≤ k ≤ m, t ∈ P i,k , x t− (i) otherwise. (i, t) ∈ Z d × R (6.4) We have the following continuous-time equivalents of Lemmas 4 and 5. Lemma 37 (Minimal and maximal trajectories) Let P be a realization of the Poisson processes defined in (6.3). Then there exist trajectories x and x that are uniquely characterised by the property that each trajectory x of P satisfies x ≤ x ≤ x (pointwise). Lemma 38 (The lower and upper invariant laws) Let ϕ 0 , . . . , ϕ m be monotonic functions, let r 0 , . . . , r m be nonnegative rates, and let ν and ν denote the lower and upper invariant laws of the corresponding monotone interacting particle system. Let P = P (i,k) i∈Z d ,0≤k≤m be a family of independent Poisson processes, each with rate r k , and let X and X be the minimal and maximal trajectories of P. Then for each t ∈ R, the random variables X t and X t are distributed according to the laws ν and ν, respectively. We omit the proofs, as they go along the same lines as that of the discrete time statements. From now on, we fix a realization P of the Poisson processes such that the sets P i,k are pairwise disjoint. Recall the definition of A(ϕ k ) in (1.8). We fix an integer σ ≥ 2 and for each 1 ≤ k ≤ m and 1 ≤ s ≤ σ we choose a set A s (ϕ k ) ∈ A(ϕ k ). (6.5) Definition 39 A continuous Toom contour (V, E, v • , ψ) with σ charges is present in the realization of the Poisson processes P = P i,k i∈Z d ,0≤k≤m if: (i) ψ d+1 (v) ∈ P ψ(v),0 if and only if v ∈ V * , (ii) ψ d+1 (v) ∈ ∪ m k=1 P ψ(v),k for all v ∈ V hor ∪ (V • \{v • }), (iii) ψ d+1 (v) ∈ P ψ(v),k for some 1 ≤ k ≤ m such that A s (ϕ k ) = {(0, 0)} for all v ∈ V s ∩ V vert (1 ≤ s ≤ σ), (iv) P ψ(v),k ∩ ψ d+1 (w), ψ d+1 (v) = ∅ for all (v, w) ∈ E s such that w ∈ V vert and for all 1 ≤ k ≤ m such that (0, 0) / ∈ A s (ϕ k ) (1 ≤ s ≤ σ), (v) ψ(w) − ψ(v) ∈ A s (ϕ k ) if ψ d+1 (v) ∈ P ψ(v),k for some 1 ≤ k ≤ m, for all (v, w) ∈ E * with v ∈ V hor (1 ≤ s ≤ σ), (vi) ψ(w) − ψ(v) ∈ σ s=1 A s (ϕ k ) if ψ d+1 (v) ∈ P ψ(v),k for some 1 ≤ k ≤ m, for all (v, w) ∈ E • , where E • and E * are defined in (2.5). Condition (i) says that sinks and only sinks are mapped to defective points. Together with condition (iv) of Definition 35 of a continuous embedding this implies that we cannot encounter any defective point along a vertical segment of the contour. Condition (ii) says that vertices in V hor and sources (except for the root) are mapped to type k arrival points with 1 ≤ k ≤ m. As the other endpoint of the horizontal segment is not an arrival point, the consecutive segment must be vertical, furthermore, together with (i) this implies that there cannot be a defective point at either end of a horizontal segment. Condition (iii) says that internal vertices with charge s in V vert are mapped to type k arrival points with A s (ϕ k ) = {(0, 0)}. Condition (iv) says that we can only encounter type k arrival points with (0, 0) ∈ A s (ϕ k ) along a vertical segment in ψ( E s ) (1 ≤ s ≤ σ). Condition (v) says that if ψ((v, w)) is a horizontal segment such that v is an internal vertex with charge s or the root that is mapped into a type k arrival point (1 ≤ k ≤ m), then (v, w) is mapped to a pair of space-time points of the form (i, t), (i + j, t) with j ∈ A s (ϕ k ). Condition (vi) is similar, except that if v is a source different from the root, then we only require that j ∈ σ s=1 A s (ϕ k ). Again, we can strengthen this definition for the σ = 2 case. Definition 40 A continuous Toom contour (V, E, v • , ψ) with 2 charges is strongly present in the realization of the Poisson processes P = P i,k i∈Z d ,0≤k≤m if in addition to conditions (i)-(vi) of Definition 39, for each v ∈ V • \{v • } and w 1 , w 2 ∈ V with (v, w s ) ∈ E s,out (v) (s = 1, 2), one has: (vii) ψ(w i ) − ψ(v) ∈ A 3−i (ϕ k ) if ψ d+1 (v) ∈ P ψ(v),k for some 1 ≤ k ≤ m (i = 1, 2), (viii) ψ(w 1 ) = ψ(w 2 ). Our aim is to show that x 0 (0) implies the existence of a continuous Toom contour rooted at (0, 0) present in P. To that end, we define "connected components" of space-time points in state 0, that will play the role of explanation graphs in continuous time. We first define oriented paths on the space-time picture of the process. For each t ∈ P i,k (i ∈ Z d , 1 ≤ k ≤ m) such that x t (i) = 0 place an arrow (an oriented edge) pointing from (i, t) to each (j, t) ∈ A s (ϕ k ) such that x t (j) = 0 (1 ≤ s ≤ σ). It is easy to see that we place at least one arrow pointing to each set A s (ϕ k ), otherwise site i would flip to state 1 at time t. Furthermore, for each t ∈ P i,0 place a death mark at (i, t). A path moves in the decreasing time direction without passing through death marks and possibly jumping along arrows in the direction of the arrow. More precisely, it is a function γ : [t 1 , t 2 ] → Z d which is left continuous with right limits and satisfies, for all t ∈ (t 1 , t 2 ), t / ∈ P γ(t),0 and γ(t) = γ(t+) implies t ∈ P γ(t),k , γ(t+) − γ(t) ∈ A s (ϕ k ) and x t (γ(t+)) = 0 for some 1 ≤ k ≤ m, 1 ≤ s ≤ σ. We say that two points (i, t), (j, s) with t > s are connected by a path if there exists a path γ : [s, t] → Z d with γ(t) = i and γ(s) = j. Define Γ (i,t) := {(j, s) : (i, t) and (j, s) are connected by a path} (6.6) and Γ T (i,t) := Γ (i,t) ∩ Z d × [t − T, t] . If x 0 (0) = 0, then by the definition of the paths and arrows we have x s (j) = 0 for all (j, s) ∈ Γ (0,0) . Theorem 41 (Presence of a continuous Toom contour) Let ϕ 0 , . . . , ϕ m be monotonic functions where ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero, and let r 0 , . . . , r m be nonnegative rates. Let P be a realization of the Poisson processes defined in (6.3), and denote its maximal trajectory by x. Let σ ≥ 2 be an integer and for each 1 ≤ s ≤ σ and 1 ≤ k ≤ m, let A s (ϕ k ) ∈ A(ϕ k ) be fixed. Then, if Γ T (0,0) is bounded for all T > 0, x 0 (0) = 0 implies that with respect to the given choice of σ and the sets A s (ϕ k ), there is a continuous Toom contour (V, E, v • , ψ) rooted at (0, 0) present in P for σ ≥ 2, and strongly present in P for σ = 2. The monotone interacting particle systems we consider here have the property that Γ T is bounded for all T > 0 (see for example Chapter 4 of the lecture notes [Swart17] ), if m k=0 r k < ∞, m k=0 r k (\| ∪ A∈A A\| − 1) < ∞. (6.7) Proof As Γ T (0,0) is bounded, the set Γ T (0,0) ∩ ∪ i∈Z d ,0≤k≤m {i} × P i,k is finite for all T > 0, therefore we can order the arrival points in Γ (0,0) in decreasing order. Denote by (i l , t l ) its elements with 0 ≥ t 1 > t 2 > . . . , and let t 0 := 0. We define a monotonic flow φ in Z d+1 as follows. For all (i, t) ∈ Z d+1 we let φ (i,t) := ϕ k if (i, t) = (i l , −2l) for some t l ∈ P i l ,k (0 ≤ k ≤ m), ϕ id otherwise, (6.8) where ϕ id is the identity map defined in (1.6). Denoting by x the maximal trajectory of this monotonic flow, it is easy to see that x 0 (0) = 0, thus Theorem 7 implies the existence of a Toom contour (V , E , v • , ψ ) rooted at (0, 0) present in φ with respect to the given choice of σ and the sets A s (ϕ k ). We use this discrete-time contour to define the continuous-time one. For all v ∈ V such that ψ (v) = (i, −l) we let ψ(v) := ψ(w 1 ) if ∃w 1 , w 2 : (w 1 , v), (v, w 2 ) ∈ E and ψ (w 1 ) = ψ (w 2 ) = ψ (v), (i, t l/2 ) otherwise. (6.9) Recall that for v, w ∈ V , we write v E w when we can reach w from v through directed edges of E . We define W(v) := {w ∈ V : v E w and ψ(w) = ψ(v)} ∀v ∈ V . (6.10) Note that W(v) = {v} for all v ∈ V * . Set V := ∪ σ s=1 V s ∪ V • ∪ V * with V • := {W(v) : v ∈ V • }, V * := {W(v) : v ∈ V * }, V s := {W(v) : v ∈ V s \ ∪ w∈V • W(w)} (1 ≤ s ≤ σ). (6.11) For all W ∈ V we let ψ(W ) := ψ(w) for some w ∈ W . We further define E s := {(W 1 , W 2 ) ∈ V × V : ∃w i ∈ W i such that (w 1 , w 2 ) ∈ E s } (1 ≤ s ≤ σ). (6.12) Letting v • be the set W ⊂ V containing v • , we claim that (V, E, v • , ψ) is a continuous Toom contour rooted at (0, 0) present in P for σ ≥ 2, and strongly present in P for σ = 2. (See Figure 6 for an example of the construction.) Let us start with some simple observations. By definition, in φ at each height −2l (1 ≤ l ≤ n) there is exactly one site (i, −2l) such that φ (i,−2l) = ϕ id , every other site of Z d+1 applies identity map. By the construction of the Toom contour a site with the identity map cannot be the image of a source, furthermore any edge in ψ ( E ) starting at such a site is vertical. Any edge starting at a site with ϕ k (1 ≤ k ≤ m) has the form (i, t), (j, t − 1) for some t ∈ 2Z, i, j ∈ Z d . We call these edges diagonal, if i = j. Thus, ψ ( E ) is the union of vertical and diagonal edges, such that each diagonal edge points from an even height to an the charge is diverted at this point in a horizontal direction, so it is necessarily the endpoint of that vertical segment. Finally, conditions (v) and (vi) are immediate from conditions (iii) and (iv) of Definition 6. Since moreover (V , E , v • , ψ ) satisfies Definition 8 for σ = 2, the defining properties of Definition 40 hold for (V , E , v • , ψ ). Remark 42 We have observed before, that in the image under ψ of a type s charge (1 ≤ s ≤ σ) horizontal segments are always followed by vertical segments. The construction of the continuous Toom contour described above also ensures that vertical segments either end at a defective point, or are followed by a horizontal segment. Thus, starting from the image of the source, we have an alternating sequence of horizontal and vertical edges ending with a vertical edge at the image of the sink. Furthermore, if (0, 0) / ∈ ∪ m k=0 P 0,k , then φ (0,0) = ϕ id , so every (v • , w) ∈ ψ ( E ) is vertical. (6.9) then implies that every segment in the continuous contour starting at ψ(v • ) is also vertical. Explicit bounds Sexual contact process on Z d (d ≥ 1) Recall from Subsection 2.6 that we define A 1 := {0} and A 2 := {e 1 , . . . , e d } and we have A(ϕ coop,d ) = A 1 , A 2 . (6.15) We set σ := \|A(ϕ coop,d )\| = 2, and for the sets A s (ϕ k ) in (2.9) we make the choices A 1 (ϕ coop,d ) := A 1 , A 2 (ϕ coop,d ) := A 2 , (6.16) that is we have A s (ϕ 1 ) = A 1 only for s = 2. Let P = P (i,k) i∈Z d ,k=0,1 be a family of independent Poisson processes such that for each i P (i,0) has rate 1 and P (i,1) has rate λ. In line with the terminology used for contact processes, we will call type 0 arrival points death marks and type 1 arrivel points birth marks. Then Theorem 41 implies the Peierls bound: 1 − ρ = P[X 0 (0) = 0] ≤ P a Toom contour rooted at 0 is strongly present in P . In what follows, we give an upper bound on this probability. Definitions 39 and 40 imply that a continuous Toom contour is strongly present in P if and only if the following conditions are satisfied: (i) ψ(v) is a death mark for all v ∈ V * , (ii) ψ(v) is a birth mark for all v ∈ V hor , (iii) There are no death marks along vertical segments of ψ( E), (iv) There are no birth marks along vertical segments of ψ( E 2 ), (v) v ∈ V 2 ∪ V • for all v ∈ V hor , (vi) Horizontal and vertical segments alternate along each path between a source and a sink, (viia) If (v, w) ∈ E • 1 , then ψ (v, w) is a vertical segment, (viib) If (v, w) ∈ E • 2 , then ψ (v, w) is a horizontal segment, (viii) If (v, w) ∈ E with w ∈ V * , then ψ (v, w) is a vertical segment, where E • i is defined in (2.5). As horizontal segments cannot start at the image of a type 1 internal vertex and they alternate with vertical segments along each path between a source and a sink, this implies that the image of a type 1 charge starting at a source and ending at a sink is either a single vertical segment (that is there is no internal vertex along the path), or a horizontal segment followed by a vertical segment (that is there is exactly one internal vertex along the path). Furthermore, by Remark 42, P (1,λ) -a.s. the type 1 path starting at v • consists of a single vertical segment. We now can argue similarly as in the discrete time case in Section 5. If we reverse the direction of edges of charge 2, then the Toom graph becomes a directed cycle with edge set E 1 ∪ E 2 . We then call vertical segments in ψ( E 1 ) upward and in ψ( E 2 ) downward, and horizontal segments in ψ( E 1 ) outward and in ψ( E 2 ) inward. As \|A 2 \| = d we distinguish d types of outward and inward segments: we say that ψ (v, w) is type i, if \| ψ(w) − ψ(v)\| = e i . If (V, E, v • , ψ) is a continuous Toom contour rooted at 0 that is strongly present in P, then we can fully specify ψ by saying for each (v, w) ∈ E 1 ∪ E 2 whether ψ (v, w) is an upward, a downward, an outward or an inward segment, and its length in the former two and type in the latter two cases. In other words, we can represent the contour by a word of length n consisting of the letters from the alphabet {o 1 , . . . , o d , u, d, i 1 , . . . , i d }, which represents the different kinds of steps the cycle can take, and a vector l that contains the length of each vertical segment along the cycle in the order we encouner them. Then we can obtain a word consisting of these letters that must satisfy the following rules: • The first step is an upward segment. • Each outward segment must be immediately preceded by a downward segment and followed by an upward segment. • Between two occurrences of the string Do · U , and also before the first and after the last occurrence, we see a sequence of the string Di · of length ≥ 0. • The last step is a downward segment. Notice that the structure of a possible word is exactly the same as in (5.4). Then the contour in the bootom left of Figure 6 is described by the following word: • \|U * \| D • \|o 2 1 \|U * \| D • \|o 1 1 \|U * \| D 2 \|i 2 2 \| D 2 \|i 1 2 \|D • \|. (6.18) For any continuous Toom contour T denote by W (T ) the corresponding word satisfying these rules and by W the set of all possible words satisfying these rules. We then can bound P[X 0 (0) = 0] ≤ W ∈W P a Toom contour T with W (T ) = W rooted at 0 is strongly present in P . (6.19) From this point on, we can count the number of possible words and assign probabilities to each following the same line of thought (adapted to continuous time) as in Section 5 for the discrete-time monotone cellular automaton that applies the cooperative branching and the identity map. We then recover the following Peierls bound: P[X 0 (0) = 0] ≤ 1 1 + λ ∞ i=0 16dλ (2d + 1)λ + 1 (λ + 1) 3 i . (6.20) The argument is similar to that of [Gra99, Lemma 8 and 9]. Presenting it would be long and technical, but not particularly challenging, so we will skip it. As we have mentioned earlier, we can think of this process as the limit of the random cellular automaton with time steps of size ε where the maps ϕ 0 , ϕ coop,d and ϕ id are applied with probabilities ε, ελ, and 1 − ε(1 + λ), respectively. Observe that we recover the exact same Peierls bound by substituting p = ε, r = ελ, and q = 1 − ε(1 + λ) into (5.10) and letting ε → 0. In particular for d = 1 we obtain the bound In this section, we explain how explanation graphs, whose definition looks somewhat complicated at first sight, naturally arise from a more elementary concept, which we will call a minimal explanation. Our definition of a minimal explanation will be similar to, though different from the definition of John Preskill [Pre07]. We introduce minimal explanations in Subsection 7.1 and then discuss their relation to explanation graphs in Subsection 7.2. where 1 A denotes the indicator function of A and hence 1 − 1 Z is the configuration that is zero on Z and one elsewhere. Clearly, A ↑ (ϕ) is an increasing set in the sense that A ↑ (ϕ) A ⊂ A implies A ∈ A(ϕ). Likewise Z ↑ (ϕ) is increasing. We say that an element A ∈ A ↑ (ϕ) is minimal if A, A ∈ A ↑ (ϕ) and A ⊂ A imply A = A. We define minimal elements of Z ↑ (ϕ) in the same way and set For monotonic maps ϕ and ϕ defined on {0, 1} Z d , we write ϕ ≤ ϕ if ϕ(x) ≤ ϕ (x) ∀x ∈ {0, 1} Z d . Moreover, we write ϕ ≺ ϕ if and only if Z(ϕ) ⊂ Z(ϕ ). Finite explanations (7.7) Note that ϕ ≺ ϕ implies that ϕ ≥ ϕ . For monotonic flows φ and ψ, we write φ ≤ ψ (resp. φ ≺ ψ) if φ (i,t) ≤ ψ (i,t) (resp. φ (i,t) ≺ ψ (i,t) ) for all (i, t) ∈ Z d+1 . We let x φ denote the maximal trajectory of a monotonic flow φ. By definition, a finite explanation for (0, 0) is a monotonic flow ψ such that: (i) x ψ 0 (0) = 0, (ii) ψ (i,t) = ϕ 1 for finitely many (i, t) ∈ Z d+1 . By definition, a minimal explanation for (0, 0) is a finite explanation ψ that is minimal with respect to the partial order ≺, i.e., ψ has the property that if ψ is a finite explanation for (0, 0) such that ψ ≺ ψ, then ψ = ψ. Lemma 43 (Existence of a minimal explanation) Let φ be a monotonic flow. Then x φ 0 (0) = 0 if and only if there exists a minimal explanation ψ for (0, 0) such that ψ ≺ φ. Proof Assume that there exists a minimal explanation ψ for (0, 0) such that ψ ≺ φ. Then ψ ≥ φ and hence 0 = x ψ 0 (0) ≥ x φ 0 (0). To complete the proof, we must show that conversely, x φ 0 (0) = 0 implies the existence of a minimal explanation ψ for (0, 0) such that ψ ≺ φ. We first prove the existence of a finite explanation ψ for (0, 0) such that ψ ≺ φ. For each s ∈ Z, we define x s as in (3.1). Then (3.2) implies that x −n 0 (0) = 0 for some 0 ≤ n < ∞. For each (i, t) ∈ Z d+1 , let U (i, t) := (j, t − 1) : j ∈ A for some A ∈ A(φ (i,t) ) (7.8) denote the set of "ancestors" of (i, t). For any Z ⊂ Z d+1 , we set U (Z) := {U (z) : z ∈ Z} and we define inductively U 0 (Z) := Z and U k+1 (Z) := U (U k (Z)) (k ≥ 0). Then n k=0 U k (0, 0) is a finite set. Since x −n 0 (0) = 0, it follows that setting ψ (i,t) := φ (i,t) if (i, t) ∈ n k=0 U k (0, 0), ϕ 1 otherwise (7.9) defines a finite explanation ψ for (0, 0) such that ψ ≺ φ. We observe that for a given monotonic map ϕ, there are only finitely many monotonic maps ϕ such that ϕ ≺ ϕ. Also, since Z(ϕ 1 ) = ∅, the only monotonic map ϕ such that ϕ ≺ ϕ 1 is ϕ = ϕ 1 . Therefore, since ψ (i,t) = ϕ 1 for finitely many (i, t) ∈ Z d+1 , there exist ony finitely many monotonic flows ψ such that ψ ≺ ψ. It follows that the set of all finite explanations ψ for (0, 0) that satisfy ψ ≺ ψ must contain at least one minimal element, which is a minimal explanation for (0, 0) such that ψ ≺ φ. The following lemma gives a more explicit description of minimal explanations. In Figure 3 on the right, a minimal explanation ψ for (0, 0) is drawn with ψ ≺ φ, where φ is a monotonic flow that takes values in {ϕ 0 , ϕ coop }. For each (i, t) ∈ Z d+1 such that ψ (i,t) = ϕ 1 , thick black lines join (i, t) to the points (j, t − 1) with j ∈ Z (i,t) , where Z (i,t) is the set defined in point (v) below. Orange stars indicate points (i, t) where ψ (i,t) = ϕ 0 . The minimal explanation drawn in Figure 3 has the special property that even if we replace ψ (i,t) by φ (i,t) in all points except for the defective points of φ, then it is still true that removing any of the defective points of ψ results in the origin having the value one. This means that the set of defective points drawn in Figure 3 corresponds to a "minimal explanation" in the sense defined by John Preskill in [Pre07], which is a bit stronger than our definition. Lemma 44 (Minimal explanations) Let ψ be a finite explanation for (0, 0). Then ψ is a minimal explanation for (0, 0) if and only if in addition to conditions (i)-(iii) of the definition of a finite explanation, one has: (iv) ψ (i,t) = ϕ 1 for all (i, t) ∈ Z d+1 \{(0, 0)} such that t ≥ 0, (v) for each (i, t) ∈ Z d+1 such that ψ (i,t) = ϕ 1 , there exists a finite Z (i,t) ⊂ Z d such that Z(ψ (i,t) ) = {Z (i,t) }, (vi) for each (i, t) ∈ Z d+1 \{(0, 0)} such that ψ (i,t) = ϕ 1 , there exists a j ∈ Z d such that ψ (j,t+1) = ϕ 1 and i ∈ Z (j,t+1) . Moreover, each minimal explanation ψ for (0, 0) satisfies: (vii) x ψ t (i) = 0 for each (i, t) ∈ Z d+1 such that ψ (i,t) = ϕ 1 , Proof We first show that a finite explanation ψ for (0, 0) satisfying (iv)-(vi) is minimal. By our definition of minimal explanations, we must check that if ψ is a finite explanation such that ψ ≺ ψ, then ψ = ψ. Imagine that conversely, ψ (i,t) = ψ (i,t) for some (i, t) ∈ Z d+1 . Then by (v) and the fact that ψ (i,t) ≺ ψ (i,t) , we must have that Z(ψ (i,t) ) = ∅ and hence ψ (i,t) = ϕ 1 . Since ψ (i,t) = ψ (i,t) , it follows that ψ (i,t) = ϕ 1 . By (iv), this implies that either (i, t) = (0, 0) or t < 0. Let n := −t. Using (vi), we see that there exist i = i 0 , . . . , i n such that ψ (i k ,t+k) = ϕ 1 (0 ≤ k ≤ n) and i k−1 ∈ Z (i k ,t+k) (0 < k ≤ n). By (iv), we must have i n = 0. Since ψ (i,t) = ϕ 1 we have x ψ t (i) = 1. Using the fact that ψ ≺ ψ and i k−1 ∈ Z (i k ,t+k) (0 < k ≤ n), it follows that x ψ t+k (i k ) = 1 for all 0 ≤ k ≤ n. In particular, this shows that x ψ 0 (0) = 1, contradicting the fact that ψ is a finite explanation for (0, 0). We next show that each minimal explanation ψ for (0, 0) satisfies (iv)-(vii). Property (iv) follows from the fact that if ψ (i,t) = ϕ 1 for some (i, t) ∈ Z d+1 \{(0, 0)} such that t ≥ 0, then setting ψ (i,t) := ϕ 1 and ψ (j,s) := ψ (j,s) for all (j, s) = (i, s) defines a finite explanation ψ ≺ ψ. Property (vii) follows in the same way: if x ψ t (i) = 1 for some (i, t) ∈ Z d+1 such that ψ (i,t) = ϕ 1 , then we can replace ψ (i,t) by ϕ 1 without changing the fact that ψ is a finite explanation. To prove (v), we first observe that if ψ (i,t) = ϕ 1 for some (i, t) ∈ Z d+1 , then x ψ t (i) = 0 by (vii). It follows that there exists some Z ∈ Z(ψ (i,t) ) such that x ψ t−1 (j) = 0 for all j ∈ Z. (Note that this in particular includes the case that ψ (i,t) = ϕ 0 and Z(ψ (i,t) ) = {∅}.) If Z(ψ (i,t) ) contains any other elements except for Z, then we can remove these without changing the fact that ψ is a finite explanation. Therefore, by minimality, we must have Z(ψ (i,t) ) = {Z}, proving (v). To prove (vi), finally, assume that (i, t) ∈ Z d+1 \{(0, 0)} and ψ (i,t) = ϕ 1 , but there does not exist a j ∈ Z d such that ψ (j,t+1) = ϕ 1 and i ∈ Z (j,t+1) . Then we can replace ψ (i,t) by ϕ 1 without changing the fact that ψ is a finite explanation, which contradicts minimality. This completes the proof. Explanation graphs revisited We claim that in the proof of many of our results, such as Theorems 7 and 9, we can without loss of generality assume that A(ϕ k ) = A s (ϕ k ) : 1 ≤ s ≤ σ (1 ≤ k ≤ m). (7.10) To see this, let φ be a monotonic flow on {0, 1} Z d taking values in {ϕ 0 , . . . , ϕ m }, where ϕ 0 = ϕ 0 is the constant map that always gives the outcome zero and ϕ 1 , . . . , ϕ m are non-constant. Let σ ≥ 2 be an integer and for each 1 ≤ k ≤ m and 1 ≤ s ≤ σ, fix A s (ϕ k ) ∈ A(ϕ k ). We let φ * = (φ * (i,t) ) (i,t)∈Z d+1 denote the image of φ under the map from {ϕ 0 , . . . , ϕ m } to {ϕ * 0 , . . . , ϕ * m } defined by setting ϕ * 0 := ϕ 0 and ϕ * k (x) := σ s=1 i∈As(ϕ k ) x(i) 1 ≤ k ≤ m, x ∈ {0, 1} Z d . (7.11) We set A s (ϕ * k ) := A s (ϕ k ) (1 ≤ k ≤ m, 1 ≤ s ≤ σ). We make the following simple observations. Lemma 45 (Modified monotonic flow) The modified monotonic flow φ * has the following properties: (i) φ * satisfies (7.10), (ii) φ * ≥ φ, (iii) x φ * ≥ x φ , (iv) an explanation graph is present in φ * if and only if it is present such that ψ ≺ φ, (v) a Toom contour is (strongly) present in φ * if and only if it is present such that ψ ≺ φ. Proof Property (iii) is a direct consequence of (ii) and all other properties follow directly from the definitions. Because of Lemma 45, in the proof of results such as Theorems 7 and 9 about the (strong) presence of Toom contours or Lemma 22 about the presence of an explanation graph, we can without loss of generality assume that (7.10) holds. Indeed, by part (iii) of the lemma, x φ 0 (0) = 0 implies x φ * 0 (0) = 0 so replacing φ by φ * , in view of parts (iv) and (v), it suffices to prove the presence of an explanation graph or the (strong) presence of a Toom contour in φ * . We now come to the main subject of this subsection, which is to link minimal explanations to explanation graphs. We start with a useful observation. Lemma 46 (Presence of an explanation graph) Assume that φ satisfies (7.10). Then properties (ii) and (iii) of Definition 21 imply property (i). Proof Property (ii) of Definition 21 implies that x t (i) = 0 ∀(i, t) ∈ U * . (7.12) We next claim that for (i, t) ∈ U \U * , x t−1 (j) = 0 ∀ (i, t), (j, t − 1) ∈ H implies x t (i) = 0. (7.13) Indeed, if x t−1 (j) = 0 for all (i, t), (j, t − 1) ∈ H, then by property (iii) of Definition 21, for each 1 ≤ s ≤ σ, there is a k ∈ A s (φ (i,t) ) such that x t−1 (i + k) = 0, which by (7.10) implies that x t (i) = 0. Define inductively U 0 := U * and U n+1 := {u ∈ U : v ∈ U n ∀(u, v) ∈ H}. Then (7.12) and (7.13) imply that x t (i) = 0 for all (i, t) ∈ ∞ n=0 U n = U . We now make the link between minimal explanations and the presence of explanation graphs as defined in Definitions 20 and 21. As before, φ is a monotonic flow on {0, 1} Z d taking values in {ϕ 0 , . . . , ϕ m }, where ϕ 0 = ϕ 0 and ϕ 1 , . . . , ϕ m are non-constant. Moreover, we have fixed an integer σ ≥ 2 and for each 1 ≤ k ≤ m and 1 ≤ s ≤ σ, we have fixed A s (ϕ k ) ∈ A(ϕ k ). Lemma 47 (Minimal explanations and explanation graphs) Assume that φ satisfies (7.10) and that ψ is a minimal explanation for (0, 0) such that ψ ≺ φ. For each (i, t) ∈ Z d+1 such that ψ (i,t) = ϕ 1 , let Z (i,t) be as in point (v) of Lemma 44. Then there is an explanation graph (U, H) for (0, 0) present in φ such that: U = (i, t) ∈ Z d+1 : ψ (i,t) = ϕ 1 , U * = (i, t) ∈ U : ψ (i,t) = ϕ 0 , and H = (i, t), (j, t − 1) : (i, t) ∈ U, j ∈ Z (i,t) . (7.14) Proof Let U and U * be defined by (7.14). Let (i, t) ∈ U \U * . Since ψ ≺ φ we have Z(ψ (i,t) ) ⊂ Z(φ (i,t) ) and hence Z (i,t) ∈ Z(φ (i,t) ), so by (7.6), for each 1 ≤ s ≤ σ, we can choose some j s (i, t) ∈ Z (i,t) ∩ A s (φ (i,t) ). We claim that Z (i,t) = {j 1 (i, t), . . . , j σ (i, t)}. To see this, set Z (i,t) := {j 1 (i, t), . . . , j σ (i, t)}. Then Z (i,t) ⊂ Z (i,t) and (7.10) implies that Z (i,t) ∩ A = ∅ for all A ∈ A(φ (i,t) ), which by (7.6) implies that Z (i,t) ∈ Z ↑ (φ (i,t) ). By (7.2), Z (i,t) is a minimal element of Z ↑ (φ (i,t) ), so we conclude that Z (i,t) = Z (i,t) . We claim that setting H s := (i, t), (j s (i, t), t − 1) : (i, t) ∈ U \U * (1 ≤ s ≤ σ) (7.15) now defines an explanation graph that is present in φ. Properties (i), (ii), (iv) and (v) of Definition 20 follow immediately from our definitions and the fact that ψ (0,0) = ϕ 1 since ψ is a mininal explanation for (0, 0). Property (iii) follows from Lemma 44 (vi). This proves that (U, H) is an explanation graph. To see that (U, H) is present in φ, we must check conditions (i)-(iii) of Definition 21. Condition (i) follows from Lemma 44 (vii) and conditions (ii) and (iii) are immediate from our definitions. Discussion As before, let φ be a monotonic flow on {0, 1} Z d taking values in {ϕ 0 , . . . , ϕ m }, where ϕ 0 = ϕ 0 and ϕ 1 , . . . , ϕ m are non-constant. Let σ ≥ 2 and for each 1 ≤ k ≤ m and 1 ≤ s ≤ σ, let A s (ϕ k ) ∈ A(ϕ k ) be fixed. Consider the following conditions: (i) x φ 0 (0) = 0, (ii) there exists a minimal explanation ψ for (0, 0) such that ψ ≺ φ, (iii) there is an explanation graph (U, H) for (0, 0) present in φ, (iv) there is a Toom contour (V, E, v • , ψ) rooted at (0, 0) present in φ. Theorem 7 and Lemmas 22 and 43 say that conditions (i)-(iii) are equivalent and imply (iv). As the example in Figure 3 showed, (iv) is strictly weaker than the other three conditions. This raises the question whether it is possible to prove Toom's stability theorem using a Peierls argument based on minimal explanations, as suggested in [Pre07]. Let us say that (i, t) is a defective site for a finite explanation ψ if ψ (i,t) = ϕ 0 . Let ϕ be an eroder and let M n denote the number of minimal explanations ψ for (0, 0) with n defective sites that satisfy ψ (i,t) ≺ ϕ whenever (i, t) is not defective. We pose the following open problem: Do there exist finite constants C, N such that M n ≤ CN n (n ≥ 0)? If the answer to this question is affirmative, then it should be possible to set up a Peierls argument based on minimal explanations, rather than Toom contours. In principle, such an argument has the potential to be simpler and more powerful than the Peierls arguments used in this article, but as we have seen the relation between minimal explanations and Toom contours is not straightforward and finding a good upper bound on the number of minimal explanations with a given number of defective sites seems even harder than in the case of Toom contours. [X n (i) = 1] = ν(dx)x(i) and ρ := lim n→∞ P 1 [X n (i) = 1] = ν(dx)x(i) (1.4) denote the intensities of the lower and upper invariant laws. Completely analogue statements hold in the continuous-time setting [Lig85, Thm III.2.3 For concreteness, to have some examples at hand, we consider three further, nontrivial examples of monotonic functions. For simplicity, we restrict ourselves to two dimensions. We will be interested in the functions ϕ NEC (x) := round (x(0, 0) + x(0, 1) + x(1, 0))/3 , ϕ NN (x) := round (x(0, 0) + x(0, 1) + x(1, 0) + x(0, −1) + x(−1, 0))/5 , ϕ coop (x) := x(0, 0) ∨ x(0, 1) ∧ x(1, 0) , (1.7) Figure 1 : 1Density ρ of the upper invariant law of two monotone cellular automata as a function of the parameters, shown on a scale from 0 (white) to 1 (black). On the left: a version of Figure 2 : 2Example of a Toom graph with three charges. Sources and sinks are indicated with solid dots and internal vertices are indicated with open dots. Note the isolated vertex in the lower right corner, which is a source and a sink at the same time. Figure 3 : 3On the left: a Toom graph with two charges. Middle: embedding of the Toom graph on the left, with time running downwards. The connected component containing the root v • forms a Toom contour rooted at the origin (0, 0, 0) 0), (0, 1)}, A 2 := {(0, 0), (1, 0)}, and A 3 := {(0, 1), (1, 0)}. Lemma 22 ( 22Presence of an explanation graph) The maximal trajectory x of a monotonic flow φ satisfies x 0 (0) = 0 if and only if there is an explanation graph (U, H) for (0, 0) present in φ. {v i } ∪ P s (a i,s ) = a 0 i,s , . . . , a m(i,s) i,s ,(3.14) Figure 4 : 4The process of exploration and loop erasion. Lemma 33 (The eroder property) If a non-constant monotonic function ϕ : {0, 1} Z d → {0, 1} satisfies (2.12), then ϕ is an eroder. Figure 5 : 5The Toom cycle ψ described in the proof of Proposition 18. so we can choose r sufficiently large such that Let T be a Toom contour strongly present in Φ rooted at 0. Then W (T ), l + (T ) and l − (T ) uniquely determine (V, E, 0, ψ). j) denote the number of different words that have i inward steps and j inward steps with the superscript • made from the alphabet {o 1 , . . . , o d , U, D, i 1 , . . . , i d , i • 1 , . . . , i • d } that satisfy our rules. L (W) := {(l + (T ), l − (T )) : W (T ) = W} (5.11) Recall that by Claim 1 W (T ) = W, l + (T ) and l − (T ) uniquely specify the Toom contour T . We then have T ∈T 0 :W (T )=W P T is strongly present in Φ ≤ p i+1 r 2i−j (l + ,l − )∈L(W) q l + 1 −i . (5.12) Figure 6 : 6Top left: A realization of P that applies the maps ϕ 0 and ϕ coop with rates r 0 and r 1 respectively. The points marked with a star are defective, ensuring that the origin (0,0,0) is in state 0. The connected component Γ (0,0,0) of the origin is marked by black. Right: The monotone cellular automaton φ defined in (6.8) and the corresponding Toom contour rooted at (0,0,0). The sites marked with a star and open dot apply ϕ 0 and ϕ coop respectively, every other site applies the identity map. The origin in state zero. Middle: The Toom graph corresponding to the Toom contour on the right. The green sets correspond to the vertices of the Toom graph of the continuous contour, defined in (6.11). Bottom left: The Toom contour corresponding to the realization of P on the top left. λ c ( 1 ) 1≤ 49.3242 . . . , (6.21) and for d = 2 the bound λ c (2) ≤ 161.1985 . . . . of Theorem 7 started with Lemma 22, which shows that if x 0 (0) = 0, then there is an explanation graph present in φ, in the sense of Definitions 20 and 21. each monotonic map ϕ : {0, 1} Z d → {0, 1}, we define A ↑ (ϕ) := A ⊂ Z d : ϕ(1 A ) = 1 , Z ↑ (ϕ) := Z ⊂ Z d : ϕ(1 − 1 Z ) = 0 , (7.1) A (ϕ) := A ∈ A ↑ (ϕ) : A is minimal and Z(ϕ) := Z ∈ Z ↑ (ϕ) : Z is minimal . (7.2)Since monotonic maps are local (i.e., depend only on finitely many coordinates), it is not hard to see thatA ↑ (ϕ) := A ⊂ Z d : A ⊃ A for some A ∈ A(ϕ) , Z ↑ (ϕ) := Z ⊂ Z d : Z ⊃ Z for some Z ∈ Z(ϕ). , our present definition of A(ϕ) coincides with the one given in (1.8). We note that A(ϕ 0 ) = ∅ and A(ϕ 1 ) = {∅}, and similarly Z(ϕ 0 ) = {∅} and Z(ϕ 1 ) = ∅. One hasA ∈ A ↑ (ϕ) if and only if A ∩ Z = ∅ ∀Z ∈ Z ↑ (ϕ), (7.5) and by (7.3) the same is true with Z ↑ (ϕ) replaced by Z(ϕ). Similarly, Z ∈ Z ↑ (ϕ) if and only if Z ∩ A = ∅ ∀A ∈ A(ϕ). (7.6) Unfortunately, theirFigure 6contains a small mistake, in the form of an arrow that should not be there. In particular,[Gac95] needs p < 2 −21 3 −8 . Lemmas 12 and 13 are similar to [Too80, Lemmas 1 and 2]. The main difference is that in Toom's construction, the number of incoming edges of each charge equals the number of outgoing edges of that charge at all vertices of the contour, i.e., there are no sources and sinks. Since Ponselet discusses stability of the all-zero fixed point while we discuss stability of the all-one fixed point, in[Pon13], the roles of zeros and ones are reversed compared to our conventions. N n ≤ n σ/2 −1 ( 1 2 σ + 1) (σ+1)n−1 M σn . (4.7)5 This is a simple variation of the "Bridges of Königsberg" problem that was solved by Euler. AcknowledgmentWe thank Anja Sturm who was involved in the earlier phases of writing this paper for her contributions to the discussions. We thank Ivailo Hartarsky for useful discussions. The first author is supported by grant 20-08468S of the Czech Science Foundation (GA CR). The second and third authors are supported by ERC Starting Grant 680275 "MALIG". ∈ ψ (V s ∪ V * ) for each (i, t), (j, t − 1) ∈ ψ ( E s ) with i = j (1 ≤ s ≤ σ). , t) / ∈ ψ (V s ∪ V * ) for each (i, t), (j, t − 1) ∈ ψ ( E s ) with i = j (1 ≤ s ≤ σ). Assume that (j, t) = ψ (v) for some v ∈ V s , then there is a w ∈ V s such that (v, w) ∈ E and ψ ((v, w)) is vertical. This means that ψ (w) = (j, t − 1), that is a type s vertex overlaps with another type s vertex, contradicting property (iii) of Definition 2. Let us now examine the image of (V , E ) under ψ. By definition, for each (v, w) ∈ E such that ψ ((v, w)) is diagonal we have ψ d+1 (v) = ψ d+1 (w). Furthermore, ψ(v) = ψ (v) for all v ∈ V , implying that ψ( E ) is the union of horizontal and vertical segments. Observe that for any sequence of vertices v 1 , . . . , v n ∈ V s (1 ≤ s ≤ σ) such that ψ ((v i , v i+1 )) is vertical for each 1 ≤ i ≤ n − 1 the embedding ψ maps v 2. As ϕ (i,t) = ϕ id , we must have ϕ (j,t) = ϕ id , furthermore, we have the identity map at every , clearly (j, t) / ∈ ψ. w)) = (ψ(v), ψ(w)) = ∅ if ψ(v) = ψ(w), we have ψ(V • ) = ψ(V • ), ψ(v • ) = ψ(v • ), ψ(V * ) = ψ(V * ), ψ. V s ) = ψ(V s ) \ ψ(V • ), ψ( E s ) = ψ( E s ), (1 ≤ s ≤ σAs ϕ (i,t) = ϕ id , we must have ϕ (j,t) = ϕ id , furthermore, we have the identity map at every , clearly (j, t) / ∈ ψ (V * ). Assume that (j, t) = ψ (v) for some v ∈ V s , then there is a w ∈ V s such that (v, w) ∈ E and ψ ((v, w)) is vertical. This means that ψ (w) = (j, t − 1), that is a type s vertex overlaps with another type s vertex, contradicting property (iii) of Definition 2. Let us now examine the image of (V , E ) under ψ. By definition, for each (v, w) ∈ E such that ψ ((v, w)) is diagonal we have ψ d+1 (v) = ψ d+1 (w). Furthermore, ψ(v) = ψ (v) for all v ∈ V , implying that ψ( E ) is the union of horizontal and vertical segments. Observe that for any sequence of vertices v 1 , . . . , v n ∈ V s (1 ≤ s ≤ σ) such that ψ ((v i , v i+1 )) is vertical for each 1 ≤ i ≤ n − 1 the embedding ψ maps v 2 , . . . , v n−1 that, with the convention that ψ((v, w)) = (ψ(v), ψ(w)) = ∅ if ψ(v) = ψ(w), we have ψ(V • ) = ψ(V • ), ψ(v • ) = ψ(v • ), ψ(V * ) = ψ(V * ), ψ(V s ) = ψ(V s ) \ ψ(V • ), ψ( E s ) = ψ( E s ), (1 ≤ s ≤ σ). is diagonal or v ∈ V • we have φ ψ (v) = ϕ k (1 ≤ k ≤ m), thus ψ d+1 (v) is an arrival point of P ψ(v),k . Finally, for each v ∈ V * we have φ ψ (v) = ϕ 0 , thus ψ d+1 (v) is a defective point. For any (v, w) ∈ E such that ψ. We are now ready to show that (V, E, v • , ψ) is a continuous Toom contour rooted at (0, 0)For any (v, w) ∈ E such that ψ ((v, w)) is diagonal or v ∈ V • we have φ ψ (v) = ϕ k (1 ≤ k ≤ m), thus ψ d+1 (v) is an arrival point of P ψ(v),k . Finally, for each v ∈ V * we have φ ψ (v) = ϕ 0 , thus ψ d+1 (v) is a defective point. We are now ready to show that (V, E, v • , ψ) is a continuous Toom contour rooted at (0, 0). As ψ satisfies Definition 2, its properties (ii) and (iii) together with (6.9) and (6.13) easily yield conditions (ii) and (iii). Finally, assume that (iv) internal vertex of type s. This contradicts conditions (ii) and (iii) of Definition 2, therefore condition (iv) must hold. By Defintion 3 and the definition of ψ we have ψ d+1 (v) ≤ 0 for all v ∈ V (hence for all v ∈ V as well), and ψ(v • ) = ψ(v • ) = (0, 0). By (6.11) any vertex v ∈ V. We have already seen that ψ satisfies condition (i) of Definition 35 of a continuous embedding. such that ψ(v) = (0, 0) is contained in some W ∈ V • , thus (V, E, v • , ψ) satisfies the defining property of Definition 36 of a continuous Toom contour rooted at v • . We are left to show that this contour is (strongly) present in P. As (V , E , v • , ψ ) is a Toom contour rooted at (0, 0) present in φ, it satisfies Definition 6. We now check the conditions of Definition 39. We have already seen that conditions (i) and (ii) hold. Condition (iiiAs (V , E ) is a Toom graph, it is straightforward to check that (V, E) is a Toom graph as well. We have already seen that ψ satisfies condition (i) of Definition 35 of a continuous embedding. As ψ satisfies Definition 2, its properties (ii) and (iii) together with (6.9) and (6.13) easily yield conditions (ii) and (iii). Finally, assume that (iv) internal vertex of type s. This contradicts conditions (ii) and (iii) of Definition 2, therefore condition (iv) must hold. By Defintion 3 and the definition of ψ we have ψ d+1 (v) ≤ 0 for all v ∈ V (hence for all v ∈ V as well), and ψ(v • ) = ψ(v • ) = (0, 0). By (6.11) any vertex v ∈ V such that ψ(v) = (0, 0) is contained in some W ∈ V • , thus (V, E, v • , ψ) satisfies the defining property of Definition 36 of a continuous Toom contour rooted at v • . We are left to show that this contour is (strongly) present in P. 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[ "Exceptional points in the scattering continuum", "Exceptional points in the scattering continuum" ]	[ "J Oko Lowicz ¶ \nInstitute of Nuclear Physics\nPolish Academy of Sciences\nRadzikowskiego 152PL-31342KrakówPoland\n", "M P Loszajczak $ \nGrand Accélérateur National d'Ions Lourds (GANIL)\nCEA/DSM\nCNRS/IN2P3\nBP 5027F-14076Caen Cedex 05France\n" ]	[ "Institute of Nuclear Physics\nPolish Academy of Sciences\nRadzikowskiego 152PL-31342KrakówPoland", "Grand Accélérateur National d'Ions Lourds (GANIL)\nCEA/DSM\nCNRS/IN2P3\nBP 5027F-14076Caen Cedex 05France" ]	[]	The manifestation of exceptional points in the scattering continuum of atomic nucleus is studied using the real-energy continuum shell model. It is shown that low-energy exceptional points appear for realistic values of coupling to the continuum and, hence, could be accessible experimentally. Experimental signatures are proposed which include the jump by 2π of the elastic scattering phase shift and a salient energy dependence of cross-sections in the vicinity of the exceptional point.	10.1103/physrevc.80.034619	[ "https://arxiv.org/pdf/0909.2766v1.pdf" ]	119,221,451	0909.2766	33904b60fd087791a8b457c1071a4b286d4683ef	Exceptional points in the scattering continuum 15 Sep 2009 J Oko Lowicz ¶ Institute of Nuclear Physics Polish Academy of Sciences Radzikowskiego 152PL-31342KrakówPoland M P Loszajczak $ Grand Accélérateur National d'Ions Lourds (GANIL) CEA/DSM CNRS/IN2P3 BP 5027F-14076Caen Cedex 05France Exceptional points in the scattering continuum 15 Sep 2009(Dated: September 15, 2009)numbers: number(s): 2570Ef0365Vf2160Cs2540Cm2540Ep The manifestation of exceptional points in the scattering continuum of atomic nucleus is studied using the real-energy continuum shell model. It is shown that low-energy exceptional points appear for realistic values of coupling to the continuum and, hence, could be accessible experimentally. Experimental signatures are proposed which include the jump by 2π of the elastic scattering phase shift and a salient energy dependence of cross-sections in the vicinity of the exceptional point. INTRODUCTION The structure of loosely bound and unbound nuclei is strongly impacted by many-body correlations and nonperturbative coupling to the external environment of scattering states and decay channels [1,2]. This is particularly important in exotic nuclei where new phenomena, at the borderline of nuclear structure and nuclear reactions, are expected. Some of them, like the halos [3], the segregation of time scales in the context of non-Hermitian Hamiltonians [4], the alignment of near-threshold states with decay channels [5], and the resonance crossings [6,7] appear in various open mesoscopic systems. Their universality is the consequence of the non-Hermitian nature of an eigenvalue problem in open quantum systems. Resonances are commonly found in quantum systems, independently of their interactions, building blocks and energy scales involved. Much interest is concentrated on resonance degeneracies, the so-called exceptional points (EPs) [6]. Their connection to avoided crossings and spectral properties of Hermitian systems [8,9] as well as the associated geometric phases have been discussed in simple models in considerable detail [10]. The interesting question is their manifestation in nuclear scattering experiments. Here, a much studied case was the 2 + doublet in 8 Be [11,12,13,14,15]. Based on this example, von Brentano [16] discussed the width attraction for mixed resonances, and Hernandéz and Mondragón [17] showed that the true crossing of resonances can be obtained by the variation of two parameters in the Jordan block of rank two. In this latter analysis, it was shown that the resonating part of the scattering matrix (S-matrix) for one open channel and two internal states is compatible with the two-level formula of the R-matrix theory used in the experimental analysis of excitation functions of elastic scattering 4 He(α, α 0 ) 4 He [15] and, hence, the 2 + doublet in 8 Be may actually be close to the true resonance degeneracy. Properties of atomic nucleus around the continuum threshold change rapidly with the nucleon number, the excitation energy and the coupling to the environment of scattering states. A consistent description of the interplay between scattering and resonant states requires an open system formulation of the nuclear shell model (see [1,2,18] for recent reviews). The real-energy continuum shell model [19,20,21] provides a suitable unified framework with the help of an effective non-Hermitian Hamiltonian. In this work, for the first time we focus on a realistic model of an unbound atomic nucleus to see whether one or more EPs can appear in the low energy continuum for sensible parameters of the open quantum system Hamiltonian. In particular, we discuss possible experimental signatures of the EPs and show the evolution of these signatures in the vicinity of the EP. Finally, on the example of spectroscopic factors we demonstrate the entanglement of resonance wave functions close to the EP. FORMULATION OF THE CONTINUUM SHELL MODEL Let us briefly review the Shell Model Embedded in the Continuum (SMEC) [21], which is a recent realization of the real-energy continuum shell model. The total function space of an A−particle system consists of the set of square-integrable functions Q ≡ {ψ A i }, used in the standard nuclear Shell Model (SM), and the set of embedding scattering states P ≡ {ζ c E }. These two sets are obtained by solving the Schrödinger equation, separately for discrete (SM) states (the closed quantum system) and for scattering states (the environment). Decay channels 'c' are determined by the motion of an unbound particle in a state l j relative to the A − 1 nucleus with all nucleons on bounded single-particle (s.p.) orbits in the SM eigenstate ψ A−1 j . Using these function sets, one defines projection operators: i } which contains the continuation of any SM eigenfunction ψ A i in P, and then construct the complete solution in Q + P [1]. Recently, this approach has been extended to describe the two-proton radioactivity with the twoparticle continuum [22]. Q = N i=1 \|ψ A i ψ A i \| ;P = ∞ 0 dE\|ζ E ζ E \|and Open quantum system solutions in Q, which include couplings to the environment of scattering states and decay channels, are obtained by solving the eigenvalue problem for the energy-dependent effective Hamiltonian: H QQ (E) = H QQ + H QP G (+) P (E)H P Q , where H QQ is the closed system Hamiltonian, G (+) P (E) is the Green function for the motion of a single nucleon in P subspace and E is the energy of this nucleon (the scattering energy). Index '+' in G The scattering function Ψ c E is a solution of a Schrödinger equation in the total function space: Ψ c E = ζ c E + α a αΦα , where a α ≡ Φ α \|H QP \|ζ c E /(E − E α ) , andΦ α ≡ (1 + G (+) P H P Q )Φ α . Inside of an interaction region, the dominant contributions to Ψ c E are given by eigenfunctions Φ α of the effective non-Hermitian Hamiltonian [1]: Ψ c E ∼ α a α Φ α . For bounds states, eigenvalues E α (E) of H QQ (E) are real and E α (E) = E. For unbound states, physical resonances can be identified with the narrow poles of the S-matrix [2,23], or using the Breit-Wigner approach which leads to a fixed-point condition [1,18,24]: E α = Re (E α (E)) \| E=Eα ; Γ α = −2 Im (E α (E)) \| E=Eα (1) Here it is assumed that the origin of Re (E) is fixed at the lowest particle emission threshold. An EP is a generic phenomenon in Hamiltonian systems. In our case, the EP can appear as a result of the continuum-coupling term H QP G (+) P (E)H P Q for energies above the first particle emission threshold (E > 0). The eigenvalue degeneracies are indicated by common roots of the two equations [6]: ∂ (ν) ∂E det [H QQ (E; V 0 ) − EI] = 0 ν = 0, 1 (2) Single-root solutions of Eq. (2) correspond to EPs associated with decaying states. The maximal number of those roots is M max = n(n − 1), where n is the number of states of given angular momentum J and parity π. In quantum integrable models with at least two parameterdependent integrals of motion one finds also double-root solutions which correspond to non-singular crossing of two levels with two different wave functions. Hence, the actual number of EPs in these systems is always smaller than M max [9]. The position of EPs in the spectrum of eigenvalues of H QQ depends both on the chosen interaction and the energy E of the system. In general, eigenvalues of the energy-dependent effective Hamiltonian H QQ (E) need not satisfy the fixed-point condition (1) and hence need not correspond to poles of the S-matrix (resonances). In the following, we shall consider uniquely the case where EPs are identical with double-poles of the S-matrix. EXCEPTIONAL POINTS IN THE SCATTERING CONTINUUM OF 16 Ne Let us investigate properties of EPs on the example of 16 Ne. SM eigenstates in this nucleus correspond to a complicated mixture of configurations associated with the dynamics of the 16 O core. Our goal is to see if EPs can be possibly found in the scattering continuum of atomic nucleus at low excitation energies and for physical strength of the continuum coupling. SMEC calculations are performed in p 1/2 , d 5/2 , s 1/2 model space. For H QQ we take the ZBM Hamiltonian [25] which correctly describes the configuration mixing around N = Z = 8 shell closure. The residual coupling H QP between Q and the embedding continuum P is generated by the contact force: H QP = H P Q = V 0 δ(r 1 − r 2 ) . For each J π , the SM states \|ψ i (J π ) of the closed quantum system are interconnected via the coupling to common decay channels [ 15 F(K π ) ⊗ p lj ] J π E ′ with K π = 1/2 + , 5/2 + , and 1/2 − which have the thresholds at E = 0 (the elastic channel), 0.67 MeV, and 2.26 MeV, respectively. In the ZBM model space, these are all possible one-proton (1p) decay channels in 16 Ne. The size of a non-Hermitian correction to H QQ depends on two real parameters: the strength V 0 of the continuum coupling in H QP (H P Q ) and the system energy E. The range of relevant V 0 values can be determined, for example, by fitting decay widths of the lowest states in 15 F. For the present Hamiltonian, experimental decay widths of the ground state 1/2 + 1 and the first excited state 5/2 + 1 in 15 F are reproduced using V 0 = −3500 ± 450 MeV·fm 3 and V 0 = −1100 ± 50 MeV·fm 3 , respectively. The error bars in V 0 reflect experimental uncertainties of those widths. The weak dependence of 1p decay widths on the sign of V 0 is generated by the channel-channel coupling and disappears in a single-channel case. E [MeV] V 0 [MeV·fm 3 ] J π = 1 - FIG. 1: The map of J π = 1 − exceptional points in the continuum of 16 Ne as found in SMEC. For more details, see the description in the text. Fig. 1 shows energies E and strengths V 0 which correspond to J π = 1 − EPs in the scattering continuum of 16 Ne. Decay channels [ 15 F(K π ) ⊗ p lj ] 1 − E ′ with K π = 1/2 + , 5/2 + , and 1/2 − have been included with proton partial waves: p 1/2 , p 3/2 for K π = 1/2 + , p 3/2 , f 5/2 , f 7/2 for K π = 5/2 + , and s 1/2 , d 3/2 for K π = 1/2 − . The number of 1 − SM states is 3 and, hence, the maximal number of 1 − EPs in SMEC could be 6. Indeed, all of them exist at E < 20 MeV in a physical range of V 0 values (1100 MeV·fm 3 < \|V 0 \| < 3500 MeV·fm 3 ). They have been found by scanning the energy dependence of all eigenvalues over a certain range of V 0 , searching for all real-energy crossings or width crossings (avoided crossings). Once found, we have tuned V 0 to find out whether these crossings evolve into EPs at some combination of V 0 and E. One should stress that the passage through EP always occurs if, e.g., the real-energy crossing moves towards E = 0. Since such a crossing cannot move into the region E < 0, therefore it converts into an avoided crossing via the formation of an EP. The lowest EP in Fig. 1 is seen at V The upper part of Fig. 2 shows the phase shifts δ lj for p+ 15 F elastic scattering as a function of the proton energy for p 1/2 (dashed-dotted line) and p 3/2 (dashed line) partial waves. In the partial wave p 1/2 , the elastic scattering phase shift exhibits a jump by 2π at the EP with J π = 1 − . This unusual jump in the elastic scattering phase shift is an unmistakable and robust signal of a double-pole of the S-matrix (EP) which persists also in its neighborhood, as shall be discussed below. istic double-hump shape [26] with asymmetric tails in energy. The inelastic cross section in this case exhibits a single peak. Both inelastic channels [ 15 F(5/2 + ) ⊗ p lj ] 1 − E ′ and [ 15 F(1/2 − ) ⊗ p lj ] 1 − E ′ are opened at the EP. Substantial background contribution to both cross sections comes from broad resonances, mainly 0 + and 2 + . A sharp peak at E ≃ 1.65 MeV corresponds to an ordinary resonance 2 − . The above discussion of the double-poles of the Smatrix (EPs) and their manifestation in the many-body scattering continuum concerns 1 − states. The same analysis for J π = 0 + , 2 + states of 16 Ne gives qualitatively similar results. Also in these two cases, the number of EPs is maximal but only a fraction of them appears in the relevant range of E and V 0 values. Behavior of scattering wave functions in the vicinity of the exceptional point A true crossing of two resonant states is accidental and, hence, improbable in nuclear scattering experimentation. In this section, we will investigate the behavior of scattering states in the vicinity of an EP (the double-pole of the S-matrix) as the observation of such a situation is more plausible. (Fig. 6). The case shown in Fig. 5 corresponds to a subcritical coupling where two resonances cross freely in energy and repel in width [27]. In this regime, the scattering energy E corresponding to the closest approach of 1 − eigenvalues in the complex plane (E ≃ 2.47 MeV) is higher than the scattering energy corresponding to the EP at a critical coupling V (cr) 0 =-1617.4 MeV·fm 3 . Nevertheless, the elastic scattering phase shift shows the jump by 2π at the position of the EP and not at the point of the closest approach of eigenvalues. Fig. 6 shows the situation corresponding to an overcritical coupling where two resonances exhibit level repulsion in energy and a free crossing of their widths [27]. In this case, the point of the closest approach of 1 − eigenvalues in the complex plane is found at the scattering energy (E = 2.13 MeV) which is lower than than the corresponding energy for the EP. Again, the elastic scattering phase shift shows the jump by 2π at the position Next two figures show the elastic and inelastic cross sections for 15 F(p, p ′ ) in the vicinity of the EP with J π = 1 − in the subcritical (Fig. 7) and overcritical (Fig. 8 couplings remain same as for the critical coupling (see Fig. 3). In both cases, one see a double-hump shape in the elastic cross sections and a single-hump shape in the inelastic cross section. One observes also a strong asymmetry in widths and heights of two peaks and a small shift of the position of the interference minimum in between the two peaks with respect to the energy which the EP is found for a critical coupling. Entangled eigenstates of the effective Hamiltonian Complex and biorthogonal eigenstates of the effective non-Hermitian Hamiltonian provide a convenient basis in which the resonant part of the scattering function can be expressed. These eigenstates are obtained by an orthogonal and, in general, non-unitary transformation of SM eigenstates [1] which is a consequence of their mixing via coupling to common decay channels. The same coupling is responsible for the entanglement of two eigenstates involved in building of an EP, as illustrated in Fig. 9 on the example of spectroscopic factors. Φ(1 − 1 )(E) (Φ(1 − 2 )(E) ) eigenvalues of the effective Hamiltonian H QQ (E) as a function of the scattering energy E. For a critical coupling (plot (b)), the spectroscopic factors for Φ(1 − 1 ) and Φ(1 − 2 ) wavefunctions diverge at the EP (the double-pole of the S-matrix) but their sum (longdashed line in Fig. 9) remains finite and constant over a whole region of scattering energies surrounding the EP. In that sense, Φ(1 − 1 ) and Φ(1 − 2 ) resonance wavefunctions form an inseparable doublet of eigenfunctions with entangled spectroscopic factors. This entanglement is a direct consequence of the energy dependence of coefficients b αi : \|Φ α = i b αi (E)\|ψ i , in a decomposition of H QQ (E) eigenstates in the basis of SM eigenstates. One may notice that the energy dependence of Re(S 2 ) in the vicinity of the double-pole for 1 − 1 and 1 − 2 eigenstates is quite different in all three regimes of the continuum coupling. In particular, in the overcritical regime of coupling, an EP yields entangled states in a broad range of scattering energies. The strongest entanglement is found at the scattering energy which corresponds to the point of the closest approach of eigenvalues in the complex plane for all regimes of coupling. Obviously, the entanglement of resonance eigenfunctions in the vicinity of an EP is a generic phenomenon in open quantum systems which is manifested in matrix elements and expectation values for any operator which does not commute with the Hamiltonian. CONCLUSIONS In conclusion, we have shown in SMEC studies of the one-nucleon continuum that EPs exist for realistic values of the continuum coupling strength. In the studied case of 16 Ne, few of those EPs appear at sufficiently low excitation energies to be seen in the excitation function as individual peaks associated with a jump by 2π of the elastic scattering phase shift. The occurrence of an EP leaves also characteristic imprints in its neighborhood, i.e. for avoided crossing of resonances. In all closeto-critical regimes of the continuum coupling where real and/or imaginary parts of the two eigenvalues coincide, one finds qualitatively similar features of the elastic scattering phase shift and the elastic cross-section as found for the critical coupling at around the EP (the doublepole of the S-matrix). This gives a real chance that EPs or their traces may actually be searched for experimentally in the atomic nucleus. The well-known case of 2 + doublet in 8 Be, where resonance energies and widths are 16623±3 keV, 107±0.5 keV and 16925±3 keV, 74.4±0.4 keV, respectively [15], nearly satisfies the resonance conditions in the close-to-critical regime of couplings. Various situations in this regime have been studied experimentally in the microwave cavity [27]. Avoided crossing of two resonances with the same quantum numbers provide the valuable information about the configuration mixing in open quantum systems. As the formation of any EP in the scattering continuum depends on a subtle interplay between internal Hamiltonian (H QQ ) and the coupling to the external environment of decay channels, its finding provides a stringent test of an effective nucleon-nucleon interaction and the configuration mixing in the open quantum system regime. Such tests are crucial for a quantitative description of atomic nuclei in the vicinity of drip lines. We wish to thank W. Nazarewicz for stimulating discussions and suggestions. projected Hamiltonians:QHQ ≡ H QQ ,P HP ≡ H P P ,QHP ≡ H QP ,P HQ ≡ H P Q . Assuming Q + P = I, one can determine the third set of functions {ω (+) the outgoing boundary in the scattering problem. H QQ is non-Hermitian for unbound states and its eigenstates \|Φ α are linear combinations of SM eigenstates \|ψ i . The eigenstates of H QQ are biorthogonal; the left \|Φ α and right \|Φᾱ eigenstates have the wave functions related by the complex conjugation. The orthonormality condition in the biorthogonal basis reads: Φᾱ\|Φ β = δ α,β . Similarly, the matrix element of an operatorÔ is O αβ = Φᾱ\|Ô\|Φ α . FIG. 2 : 2MeV·fm 3 and E = 2.33 MeV. This EP corresponds to a degeneracy of the first two 1 − eigenvalues of H QQ for V 0 < 0. Energy E i and width Γ i of 1 − 1 and 1 − 2 eigenvalues are shown in Fig. 2 as a function of the scattering energy. For E > 2.33 MeV, width of these two eigenvalues grow apart very fast. E 1 (E) (solid line) and E 2 (E) (dotted line) cross again for E ≃ 3.2 MeV. At this energy, The upper plot exhibits the elastic scattering phase shifts δp 1/2 (dashed-dotted line) and δp 3/2 (dashed line) for p+ 15 F reaction in 1 − partial waves at around the EP (the double-pole of the S-matrix) with J π = 1 − . Lower plots show real and imaginary parts of 1 − 1 (solid line) and 1 − 2 (dotted line) eigenvalues of the effective Hamiltonian HQQ(E) as a function of the scattering energy E. For other details, see the description in the text. Γ 1 and Γ 2 are different and, hence, the corresponding eigenfunctions are different as well. Fig. 3 FIG. 3 : 33shows the elastic and inelastic cross sections for 15 F(p, p ′ ) in the vicinity of an EP. The solid line represents a sum of different partial contributions of both parities with J ≤ 5 whereas the dashed line shows the resonance part of 1 − contribution in these cross sections. The cross sections are plotted as a function of center of mass scattering energy for V (cr) 0 = −1617.4 MeV·fm 3 . The elastic cross section at the EP shows a character-Elastic and inelastic cross-sections in the reaction 15 F(p, p ′ ) as a function of the proton energy E at around the EP (the double-pole of the S-matrix) with J π = 1 − for 1 − resonances only (dashed line) and for all resonances with J ≤ 5 (solid line). For more details, see the description in the text. Fig. 4 FIG. 4 : 44exhibits the phase shifts δ lj for p+15 F elastic scattering as a function of the proton energy for The elastic scattering phase shifts δp 1/2 for p+15 F reaction in 1 − partial waves at around the EP (the doublepole of the S-matrix) with J π = 1 − at V MeV·fm 3 (solid line). Different curves correspond to different strength V0 of the continuum coupling: V0=-1800 MeV·fm 3 (long-dashed line), -1700 MeV·fm 3 (dashed-dotted line), -1500 MeV·fm 3 (short-dashed line) and -1430 MeV·fm 3 (dotted line). various values of the strength V 0 (V 0 =-1800 MeV·fm 3 (long-dashed line), -1700 MeV·fm 3 (dashed-dotted line), -1617.4 MeV·fm 3 (solid line), -1500 MeV·fm 3 (shortdashed line) and -1430 MeV·fm 3 (dotted line)) of the residual coupling H QP = H P Q = V 0 δ(r 1 − r 2 ) between Q and P subspaces. The characteristic change by a 2π of the elastic phase shift is seen in a broad interval -1800 MeV·fm 3 ≤ V 0 ≤ -1500 MeV·fm 3 of the continuum coupling strength. Fig. 5 and 6 show energies E i and widths Γ i of 1 − 1 and 1 − 2 eigenvalues as a function of the scattering energy for two values of V 0 : -1560 MeV·fm 3 (Fig. 5) and -1680 MeV·fm 3 FIG. 5 : 5The same as in Fig. 2 but in the subcritical regime of coupling (V0 = −1560 MeV·fm 3 ). For more details, see the caption of Fig. 2 and the description in the text. of the double-pole. From these two examples, one can see that the characteristic jump by 2π of the elastic scattering phase shift remains a robust signature of the EP in all close-to-critical regimes of the coupling to the continuum: the subcritical coupling (\|V 0 \| < \|V (cr) 0 \|), the critical coupling (\|V 0 \| = \|V (cr) 0 \|), and the overcritical coupling (\|V 0 \| > \|V (cr) 0 \|), where real and/or imaginary parts of two eigenvalues coincide. ) regimes of the continuum coupling. The curves shown by solid lines in Figs. 7,8 represent a sum of different partial contributions of both parities with J ≤ 5. The curves shown by dashed lines exhibit the resonance part of 1 − contribution in these cross sections. The qualitative features of the cross sections for the subcritical (V 0 = −1560 MeV·fm 3 ) and overcritical (V 0 = −1680 MeV·fm 3 ) FIG. 6 : 6The same as in Fig. 2 but in the overcritical regime of coupling (V0 = −1680 MeV·fm 3 ). For more details, see the caption of Fig. 2 and the description in the text. Fig. 9 FIG. 7 : 97exhibits the real part of the spectroscopic factor Re(S 2 )=Re 16 Ne(1 − n )\|[ 15 F(1/2 + 1 ) ⊗ p(0p 1/2 )] 1 − 2 in 16 Ne in three regimes of continuum coupling: (a) the subcritical regime (V 0 = −1560 MeV·fm 3 ), (b) the critical regime (V (cr) 0 = −1617.4 MeV·fm 3 ), and (c) the overcritical regime (V 0 = −1680 MeV·fm 3 ). The solid (short-dashed) lines show the spectroscopic factors for The same as in Fig. 3 but in the subcritical regime of coupling (V0 = −1560 MeV·fm 3 ). For more details, see the caption of Fig. 2 and the description in the text. FIG. 8 : 8The same as in Fig. 3 but in the overcritical regime of coupling (V0 = −1680 MeV·fm 3 ). 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[ "Oblivion of Online Reputation: How Time Cues Improve Online Recruitment", "Oblivion of Online Reputation: How Time Cues Improve Online Recruitment" ]	[ "Alexander Novotny \nInstitute for Management Information Systems\nVienna University of Economics and Business Administration\nWelthandelsplatz 1A-1020Vienna\n", "Sarah Spiekermann [email protected] \nInstitute for Management Information Systems\nVienna University of Economics and Business Administration\nWelthandelsplatz 1A-1020Vienna\n" ]	[ "Institute for Management Information Systems\nVienna University of Economics and Business Administration\nWelthandelsplatz 1A-1020Vienna", "Institute for Management Information Systems\nVienna University of Economics and Business Administration\nWelthandelsplatz 1A-1020Vienna" ]	[]	In online crowdsourcing labour markets, employers decide which job-seekers to hire based on their reputation profiles. If reputation systems neglect the aspect of time when displaying reputation profiles, though, employers risk taking false decisions, deeming an obsolete reputation to be still relevant. As a consequence, job-seekers might be unwarrantedly deprived of getting hired for new jobs and can be harmed in their professional careers in the long-run. This paper argues that exposing employers to the temporal context of job-seekers' reputation leads to better hiring decisions. Visible temporal context in reputation systems helps employers to ignore a job-seeker's obsolete reputation. An experimental lab study with 335 students shows that current reputation systems fall short of making them aware of obsolete reputation. In contrast, graphical time cues improve the social efficiency of hiring decisions.Biographical notes: Alexander Novotny is an information privacy and security specialist. He works as an information security risk manager in the utilities industry. Moreover, he is a lecturer at the Institute for Management Information Systems at the Vienna University of Economics and Business. He has been teaching lectures on information privacy and security, ethical computing, and the foundations of information and communication technology. Alexander holds a Ph.D. in the economic and social sciences with a major in business information systems. His research interests focus on electronic privacy, information security and ethical computing. He served as a standardization expert for digital marketing and privacy at the Austrian Standards Institute.	10.1504/ijeb.2017.10003869	[ "https://arxiv.org/pdf/2005.06302v1.pdf" ]	31,501,607	2005.06302	0b244261b83213695505935d123fe865c8673f1c	Oblivion of Online Reputation: How Time Cues Improve Online Recruitment Alexander Novotny Institute for Management Information Systems Vienna University of Economics and Business Administration Welthandelsplatz 1A-1020Vienna Sarah Spiekermann [email protected] Institute for Management Information Systems Vienna University of Economics and Business Administration Welthandelsplatz 1A-1020Vienna Oblivion of Online Reputation: How Time Cues Improve Online Recruitment 1 Corresponding author Sarah Spiekermann is a professor for Information Systems and chairs the Institute for Management Information Systems at Vienna University of Economics and Business. Before tenured in Vienna, she was assistant professor at the Institute of Information Systems at Humboldt University Berlin, Germany and held an Adjunct Professor positions with Carnegie Mellon University. Sarah is best known for her work on electronic privacy and electronic marketing. The key goal of her work is to investigate 2 the importance of behavioral constructs and social values for IT design and to refine the concept of ethical computing in an e-Society. She co-founded the Privacy & Sustainable Computing Lab at the Vienna University of Economics and Business.obsolete reputationonline crowdsourcing labour marketsright to be forgotten In online crowdsourcing labour markets, employers decide which job-seekers to hire based on their reputation profiles. If reputation systems neglect the aspect of time when displaying reputation profiles, though, employers risk taking false decisions, deeming an obsolete reputation to be still relevant. As a consequence, job-seekers might be unwarrantedly deprived of getting hired for new jobs and can be harmed in their professional careers in the long-run. This paper argues that exposing employers to the temporal context of job-seekers' reputation leads to better hiring decisions. Visible temporal context in reputation systems helps employers to ignore a job-seeker's obsolete reputation. An experimental lab study with 335 students shows that current reputation systems fall short of making them aware of obsolete reputation. In contrast, graphical time cues improve the social efficiency of hiring decisions.Biographical notes: Alexander Novotny is an information privacy and security specialist. He works as an information security risk manager in the utilities industry. Moreover, he is a lecturer at the Institute for Management Information Systems at the Vienna University of Economics and Business. He has been teaching lectures on information privacy and security, ethical computing, and the foundations of information and communication technology. Alexander holds a Ph.D. in the economic and social sciences with a major in business information systems. His research interests focus on electronic privacy, information security and ethical computing. He served as a standardization expert for digital marketing and privacy at the Austrian Standards Institute. the importance of behavioral constructs and social values for IT design and to refine the concept of ethical computing in an e-Society. She co-founded the Privacy & Sustainable Computing Lab at the Vienna University of Economics and Business. Introduction A good online reputation is increasingly vital for people. Online reputation systems retain the performance histories of people for an indefinite time and determine to a large extent "what is generally said or believed about a person's or thing's character or standing" (Jøsang et al., 2007, p.620). Take the case of Med Express vs. Amy Nichols (Wolford, 2013). Following a dispute on shipping costs, Mrs. Nichols left a negative review on eBay seller Med Express. Since Med Express was aware that the negative feedback would persistently stay in its reputation profile, it sued Nichols for having caused irreparable damage of lost customer value and revenue. Indeed, market actors with a low reputation receive substantially lower prices (Depken Ii and Gregorius, 2010). Eventually, Med Express dismissed the lawsuit. While online reputational feedback is indefinitely retained online, people's true being and careers are marked by progress, disruptions and shifts. On average, U.S. citizens have 11 different jobs throughout their working lives (BLS, 2012) and move their residences 12 times during a lifetime (USCensus, 2007). With bankruptcy filers increasing from 0.15 to 5.3 per 1,000 people and year over the last century, people's financial lives are increasingly coined by crisis and restart (Garrett, 2006). Constant human change is not only limited to such externally observable professional life changes. Aristotle argues in his "Physics" that no time can pass without humans' inner change (Coope, 2001). Ricoeur's (1988) phenomenological philosophy comes to a similar conclusion. In his narrative model, he argues that novel experiences as well as new connections and relationships between actors surfacing over time lead people to a constant reframing of their histories and pave the way for personal evolvement. This constant human evolution stands in sharp contrast to the static and almost timeless online presence of people. A problem arises when statically retained online information diminishes a person's true reputation and arbitrarily harms him or her. To protect people from such a damaged reputation, EU policy makers decided to include a "right to be forgotten" in Europe's Data Protection Regulation (De Hert and Papakonstantinou, 2012, p.136). The right aims at enabling a person to "determine the development of his life in an autonomous way, without being perpetually or periodically stigmatized as a consequence of a specific action performed in the past" (Mantelero, 2013, p.230). The legal metaphor of "forgetting" on the Internet refers to how prominently and easily a person's online information can be accessed. Of course, society should never forget some of people's historic actions, for example the atrocities by some dictators and their henchmen. But apart from such consciously unforgotten crimes, personal information retained online may be forgotten after some time. In this article we investigate the problem of obsolete online reputation for online recruiting, in particular in scope of online crowdsourcing labour markets. On globally operating crowdsourcing labour market platforms, job-seekers are domiciled in different countries. Particularly through the job-seeker's reputation profiles on these platforms, employers become acquainted with job-seekers and take a decision whether to hire them. Examples are platforms such as upwork.com, elance.com and guru.com. They use an "open call format" to request work from a "large network of potential labourers" (Howe and Robinson, 2006). Typically, the work requested is on a project basis. They may involve the creation of software, graphic works and advertising texts, for instance. One sort of software works sourced from online crowdsourcing labour markets are app designers developing mobile apps customized to the needs of their employing principals. Apps are software programs that are particularly designed to run on mobile devices such as smartphones and tablet PCs. The demand for developing apps facing customers or for enterprise purposes is rising (GoodTechnology, 2014). Customer apps, for example, allow browsing product catalogs and finding the next store while on the go. Enterprise apps may support travelling employees with access to documents and the company's customer database. Compared to traditional offline recruitment channels, online crowdsourcing labour markets offering job-seeking app designers recruitable for app development projects typically address employers who are more technology-affine. One could argue that online crowdsourcing labour markets in particular should strive for providing employers with the timeliest information about potential job-seekers. Temporal context information about how a job seeker is today versus how he or she was in the past is crucial. Avoiding that employers hire based on obsolete reputation thereby harming job-seekers and society, studies advocate for employers disregarding obsolete reputation: "since the reputation values are associated with human individuals and humans change their behaviour over time, it is desirable to disregard very old ratings" (Zacharia et al., 2000, p.376). Also, older online reviews are perceived to be less helpful for making decisions on online market transactions (Cao et al., 2011;Hu et al., 2008). However, today's online crowdsourcing labour markets make the importance of time little visually salient yet. Employers are easily led to neglect the necessity to disregard obsolete reputation when they judge on potential new hires. The reason for this neglect is that current user interfaces of crowdsourcing labour markets' reputation systems do not make employers aware of reviews that are outdated. In previous work (source blinded for peer-review), 16 generic categories of design alternatives were found that help visualizing time context of person-related information in user interfaces. Design alternatives for time visualization, for example, include graphical timelines and visual decay. Timelines display the time-related dimension of information on horizontal screen-space. Interfaces drawing on the visual decay metaphor present outdated information in a progressively dissolved state. Current user interfaces of crowdsourcing labour markets' reputation systems, though, only use graphical time visualizations assigned to one category: text-based symbol of time. These interfaces, for instance, add a timestamp to reviews and display text-based listings of the feedback received within the last 12 months. They only contain tiny textbased cues to the publication date, mostly printed in small font size and in grey letters. None of them, though, uses any graphical time visualizations (see Table 6 in the Appendix). HCI studies have shown for long that text cues have little salience in interfaces and impose an additional cognitive load on profile viewers (Hong et al., 2007). As a result, employers can probably hardly notice the temporal context of reviews. In addition to textual time cues, current reputation systems optionally sort reviews by publication time (see Table 6 in the Appendix). Empirical results show that chronologically ordering reviews increases their helpfulness (Otterbacher, 2009). Like on a job-seeker's curriculum vitae (CV), recent occurrences are displayed first, making them immediately visible to employers. eBay's reputation system, for example, provides sophisticated options for sorting and aggregating feedback by time. Feedback can be filtered by recentness (last 1, 6 and 12 months) keeping all feedback still accessible. But recency filters can be misleading as well. If someone has mostly performed well in the past, but recently failed to meet some demands, chances are high that job seekers immediately suffer penalties, because their history is not viewed holistically. Against this background, we argue in this paper that reputation systems should more saliently visualize time. Visibility is generally a key design principle of user interfaces aiming to keep information recipients on top of the current status of a system, process, action or entity (Nielsen, 1994, p.154;Norman, 1988). We argue that a higher visibility of reviews' temporal aspects makes future employers aware of outdated reputational information and helps them to avoid it. Potential future employers are less likely to base their judgments on obsolete reputation. Aiming to address the call for a "right to be forgotten" from a system design angle, we ask: How should reputation systems display obsolete information? Would a higher visibility of reputation's temporal context encourage employers to avoid obsolete reputation and focus on job-seekers' timely achievements? And would first-rate job-seekers who are still plagued by a negative, outdated reputation obtain a second chance of getting hired? To elucidate these questions, an experimental laboratory study was conducted. It manipulated visual time cues in the reputation system user interface of a fictitious online crowdsourcing labour market. The experiment manipulated two types of visual time cues: First (and similar to how current online crowdsourcing labour markets' reputation systems organize time) the temporal order of reviews in reputation profiles was manipulated. Second, we tested the effects of a new salient graphical timelines cue in reputation profiles. We find that graphical timelines encourage employers to assess job-seekers in a more recent light and to discount job-seekers' obsolete reputation. In contrast, we could not find this effect for ordering reviews by time which is the current standard for job-seekers' reputation profiles. The remainder of this paper is structured as follows. Section 2 presents the theoretical background and hypotheses. Section 3 describes the method of the experimental study. Section 4 outlines the results which are discussed in Section 5. The final Section 6 draws conclusions and points to future work. Theoretical background In online crowdsourcing labour markets, employers face the problem of hiring jobseekers without having knowledge of their skills, traits, experience and quality of work. Information asymmetry (Akerlof, 1970) reigns between employers and job-seekers making hiring decisions risky. This section reviews how online reputation systems help reducing this information asymmetry and handle temporal aspects of job-seekers' reputation. Then, we outline our hypotheses how time cues for reputation can support employers with disregarding obsolete reputation. Online reputation systems and market transparency To reduce uncertainty about job-seekers' quality, online crowdsourcing labour markets operate reputation systems (Jøsang et al., 2007). "A reputation system collects, distributes, and aggregates feedback about [market] participants' past behaviour" (Resnick et al., 2000). A good online reputation serves as a trust mark reducing transaction risks and remediating information asymmetry (Ba and Pavlou, 2002). The trust-building effect and transparency for employers is highest if they have access to the full reputation history of job-seekers. Maximum market transparency is achieved if no incidents in a job-seeker's reputation history are omitted. Hence, deleting older reviews in job-seekers' reputation profiles would reduce transparency. Also, allowing job-seekers to delete reviews could incentivize them to manipulate their reputation and cast themselves in a potentially falsified positive light. Such feedback manipulation undermines the very goal of online reputation systems, that is to present a person's true background (Dini and Spagnolo, 2009). It would also censor past employers' opinions and their right to free speech online (Rosen, 2012). Yet, one dimension of transparency is also to consider the "appropriateness" of information (Spiekermann, 2015;Turilli and Floridi, 2009). It may be that reviews about job-seekers were entered into a reputation system long ago and are by now obsolete. The person concerned may have changed. If deleting such obsolete reviews is no viable option for the reasons given above, how can job-seekers safeguard themselves from being eternally reproached for their past? For employers appropriately basing their hiring decision on information about the up-todate current skills and experience of job-seekers, they should disregard job seekers' obsolete past behaviour and "forget" it at some point in time. Zacharia et al. (2000, p.376) argue: "Reputation values are associated with human individuals and humans change their behaviour over time, it is desirable to disregard very old ratings." At the organizational level, employers are therefore well advised to disregard those parts of a job-seeker's reputation which seems to be obsolete. Ignoring the need to disregard obsolete reputation bears negative consequences at the societal level. Employers can run the risk of seeing job-seekers in a "false light" (Prosser, 1960, p.398). They assess job-seekers based on outdated reputational information that does not accurately reflect the job-seeker's current skills and quality. If reputation in electronic markets is perceived to be low, lower market prices (i.e., wages) can result (Depken Ii and Gregorius, 2010). In extreme cases, actors with a low, but obsolete online reputation may get "stigmatized" (Solove, 2006, p.547). Such jobseekers may never get hired in the online crowdsourcing labour market again. Rebuilding a positive reputation for the future is tedious and costly (Matzat and Snijders, 2012). Eventually, despite their good present work quality that would rehabilitate an obsolete negative reputation, they may have to drop out of the job market. Consequently, not disregarding obsolete reputation may create social costs in the crowdsourcing labour market. If we embrace the argument that obsolete reputation should be disregarded and "forgotten" online, then it would make sense to support employers with this task. Reputation systems should then facilitate and remind employers to disregard outdated reputation. User interface cues may help employers to identify and ignore obsolete reputation. User interface cues were found to influence the behaviour of actors in electronic market environments: for example, visually cueing to the human face of sales personnel was found to increase consumer trust on e-commerce websites (Aldiri et al., 2008). Also, the design of reputation systems was shown to influence the effect of a market actor's reputation on a transaction (Klein et al., 2009;Zhu and Zhang, 2010). In online political discussion fora, visual interface cues were found to make online community members listen better to each other's opinions (Manosevitch et al., 2014). The design of online reputation systems may also cue reviewers to rethink their behaviour (Ekstrom et al., 2005). Obsolescence of reputation If user interfaces should cue employers to obsolete reputation, when is a job-seeker's online reputation obsolete in the particular context under investigation? In an online survey about reputation in crowdsourcing labour markets (n=494, the participants were sampled from the same student population as in the experiment), we found that 82.8% of reputation system users believe that reviews about job-seekers are outdated if these were published three years or longer ago (see Figure 1). Individual employers gradually deem reviews to get obsolete based on various factors such as job seekers' ability to learn, gain experience and their changing motivation and personality. Even though individual perceptions of review obsolescence vary depending on these factors, there is a community consensus on when reviews shall be considered as outdated and not be further used to judge job-seekers. In our analysis presented below we make the conservative assumption that reviews published more than four years ago are obsolete (even 93.3% of reputation system users deem five year old reviews to be obsolete). Hence, employers should mainly consider reviews which were published less than four years ago to get a rather recent and accurate picture of job-seekers. To support employers with this task, reputation systems should highlight the temporal context of reputation. Hypotheses Despite the importance of disregarding obsolete reputation for hiring, some current online crowdsourcing labour market platforms tend to prefer highlighting the best achievements of app designers over their most recent ones (see Table 6). Consequently, these platforms display reviews by ordering the highest rated ones first in reputation profiles. Employers are advertised that chances of finding a well-rated job-seekers are high on the respective platform. In line with our argumentation, though, we expect that this practice disobeys time aspects and does not encourage employers to disregard obsolete reputation. H1: Reputation profile interfaces ordering the reviews by star rating do not influence employer disregard of obsolete reputation. On the contrary, this paper proposes that visually cueing to reviews' temporal context in a reputation system's user interface (time cues) hints employers to disregard obsolete reputation. Because of their higher awareness of the reviews' temporal context, they can become more aware that obsolete reviews are irrelevant and inappropriate for their hiring decisions. We hypothesize that two types of visual time cues that reputation systems may include increase employers' disregard of obsolete reputation: temporal order and graphical timelines. The temporal order cue represents the current practice of how online crowdsourcing labour markets' reputation systems focus employers on time (see Table 6 in the Appendix). Following this current practice, we expect temporal order cues encourage employers to disregard obsolete reputation when hiring: H2: Reputation profile interfaces ordering the reviews by time increase employer disregard of obsolete reputation. Chronologic CVs used by job seekers to traditionally apply for jobs follow the convention of being in descending temporal order. Employers start reading CVs from the top, thereby first encountering a job seeker's most recent experiences. Also, narrating job-seekers' reputation histories in reputation profiles can either start with the most recent or the oldest event. We expect that online crowdsourcing labour market platforms sorting the most recent reviews first in reputation profiles guide employers' attention closer to job-seekers' recent professional conduct. As a result, employers are more likely to disregard obsolete reputation when making hiring decisions. H3. Reviews in descending temporal order more strongly encourage employers to disregard obsolete reputation than reviews in ascending temporal order. Beyond mere time order, this paper suggests a novel time cue for reputation systems: timelines that graphically visualize the passage of time in reputation profiles. Online advertising effectiveness research shows that graphical interface cues attract more attention than text-based or mixed cues (Hsieh and Chen, 2011). Visualizing time is a largely unexplored topic in human-computer interaction (HCI) research (Lindley et al., 2013, p.3212). Keeping with how Western cultures read from left to right, proposals use timelines mapping the past to the left and the future to the right (Santiago et al., 2007). Timelines are visual metaphors capable of narrating personal histories (Plaisant et al., 1996;Thiry et al., 2013) and could also be used to tell job-seekers "reputation histories". The timelines cue was selected over other approaches for graphically visualizing time ([source blinded for peer-review], p. 552) for three reasons. First, timelines are suitable for representing a reviews' publication date with high granularity. Second, they are able to clearly arrange high amounts of reviews contained in reputation profiles. And they can be dynamically created within the user interface ([source blinded for peer-review], p. 553). The idea of graphical timelines is taken up to visualize when reviews were published within reputation profiles. We introduce a timelines cue representing reviews as dots on graphical timelines progressing from left to right (see Figure 2). The dots contain the review's rating. Timelines aim focusing employers towards recent reputational information about job-seekers' when assessing job-seekers and pointing employers towards disregarding obsolete reputation: H4: Reputation profile interfaces displaying reviews on graphical timelines increase employer disregard of obsolete reputation. Method A computer lab experiment was conducted in German with students who were all online reputation system users. Participants were recruited via a university mailing list. Those who are not reputation system users and who already participated in the preceding survey were denied registration. Participants were paid 10 euros in cash. They completed the experiment on identical screens which were sheltered from other participants' gazes. Median completion time was 20.92 minutes. Participants were situated into an identical scenario. They were told that their university wants to develop a new mobile app called the "CampusApp". This app would enable spontaneous orientation on campus and provide ad-hoc directions to currently vacant student project rooms. The interface of a fictitious online crowdsourcing labour market platform showed the participants the reputation profiles of four app designers (hereafter denoted AD1 to AD4) who were offering their support to develop this app. The participants were requested to suggest hiring one out of the four app designers. We asked them: "Who would be the best app designer for the CampusApp?" A choice set of four profiles reflected hiring decisions between multiple job-seekers on real online crowdsourcing labour market platforms while keeping the selection task manageable for participants. Participants were told that they are in charge of hiring an app designer for the university's app design project. Ensuring incentive compatibility, participants could win additional 20 euros for making a selection that fits the university's app design project well. Participants had free choice and were not restricted to follow their own preferences and judgements while making a decision. Online reputation profiles The reputation profiles had no systematic differences except for the average rating score of the reviews which were regarded to be still relevant and not obsolete (see Table 1). Each profile contained eight reviews that were published in the 5-year time frame between 2010 and 2014. Online reviews for the profiles were selected from 120 review texts. The reviews were written in a professional, evaluative and non-technical language. The reviews' length was 25 to 35 words. Based on Goldberg's (1990Goldberg's ( , p.1224 personality trait III+ scheme all reviews made statements about an app designer's conscientiousness and motivation while doing design work. For instance, app designers were described to be dependable, punctual, disorganized, or aimless. The reviews were pre-tested for valence (on a scale from 1.00 to 5.00 stars with 5.00 being the best rating), text quality (readability, helpfulness, understandability) and congruence between headline and text. The pre-test data was based on 3,123 rating points provided by 494 participants (see Section 2). Reputation profiles were compiled to exhibit no statistical differences (n=812) of perceived valence (F=0.459, df=3, p=0.711), text quality (F=0.216, df=3, p=0.885) and congruence between headlines and text (F=2.115, df=3, p=0.097). In order to avoid bias from identifying profile characteristics, randomly generated pseudonyms were used as app designers' names (e.g., "Wtmfc") and former employers (e.g., "uoweoi898"). Profiles had identical dummy profile pictures showing a person's grey silhouette. The grey silhouettes ensured that choice was not biased by the sympathy and appearance of the app designers. An example profile is depicted in Figure 3. The app designers' total profile rating scores randomly varied within the range of 4.01 to 4.19 stars. Random variation of total rating scores made the hiring task more realistic because job-seekers do not have equal reputation scores in real-world online crowdsourcing labour markets either. Randomization was stochastically independent of the random assignment of participants to interface manipulations. Multinomial logistic regressions confirmed that the random total rating scores did not systematically influence app designer choice. Dependent measure We operationalise employer disregard of obsolete reviews as follows: reviews published in 2010 were deemed to be outdated an obsolete by employers because they are more than four years old (see above). In contrast, the reviews published between 2011 and 2014 are still regarded to be relevant. Thus, we only use the reviews published between 2011 and 2014 to calculate an average rating score of the non-obsolete reviews for each reputation profile. To calculate the score of a profile, the ratings are added and divided by the number of reviews as depicted in Figure 4 (e.g., for AD3: (4.99+4.86+4.81+4.94+1.87+4.92)/6=4.398). This score is the only systematic difference between the four app designers' reputation profiles (see Table 1). The employers' disregard of the obsolete reviews is then measured by the score of the app designer's profile that was selected by the participant. Employers who select a reputation profile assigned a higher score disregard obsolete reputation to a higher degree. Independent variables: manipulations of interface time cues In a between-subjects design, the user interface displaying the reputation profiles was manipulated to include time cues. Avoiding order bias, the reputation profiles were displayed in random order to participants. Participants were randomly assigned to one of six conditions. All conditions displayed text dates when the reviews were published. Beyond text dates, three treatment conditions (TC1, TC2, TC3) were using visual time cues. TC1 gives a temporal order cue. The most recent reviews in the profiles are ordered first. TC2 combines the temporal order cue with the timelines cue (see Figure 3). Recent reviews are ordered first and graphical timelines of the profiles are displayed. The timelines cue's colour combination (dark yellow/ultramarine blue) had a neutral colour weight on the Kobayashi colour image scale. The timelines cue is always combined with the temporal order cue because items on a timeline are in a natural temporal order. TC3 is identical to TC2, except that the oldest reviews are ordered first. This condition was added to investigate the influence of a particular direction of temporal order (ascending or descending). Three control conditions did not display visual time cues (CC1, CC2, CC3). The control conditions provide no visual time cues. CC1 sorted the reviews randomly. CC2 and CC3 order the highest rated reviews first. CC3 was added to control for bias resulting from the mere presence of graphical cues in the user interface. It orders the highest rated reviews first and displays graphical profile representations that do not contain any temporal information (see Figure 5). The visual interface cues were pretested. Post-questionnaire After participants saw the manipulated reputation system interfaces and made an app designer choice, they answered a questionnaire. First, they judged the correctness of six statements on time-related information testing whether visual time cues actually made them focus on time (manipulation checks). As a second check, participants were asked whether they focused on the reputation development history of the individual app designers while making their choice. Answers were provided on a five-point Likert scale ("did incorporate strongly" to "did not incorporate at all"). Then, participants answered control variables and demographics. They rated how often they normally read and author reviews online ("never" to "very frequent"). Two types of participant involvement were controlled for. Involvement into the CampusApp scenario (scenario involvement) was measured by five 7-point semantic differentials adopted from the personal involvement inventory (Zaichkowsky, 1985, p.350). Involvement into the task of selecting an app designer (choice involvement) was measured by six items adapted from the new involvement profile (Jain and Srinivasan, 1990, p.597). Some participants may condone mistakes that app designers made in the past. Others may be less lenient and judge app designers who made mistakes as enduringly negative. Hence, participants' forgiveness -whether they have a generally merciful personality -was measured by five items adapted from the "Heartland forgiveness of others" subscale (Thompson et al., 2005, p.358). Participants' rational thinking style while cognitively processing the reputation profiles was controlled for using three items from the rational situation-specific thinking style scale (Novak and Hoffman, 2009, p.60). The control variables exhibit satisfactory psychometric characteristics (see Table 7 in the Appendix). Other control variables did not explain a relevant amount of covariance and were not included into the analysis. Results 382 participants completed the experiment. 47 participants were excluded from the sample: One was no reputation system user. Eight were careless responders finishing the study in less than the median time minus 1.3 times the interquartile range. 38 participants wrongfully answered a check on review order in the reputation profiles. Eventually, a sample of 335 participants remained. Descriptive results Laboratory participants were pre-screened to be online users with experience of reputation systems. They were young (mean age 23.31 years), well-educated (97.6% university students), familiar with online marketplaces (87.4% transact at least once a year online), and mobile phone apps (88.9% use apps at least often, see Table 2). These characteristics are typical for users affine to online recruitingand were reflected in the design of the experimental scenario. The raw data (n=382) and cleaned sample (n=335) show akin demographic characteristics (see Table 2). The further analysis is conducted on the cleaned sample. The manipulation checks confirmed that participants in the TCs had a higher awareness of temporal information contained in the profiles than those in the CCs (ΔM min =0.948, SE=0.351, p=0.007 (CC3-TC1); ΔM max =2.536, SE=0.357, p<0.001 (CC2-TC2)), except for the contrast between TC1 and CC1 which closely missed significance (ΔM=0.724, SE=0.380, p=0.059). This difference is significant for participants in the TCs who put more focus on the reputation development history of the app designers than those in the CCs (ΔM min =0.690, SE=0.263, p=0.009 (CC1-TC1); ΔM max =2.059, SE=0.249, p<0.001 (CC2-TC2)). The most effective manipulations are TC2 followed by TC3 which both combine temporal order and timelines cues. Further, a linear regression confirmed that the manipulations were not confounded with an increase in participants' confidence of their app designer choice (Timelines cue: B=-0.071, SE=0.059, p=0.232; Temporal order cue: B=-0.025, SE=0.060, p=0.671). Employer disregard of obsolete reputation was consistently higher in the TCs than in the CCs. In the CCs, it similarly ranged between 4.392 (SD=0.335) and 4.427 (SD=0.377). The temporal order cue (TC1) increased disregard of obsolete reputation to 4.534 (SD=0.400). Combining temporal order and timelines cues (TC2) caused the highest disregard of obsolete reputation (M=4.760, SD=0.237). Reversing the direction of temporal order (TC3) resulted in a slightly lower disregard of obsolete reputation (M=4.660, SD=0.310). Table 3 summarizes the number of subjects as well as the means and standard deviations of their disregard of obsolete reputation across the conditions. Both the cleaned sample and raw data (asterisked values in Table 3) exhibit stable means and standard deviations of disregard of obsolete reputation. Because group sizes were unbalanced (see Table 3), linear regression models serving as unbalanced groups ANOVA were used to examine the differences between the conditions. Main effects were effect coded and simple effects (i.e., contrasts between the conditions) were dummy coded. The effect of ordering the reviews by star rating on disregarding obsolete reputation The mere presence of a graphical element in the interface (comparing CC2 to CC3) did not influence employers' disregard of obsolete reputation (ΔM=0.035, SE=0.064, p=0.583). In a step-wise linear regression accounting for the control variables, sorting by highest rated reviews first in reputation profiles did not increase disregard of obsolete reputation compared to a random review order (B=0.006, SE=0.057, ΔF=0.012, p=0.912). There were no differences in employers disregarding obsolete reputation between the three CCs, neither on an overall level (F=0.155, df=2, p=0.856), nor when each CC was contrasted with each other (CC1-CC2: ΔM=-0.012, SE=0.066, p=0.852; CC1-CC3: ΔM=0.023, SE=0.070, p=0.747; CC2-CC3: ΔM=0.035, SE=0.064, p=0.583). Hence, to simplify the further analysis, all CC were combined into one control pool (CP). The effect of time cues on disregarding obsolete reputation Using a step-wise regression procedure, the main effects of the timelines and temporal order cues were tested (see Table 4). Residuals were normally distributed and error variances were homoscedastic. Multicollinearity of the predictors was low (VIF max =2.853; variance inflation factors should be below 10). The timelines cue increased disregard of obsolete reputation and explained the biggest share of variance (see step 5.a, B=0.113, SE=0.033, ΔF=36.800, p=0.001). Comparing TC1 to TC2, the effect of the timelines cue is robust if the temporal order cue is held constant (B=0.201, SE=0.061, ΔF=10.681, p=0.001). In contrast, the main effect of temporally ordering by recent reviews was not significant (B=0.010, SE=0.033, ΔF=0.089, p=0.766). Also when the particular direction of temporal order is ignored, the temporal order cue is not significant (see step 5.b, B=-0.027, SE=0.048, ΔF=0.307, p=0.580). Because timelines naturally order reviews by time, the "timelines" and "temporal order" predictors are moderately multicollinear (VIF of timelines = 7.650, results robust if VIF<10). Moreover, even though significance is only marginally missed, there was no difference in disregard of obsolete reputation between reviews sorted in descending temporal order (TC2) and in ascending temporal order (TC3) (B=0.094, SE=0.049, ΔF=3.678, p=0.058). Employers forgiving workers a negative obsolete reputation were more disregarding obsolete reputation (B=0.062, SE=0.019, p=0.001). However, neither the timelines cue (B=0.079, SE=0.160, p=0.620) nor the temporal order cue (B=0.036, SE=0.143, p=0.804) influenced employers' forgiveness. Also, participants who addressed the profiles with a rational thinking style were more disregarding of obsolete reputation (B=0.150, SE=0.030, p<0.001). By systematically comparing the profiles they were recognizing that obsolete reputation should be ignored. Online crowdsourcing labour market platforms ordering reviews by star rating do not contribute to hint employers to disregard obsolete reputation when making hiring decisions (H1 supported, see Table 5). However, the results provide evidence that visual time cues may increase disregard of obsolete reputation. The effect depends on the types of visual time cues provided in reputation system interfaces. Temporal order cues do not significantly hint employers towards the need to disregard obsolete reputation (H2 rejected), regardless whether reviews are ordered in ascending or descending order (H3 rejected). The timelines cue, in contrast, makes employers aware of disregarding obsolete reputation when hiring (H4 supported). Discussion and limitations The findings show that current online reputation systems tolerate hiring in online crowdsourcing labour markets which are based on obsolete reputation. Textual time cues (i.e., reviews publication dates) and sorting reviews by star rating downplay temporal aspects in job-seeker's reputation profiles. They hardly raise employers' awareness for forgotten need to disregard obsolete reputation. Designers of online crowdsourcing labour market platforms should therefore follow a few guidelines for displaying job-seekers' reputation histories: Most importantly, the reputation systems on these platforms should provide graphical time cues. The timeline metaphor turned out to be useful for graphically visualizing time. It evokes the basic mental picture of how people imagine time in Western societies. Temporal order cues did not influence employers to disregard obsolete reputation, irrespective whether new or obsolete reviews are put to the front. Sorting by best reviews has similarly no effect on disregard of obsolete reputation than a completely random review order. Hence, reputation systems should avoid this order to be the default option. Interface designers using visual time cues, though, need to pay attention to usability. Despite no effect of temporal order cues was found, the possibility to sort reviews by time is a feature that reputation system users expect from familiar reputation systems. This feature may be provided additionally in user interfaces. If user interfaces were designed by these guidelines for presenting obsolete reputation, we argue that job-seekers would regain new chances for job opportunities, despite their profiles containing a negative, obsolete reputation. As a result, online crowdsourcing labour markets would grant job-seekers a fair equality of employment opportunities over the course of their reputational online life. An obsolete reputation would not further irreversibly "knock out" job-seekers from online crowdsourcing labour markets. Job-seekers would have no need to find strategies for declaring "reputation bankruptcy" (Zittrain, 2010, p.228) harming market transparency. Honest job-seekers would not be pressurised migrating to other platforms where their reputation is unknown. And crooked job-seekers would have a lower incentive to delete their accounts containing their reputation profile and re-registering using a false identity. Desires for deleting entire reputation histories would diminish. For ensuring an effective reputation system, only defamatory and denigratory reviews would require deletion. The visibility created around time-related aspects of reputation also makes job-seekers' recent performance more transparent. Job-seekers whose performance dropped recently may oppose this new transparency because they would need to take over accountability for their recent performance. Our position is that online crowdsourcing labour markets should not support job-seekers with hiding low professional performance that still has relevance for employers in the presence. It is important that job-seekers are not eternally reproached for an obsolete performance that is not further relevant. If employers overestimate a negative, but obsolete reputation of job-seekers, then these job-seekers will not get hired despite their good present work quality. Job-seekers are hindered from recovering from obsolete reputation and eventually are lost for the market. As a result, online crowdsourcing labour markets forfeit a share of their efficiency. An unanticipated result is that employers' forgiveness plays a key role when jobseekers' reputation is retained forever. Forgiving and merciful employers were more disregarding obsolete reputation contained in the profiles. Forgiveness, though, is a predisposition of employers and was not influenced by the interface cues. This finding can be interpreted as more merciful employers ignoring differences in job-seekers' individual reputation histories after some time. For merciful employers, a job-seeker's obsolete reputation is no longer relevant. Limitations Our study has several limitations. The university student sample of reputation system users enables drawing conclusions about a young, technology-affine population who is familiar with online channels for reputation. It does not, though, allow generalizing the results to other settings, particularly to offline contexts. The visual time cues were tested using a Western culture sample. Reputation system users from other cultural areas may have a different mental picture of time. Moreover, our results on the time period until obsolescence of online reputation is highly context-dependent within the setting of reputation systems on online crowdsourcing labour market platforms. In other contexts, such as on e-shopping or online auction platforms, the appropriate time to obsolescence of reputation may be shorter or longer. Conclusions and future research On online crowdsourcing labour market platforms, job-seekers' reputation histories remain accessible for unlimited time. In these markets' reputation systems, an experimental lab study examined the presentation of job-seekers' obsolete reputation and how it influences hiring. The findings show that current reputation systems hardly visualize obsolete reputation and scarcely encourage employers to disregard it. As a consequence, employers risk misjudging job-seekers' true current qualification. Jobseekers afflicted by an inadequately presented obsolete reputation get less likely hired resulting in undesirable market outcomes. Empowering employers to ignore job-seekers' obsolete reputation, online crowdsourcing labour market platforms should graphically visualize time in reputation profiles. Time cues based on the graphical timeline metaphor proofed to make employers better aware of the need to disregard obsolete reputation compared to the temporal order cues implemented on current platforms. The paper contributed by issuing recommendations on improving the display of obsolete reputation in reputation systems and discussing the economic and social benefits of visible time display in online crowdsourcing labour markets. Future research should focus on designing additional visual time cues and develop reusable graphical user interface (GUI) library components. GUI components could simplify the widespread deployment of visual time cues across online contexts presenting people's online reputation. As reputation histories are retained online increasingly longer, future work should devote more attention to the economic and social impacts of obsolete online reputation. Figure 1. Employer consensus about time to review obsolescence. university student (97.6/97.1%), high-school student (0.3%/0.3%), part-time employed (28.7%/28.3%), full-time employed (2.7%/2.4%), unemployed (0.6%/0.8%), other occupation (4.2%/3.9%) Online market transactions never (6.0%/5.2%), less than 1x/year (6.6%/6.8%), at least 1x/year (49.2%/48.9%), at least 1x/month (30.4%/30.9%), at least 1x/week (7.8%/8.2%) App usage 2 very often (70.7%/70.7%), often (18.2%/17.9%), sometimes (7.0%/6.7%), rarely (3.8%/3.9%), never (0.3%/0.8%) 1 multiple selection possible, 2 of those 314/358* participants owning a smartphone or tablet PC, * … asterisked values represent the untrimmed sample (n=382) before data cleaning. Figures Tables Reputation profile H1 Reputation profile interfaces ordering the reviews by star rating do not influence employer disregard of obsolete reputation. supported H2 Reputation profile interfaces ordering the reviews by time increase employer disregard of obsolete reputation. rejected H3 Reviews in descending temporal order more strongly encourage employers to disregard obsolete reputation than reviews in ascending temporal order. rejected H4 Reputation profile interfaces displaying reviews on graphical timelines increase employer disregard of obsolete reputation. supported F2 Although an app-designer received negative reviews a couple of years ago, I am eventually able to see him or her as a good designer. -0.010 0.155 0.053 0.775 F3 I forgive an app designer the mistakes which were described in a couple of years old reviews. 0.057 -0.009 0.107 0.802 F4 I do not opt for an app designer who received negative reviews a couple of years ago. Item dropped F5 If an app designer received negative reviews a couple of years ago, I continue to appraise him or her badly. Item dropped Bartlett's test on sphericity (df = 91) p < 0.001*, Kaiser-Meyer-Olkin = 0.786 Table 7. Scales of control variables and their psychometric characteristics. Figure 2 . 2Timelines cue. Figure 3 . 3Excerpt of an app designer's reputation profile. Figure 4 . 4Operationalization of employer disregard of obsolete reputation. Figure 5 . 5Graphical representations of profiles containing no time information (CC3). Table 1. Obsolescence of reviews contained in reputation profiles. Age (years) M = 23.31/23.34, SD = 4.13/4.236, min = 18/18, max = 53/53* Sex female = 56.4%/55.5%, male = 43.6%/44.5% Occupation 1Employer consensus of review obsolescence Year AD1 AD2 AD3 AD4 Date Rating Date Rating Date Rating Date Rating 2010 04/18 4.78 05/16 1.80 04/29 4.79 03/22 1.73 Obsolete 08/21 4.89 09/20 4.98 10/22 1.62 11/06 1.83 2011 03/10 4.96 10/15 4.95 05/02 4.99 08/06 4.77 Not obsolete 2012 03/18 4.85 04/19 4.88 10/22 4.86 08/08 4.95 2013 04/28 4.94 03/03 4.84 02/03 4.81 02/10 4.89 08/18 4.97 10/27 4.70 10/07 4.94 09/22 4.86 2014 02/01 1.77 02/16 4.93 03/10 1.87 01/24 4.90 03/11 1.64 04/21 1.72 04/12 4.92 04/02 4.87 Avg. total rating 4.10 4.10 4.10 4.10 All reviews Avg. non- obsolete rating 3.855 4.337 4.398 4.873 Only not obsolete reviews (2011-2014) Table 2 . 2Demographic characteristics of the sample. TC2 (n=54, n=61) M=4.760, SD=0.237 M=4.724, SD=0.268 CC2 (n=61, n=63) M=4.392, SD=0.335 M=4.400, SD=0.335 CC3 (n=65, n=69) M=4.427, SD=0.377 M=4.423, SD=0.376 … Asterisked values represent the untrimmed sample (n=382) before data cleaning.Treatment conditions (TC) Timelines cue Temporal order cue no yes recent review first TC1 (n=53, n=63) M=4.534, SD=0.400 M=4.499, SD=0.392 oldest review first - TC3 (n=53, n=61) M=4.660, SD=0.310 M=4.664, SD=0.300 Control conditions (CC) Other graphical interface representation Reviews in other order no yes random order CC1 (n=49, n=65) M=4.404, SD=0.358 M=4.410, SD=0.346 - highest rated review first Table 3 . 3Distribution of employer disregard of obsolete reputation across the conditions.Step 1 2 3 4 5.a (TC1, 2) 5.b (TC1,2,3) Predictor β, (B), [SE] β, (B), [SE] β, (B), [SE] β, (B), [SE] β, (B), [SE] VIF β, (B), [SE] VIF Scenario involvement -0.022 -0.043 -0.068 -0.078 -0.078 1.074 -0.079 1.074 (-0.009) (-0.017) (-0.026) (-0.030) (-0.030) (-0.031) [0.022] [0.021] [0.020] [0.019] [0.019] [0.019] Choice involvement 0.040 -0.006 -0.074 -0.054 -0.054 1.138 -0.053 1.139 (0.020) (-0.003) (-0.037) (-0.027) (-0.027) (-0.027) [0.028] [0.028] [0.027] [0.026] [0.026] [0.026] Reputation system experience 0.084 0.071 -0.013 -0.008 -0.017 1.114 -0.015 1.117 (0.039) (0.033) (-0.006) (0.023) (-0.008) (-0.007) [0.026] [0.025] [0.025] [-0.017] [0.023] [0.023] Forgiveness 0.293* 0.205* 0.172 0.172 1.152 0.172 1.152 (0.106) (0.074) (0.062) (0.062) (0.063) [0.019] [0.019] [0.019] [0.019] [0.019] Rational thinking style 0.323* 0.277* 0.277* 1.352 0.276* 1.353 (0.175) (0.150) (0.150) (0.150) [0.031] [0.030] [0.030] [0.030] Timelines 0.297* 0.277 2.853 0.365 7.650 (0.121) (0.113) (0.149) [0.020] [0.033] [0.054] Temporal order 0.024 2.813 -0.073 7.604 (0.010) (-0.027) [0.033] [0.048] Constant (4.325) * * [0.205] [0.209] [0.201] [0.193] [0.193] [0.193] Corr. R2 0.0% 8.0% 15.7% 24.0% 23.8% 23.8% ΔR2 0.4% 8.0% 7.7% 8.3% -0.2% -0.2% ΔF Sig. of step 2.306 29.908 *** Table 5 . 5Summary of empirical support for hypotheses. component matrix: Principal component analysis with varimax rotation and Kaiser-normalization. Scenario involvement Cronbach's α=0.844, CR=0.860, AVE=0.555 Such a CampusApp… Rational thinking style Cronbach's α=0.763, CR=0.765, AVE=0.522 Choice involvement Cronbach's α=0.725, CR=0.734, AVE=0.486 The right choice of the app designer is… CI1 essential -non-essential 0.101 0.128 0.810 0.091 CI2 beneficial -not beneficial 0.047 0.017 0.740 0.105 CI3 needed -not needed 0.115 0.202 0.808 -0.020 Forgiveness Cronbach's α=0.719, CR=0.719, AVE=0.468F1With time I move past negative reviews an app designer received a couple of years ago.Rotated Item no. Item Table 4.Step-wise linear regression results.Table 6. Visual time cues on crowdsourcing labour market platforms. The market for "lemons": Quality uncertainty and the market mechanism. 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[ "Structure of Flux Line Lattices with Weak Disorder at Large Length Scales", "Structure of Flux Line Lattices with Weak Disorder at Large Length Scales" ]	[ "Philip Kim \nDivision of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA\n", "Zhen Yao \nDivision of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA\n", "Cristian A Bolle \nDivision of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA\n", "Charles M Lieber \nDivision of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA\n" ]	[ "Division of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA", "Division of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA", "Division of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA", "Division of Engineering and Applied Sciences\nHarvard University\n02138CambridgeMA" ]	[]	Dislocation-free decoration images containing up to 80,000 vortices have been obtained on high quality Bi2Sr2CaCu2O8+x superconducting single crystals. The observed flux line lattices are in the random manifold regime with a roughening exponent of 0.44 for length scales up to 80-100 lattice constants. At larger length scales, the data exhibit nonequilibrium features that persist for different cooling rates and field histories.	10.1103/physrevb.60.r12589	[ "https://arxiv.org/pdf/cond-mat/9910184v2.pdf" ]	108,291,358	cond-mat/9910184	e77f5ee2652373d8378a2b463468535c6b1eefda	Structure of Flux Line Lattices with Weak Disorder at Large Length Scales 15 Oct 1999 Philip Kim Division of Engineering and Applied Sciences Harvard University 02138CambridgeMA Zhen Yao Division of Engineering and Applied Sciences Harvard University 02138CambridgeMA Cristian A Bolle Division of Engineering and Applied Sciences Harvard University 02138CambridgeMA Charles M Lieber Division of Engineering and Applied Sciences Harvard University 02138CambridgeMA Structure of Flux Line Lattices with Weak Disorder at Large Length Scales 15 Oct 1999numbers: 7460Ge7510Nr7460Ec7472Hs Dislocation-free decoration images containing up to 80,000 vortices have been obtained on high quality Bi2Sr2CaCu2O8+x superconducting single crystals. The observed flux line lattices are in the random manifold regime with a roughening exponent of 0.44 for length scales up to 80-100 lattice constants. At larger length scales, the data exhibit nonequilibrium features that persist for different cooling rates and field histories. Recent studies of high temperature superconductors have shown richness in the phase diagram due to the presence of weak quenched disorder [1]. Larkin first showed that arbitrarily weak disorder destroys the long range translational order of flux lines (FLs) in a lattice [2]. It was recently pointed out that the Larkin model, which is based on a small displacement expansion of the disorder potential, cannot be applied to length scales larger than the correlated volume of the impurity potential termed the Larkin regime [3][4][5]. Beyond the Larkin regime, the behavior of FLs in the absence of dislocations has been considered using elastic models [3][4][5]. First, FLs start to behave collectively as an elastic manifold in a random potential with many metastable states (the random manifold regime) [3]. In this random manifold regime, the translational order decreases as a stretched exponential, whereas there is a more rapid exponential decay in the Larkin regime. At even larger length scales, when the displacement correlation of FLs become comparable to the lattice spacing, the random manifold regime transits to a quasiordered regime where the translational order decays as a power law [4,5]. Experimentally, neutron diffraction [6] and local Hall probe measurements [7] have shown the existence of an order-disorder phase transition with increased field, although the microscopic details of these phases are not clear. Theoretical progress describing FLs in the presence of weak disorder has been made within elastic theory, which proposes the absence of dislocations at equilibrium [5,8,9]. To date, however, there has been no experimental work addressing the structure of dislocation-free FL lattices at large length scales. Previous magnetic decoration studies [10,11] showed that the dislocation density decreases and the translational order increases with increasing magnetic field. However, only relatively shortrange translational order could be probed in the previous work due to the finite image size and relatively low applied fields. In this paper, we report the first large length scale structural studies of FLs with measurements extending up to ∼ 300 lattice constants and fields up to 120 G. Realspace images show dislocation free regions containing up to the order of 10 5 FLs. A very low density of dislocations was also observed, although detailed analysis suggests that the dislocations are not equilibrium features. The translational correlation function and displacement correlator have been calculated from dislocation free data to examine quantitatively the decay of order. These results show a stretched exponential decay of the translational order indicating that FLs are in the random manifold regime. The experimentally determined roughening exponent in the random manifold regime agrees well with theoretical predictions. High quality single crystals of Bi 2 Sr 2 CaCu 2 O 8+x (BSCCO) were grown as described elsewhere [12]. Typically, crystals of ∼ 1mm × 1mm × 20µm size were mounted on a copper cold-finger and decorated with thermally evaporated iron clusters at 4 K. The samples were cooled down to 4 K using different thermal cycles to test nonequilibrium effects and to achieve as close an equilibrium configuration of FLs as possible within the experimental time scale. The FL structure was imaged after decoration using a scanning electron microscope equipped with a 4096 x 4096 pixel, 8-bit gray-scale image acquisition system. Nonlinearity in the system was eliminated using grating standards. This high-resolution system enabled us to acquire images containing nearly 10 5 FLs, while maintaining a similar resolution (∼ 14 pixels between vortices) to previous studies of 10 3 FLs. In addition, an iterative Voronoi construction [13] was used to reduce the positioning inaccuracy to 3 % of a lattice constant. Samples were decorated in fields of 70, 80 and 120 G parallel to the c axis of BSCCO single crystals. In contrast to the previous decoration experiments at lower fields [10,11], we find that the dislocations are rare at these fields. The density of dislocations was 1.7 × 10 −5 , 1.4 × 10 −5 and 3.1 × 10 −5 for 70, 80 and 120 G, respectively, where the total number of vortices is ∼ 240, 000 for each field. It is thus trivial to find many large 100 × 100 µm 2 dislocation free regions in the decorated samples. The size of the largest dislocation free image, which was obtained in a field of 70 G, is 152 × 152 µm 2 with 78,363 vortices [14]. Although a small number of dislocations are detected in our FL images, this does not imply that they are energetically favorable at equilibrium. On the contrary, we believe that the large dislocation-free areas observed in the images provide a lower bound for the length scale of equilibrium dislocation loops. We discuss this point below after presenting a quantitative analysis of the translational order. To study quantitatively the FL lattice order, we proceed as follows. First, a perfect lattice is constructed and registered to the FL positions obtained from an experimental image. The initial lattice vectors used to construct the perfect lattice were obtained from the Fourier transform of the vortex positions. When an image contains a dislocation, the continuum approximation is used to construct the perfect lattice with the dislocation [15]. Next we minimized the root mean square displacement between the underlying perfect lattice and the real FL lattice by varying the position and orientation of the two lattice vectors of the perfect lattice. The displacement vector u(r) associated with each of the vortices positioned at r relative to the perfect lattice was then computed. Fig 1 displays a color-representation of the displacement field for a typical dislocation-free image and an image containing three dislocations. In Fig 1(a), the average displacement is 0.22 a 0 , where a 0 is the lattice constant. Qualitatively, the map consists of several intermixed domain-like structures, within which the displacement fields are correlated. These uniformly dispersed domain-like structures of the displacement field produce sharp Bragg peaks in Fourier space (see Fig 3(b) later). We also believe that u(r) provides a quick indication of nonequilibrium effects. For example, Fig 1(b) exhibits large domains of correlated displacements that are sheared relative to each other; that is, the blue-greenblue coded domains. We believe that this larger scale distortion is a manifestation of a nonequilibrium structure that may arise from quenched dynamics of FLs during our field-cooling process (see below). To compare our data directly with theoretical predictions [5], we have calculated the displacement correlator, B(r), and translational correlation function, C G (r). B(r) and C G (r) are defined as [u(r) − u(0)] 2 /2 and e iG[u(r)−u(0)] , respectively, where is the average over thermal fluctuations and quenched disorder, and G is one of the reciprocal lattice vectors. Theoretically [5], we expect B(r) will show three distinct behaviors as r increases: B(r) ∼ r in the Larkin regime, where B(r) is less than the square of ξ, the in-plane coherence length. As r increases further, FLs are in the random manifold regime where ξ 2 < B(r) < a 2 0 . In this regime B(r) ∼ r 2ν with the roughening exponent 2ν (< 1). Finally, at the largest length scales (the quasiordered regime) where a 2 0 < B(r), B(r) ∼ ln r. Since the in-plane ξ of BSCCO is only ∼ 20Å, the Larkin regime is irrelevant in our experiment (i.e., a 0 ≫ ξ) . Fig 2(a) shows the behavior of B(r) calculated from the data in Fig 1(a). For r < 80a 0 , B(r) can be fit well with a power law, B(r) ∼ r 2ν , with 2ν = 0.44. Thus our experiment is probing the ran-dom manifold regime at least up to this scale. Indeed, B(r) grows only up to 0.05 a 2 0 at r = 80a 0 , well below the expected crossover to the quasiordered regime, i.e. B(r) ∼ a 2 0 . A naive extrapolation to B(r) = a 2 0 suggests the crossover at r ∼ 10, 000a 0 (∼ 4 mm), which is far beyond our experimental limit. Samples with such a large clean area, and direct imaging of ∼ 10 8 vortices would be required to observe the logarithmic roughening of FLs. The roughening exponent 2ν is found to be independent of the field (70 -120 G) and consistent with the estimate 2ν = 2/5 obtained by Feigelman et al. using a scaling argument [3]. As shown in Fig 2(b), C G (r) and e −G 2 B(r)/2 overlap with each other for r < L * , where the measured L * is ∼ 80a 0 . These results support the Gaussian approximation, C G (r) ≈ e −G 2 B(r)/2 , which has been simply assumed for the equilibrium FLs lattice [5] within this length scale. For r > L * , however, B(r) deviates strongly from expected behavior; that is,B(r) saturates and even decreases as r increases. In addition, the Gaussian approximation breaks down for r > L * as evidenced by the difference between C G (r) and e −G 2 B(r)/2 . We believe that this behavior can be attributed to nonequilibrium FL structures at the larger length scales of our experiment. To examine this point further, we decompose B(r) into its longitudinal [B L (r)] and transverse [B T (r)] parts: B(r) = (B L (r) + B T (r))/2, where B L (r) = u(r) − u(0) · r r 2 . (1) It is worth noting that in the random manifold regime, the ratio of B T (r) and B L (r) is predicted to be 2ν +1 [5], and thus an independent estimate of the roughening exponent. The average value of this ratio measured from our data (inset to Fig 3(a)) is 1.40, which is consistent with the value of 2ν obtained from B(r). As shown in Fig 3(a), both B L and B T (r) are described well with the power law behavior up to r ∼ L * . Beyond this range, however, the transverse displacement B T (r) first deviates from power law causing deviations in B(r). Thus, we infer that shear motion of FL lattice should be responsible for the abnormal behavior of B(r). Since the shear modulus of FL lattice is much smaller in magnitude than the compressional modulus [1], B T (r) is always larger than B L (r), and the shear motion dominates the relaxation of the FL lattice during the field cooling process. As temperature decreases, the long wavelength component of shear motion is frozen out. We believe that the domain-like structures seen in Fig 1 are a This issue can also be addressed through Fourier space analysis. Fig 3(b) displays a blow-up of one Bragg peak. Several small satellite peaks appear around the relatively sharp main peak; these satellite peaks indicate a largescale modulation of the FL lattice. If the FLs were in equilibrium, only one main peak should be expected. The corresponding real space distance between the main and satellite peaks is, again, ∼ L * . Hence these satellite peaks provide another evidence of the frozen-in dynamics beyond the equilibrium length scale L * . In addition, we have prepared FL lattices in different ways to address the nonequilibrium structures. For example, we cooled the samples in the absence of a field to 65 K, applied a field 70 G, and then cooled slowly (0.1 K/min) to 4 K. Significantly, we find a similar density of dislocations and FL structure compared to the rapid (10 K/s) field-cooled samples. Since 65 K is far below the melting temperature [16], this observation suggests that the nonequilibrium structures originate from the frozen-in dynamics far below the melting temperature. Although we can probe FLs up to a length scale of ∼ 300 a 0 , there is a much smaller length scale L * that prohibits direct application of the theory derived for an equilibrium FLs. Further studies should address this important issue. Finally, we consider the origin of dislocations observed in our experiments, since nonequilibrium vs. equilibrium nature of dislocations is critical to the existence of the Bragg glass phase. We believe that our data, which exhibit the small numbers of dislocations, in fact, favors nonequilibrium nature of dislocation in the FL lattice we probed by following reasons. First, it is found that most dislocations are pinned in between domain boundaries (see Fig 1(b) for example). If there were a dislocation within the domain-like structures where FLs are locally in equilibrium, the dislocation should be an equilibrium feature. Second, L * ≪ L d = n −1/2 d ∼ 250a 0 , where L d and n d are the average distance between dislocations and the density of dislocations, respectively. If dislocations were energetically favorable in an equilibrium FL lattice, large dislocation loops should proliferate beyond the equilibrium length scale L * . In addition, if some dislocations drift within domains, and are pinned at domain boundaries, we should have L d < ∼ L * . Therefore, our experiment (L * ≪ L d ) suggests that dislocations are not equilibrium features in the FL lattice. Together, our data provide a lower bound for the length scale of equilibrium dislocation loop in the FL lattice. In summary, we have obtained large scale dislocationfree images of the FL lattice in high quality BSCCO superconductors. Quantitative analyses of the translational order indicate that the system is in equilibrium for length scales up to ∼ 80a 0 , and that FLs are in the random manifold regime with a roughening exponent 2ν = 0.44. We suggest that the very small density of dislocations observed in our data is an out-of-equilibrium feature due to the short time scales involved in our field-cooled experiments. We thank D. R. Nelson, D. S. Fisher, P. Le Doussal, and T. Giamarchi snap shot of these frozen long wavelength shear motions. Note that the characteristic length scale of these domain like structures in Fig 1 is again ∼ L * , which explains the deviations in B(r) for r > L * . Therefore, L * is the equilibrium length scale within which FLs can relax to the local equilibrium during our experimental time scale. FIGFIG map correspond to smaller (larger) displacements. Different colors correspond to vortex displacements in different directions, as shown in the inserted color wheel. The two solid lines, inner and outer circles in the color wheel correspond to two basis vectors of the lattice, displacements of 0.5 a0 and a0 respectively. Samples were decorated at 70 G.(a) dislocation-free image containing 37003 vortices. The edge of the image correspond to 106 µm. The lower inset shows a part of both real FL image and a perfect lattice (yellow) with displacement vectors (red). (b) larger scale image containing three dislocations (highlighted by red dots and circles) and large scale shearing. The image contains 78385 vortices in the 160 x 160 µm 2 area. Transverse (open circle) and longitudinal (solid circle) displacement correlators, i.e. B T (r) and B L (r), as a function of distance calculated from Fig. 1(a). The insert shows the ration of the two quantities. (b) The Fourier space image showing all six first order Bragg peaks. (c) Detailed of one of the Fourier peaks calculated from the same image (inverted gray scale). Dark arrows highlight two satellite peaks. for helpful discussion. CML acknowledges support of this work by the NSF Division of Materials Research. * Present address: Department of Applied Physics and DIMES, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands. Lucent technologies. 700 Mountain Avenue, Murray Hill, NJ 07974Bell Laboratoriesaddress: Bell Laboratories, Lucent technologies, 700 Mountain Avenue, Murray Hill, NJ 07974. . G Blatter, M V Feigelman, V M Genshkenbein, A I Larkin, V M Vinokur, Rev. Mod. Phys. 481125G. Blatter, M. V. Feigelman, V. M. Genshkenbein, A. I. Larkin, and V. M. 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[ "On the integrability of a lattice equation with two continuum limits", "On the integrability of a lattice equation with two continuum limits" ]	[ "R N Garifullin ", "R I Yamilov " ]	[]	[]	We study a new example of lattice equation being one of the key equations of a recent generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits which are the wellknown fifth order partial-differential equations, namely, the Sawada-Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an L − A pair and a hierarchy of conservation laws.	10.1007/s10958-020-05160-x	[ "https://arxiv.org/pdf/1708.03179v1.pdf" ]	119,240,494	1708.03179	3ea6ec808fcc602d667421cc5f82893b4dae1698	On the integrability of a lattice equation with two continuum limits 10 Aug 2017 August 11, 2017 R N Garifullin R I Yamilov On the integrability of a lattice equation with two continuum limits 10 Aug 2017 August 11, 2017 We study a new example of lattice equation being one of the key equations of a recent generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits which are the wellknown fifth order partial-differential equations, namely, the Sawada-Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an L − A pair and a hierarchy of conservation laws. Introduction We consider the differential-difference equation u n,t = (u n + 1) u n+2 u n (u n+1 + 1) 2 u n+1 − u n−2 u n (u n−1 + 1) 2 u n−1 + (2u n + 1)(u n+1 − u n−1 ) , (1) where n ∈ Z, while u n (t) is an unknown function of one discrete variable n and one continuous variable t, and the index t in u n,t denotes the time derivative. Equation (1) is obtained as a result of the generalized symmetry classification of five-point differentialdifference equations u n,t = F (u n+2 , u n+1 , u n , u n−1 , u n−2 ), (2) carried out in [8][9][10]. Equation (1) coincides with the equation [9, (E16)]. Even earlier this equation has been obtained in [2]. Equations of the form (2) play an important role in the study of four-point discrete equations on the square lattice, which are very relevant for today, see e.g. [1,5,6,17]. At the present time there is very little information on equation (1). It has been shown in [9] that equation (1) possesses a nine-point generalized symmetry of the form u n,θ = G(u n+4 , u n+3 , . . . , u n−4 ). As for relations to the other known integrable equations of the form (2), nothing useful from the viewpoint of constructing solutions is known, see details in the next Section. However, this equation occupies a special place in the classification [8][9][10]. In particular, it possesses a remarkable property discovered in [7]. This equation has two different continuum limits being the well-known Kaup-Kupershmidt and Sawada-Kotera equations, details will be given below. For this reason, equation (1) deserves a more detailed study. In Section 2 we discuss the known properties of equation (1). In order to justify the integrability of (1), we construct an L − A pair in Section 3 and show that it provides an infinity hierarchy of conservation laws in Section 4. 2 Special place of equation (1) in the classification [8][9][10] In two lists of integrable equations of the form (2) presented in [9,10], the following four equations occupy a special place: those are (1) and u n,t = (u 2 n − 1)(u n+2 u 2 n+1 − 1 − u n−2 u 2 n−1 − 1),(3)u n,t = u 2 n (u n+2 u n+1 − u n−1 u n−2 ) − u n (u n+1 − u n−1 ),(4)u n,t = u n+1 u 3 n u n−1 (u n+2 u n+1 − u n−1 u n−2 ) − u 2 n (u n+1 − u n−1 ).(5) Equations (3)(4)(5) correspond to equations (E17), (E15) of [9] and (E14) of [10], respectively. Equation (4) is known for a long time [19]. All other equations of [9,10] go over in the continuum limit into the third order equations of the form U τ = U xxx + F (U xx , U x , U),(6) where the indices τ and x denote τ and x partial derivatives, and mainly into the Korteweg-de Vries equation. These four equations correspond in the continuum limit to the fifth order equations of the form: U τ = U xxxxx + F (U xxxx , U xxx , U xx , U x , U).(7) For all the four equations (1,3-5) we get in the continuum limit one of the two well-known equations. One of them is the Kaup-Kupershmidt equation [4,12]: U τ = U xxxxx + 5UU xxx + 25 2 U x U xx + 5U 2 U x ,(8) and the second one is the Sawada-Kotera equation [18]: U τ = U xxxxx + 5UU xxx + 5U x U xx + 5U 2 U x .(9) Using the substitution u n (t) = √ 3 3 + √ 3 2 ε 2 U τ − 18 5 ε 5 t, x + 4 3 εt , x = εn,(10) in equation (5), we get at ε → 0 the Sawada-Kotera equation (9). All other continuum limits are known, see [1] for (4) and [7] for (1) and (3). Here we explicitly replicate substitutions for equation (1) under study which has two different continuum limits. The substitution u n (t) = − 4 3 − ε 2 U τ − 18 5 ε 5 t, x + 4 3 εt , x = εn,(11) in (1) leads to equation (8), while the substitution u n (t) = − 2 3 + ε 2 U τ − 18 5 ε 5 t, x + 4 3 εt , x = εn,(12) leads to equation (9). The link between these discrete and continuous equations is shown in the following diagram: (3) (1) (4) (8) (9) (5) ℄ (11) ℄ (12) Ù (10) We see that equation (8) has two different integrable approximations, while equation (9) has three approximations. As far as we know, there are no relations between (1,3-5) and other known equations of the form (2) presented in [9,10]. More precisely, we mean relations in the form of the transformationsû n = ϕ(u n+k , u n+k−1 , . . . , u n+m ), k > m,(13) and their compositions, see a detailed discussion of such transformations in [8]. As for relations among (1,3-5), equation (5) is transformed into (4) byû n = u n+1 u n , i.e. (5) is a simple modification of (4). There is a complicated relation between equations (1) and (4) found in [2]. As it is shown in [9], it is a composition of two Miura type transformations. It is very difficult to use that relation for the construction of solutions because the problem is reduced to solving the discrete Riccati type equations [9]. There is a complete list of integrable equations of the form (7), see [3,13,16]. Equations (8) and (9) play the key role in that list, since all the other are transformed into these two by transformations of the form: U = Φ(U, U x , U xx , . . . , U x...x ). L − A pair As the continuum limit shows, equation (1) should be close to equations (3,4) in its integrability properties, and these equations (3,4) have the L − A pairs defined by 3 × 3 matrices [1,7]. Here we construct an L − A pair for equation (1) following [7]. We look for an L − A pair of the form L n ψ n = 0, ψ n,t = A n ψ n with the operator L n of the form L n = T 2 + l (1) n T + l (0) n + l (−1) n T −1 ,(15) where l (k) n with k = −1, 0, 1 depend on a finite number of the functions u n+j . Here T is the shift operator: T h n = h n+1 . In the case of (15) the operator A n can be chosen as: A n = a (1) n T + a (0) n + a (−1) n T −1 . The compatibility condition for system (14) has the form d(L n ψ n ) dt = (L n,t + L n A n )ψ n = 0(16) and it must be satisfied in virtue of the equations (1) and L n ψ n = 0. If we suppose that the coefficients l (k) n depend on u n only, then we can check that a (k) n have to depend on u n−1 , u n only. However, in this case the problem has no solution for equation (1). Therefore we proceed to the case when the functions l (k) n depend on u n , u n+1 . Then the coefficients a (k) n must depend on u n−1 , u n , u n+1 . In this case we have managed to find operators L n and A n with one irremovable arbitrary constant λ playing the role of the spectral parameter here: L n = T 2 − U n+1 u n+1 T + λ U n+1 u n 1 − u n U n T −1 ,(17)A n = u n U n (λT −1 − λ −1 T ) + u n U 2 n (u n−1 + u n+1 T −1 )(T − 1),(18) where U n = u n 1 + u n . The L − A pair (14,17,18) can be rewritten in the standard matrix form with 3 × 3 matricesL n ,Ã n : Ψ n+1 =L n Ψ n , Ψ n,t =Ã n Ψ n , where Ψ n is a spectral vector-function, whose standard form is Ψ n =   ψ n+1 ψ n ψ n−1   . Here we slightly change Ψ by gauge transformation to simplify the matricesL n ,Ã n : Ψ n =   U n (λψ n+1 + 1 un ψ n ) ψ n ψ n−1   ,A n =    u n−1 u n−2 U 2 n−1 − u n+1 un U 2 n − u n + u n−1 − u n−1 λU n−1 − u n−1 u n−2 λU 2 n−1 u n+1 un+u n+1 +un Un λ(1 + u n )u n−1 − u n u n+1 un+u n+1 −unu n−1 Un − 1 λ λun Un − u n+1 un U 2 n λu n−1 − (1 + u n−1 )u n u n−1 u n−2 U 2 n−1 − u n−1 λU n−1 λ − u n−1 u n−2 U 2 n−1 + unu n−1 U n−1    .(20) (21) In this case, unlike (16), the compatibility condition can be represented in a form which does not use the spectral vector-function Ψ n . It will be the following matrix form in terms of 3 × 3 matrices:L n,t =Ã n+1Ln −L nÃn . (22) Conservation laws As far as we know, there exist two methods to construct the conservation laws by using the matrix L − A pair (22), see [5,11,14]. However, we do not see how to apply those methods in the case of matrices (20) and (21). Here we will use a different scheme, presented in [7], for deriving conservation laws from the L − A pair (14). In [7] that scheme was applied to one equation (3) only. Here we check it again by example of one more equation (1). The structure of operators (17,18) allows us to rewrite the L − A pair (14) in form of the Lax pair. The operator L n has a linear dependence on λ: L n = P n − λQ n ,(23) where P n = T 2 − U n+1 u n+1 T, Q n = − U n+1 u n 1 − u n U n T −1 , and U n is defined by (19). IntroducingL n = Q −1 n P n we get an equation of the form: L n ψ n = λψ n . The functions λψ n and λ −1 ψ n in the second equation of (14) can be expressed in terms ofL n and ψ n , using (24) and its consequence λ −1 ψ n =L −1 n ψ n . As a result we have: ψ n,t =Â n ψ n ,(25)whereÂ n = u n U n (T −1 Q −1 n P n − T P −1 n Q n ) + u n U 2 n (u n−1 + u n+1 T −1 )(T − 1). It is important that the new operatorsL n andÂ n in the L − A pair (24,25) do not depend on the spectral parameter λ. For this reason, the compatibility condition can be written in the operator form, without using the ψ-function: L n,t =Â nLn −L nÂn = [Â n ,L n ],(26) and this is nothing but the Lax equation. The difference between this L − A pair and the well-known Lax pairs for the Toda and Volterra equations is that the operatorsL n and A n are nonlocal. Nevertheless, using the definition of inverse operators: P n P −1 n = P −1 n P n = 1, Q n Q −1 n = Q −1 n Q n = 1 (27) and the fact that they are linear, we can check that (26) is true by direct calculation. The conservation laws of equation (1), which are expressions of the form ρ (k) n,t = (T − 1)σ (k) n , k ≥ 0, can be derived from the Lax equation (26), notwithstanding the nonlocal structure of L n ,Â n , see [20]. For this, first of all, we have to represent the operatorsL n ,Â n as formal series in powers of T −1 : H n = k≤N h (k) n T k .(28) Formal series of this kind can be multiplied according the rule: (a n T k )(b n T j ) = a n b n+k T k+j . The inverse series of the form (28) can be easily obtained by definition (27), for instance: Q −1 n = −(1 + q n T −1 + (q n T −1 ) 2 + . . . + (q n T −1 ) k + . . .) u n U n+1 , q n = u n U n . The seriesL n has the second order: L n = k≤2 l (k) n T k = − u n U n+1 T 2 + u n u n−1 U 2 n − 1 u n+1 T + u n−1 U n u n u n−2 U 2 n−1 − 1 T 0 + . . . . The conserved densities ρ (k) n of equation (1) can be found as: ρ (0) n = log l (2) n , ρ (k) n = resL k n , k ≥ 1,(29) where the residue of formal series (28) is defined by the rule: res H n = h (0) n , see [20]. Corresponding functions σ (k) n can easily be found by direct calculation. The conserved densitiesρ (k) n below have been found in this way and then simplified in accordance with the rule:ρ (k) n = c k ρ (k) n + (T − 1)g (k) n , where c k is a constant and g (k) n is a function. The first three densities of equation (1) read: ρ (0) n = log(u n + 1), ρ (1) n = V n+1 u n−1 U n , ρ (2) n = u n+2 u n+1 u 2 n u n−1 u n−2 U 2 n+1 U n U 2 n−1 + u n+1 u n−2 (V 2 n − u n u n−1 ) U n U n−1 + u 2 n+1 u 2 n u n−1 2U 2 n+1 U 3 n − u n+1 u n u n−1 U n+1 U n + u n+1 u n−1 (V n+1 − 1)V n 2U n+1 U 2 n + u 2 n−1 2U 2 n , where V n = u n u n−1 + u n + u n−1 . We can easily check that ∂ 2ρ (1) n ∂u n+1 ∂u n−1 = 0, ∂ 2ρ (2) n ∂u n+2 ∂u n−2 = 0. Therefore, in accordance with a theory of the review [20], the conserved densitiesρ (0) n ,ρ(1) n ,ρ (2) n have the orders 0, 2, 4 respectively. This means that we have got three conserved densities, which are nontrivial and essentially different. . V E Adler, arXiv:1103.5139nlin.SIV.E. Adler. 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Yamilov. On the integrability of a discrete analogue of the Kaup-Kupershmidt equation, in press in Ufa Mathematical Journal (2017), arXiv:1612.03652v1 [nlin.SI]. Non-invertible transformations of differential-difference equations. R N Garifullin, R I Yamilov, D Levi, J. Phys. A: Math. Theor. 4912ppR.N. Garifullin, R.I. Yamilov and D. Levi. Non-invertible transformations of differential-difference equations // J. Phys. A: Math. Theor. 49 (2016) 37LT01 (12pp). Classification of five-point differentialdifference equations. R N Garifullin, R I Yamilov, D Levi, J. Phys. A: Math. Theor. 5012520127pp)R.N. Garifullin, R.I. Yamilov, D. Levi. Classification of five-point differential- difference equations // J. Phys. A: Math. Theor. 50 (2017) 125201 (27pp). Classification of five-point differentialdifference equations II. R N Garifullin, R I Yamilov, D Levi, arXiv:1708.02456nlin.SIR.N. Garifullin, R.I. Yamilov, D. Levi. 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[ "COMMON KINGS OF A CHAIN OF CYCLES IN A STRONG TOURNAMENT", "COMMON KINGS OF A CHAIN OF CYCLES IN A STRONG TOURNAMENT" ]	[ "Logan ", "Zeyu Zheng " ]	[]	[]	It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge.	null	[ "https://arxiv.org/pdf/2206.04154v1.pdf" ]	249,538,528	2206.04154	479b842b8642432114b60660bebb781326787642	COMMON KINGS OF A CHAIN OF CYCLES IN A STRONG TOURNAMENT Logan Zeyu Zheng COMMON KINGS OF A CHAIN OF CYCLES IN A STRONG TOURNAMENT It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge. Introduction A tournament is a directed graph, obtained by assigning an orientation to each edge of a complete graph. A king is a vertex k such that each other vertex is reachable from k by a directed path of length at most 2 [1]. A strong tournament has the property that every vertex is reachable from every other by a directed path [2]. A chain of cycles is a sequence C 1 , C 2 , . . . , C n such that for i ∈ [1, n − 1], the cycle C i+1 is obtained from C i by replacing some edge (x, y) of C i by two edges (x, z) and (z, y) where z is not on C i . It is known that every tournament has a king [1], and it is known that every strong tournament T n has directed cycles of length 3, ..., n [2]. The proof can be modified so that all cycles contain a common vertex which can be any king in the whole tournament, but not necessarily of the subtournaments determined by the vertices of the cycles. In this note, we show that: Theorem 1.1. Given a strong tournament T n on n vertices and any king k of T n , there exist a chain of cycles C 3 , C 4 , . . . , C n where C i is of length i such that k is a king in all subtournaments induced by the vertex sets of the cycles in the chain. Proof of Theorem 1.1 Let k be a king of T n . We use A to denote N + (k) and use B to denote N − (k). A result in [3] shows that we can partition A into A 1 , A 2 , . . . , A r such that A i determines a strong subtournament for all integer i ∈ [1, r], and for all i < j, all directed edges between A i and A j terminates in A j . Claim 1. There exists a * ∈ A r , b * ∈ B such that (a * , b * ) is an edge in T n . For a chosen a ∈ A r , as T n is strong, there exists a directed path from a to k. Choose the shortest such path. This path will look like a → · · · a * → b * → k for some b * ∈ B. a * ∈ A r because A r is fully beaten by A i for all i < r, so this path resides in A r and no other elements of B are included because this would imply a shorter path. This satisfies Claim 1. We see that the subtournament on A 1 ∪ A 2 ∪ ... ∪ A r−1 has Hamiltonian path P 1 . Also, A r is strong, so it has a Hamiltonian cycle, in which there is a Hamiltonian path P 2 ending at a * . Concatenating P 1 and P 2 , this satisfies Claim 2. We denote this path by P = a 1 → a 2 → · · · → a d = a * where d = deg + k. Using Claim 1 and Claim 2, we can construct length-i directed cycles C i of which k is the king for i ∈ [3, n]. For any i ∈ [1, d], we see that C i+2 = k → a d−i+1 → · · · → a d → b → k is a directed cycle of length i + 2 and k is the king of the subtournament on V (C i+2 ). This gives us cycles of length from 3 to d + 2. Given a directed cycle C i−1 of length i − 1 where i ∈ [d + 3, n] and (A ∪ {k}) ⊂ V (C i−1 ) . We now show that we can always find a cycle of length i and its vertex set still contains A ∪ {k}. Since T n is strong we know that there exists an edge (w, z) ∈ T n so that w ∈ C i−1 and z ∈ V (T n ) \ V (C i−1 ). Note that (V (T n ) \ V (C i−1 )) ⊂ B, so (z, k) is an edge in T n . Therefore, there exists an edge (x, y) on the cycle C i−1 such that both (x, z) and (z, y) are edges in T n . We can insert z in the cycle C i−1 between x and y to get a directed cycle C i of length i, and (A ∪ {k}) ⊆ V (C i ). This recursively gives us cycles of length from d + 3 to n. Now we have created a collection of cycles C 3 , C 4 , . . . , C n in T n of which k is the king of subtournament induced by vertices of each of them. Note as an added fact that C i+1 can be obtained from C i by replacing some edge (x, y) of C i by two edges (x, z) and (z, y) where z ∈ V (T n ) \ V (C i ) for all integer i ∈ [3, n − 1], which finishes the proof. Claim 2 . 2There is a Hamiltonian path through A ending at a * .1 arXiv:2206.04154v1 [math.CO] 8 Jun 2022 AcknowledgementsThe authors thank Professor András Gyárfás for proposing the research problem we solve here in his course Advanced Combinatorics and for his valuable suggestions. The work have been conducted under the auspices of the Budapest Semesters in Mathematics Program during the Spring of 2022.Yaobin Chen, Professor Jie Ma, Professor Hehui Wu have provided different proofs to the main theorem. The second author would like to thank them for fruitful early discussions on this problem. On dominance relations and the structure of animal societies: III The condition for a score structure. Hg Landau, The bulletin of mathematical biophysics. 15HG Landau. "On dominance relations and the structure of animal societies: III The condition for a score structure". In: The bulletin of mathematical biophysics 15.2 (1953), pp. 143-148. Topic on Tournaments, Holt, Rinehart and Winston. Jw Moon, IncNew YorkJW Moon. "Topic on Tournaments, Holt, Rinehart and Winston". In: Inc., New York (1968). Tournaments, Selected Topics in Graph Theory. K B Reid, W Lowell, Beineke, Academic PressLondonKB Reid and Lowell W Beineke. "Tournaments, Selected Topics in Graph Theory". In: Academic Press, London (1978), pp. 169-204.	[]
[ "Deterministic preparation of arbitrary quantum state by a shaped photon pulse in one-dimensional continuum", "Deterministic preparation of arbitrary quantum state by a shaped photon pulse in one-dimensional continuum" ]	[ "Zeyang Liao \nInstitute for Quantum Science and Engineering (IQSE)\nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College StationTXUSA\n", "M Suhail Zubairy \nInstitute for Quantum Science and Engineering (IQSE)\nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College StationTXUSA\n" ]	[ "Institute for Quantum Science and Engineering (IQSE)\nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College StationTXUSA", "Institute for Quantum Science and Engineering (IQSE)\nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College StationTXUSA" ]	[]	We propose a method to deterministically prepare an arbitrary quantum state in a one-dimensional (1D) continuum by a shaped photon pulse. This method is based on time-reverse of the quantum emission process. We show that the desired quantum state such as Dicke or timed-Dicke state can be successfully prepared with very high fidelity even if the dissipation to the environment is nonnegligible and the pulse shaping is not perfect. This method is experimentally feasible in 1D waveguide-QED or circuit-QED system.	10.1103/physreva.98.023815	[ "https://arxiv.org/pdf/1804.07390v2.pdf" ]	59,151,742	1804.07390	170c26d5ac88ada1ff72badd9e7c3933e340d816	Deterministic preparation of arbitrary quantum state by a shaped photon pulse in one-dimensional continuum 19 Apr 2018 Zeyang Liao Institute for Quantum Science and Engineering (IQSE) Department of Physics and Astronomy Texas A&M University 77843-4242College StationTXUSA M Suhail Zubairy Institute for Quantum Science and Engineering (IQSE) Department of Physics and Astronomy Texas A&M University 77843-4242College StationTXUSA Deterministic preparation of arbitrary quantum state by a shaped photon pulse in one-dimensional continuum 19 Apr 2018arXiv:1804.07390v1 [quant-ph] We propose a method to deterministically prepare an arbitrary quantum state in a one-dimensional (1D) continuum by a shaped photon pulse. This method is based on time-reverse of the quantum emission process. We show that the desired quantum state such as Dicke or timed-Dicke state can be successfully prepared with very high fidelity even if the dissipation to the environment is nonnegligible and the pulse shaping is not perfect. This method is experimentally feasible in 1D waveguide-QED or circuit-QED system. I. INTRODUCTION Preparation of an arbitrary quantum state such as highly entangled state has important applications in quantum information, quantum simulation, and quantum metrology [1]. One of the most widely used methods to prepare a quantum state is by applying a seriers of unitary quantum gates to drive the system into the diresed state [2][3][4][5][6][7]. The realization of the quantum gates is significantly restricted by the decoherence time of the system. An alternative way to prepare a quantum state is via bath environment engineering. This method may dissipatively drive a quantum system to a desired state without worrying about the decoherence of the system [8][9][10][11][12][13][14][15]. However, the design of bath environment is usually complicated especially when the many-body interaction is non-negligible. In this paper, we propose an alternative method to prepare an arbitrary quantum state by a shaped photon pulse. This method is based on the time reversal symmetry of a closed quantum system [16][17][18]. More specifically, an arbitrary quantum state can be prepared by inverting a quantum emission process. It is known that the dynamics of a quantum system is governed by the Schrödinger equation, i.e., i ∂/∂ t \|ψ(t) = H(t)\|ψ(t) . By applying complex conjugate on both sides and taking t → −t, we have i ∂/∂ t \|ψ * (−t) = H * (−t)\|ψ * (−t) . Hence, complex conjugate of the quantum state with reverse time also satisfies the Schrödinger equation govern − −−−− → \|ψ * (0) . To prepare a quantum state \|ψ(T ) in a multi-emitter system, we can input a photon pulse with spectrum being the complex conjugate of the emission spectrum of the same quantum system prepared in the quantum state \|ψ * (T ) . Hence, the design of required photon pulse is straightforward and different from previous methods the photon number required to prepare a quantum state in current method is minimized. In the usual three-dimensional space, the time-reversal * [email protected] of an emission system is almost impratical because the photon is emitted to all directions. In contrast, the photon only emits to two directions (left or right) in the 1D waveguide-QED system [19][20][21][22][23][24][25][26]. It is therefore more feasible to reverse the emission process and prepare arbitrary quantum state in this system. It has been shown that full inversion of a two-level emitter and high effcient quantum state transfer between two emitters are possible in a 1D continumn by specially designed photon pulse [27][28][29][30][31]. Here, we show a general procedure to prepare an arbitrary quantum state in a multi-emitter system coupled to a 1D structure with many-body interactions inlcuded. To illustrate our method, here for simplicity we mainly consider the single-excitation case in the pure 1D waveguide model which is valid when the emitters mainly couple to a single mode of a quasi-1D waveguide like a line defect in a photonic crystal [32] or superconduting transmission line [33,34]. We also consider the noises like decay to the free space, imperfect pulse shaping and emitter position uncertainty and our numerical simulation shows that our method is robust against these noises. II. SCHEME The schematic setup is shown in Fig. 1 where N emitters with positions r 1 , r 2 , · · · , r N coupled to a 1D waveguide. The atoms are assumed to be identical and they have Λ-type energy structure. The waveguide photon mode can couple to the \|a ↔ \|b transition, while \|a ↔ \|c transition is driven by a classical light pulse. It is assumed that \|c state is substable with very slow decay rate. To prepare a desired quantum state, we first calculate the required photon pulse based on the time-reversal process. Then we can generate the required photon pulse by certain pulse shaping techniques. In the optics regime, pulse shaping techniques such as spatial ligth modulation [35,36], eletro-optic modulation [37][38][39][40][41], cross-phase modulation [42,43] have been demonstrated. In the microwave regime, arbitrary waveform generator and pulse shaping by tunable resonator-emitter coupling can be applied [44][45][46]. The shaped photon pulse is then input into the waveguide from both directions and it can drive the The schematic setup to generate arbitrary single-excitation quantum state in a 1D waveguide-QED system. The single photon source can be generated by the spontaneous down conversion (SPDC) process. The idler photon can trigger the function generator (FG) to control the photon modulator. The photon modulator (Mod) can shape the signal photon to the desired shape. The shaped single photon pulse is injected from both ends of the waveguide to excite the \|a ↔ \|b transition and a classical π pulse is applied to transfer the population from \|a state to \|c state. atomic transition between states \|b and \|a . At a predetermined time, we apply a coherent classical π pulse to transfer the population in state \|a to state \|c . Since state \|c has a very slow decay rate, the prepared quantum state can then be preserved for an extended period of time for further applications. The interaction Hamiltonian of the system in the rotating wave approximation is given by [47,48] H = N j=1 k g k e ikrj a k σ + j e −iδω k t + Ω(t)\|a j c\| + H.c. (1) The first term is the coupling between the waveguide photon and the \|a ↔ \|b transition with coupling strength g k = µ ab · E k (r j )/ ( µ ab is the transition dipole moment, E k (r j ) is the guided photon field strength at position r j , and is the Planck constant). The second term is the coupling between the classical driving light and the \|a ↔ \|c transition with Rabi frequency Ω(t). Here, a † k (a − k ) are the creation (annihilation) operators of the waveguide photon modes with wavevector k, and σ + j = \|a j b\| (σ − j = \|b j a\|) is the raising (lowering) operator of the jth emitter for the \|a ↔ \|b transition. δω k = (\|k\| − k a )v g is the detuning between the photon frequency and the atomic transition frequency where k a is the wave vector at frequency ω a and v g is the group velocity [49]. For a single photon excitation, the quantum state of the system at any time can be expressed as \|Ψ(t) = N j=1 [a j (t)\|a j , 0 + c j (t)\|c j , 0 ] + k β k (t)\|b, 1 k where \|a j , 0 (\|c j , 0 ) is the state that the jth atom is in the excited state \|a (\|c ) while the other atoms are in the ground state \|b with zero photon in the waveguide, \|b, 1 k is the state that all the atoms are in the ground state and one photon with wavevector k is in the waveguide mode. From the Schrödinger equation i ∂/∂ t \|ψ(t) = H\|ψ(t) , we can obtain the dynamics of the emitter system given by (see supplementary material) a j (t) =b j (t) − N l=1 ( Γ 2 e ikar jl − γδ jl )a l (t − r jl v g ) − iΩ(t)c j (t),(2)c j (t) = − iΩ(t)a j (t),(3)where b j (t) = − i 2π Γvg L 2 ∞ −∞ β k (0)e ikrj −iδω k t dk is the excitation by the incident photon with β k (0) being the spectrum of the incident photon. Γ = 2L\|g ka \| 2 /v g is the decay rate due to the waveguide photon modes, γ is the decay rate to the free space, and r jl is the distance between the jth and lth emitters. In the following, we show how to prepare the desired quantum state onto the emitter system by designing specific β k (0). For simplicity, we maily consider the case when the loss to the free space modes is negligible (γ = 0), which is reasonable because near-perfect waveguide with very small γ has been reported [50,51]. Nonetheless, in the numerical simulation, we also consider the case when γ is non-negligible and the results shows that the scheme still works well if γ is not very large. We first neglect the classical light and show how we can prepare the desired quantum state in the \|a and \|b subspace. Then we show how to apply a classical π pulse to transfer the quantum state to the \|b , \|c subspace which is more robust against decoherence. According to our theory, to prepare an arbitrary singleexcitation quantum state \|ψ = N j=1 a j \|g · · · e j · · · g in an N emitter system, the spectrum of the incident photon pulse is given by (4) where [M (δk)] jl = Γ 2 e i(ka+δk)r jl − iδω k δ jl and t 0 is the pulse propagating time to the atoms (see the appendix). β k (0) = i Γv g 2L N j,l=1 a l [M * (δk)] −1 jl e −ikrj e iδkvg t0 III. NUMERICAL EXAMPLE A. Two-emitter Let us first look at the simplest two-emitter system. Supposing that \|ψ = a 1 \|eg + a 2 \|ge to be prepared, the required photon spectrum can be calculated from Eq. (4) and its explicit form is shown in Eq. (B14) in the appendix. For example, if we want to prepare a symmetric state \|ψ S = 1 2 (\|eg + \|ge ), we can inject a photon with spectrum given by Eq. (4) with a 1 = a 2 = 1/ √ 2. The photon spectrum is shown in Fig. 2(a) where we can see that the left and right propagating photon spectrum have the same amplitude and phase. The emitter excitation amplitudes as a function of time for this input are shown in Fig. 2(b) from which we can see that the two emitters are excited with the same dynamics. At time t = 15/Γ, Re[a 1 ] = Re[a 2 ] = 0.7 ≈ 1/ √ 2 and Im[a 1 ] = Im[a 2 ] = 0 which indicates that the symmetric state \|+ = (\|eg + \|ge )/ √ 2 has been successfully approached. In practice, it is very difficult to shape the photons perfectly. We also numerically calculate the case when the required spectrum is coarse-grained sampling. We assume that the input spectrum consists of 20 discrete frequency components sampling from −2.5Γ to 2.5Γ. The results for the emitter excitation amplitudes are shown as the symbols in Fig. 2(b) from which we see that the excitation follows the curve very well. The fidelity between the evolving emitter state and the symmetric state as a function of time is shown as the black solid curve in Fig. 2(c). It is see that at t = 15/Γ the fidelity is about 1 which indicates that the symmetric state has been successfully prepared. The red dashed curve in Fig. 2(c) is the fidelity when the decay to the free space is nonzero (γ = Γ/5). Even if the decay to the free space is nonnegligible, the fidelity to prepare the symmetric state can be still very high (about 90% in this example). The dotted line is the result with coarse-grained sampling where the fidelity can also be about 87%. Similarly, if the incident photon has a spectrum given by Eq. (4) but with a 1 = −a 2 = 1/ √ 2, we can prepare the emitter system into an antisymmetric state \|− = (\|eg − \|ge )/ √ 2. The corresponding incident photon spectrum is shown in Fig. 2(d). Different from the symmetric case, the left and right propagating spectra in the antisymmetric case have opposite phase. The emitter excitation amplitude as a function of time is shown in Fig. 2(e) where we can see that the two emitters have the same excitation amplitude but opposite phase. At t = 15/Γ, Re[a 1 ] = −Re[a 2 ] = 0.7 ≈ 1/ √ 2, and Im[a 1 ] = Im[a 2 ] ≃ 0. The fidelity with respect to the antisymmetric state is about 98% which clearly shows that the antisymmetric state has been successfully prepared (see the back solid curve in Fig. 2(f)). If the decay to the free space is nonignorable, e.g. γ = Γ/5, the fidelity of the prepared state using the same pulse shown in Fig. 2(d) with respect to the antisymmetric state can still be about 90% (red dashed curve in Fig. 2(f)). From Fig. 2(e) and 2(f), we can also see the coarse-grained sampling still works very well. In addition to the symmetric and antisymmetric states, we can also drive the system to the interesting timed-Dicke state [53]. The timed-Dicke state can be probabilistically prepared by delayed-choice measurent [53] and it may be used for ultrasensitive quantum metrology [54]. However, deterministic preparation of the timed-Dicke state is still an open question. In our method, to prepare a timed-Dicke state, we can input a single photon with a spectrum given by Eq. (4) with a 1 = e ikar1 and a 2 = e ikar2 . For example, if the two emitters have separation λ/4 with r 1 = −λ/8 and r 2 = λ/8, the corresponding timed-Dicke state \|ψ T D = (e −iπ/4 \|eg + e iπ/4 \|ge )/ √ 2 with a 1 = e −iπ/4 and a 2 = e iπ/4 . The desired photon spectrum is shown in Fig. 2(g). We can see that the left and right propagating modes have different spectrum amplitudes and phases. By injecting the single photon with spectrum shown in Fig. 2(g), the emitter excitation amplitude as a function of time is shown in Fig. 2(h) where we can see that at t = 15/Γ, Re[a 1 ] = Re[a 2 ] = Im[a 2 ] = 0.492 and Im[a 1 ] = −0.492. This state has a fidelity with respect to the timed-Dicke state about 98.5% which infers that the required timed-Dicke state is successfully prepared by a single photon pulse. For imperfect waveguide such that γ = Γ/5, the fidelity to prepare the timed-Dicke state with the same pulse can still be about 90%. With coarse-grained sampling, the fidelity can still be about 87%. B. Multiple emitters This method can also be applied in a quantum system with arbitrary number of emitters. For example, to prepare a timed-Dicke state on a ten-emitter system, i.e., \|ψ T D = 1/ √ 10 10 j=1 e ikarj \|g · · · e j · · · g with r j = (−1.125 + 0.25i)λ, the required spectrum is given by Eq. (4) with a j = e i(2j−1)π/4 / √ 10 and the corresponding pulse shape in the real space is shown in Fig. 3(a). The red curve is the photon pulse incident from the right and the blue curve (is amplified by ten times for clarity) is that from the left. It is clearly seen that most of the photon pulse is coming from one direction which is the demonstraction of the directional emission of the timed-Dicke state [53]. On applying this pulse, the real part and imaginary part of the excitation amplitude for each emitter at t = 20/Γ are shown in Fig. 3(b) and 3(c), respectively. The bars with black solid filling are the expected excitation amplitudes and the bars with filling patterns are the numerical results. We can clearly see that the generated quantum state (red bars with dense pattern) by the incident pulse shown in Fig. 3(a) is very close to the expected time-Dicke state. Even if the incident spectrum and the emitter positions have 10% uncertainty, the numerical excitation amplitudes (blue bars with sparse pattern) are still very close to the expected quantum state. This can be also seen from Fig. 3(d) where the fidelity between the generated state and the expected state is shown. The red solid curve is the fidelity when the incident pulse is shown in Fig. 3(a) and the emitter positions are precisely determined, while the black dotted curve is the fidelity when noise is included, i.e., the incident pulse and the emitter positions have 10% uncertainty. At t = 20/Γ, both of them can have maximum fidelity about 96%. Thus, in our method the timed-Dicke state can be deterministically prepared in a ten-emitter system with fidelity approaching unit and it is very robust against the noises like the imperfect pulse shaping and the emitter position uncertainty. C. Robust quantum state preparation In the previous discussions, we have shown that a desired quantum state can be successfully prepared at certain time. However, after that the quantum state decay quickly. Here, we show how to preserve these quantum states in a more robust quantum state. Our idea is that we can apply a π pulse to transfer the population in the state \|a to the state \|c right after the first pulse. For example, to prepare a antisymmetric state in a two-emitter system, we can apply a single photon pulse to excite the transition from \|b to \|a and the photon spectrum is given by Eq. (4) with a 1 = −a 2 = 1/ √ 2. Meanwhile, we apply a classical light pulse with Rabi frequency Ω(t) = √ π 2δ e −(t−t0) 2 /δ 2 and δ = 0.1Γ to induce a transition from state \|a to state \|c . Since ∞ −∞ Ω(t)dt = π/2, the pulse can completely transfer the population at state \|a to state \|c . The emitter excitation amplitudes as a function of time are shown in Fig. 4(a) from which we can see that the excitation in state \|a can be transferred to the state \|c when the classical pulse is applied at around t = 10/Γ. The fidelity F a between the state a 1 (t)\|a 1 b 2 + a 2 (t)\|b 1 a 2 and the antisymmetric state (\|a 1 b 2 − \|b 1 a 2 )/ √ 2 is shown as the black solid curve in Fig. 4(b), and the fidelity F c between the state c 1 (t)\|c 1 b 2 + c 2 (t)\|b 1 c 2 and the antisymmetric state (\|c 1 b 2 − \|b 1 c 2 )/ √ 2 is shown as the red dashed curve in Fig. 4(b). As the single photon pulse is applying, F a is increasing and approaching about 1 at about t = 10/Γ. Then we apply a classical π pulse, F a decreases very quickly while F c increases rapidly and approaches about 1. The antisymmetric state is therefore successfully prepared in a more robust state because the state \|c is a substable state. IV. SUMMARY In summary, we have shown that driven by a shaped photon pulse an arbitrary quantum state such as the Dicke and timed-Dicke states in a 1D waveguide-QED system can be sucessfully prepared with fidelity approaching unit. This method is based on the time-reverse of a quantum emission process. The design of the preparation process is straightforward and the number of photon required is minimized. We also propose a method to transfer the prepared quantum state to a more robust state by applying a classical π pulse. The method shown here can be experimentally demonstrated in the circuit-QED system where strong coupling and single microwave photon pulse shaping have been successfully achieved [45,46,55]. This work may find important applications in the waveguide-QED-based quantum science. We thank S. -W. Li, J. You and X. Zeng for helpful discussions. This work is supported by a grant from the Qatar National Research Fund (QNRF) under the NPRP project 8-352-1-074. When t → ∞, Eq. (B1) can be rewritten as β k (t) = −i Γv g 2L Na j=1 e −ikrj χ j (k),(B2) where χ j (k) = ∞ −∞ e iδω k t ′ a j (t ′ )Θ(t ′ )dt ′ (B3) with Θ(t ′ ) = 1 for t ′ 0 and Θ(t ′ ) = 0 for t ′ < 0. By performing the inverse Fourier transformation a j (t)Θ(t) = 1 2π ∞ −∞ χ j (k)e −iω k t dδω k ,(B4) and using the relation (g * k e −ikrj a k σ + j e −iδω k t + g k e ikrj a † k σ − j e iδω k t ). Since the phase of g k only affect the overall phase of the quantum state, we can safely assume that g k is a real number and therefore we have g k = g * k . The only difference between H(t) and H * (−t) is that k → −k. Therefore, the spectrum emitted by a quantum state \|ψ * = N j=1 a * j \|g · · · e j · · · g under H * (−t) is given by β k = −i Γv g 2L N j,l=1 a * l [M (δk)] −1 jl e ikrj .(B10) To prepare a quantum state \|ψ = N j=1 a j \|g · · · e j · · · g , the spectrum should be β * k and it is given by β * k = i Γv g 2L N j,l=1 a l [M * (δk)] −1 jl e −ikrj ,(B11) which is Eq. (4) in the main text except the phase term e iδkvg t0 . The phase term is considering the phase accumulated when the photon pulse propagating to the atoms. by H * (−t). If \|ψ(0)H(t) − −− → \|ψ(T ) , then we also have \|ψ * (T ) H * (−t) FIG. 1: (Color online) The schematic setup to generate arbitrary single-excitation quantum state in a 1D waveguide-QED system. The single photon source can be generated by the spontaneous down conversion (SPDC) process. The idler photon can trigger the function generator (FG) to control the photon modulator. The photon modulator (Mod) can shape the signal photon to the desired shape. The shaped single photon pulse is injected from both ends of the waveguide to excite the \|a ↔ \|b transition and a classical π pulse is applied to transfer the population from \|a state to \|c state. online) Two-emitter excitation for three different single-photon pulses input. (a-c) Symmetric state preparation. (d-f) Antisymmetric state preparation. (g-i) Timed-Dicke state preparation. The first column is the spectrum of the incident photon with real part of the amplitude, and imaginary part of the amplitude shown as solid lines and thin dashed lines, respectively. The second column is the emitter excitation amplitude as a function of time with real part shown as the solid lines and the imaginary part shown as the dashed lines. The symbols are the results when the input spectrum is coarse-grained sampling. The third column is the fidelity as a function of time where the solid line is the result with γ = 0, the dashed line is the result with γ = Γ/5, and the dotted line is the result with coarse-grained sampling. The distance between the two emitters is 0.25λ and t0 = 15vg /Γ. FIG. 3 : 3(Color online) Preparation of timed-Dicke state in a ten-emitter system. (a) The incident pulse shape (the left incident pulse has been amplified by ten times for clarity). The real part (b) and imaginary part (c) of the excitation amplitude for each emitter at t = 20/Γ. The bars with solid filling are the expected excitation amplitude, the red bars with dense pattern are the numerical calculated excitation amplitudes by the incident pulse shown in (a), and the bars with sparse pattern are the numerical results when the incident spectrum and the emitter positions have 10% uncertainties. (d) The fidelity as a function of time. The red solid curve is the fidelity without considering the uncertainty and the black dotted line is the fidelity with 10% uncertainty. FIG. 4 : 4(Color online) Robust quantum state preparation. (a) The two-emitter excitation as a function of time. (b) The fidelity with respect to the symmetric state as a function of time. t0 = 10/Γ. M (δk) is an N × N matrix with matrix element given by M (δk) jl = Γ 2 e i(ka+δk)r jl − iδω k δ jl . The emission spectrum is then given by Appendix A: Dynamical equationIn this appendix, we derive Eq.(5)in the main text. From Schrödinger equation i ∂/∂ t \|ψ(t) = H\|ψ(t) with Hamiltonian given by Eq. (3) and quantum state given by Eq. (4), we haveFormally integrating Eq. (A2) and inserting the results into Eq. (A1), we obtaiṅwhere we assume that the incident photon has spectrum β k (0). The first term is excitation due to the incident photon, the second term is the interaction between atoms mediated by the waveguide photon modes, and the third term is the transition between state \|a and \|c induced by the classical pulse. The second term can be integrated using the Weisskopf-Wigner approximation and it can be calculated as −Γ/2 Na i=1 e ikar jl a i (t − r jl /v g ) where r ij = \|r j − r l \| and Γ = 2L\|g ka \| 2 /v g[47,48]. Therefore, we havėwhich is Eq. (3) in the main text.Appendix B: SpectrumIn this appendix, we calculate the emission spectrum of the waveguide-QED system described by Hamiltonian H(t) shown in Eq. (3) without the incident photon (β k (0) = 0) and the classical pulse (Ω(t) = 0). 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[ "Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover", "Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover" ]	[ "Meirav Zehavi [email protected] \nDepartment of Computer Science\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n" ]	[ "Department of Computer Science\nTechnion -Israel Institute of Technology\n32000HaifaIsrael" ]	[]	The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximal Minimal Vertex Cover (MMVC) problem, whose minimization version, Vertex Cover, is one of the most studied problems in the field of Parameterized Complexity. Our main contribution is a parameterized approximation algorithm for MMVC, including its weighted variant. We also give conditional lower bounds for the running times of algorithms for MMVC and its weighted variant.	null	[ "https://arxiv.org/pdf/1503.06438v1.pdf" ]	2,033,488	1503.06438	40c90b70be57cbcafc101fdc717d443e8e8f9ca3	Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover Meirav Zehavi [email protected] Department of Computer Science Technion -Israel Institute of Technology 32000HaifaIsrael Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximal Minimal Vertex Cover (MMVC) problem, whose minimization version, Vertex Cover, is one of the most studied problems in the field of Parameterized Complexity. Our main contribution is a parameterized approximation algorithm for MMVC, including its weighted variant. We also give conditional lower bounds for the running times of algorithms for MMVC and its weighted variant. Introduction The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. Given a maximization problem, the optimal solution to its minimization version might not only be significantly smaller, but it might also be possible to efficiently compute it by using some well-known parameterized algorithm-in such cases, one can know in advance if for a given instance of the maximization problem, the size of the optimal solution to the minimization version is a good choice as a parameter. Furthermore, assuming that an optimal solution to the minimization version can be efficiently computed, one may use it to solve the maximization problem; indeed, the optimal solution to the maximization problem and the optimal solution to its minimization version may share useful, important properties. We examine this suggestion by studying the Maximal Minimal Vertex Cover (MMVC) problem. This is a natural choice-the minimization version of MMVC is the classic Vertex Cover (VC) problem, one of the most studied problems in the field of Parameterized Complexity. In Weighted MMVC (WMMVC), we are given a graph G = (V, E) and a weight function w : V → R ≥1 . We need to find the maximum weight of a set of vertices that forms a minimal vertex cover of G. The MMVC problem is the special case of WMMVC in which w(v) = 1 for all v ∈ V . Notation: Let vc (vc w ) denote the size (weight) of a minimum vertex cover (minimum weight vertex cover) of G, and let opt (opt w ) denote the size (weight) of a maximal minimal vertex cover (maximal-weight minimal vertex cover) of G. Clearly, vc ≤ min{vc w , opt} ≤ max{vc w , opt} ≤ opt w . Observe that the gap between vc w = max{vc, vc w } and opt = min{opt, opt w } can be very large. For example, in the case of MMVC, if G is a star, then vc w = 1 while opt = \|V \| − 1. The standard notation O * hides factors polynomial in the input size. A problem is fixed-parameter tractable (FPT) with respect to a parameter k if it can be solved in time O * (f (k)), for some function f . The maximum degree of a vertex in G is denoted by ∆. Given v ∈ V , N (v) denotes the set of neighbors of v. Also, let γ denote the smallest constant such that it is known how to solve VC in time O * (γ vc ), using polynomial space. Currently, γ < 1.274 [12]. Given U ⊆ V , let G[U ] denote the subgraph of G induced by U . Finally, let M denote the maximum weight of a vertex in G. Related Work MMVC: Boria et al. [4] show that MMVC is solvable in time O * (3 m ) and polynomial space, and that WMMVC is solvable in time and space O * (2 tw ), where m is the size of a maximum matching of G, and tw is the treewidth of G. Since max{m, tw} ≤ vc (see, e.g., [19]), this shows that WMMVC is FPT with respect to vc. Moreover, they prove that MMVC is solvable in time O * (1.5874 opt ) and polynomial space, where the running time can be improved if one is interested in approximation. 1 Boria et al. [4] also prove that for any constant > 0, MMVC is inapproximable within ratios O(\|V \| − 1 2 ) and O(∆ −1 ), unless P=NP. They complement this result by proving that MMVC is approximable within ratios \|V \| − 1 2 and 3 2∆ in polynomial time. Recently, Bonnet and Paschos [3] and Bonnet et al. [2] obtained results relating to the inapproximability of MMVC in subexponential time. Furthermore, Bonnet et al. [2] prove that for any 1 < r ≤ \|V \| 1 2 , MMVC is approximable within ratio 1 r in time O * (2 \|V \| r 2 ) . Vertex Cover: First, note that VC is one of the first problems shown to be FPT. In the past two decades, it enjoyed a race towards obtaining the fastest parameterized algorithm (see [7,17,1,27,18,10,28,8,29,12]). The best parameterized algorithm, due to Chen et al. [12], has running time O * (1.274 vc ), using polynomial space. In a similar race [10,13,11,30,32,24], focusing on the case where ∆ = 3, the current winner is an algorithm by Issac et al. [24], whose running time is O * (1.153 vc ). For Weighted VC (WVC), parameterized algorithms were given in [28,21,22,31]. The fastest ones (in [31]) use time O * (1.381 vcw ) and polynomial space, time and space O * (1.347 vcw ), and time O * (1.443 vc ) and polynomial space. Kernels for VC and WVC were given in [10,14,25], and results relating to the parameterized approximability of VC were also examined in the literature (see, e.g., [5,6,20]). Finally, in the context of Parameterized Complexity, we would also like to note that vc is a parameter of interest; indeed, apart from VC, there are other problems whose parameterized complexity was studied with respect to this parameter (see, e.g., [9,23,26]). Our Contribution While it can be easily seen that WMMVC is solvable in time O * (2 vc ) and polynomial space (see Section 2), we observe that this result might be essentially tight. More precisely, we show that even if G is a bipartite graph and w(v) ∈ {1, 1 + 1 \|V \| } for all v ∈ V , an algorithm that solves WMMVC in time O * ((2− ) vcw ), which upper bounds O * ((2− ) vc ) , contradicts the SETH (Strong Exponential Time Hypothesis). We also show that even if G is a bipartite graph, an algorithm that solves MMVC in time O * ((2 − ) vc 2 ) contradicts the SETH. Then, we turn to present our main contribution, ALG, which is a parameterized approximation algorithm for WMMVC with respect to the parameter vc. More precisely, we prove the following theorem, where α is a user-controlled parameter that corresponds to a tradeoff between time and approximation ratio. Theorem 1. For any α < 1 2 − 1 M +1 such that 1 x x (1 − x) 1−x ≥ 3 1 3 where x = 1− 1−α M (2α−1)+1−α , 2 ALG runs in time O * (( 1 x x (1 − x) 1−x ) vc ) , returning a minimal vertex cover of weight at least α · opt w . ALG has a polynomial space complexity. ALG is based is on a mix of two bounded search tree-based procedures. 3 In particular, the branching vectors of one of these procedures are analyzed with respect to the size of a minimum vertex cover of a minimum vertex cover of G. Another interesting feature of this procedure is that once it reaches a leaf (of the search tree), it does not immediately return a result, but to obtain the desired approximation ratio, it first performs a computation that is, in a sense, 2 In particular, the result holds for any 1 2 < 1 2 − 1 7.35841·M +1 ≤ α < 1 2 − 1 M +1 . 3 Information on the bounded search technique, which is perhaps the most well-known technique used to design parameterized algorithms [16], is given in Section 3. an exhaustive search. We would like to note that in the design of our second procedure, we integrate rules that are part of the iterative compression-based algorithm for VC by Peiselt [29]. Since ALG can be used to solve MMVC, in which case M = 1, we immediately obtain the following corollary. Corollary 1. For any α < 2 3 such that 1 x x (1−x) 1−x ≥ 3 1 3 where x = 2 − 1 α , 4 ALG runs in time O * (( 1 x x (1−x) 1−x ) vc ) , returning a minimal vertex cover of size at least α · opt. ALG has a polynomial space complexity. Upper and Lower Bounds In this section, we give upper and conditional lower bounds related to the parameterized complexity of WMMVC and MMVC with respect to vc and vc w . Recall, in this context of the next results, that vc ≤ vc w and MMVC is a special case of WMMVC. Proof. The algorithm is as follows. First, compute a minimum vertex cover S of G in time O * (γ vc ) using polynomial space, and initialize A to S. Then, for every subset S ⊆ S, if B = S ∪ ( v∈S\S N (v)) is a minimal vertex cover of weight larger than the weight of A, update A to store B. Finally, return A. Clearly, we return a minimal vertex cover. Now, let A * be an optimal solution. Consider the iteration where we examine S = A ∩ S. Then, since A * is a vertex cover, B ⊆ A * . Suppose, by way of contradiction, that there exists v ∈ A * \ B. The vertex v does not belong to S (since A ∩ S ⊆ B). Moreover, it should have a neighbor outside A * (since A * is a minimal vertex cover). Thus, since S is a vertex cover, v has a neighbor in S \ A * . This implies that v ∈ u∈S\S N (u), which contradicts the assumption that v / ∈ B. We conclude that B = A * . Therefore, the algorithm is correct, and it clearly has the desired time and space complexities. Now, we observe that even in a restricted setting, the algorithm above is essentially optimal under the SETH. Lemma 1. For any constant > 0, WMMVC in bipartite graphs, where w(v) ∈ {1, 1 + 1 \|V \| } for all v ∈ V , cannot be solved in time O * ((2 − ) vcw ) unless the SETH fails. Proof. Fix > 0. Suppose, by way of contradiction, that there exists an algorithm, A, that solves WMMVC in the restricted setting in time O * ((2 − ) vcw ). We aim to show that this implies that there exists an algorithm that solves the Hitting Set (HS) problem in time O * ((2 − ) n ), which contradicts the SETH [15]. In HS, we are given an n-element set U , along with family of subsets of U , F = {F 1 , F 2 , . . . , F m }, and the goal is to find the minimum size of a subset 4 In particular, the result holds for any 0.53183 ≤ α < U ⊆ U that is a hitting set (i.e., U contains at least one element from every set in F). We next construct an instance (G = (V, E), w : V → R ≥1 ) of WMMVC in the restricted setting: -R 1 = {r u : u ∈ U }, and R 2 = {r c i : c ∈ {1, . . . , n + 1}, i ∈ {1, . . . , m}}. -L = {l u : u ∈ U }, and R = R 1 ∪ R 2 . -V = L ∪ R. -E = {{l u , r u } : u ∈ U }∪{{l u , r c i } : u ∈ F i , i ∈ {1, . . . , m}, c ∈ {1, . . . , n+1}}. -∀v ∈ L : w(v) = 1 + 1 \|V \| . -∀v ∈ R : w(v) = 1. An illustrated example is given in Fig. 1. It is enough to show that (1) vc ≤ n, and (2) the solution for (U, F) is q iff the solution for (G, w) is n−q \|V \| + \|R\|. Indeed, this implies that we can solve HS by constructing the above instance in polynomial time, running A in time O * ((2 − ) vc ) (since vc ≤ n), obtaining an answer of the form n−q \|V \| + \|R\|, and returning q. First, observe that L is a minimal vertex cover of G: it is a vertex cover, since every edge has exactly one endpoint in L and one endpoint in R, and it is minimal, since every vertex in L has an edge that connects is to at least one vertex in R. Therefore, vc ≤ \|L\| = n. Now, we turn to prove the second item. For the first direction, let q be the solution to (U, F), and let U be a corresponding hitting set of size q. Consider the vertex set S = {l u : u ∈ U \U }∪{r u : u ∈ U }∪R 2 . By the definition of w, the weight of S is (1+ 1 \|V \| )\|U \U \|+\|U \|+\|R 2 \| = 1 \|V \| \|U \U \|+\|R 1 \|+\|R 2 \| = n−q \|V \| +\|R\|. The set S is a vertex cover: since R 2 ⊆ S, every edge in G that does not have an endpoint in R 2 is of the form {l u , r u }, and for every u ∈ U either l u ∈ S (if u / ∈ U ) or r u ∈ S (if u ∈ U ). Moreover, S is a minimal vertex cover. Indeed, we cannot remove any vertex l u ∈ S ∩ L or r u ∈ S ∩ R 1 and still have a vertex cover, since then the edge of the form {l u , r u } is not covered. Also, we cannot remove any vertex r c i ∈ R 2 , since there is a vertex l u / ∈ S such that {l u , r c i } ∈ E (to see this, observe that because U is a hitting set, there is a vertex u ∈ U ∩ F i , which corresponds to the required vertex l u ). For the second direction, let p be the solution to (G, w), and let S be a corresponding minimal vertex cover of weight p. Clearly, p ≥ w(R) = n + m(n + 1), since R is a minimal vertex cover of G. Observe that for all u ∈ U , by the definition of G and since S is a minimal vertex cover, exactly one among the vertices l u and r u is in S. Suppose that there exists r c i ∈ R 2 \ S. Then, for all u ∈ F i , we have that l u ∈ S (by the definition of G and since S is a vertex cover), which implies that for all c ∈ {1, . . . , n + 1}, we have that r c i / ∈ S (since S is a minimal vertex cover). Thus, p = w(S ∩ (L ∪ R 1 )) + \|S ∩ R 2 \| ≤ n(1 + 1 \|V \| ) + (m − 1)(n + 1) < n + m(n + 1), which is a contradiction. Therefore, R 2 ⊆ S. Denote U = {u : r u ∈ S ∩ R 1 }. By the above discussion and the definition of w, \|S\| = \|V \R2\| 2 + \|R 2 \| = \|R 1 \| + \|R 2 \| = \|R\|, and p − \|S\| = 1 \|V \| \|S ∩ L\| = 1 \|V \| (n − \|S ∩ R 1 \|). Denoting \|U \| = q, we have that p = n−q \|V \| + \|R\|. Thus, it remains to show that U is a hitting set. Suppose, by way of contradiction, that U is not a hitting set. Thus, there exists F i ∈ F such that for all u ∈ F i , we have that u / ∈ U . By the definition of U , this implies that for all u ∈ F i , we have that l u ∈ S. Thus, N (r 1 i ) ⊆ S, while r 1 i ∈ S (since we have shown that R 2 ⊆ S), which is a contradiction to the fact that S is a minimal vertex cover. Next, we also give a conditional lower bound for MMVC. The proof is quite similar to the one above, and is thus relegated to Appendix A. Informally, the idea is to modify the above proof by adding a copy l u of every vertex l u ∈ L, which is only attached to its "mirror vertex", r u , in R 1 . While previously the weights 1 + 1 \|V \| encouraged us to choose many vertices from L, now the copies encourage us to choose these vertices. A Parameterized Approximation Algorithm In this section we develop ALG (see Theorem 1). When referring to α, suppose that it is chosen as required in Theorem 1, and define x accordingly. The algorithm is a mix of two bounded search tree-based procedures, ProcedureA and ProcedureB, which are developed in the following subsections. 5 For these procedures, we will prove the following lemmas. Having these procedures, we give the pseudocode of ALG below. The algorithm computes a minimum vertex cover U of a minimum vertex cover U , solving the given instance by calling either ProcedureA (if \|U \| ≤ vc 2 ) or ProcedureB (if \|U \| > vc 2 ) . Return ProcedureA(G, w, α, U, U , ∅, ∅). 5: else 6: Algorithm 1 ALG(G = (V, E), w : V → R ≥1 , Return ProcedureB(G, w, U, ∅, ∅). 7: end if Now, we turn to prove Theorem 1. Proof. The correctness of the approximation ratio immediately follows from Lemmas 3 and 4. Observe that the computation of U can indeed be performed in time O * (γ vc ) since U ⊆ U , and therefore \|U \| ≤ \|U \| ≤ vc. Moreover, since γ < 1.274, by Lemmas 3 and 4, the running time is bounded by O * (max{( 1 x x (1−x) 1−x ) vc , 3 vc 3 }). Since α is chosen such that ( 1 x x (1−x) 1−x ) vc ≥ 3 vc 3 , the above running time is bounded by O * (( 1 x x (1−x) 1−x ) vc ). In the rest of this section, we give necessary information on the bounded search tree technque (Section 3.1), after which we develop ProcedureA (Section 3.2) and ProcedureB (Section 3.3). The Bounded Search Tree Technique Bounded search trees form a fundamental technique in the design of recursive parameterized algorithms (see [16]). Roughly speaking, in applying this technique, one defines a list of rules of the form Rule X. [condition] action, where X is the number of the rule in the list. At each recursive call (i.e., a node in the search tree), the algorithm performs the action of the first rule whose condition is satisfied. If by performing an action, the algorithm recursively calls itself at least twice, the rule is a branching rule, and otherwise it is a reduction rule. We only consider actions that increase neither the parameter nor the size of the instance, and decrease at least one of them. Observe that, at any given time, we only store the path from the current node to the root of the search tree (rather than the entire tree). The running time of the algorithm can be bounded as follows. Suppose that the algorithm executes a branching rule where it recursively calls itself times, such that in the i th call, the current value of the parameter decreases by b i . Then, (b 1 , b 2 , . . . , b ) is called the branching vector of this rule. We say that β is the root of (b 1 , b 2 , . . . , b ) if it is the (unique) positive real root of x b * = x b * −b1 + x b * −b2 + . . . + x b * −b , where b * = max{b 1 , b 2 , . . . , b }. If b > 0 is the initial value of the parameter, and the algorithm (a) returns a result when (or before) the parameter is negative, (b) only executes branching rules whose roots are bounded by a constant c, and (c) only executes rules associated with actions performed in polynomial time, then its running time is bounded by O * (c b ). In some of the leaves of a search tree corresponding to our first procedure, we execute rules associated with actions that are not performed in polynomial time (as required in the condition (c) above). We show that for every such leaf in the search tree, we execute an action that can be performed in time O * (g( )) for some function g. Then, letting L denote the set of leaves in the search tree whose actions are not performed in polynomial time, we have that the running time of the algorithm is bounded by O * (c b + ∈L g( )). ProcedureA: The Proof of Lemma 3 The Next, we present each rule within a call ProcedureA(G, w, α, U, U , I, O). After presenting a rule, we argue its correctness (see Lemma 5). Since initially I = O = ∅, we thus have that ProcedureA guarantees the desired approximation ratio. For each branching rule, we also give the root of the corresponding branching vector (with respect to the measure above). We ensure that (1) the largest root we shall get is 2, (2) the procedure stops calling itself recursively, at the latest, once the measure drops to 0, and (3) actions not associated with leaves can be performed in polynomial time. Observe that initially the measure is vc . Thus, as explained in Section 3.1, the running time of ProcedureA is bounded by O * (2 vc + (I ,O )∈P \|{ U ⊆ U : I ⊆ U , O ∩ U = ∅, \| U \| ≥ (1 − x)vc}\|), where P is the set of all partitions of U into two sets. This running time is bounded by O * (2 vc + \|{ U ⊆ U : \| U \| ≥ (1 − x)vc}\|) = O * (2 vc + vc max i=(1−x)vc vc i ). Since x < 1 2 (by the definition of x and since α < 1 2 − 1 M +1 ), vc ≤ vc 2 and 1 x x (1−x) 1−x ≥ 3 1 3 > 2 1 2 , this running time is further bounded by O * (( 1 x x (1 − x) 1−x ) vc ) . Thus, we also have the desired bound for the running time, concluding the correctness of Lemma 5. Reduction Rule 1. [There is v ∈ O such that N (v) ∩ O = ∅] Return U . In this case there is no vertex cover that does not contain any vertex from O, and therefore it is possible to return an arbitrary minimal vertex cover. The action can clearly be performed in polynomial time. Reduction Rule 2. [There is v ∈ X such that N (v) ⊆ X, where X = I ∪ ( u∈O N (u))] Return U . Observe that any vertex cover that does not contain any vertex from O, must contain all the neighbors of the vertices in O. Thus, any vertex cover that contains all the vertices in I and none of the vertices in O, also contains the vertex v and all of its neighbors; therefore, it is not a minimal vertex cover. Thus, it is possible to return an arbitrary minimal vertex cover. The action can clearly be performed in polynomial time. Reduction Rule 3. [U = I ∪ O] Perform the following computation. Let A = I ∪ ( v∈O N (v) ∩ U ). As long as there is a vertex v ∈ A such that N (v) ∩ U ⊆ A, choose such a vertex (arbitrarily) and remove it from A. Let A be the set obtained at the end of this process. Let A = A ∪ ( v∈U \A N (v) \ U ). 3. Denote F = { U ⊆ U : I ⊆ U , O ∩ U = ∅, \| U \| ≥ (1 − x)vc}. 6 4. Initialize B = U . 5. For all F ∈ F: (a) Let B F = F ∪ ( v∈U \F N (v)) (b) If B F is a minimal vertex cover: -If the weight of B F is larger than the weight of B: Replace B by B F . 6. Return the set of maximum weight among A and B. First, observe that this rule ensure that, at the latest, the procedure stops calling itself recursively once the measure drops to 0. Furthemore, by the pseudocode, the action can be performed in time O * (\|F\|), which by the definition of F, is the desired time. It remains to prove that Lemma 3 is correct. We begin by considering the set A. Since U = I ∪ O is a vertex cover of G[U ] and the previous rules were not applied, we have that A is vertex cover of G[U ] such that I ⊆ A and O ∩ A = ∅. Thus, by its definition, A is a minimal vertex cover of G[U ] such that A ⊆ A. Thus, since U is a vertex cover, every edge either has both endpoints in U , in which case it has an endpoint in A , or it has exactly one endpoint in U and exactly one endpoint in V \ U , in which case it has an endpoint that is a vertex in A or a neighbor in V \U of a vertex in U \A . Therefore, A is a vertex cover. Moreover, every vertex in A has a neighbor in U \ A (by the minimality of A ) and every vertex in ( v∈U \A N (v) \ U ), by definition, has a neighbor in U \ A . Thus, we overall have that A is a minimal vertex cover such that A ∩ U ⊆ A. Since A is a minimal vertex cover, by the pseudocode, we return a weight, W , of a minimal vertex cover. Assume that there is a minimal vertex cover S * of weight opt w such that I ⊆ S * and O∩S * = ∅. This implies that A ⊆ S * ∩U . Now, to prove Lemma 3, it is sufficient to show that W ≥ α·opt w . Denote F * = S * ∩U . Since S * is a minimal vertex cover, we have that S * = F * ∪ ( v∈U \F * N (v)). If S * contains at least (1 − x)vc elements from U , there is an iteration where we examine F = F * , in which case B F * = S * , and therefore we return opt w . Thus, we next suppose that \|S * ∩ U \| < (1 − x)vc. Since B is initially U , to prove Lemma 3, it is now sufficient to show that max{w( A), w(U )} ≥ α · w(S * ). Since S * is a vertex cover such that I ⊆ S * and O ∩ S * = ∅, we have that A ∩ U ⊆ A ⊆ F * . By the definition of A and since S * is a minimal vertex cover, this implies that S * \ U ⊆ A \ U . Thus, overall we have that max{w(U ), w( A)} w(S * ) = max{w(U ), w( A)} w(S * \ U ) + w(S * ∩ U ) ≥ max{w(U ), w( A)} w( A \ U ) + w(S * ∩ U ) = max{w(U ), w( A)} w( A) + w(S * ∩ U ) − w( A ∩ U ) ≥ w(U ) w(U ) + w(S * ∩ U ) = 1 2 − w(U \S * ) w(U ) ≥ 1 2 − \|U \S * \| w(S * ∩U )+\|U \S * \| ≥ 1 2 − x·vc M ·(1−x)·vc+x·vc = 1 2 − x M −(M −1)x Since x = 1 − 1−α M (2α−1)+1−α , we have that the expression above is equal to α. Branching Rule 4. Let v be a vertex in U \(I∪O). Return the set of maximum weight among A and B, computed in the following branches. The correctness of Lemma 5 is preserved, since every vertex cover either contains v (an option examined in the first branch) or does not contain v (an option examined in the second branch). Moreover, it is clear that the action can be performed in polynomial time and that the branching vector is (1,1), whose root is 2. ProcedureB: The Proof of Lemma 4 This procedure is based on combining an appropriate application of the ideas used by the previous procedure (considering the fact that now the size of any vertex cover of G[U ] is larger than vc 2 ) with rules from the algorithm for VC by Peiselt [29]. Due to lack of space, the details are given in Appendix B. A Proof of Lemma 2 Fix > 0. Suppose, by way of contradiction, that there exists an algorithm, A, that solves MMVC in the restricted setting in time O * ((2 − ) vc 2 ). We aim to show that this implies that there exists an algorithm that solves HS in time O * ((2 − ) n ), which contradicts the SETH [15]. To this end, we construct an graph G = (V, E) that defines an instance of MMVC in the restricted setting: -R 1 = {r u : u ∈ U }, and R 2 = {r c i : c ∈ {1, . . . , n + 1}, i ∈ {1, . . . , m}}. -L = {l c u : u ∈ U, c ∈ {1, 2}}, and R = R 1 ∪ R 2 . -V = L ∪ R. -E = {{l c u , r u } : u ∈ U, c ∈ {1, 2}} ∪ {{l 1 u , r c i } : u ∈ F i , i ∈ {1, . . . , m}, c ∈ {1, . . . , n + 1}}. It is enough to show that (1) vc ≤ 2n, and (2) the solution for (U, F) is q iff the solution for (G, w) is (n − q) + \|R\|. Indeed, this implies that we can solve HS by constructing the above instance in polynomial time, running A in time O * ((2 − ) n ) (since vc ≤ 2n), obtaining an answer of the form (n − q) + \|R\|, and returning q. First, observe that L is a minimal vertex cover of G: it is a vertex cover, since every edge has exactly one endpoint in L and one endpoint in R, and it is minimal, since every vertex in L has an edge that connects is to at least one vertex in R. Therefore, vc ≤ \|L\| = 2n. Now, we turn to prove the second item. For the first direction, let q be the solution to (U, F), and let U be a corresponding hitting set of size q. Consider the vertex set S = {l c u : u ∈ U \ U , c ∈ {1, 2}} ∪ {r u : u ∈ U } ∪ R 2 . Observe that \|S\| = 2\|U \ U \| + \|U \| + \|R 2 \| = \|U \ U \| + \|R 1 \| + \|R 2 \| = (n − q) + \|R\|. The set S is a vertex cover: since R 2 ⊆ S, every edge in G that does not have an endpoint in R 2 is of the form {l c u , r u }, and for every u ∈ U either [l 1 u , l 2 u ∈ S (if u / ∈ U )] or [r u ∈ S (if u ∈ U )]. Moreover, S is a minimal vertex cover. Indeed, we cannot remove any vertex l u ∈ S ∩ L or r u ∈ S ∩ R 1 ) and still have a vertex cover, since then the edge of the form {l c u , r u } is not covered. Also, we cannot remove any vertex r c i ∈ R 2 , since there is a vertex l 1 u / ∈ S such that {l 1 u , r c i } ∈ E (to see this, observe that because U is a hitting set, there is a vertex u ∈ U ∩ F i , which corresponds to the required vertex l 1 u ). For the second direction, let p be the solution to the instance defined by G, and let S be a corresponding minimal vertex cover of size p. Clearly, p ≥ \|R\| = n + m(n + 1), since R is a minimal vertex cover of G. Observe that for all u ∈ U , by the definition of G and since S is a minimal vertex cover, we can assume WLOG that either [l 1 u , l 2 u ∈ S and r u / ∈ S] or [l 1 u , l 2 u / ∈ S and r u ∈ S]. Indeed, to see this, note that if l 1 u , r u ∈ S, then l 2 u / ∈ S, and thus we can replace (in S) r u by l 2 u and still have a solution. Suppose that there exists r c i ∈ R 2 \ S. Then, for all u ∈ F i , we have that l 1 u ∈ S (by the definition of G and since S is a vertex cover), which implies that for all c ∈ {1, . . . , n + 1}, we have that r c i / ∈ S (since S is a minimal vertex cover). Thus, p = \|S ∩(L∪R 1 )\|+\|S ∩R 2 \| ≤ 2n+(m−1)(n+1) < n + m(n + 1), which is a contradiction. Therefore, R 2 ⊆ S. Denote U = {u : r u ∈ S ∩ R 1 }. By the above discussion, p = \|S ∩ L\| + \|S ∩ R 1 \| + \|R 2 \| = 1 2 \|S ∩ L\| + \|R 1 \| + \|R 2 \| = (n − \|S ∩ R 1 \|) + \|R\|. Denoting \|U \| = q, we have that p = (n − q) + \|R\|. Thus, it remains to show that U is a hitting set. Suppose, by way of contradiction, that U is not a hitting set. Thus, there exists F i ∈ F such that for all u ∈ F i , we have that u / ∈ U . By the definition of U , this implies that for all u ∈ F i , we have that l 1 u ∈ S. Thus, N (r 1 i ) ⊆ S, while r 1 i ∈ S (since we have shown that R 2 ⊆ S), which is a contradiction to the fact that S is a minimal vertex cover. Lemma 6. ProcedureB returns a minimal vertex cover S that satisfies the following condition: -If there is a minimal vertex cover S * of weight opt w such that I ⊆ S * and O ∩ S * = ∅, then the weight of S is at least 1 2 − 1 M +1 · opt w . To ensure that ProcedureB runs in time O * (3 Next, we present each rule within a call ProcedureB(G, w, U, I, O). After presenting a rule, we argue its correctness (see Lemma 6). Since initially I = O = ∅, we thus have that ProcedureB guarantees the desired approximation ratio. For each branching rule, we also give the root of the corresponding branching vector (with respect to the measure above). We ensure that (1) the largest root we shall get is at most 3 1 3 , (2) the procedure stops calling itself recursively, at the latest, once the measure drops to 0, and (3) the procedure only executes rules whose actions can be performed in polynomial time. Observe that initially the measure is vc. Thus, as explained in Section 3.1, the running time of ProcedureB is bounded by O * (3 1. Let A = I ∪ ( v∈O N (v) ∩ U ). As long as there is a vertex v ∈ A such that N (v) ∩ U ⊆ A, choose such a vertex (arbitrarily) and remove it from A. Let A be the set obtained at the end of this process. Let A = A ∪ ( v∈U \A N (v) \ U ). 3. Return the set of maximum weight among A and U . Observe that this rule ensure that, at the latest, the procedure stops calling itself recursively once the measure drops to 0. We next prove that Lemma 4 is correct. In a manner similar to the explanation following Rule 3, we have that A is a minimal vertex cover such that A ∩ U ⊆ A. In particular, A ∩ U is a minimal vertex cover of G[U ], and therefore its size is larger than vc 2 (otherwise ALG would have called ProcedureA). Thus, by the pseudocode, we return a weight of a minimal vertex cover. Assume that there is a minimal vertex cover S * of weight opt w such that I ⊆ S * and O ∩ S * = ∅. This implies that A ⊆ S * ∩ U . Now, it is sufficient to show that max{w( A), w(U )} ≥ α · w(S * ). As in the explanation following Rule 3, S * \ U ⊆ A \ U . Thus, we have that Next, we denote U = U \ (I ∪ O). In the remaining (branching) rules, we first branch on neighbors of leaves in G[ U ] whose degree in this subgraph is at least two, then on leaves in G[ U ], then of vertices of degree at least three in G[ U ], and finally (in the last two rules) on the remaining vertices in G[ U ] that are not isolated in this subgraph. Although we can merge some of them, we present them separately for the sake of clarity. Branching Rule 4. [There are v, u ∈ U such that N (u) ∩ U = {v} and \|N (v) ∩ U \| ≥ 2] Return the set of maximum weight among A and B, computed in the following branches. For example, for the smallest possible α, the running time is bounded by O * (3 vc 3 ) < O * (1.44225 vc ), and for any constant α that the running time is bounded by O * ((2 − ) vc ). Observation 1 . 1WMMVC is solvable is time O * (2 vc ) and polynomial space. UFig. 1 . 1= {a, b, c}, F 1 = {a, b}, F 2 = {b, c} Construction The construction in the proof of Lemma 1. Lemma 2 . 2For any constant > 0, MMVC in bipartite graphs cannot be solved in time O * ((2 − ) vc 2 ) unless the SETH fails. Lemma 3 . 3Let U be a minimum(-size) vertex cover of G, and let U be a minimum(-size) vertex cover of G[U ]. Moreover, suppose that \|U \| ≤ vc 2 .Then, ProcedureA(G, w, α, U, U , ∅, ∅) runs in time O * ((1 x x (1−x) 1−x ) vc ), using polynomial space and returning a minimal vertex cover of weight at least α·opt w . Lemma 4 . 4Let U be a minimum(-size) vertex cover of G such that the size of a minimum(-size) vertex cover of G[U ] is larger than vc 2 . Then, ProcedureB(G, w, U, ∅, ∅) runs in time O * (3 vc 3 ), using polynomial space and returning a minimal vertex cover of weight at least 1 2 − 1 M +1 · opt w . α) 1 : 1Compute a minimum vertex cover U of G in time O * (γ vc ) and polynomial space. 2: Compute a minimum vertex cover U of G[U ] in time O * (γ vc ) and polynomial space. 3: if \|U \| ≤ vc 2 then 4: procedure ProcedureA is based on the bounded search tree technique. Each call is of the form ProcedureA(G, w, α, U, U , I, O), where G, w, α, U and U always remain the parameters with whom the procedure was called by ALG, while I and O are disjoint subsets of U to which ProcedureA adds elements as the execution progresses (initially, I = O = ∅). Roughly speaking, the sets I and O indicate that currently we are only interested in examining minimal vertex covers that contain all of the vertices in I and none of the vertices in O. Formally, we prove the following result. Lemma 5. ProcedureA returns a minimal vertex cover S that satisfies the following condition: -If there is a minimal vertex cover S * of weight opt w such that I ⊆ S * and O ∩ S * = ∅, then the weight of S is at least α · opt w . Moreover, each leaf, associated with an instance (G, w, α, U, U , I , O ), corresponds to a unique pair (I , O ), and its action can be performed in polynomial time if I ∪ O = U , and in time O * (\|{ U ⊆ U : I ⊆ U , O ∩ U = ∅, \| U \| ≥ (1 − x)vc}\|) otherwise. Let vc = \|U \|. To ensure that ProcedureA runs in time O * (( 1 x x (1−x) 1−x ) vc ), we propose the following measure: Measure: vc − \|U ∩ (I ∪ O)\|. 1 . 1A ⇐ProcedureA(G, w, α, U, U , I ∪ {v}, O). 2. B ⇐ProcedureA(G, w, α, U, U , I, O ∪ {v}). B ProcedureB: The Proof of Lemma 4 (Cont.) The procedure ProcedureB is based on the bounded search tree technique. Each call is of the form ProcedureB(G, w, U, I, O), where G, w and U always remain the parameters with whom the procedure was called by ALG, while I and O are disjoint subsets of U to which ProcedureB adds elements as the execution progresses (initially, I = O = ∅). As in the case of ProcedureA, the sets I and O indicate that currently we are only interested in examining minimal vertex cover that contains all of the vertices in I and none of the vertices in O. Formally, we prove the following result. vc 3 ), we use the measure below: Measure: vc − \|U ∩ (I ∪ O)\| − \|S(I ∪ O)\|, where S(I ∪ O) contains the vertices in U \ (I ∪ O) that do not have a neighbor in U \ (I ∪ O). 1. [There is v ∈ O such that N (v) ∩ O = ∅] Return U .Follow the explanation given for Rule 1 of ProcedureA.Reduction Rule 2. [There is v ∈ X such that N (v) ⊆ X, where X = I ∪ ( u∈O N (u))] Return U .Follow the explanation given for Rule 2 of ProcedureA. Reduction Rule 3. [U = I ∪ O ∪ S(I ∪ O)] Perform the following computation. S * \ U ) + w(S * ∩ U ) ≥ max{w(U ), w( A)} w( A \ U ) + w(S * ∩ U ) = max{w(U ), w( A)} w( A) + w(S * ∩ U ) − w( A ∩ U ) ≥ w(U ) w(U ) + w((S * \ A) ∩ U ) = w(U ) 2w(U ) − w(U \ (S * \ A)) For example, they show that one can guarantee the approximation ratios 0.1 and 0.4 in times O * (1.162 opt ) and O * (1.552 opt ), respectively. ProcedureA could also be developed without using recursion; however, relying on the bounded search tree technique simplifies the presentation, emphasizing the parts similar between ProcedureA and ProcedureB. It is not necessary to explicitly store F-we only need to iterate over it; therefore, by the pseudocode, it is clear that the space complexity of the action is polynomial. It might also contain r: once we insert v, u to I, we do not insert r to O. Branching Rule 5. [There are v, u ∈ U such that N (u) ∩ U = {v}] Return the set of maximum weight among A and B, computed in the following branches.For correctness, follow the explanation given in the previous rule. Also, since the previous rule did not apply, N (v) ∩ U = {u}. Thus, at each branch, one vertex in {v, u} is inserted to I, and the other is inserted to S(I ∪ O) ∪ O. We get the branching vector (\|{v, u}\|, \|{v, u}\|) = (2, 2), whose root is at most 3 1 3 . We note that we did not merge this rule with the previous one, since in the last rule we use the fact that Rule 4 has a branching vector better than(2,2).Return the set of maximum weight among A and B, computed in the following branches. 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Indeed, in the first branch, v is inserted to U and u is inserted to S(I ∪ O), and in the second branch, (N (v) ∩ U ) is inserted to I and v is inserted to O. The correctness of Lemma 6 is preserved since every vertex cover contains either v (an option examined in the first branch) or does not contain v, in which case it must contain all the neighbors of v (an option examined in the second branch). Since this branching vector is at least as good as (2, 3) (because \|N (v) ∩ U \| ≥ 2), we get a root that is at most 3The correctness of Lemma 6 is preserved since every vertex cover contains either v (an option examined in the first branch) or does not contain v, in which case it must contain all the neighbors of v (an option examined in the second branch). We get a branching vector that is at least as good as (\|{v, u}\|, \|(N (v) ∩ U ) ∪ {v}\|). Indeed, in the first branch, v is inserted to U and u is inserted to S(I ∪ O), and in the second branch, (N (v) ∩ U ) is inserted to I and v is inserted to O. Since this branching vector is at least as good as (2, 3) (because \|N (v) ∩ U \| ≥ 2), we get a root that is at most 3 At each branch, two vertices of the triangle are inserted to I, and the other one is inserted to S(I ∪ O). Thus, we get the branching vector (3, 3, 3and B. or it contains u, r (an option examined in the third branch. computed in the following branches. 1. A ⇐ProcedureB(G, w, U, I ∪ {v}, O)or it contains u, r (an option examined in the third branch). At each branch, two vertices of the triangle are inserted to I, and the other one is inserted to S(I ∪ O). Thus, we get the branching vector (3, 3, 3and B, computed in the following branches. 1. A ⇐ProcedureB(G, w, U, I ∪ {v}, O). . U B ⇐procedureb(g, W, ∪ {v}B ⇐ProcedureB(G, w, U, I ∪ (N (v) ∩ U ), O ∪ {v}).	[]
[ "BOTT PERIODICITY AND STABLE QUANTUM CLASSES", "BOTT PERIODICITY AND STABLE QUANTUM CLASSES" ]	[ "Yasha Savelyev " ]	[]	[]	We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on BU . This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimensional aspects of Hofer geometry of CP n , one of which we discuss here. The form of the main theorem, invites us to also speculate here on some "quantum" variants of Mumford's original conjecture (Madsen-Weiss theorem) on the cohomology of stable moduli space of Riemann surfaces.	10.1007/s00029-012-0101-7	[ "https://arxiv.org/pdf/0912.2948v6.pdf" ]	115,174,276	0912.2948	7db3bf7c07246d3bcb3b45ab551e40d0753cc5ce	BOTT PERIODICITY AND STABLE QUANTUM CLASSES 20 Jul 2011 Yasha Savelyev BOTT PERIODICITY AND STABLE QUANTUM CLASSES 20 Jul 2011 We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on BU . This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimensional aspects of Hofer geometry of CP n , one of which we discuss here. The form of the main theorem, invites us to also speculate here on some "quantum" variants of Mumford's original conjecture (Madsen-Weiss theorem) on the cohomology of stable moduli space of Riemann surfaces. Introduction In this paper we continue to investigate the theme of topology via Gromov-Witten theory and vice-versa. The topology in question is of certain important infinite dimensional spaces. Here the main such space is BU , and we show that Gromov-Witten theory (in at the moment mysterious way) completely detects its rational cohomology. The first application of this is for the study of higher dimensional aspects of Hofer geometry of Ham(CP n , ω st ), and we are able to prove a certain rigidity result for the the embedding SU (n) → Ham(CP n−1 ). Another topological application is given in [13], where we use the main result of this paper to probe topology of the configuration space of stable maps in BU, which might be the first investigation of this kind. More intrinsically, we get some new insights into Gromov-Witten invariants themselves, as through this "topological coupling" and some transcendental (as opposed to algebraic geometric in nature) methods we will compute some rather impossible looking Gromov-Witten invariants. These methods involve Bott periodicity theorem and differential geometry on loop groups. Lastly, this paper is a conceptual setup for some interesting "quantum" variants of Mumford's original conjecture on cohomology of the stabilized moduli space of Riemann surfaces, and this further fits into the above mentioned theme of topology via Gromov-Witten theory. 1.1. Outline. One of the most important theorems in topology is the Bott periodicity theorem, which is equivalent to the statement: (1.1) BU ≃ ΩSU, where SU is the infinite special unitary group. On the space BU we have Chern classes uniquely characterized by a set of axioms. It turns out that the space ΩSU also has natural intrinsic cohomology classes, characterized by axioms, but with a somewhat esoteric coefficient ring: QH(CP ∞ ), the unital completion of (ungraded) formal quantum homology ring QH(CP ∞ ), which is just a free polynomial algebra over C on one generator. Here is an indication of how it works for ΩSU (n). Consider the Hamiltonian action of the group SU (n) on CP n-1 . Using this, to a cycle f : B → ΩSU (n) we may associate a CP n-1 -bundle P f over B × S 2 , with structure group SU (n), by using f as a clutching map. Gromov-Witten invariants in the bundle P f induce cohomology classes (1.2) qc k (P f ) ∈ H 2k (B, QH(CP n-1 )). (More technically, we are talking about parametric Gromov-Witten invariants. The bundles P f always have a naturally defined deformation class of fiber-wise families of symplectic forms. Although very often the total space turns out to be Kahler in which case one can really talk about Gromov-Witten invariants and the discussion coincides.) These cohomology classes have analogues of Whitney Sum and naturality axioms as for example Chern classes. There is a also a partial normalization. We will show that these classes stabilize and induce cohomology classes on ΩSU ≃ BU . These stable quantum classes satisfy a full normalization axiom. However, with that natural normalization stable classes fail the dimension property i.e. qc k (E) does not need to vanish if E is stabilization of rank r bundle with r < 2k. This is somehow reminiscent of phenomenon of generalized (co)-homology. Thus we may think of these classes as exotic Chern classes. Theorem 1.1. The induced classes qc k on ΩSU ≃ BU are algebraically independent and generate the cohomology with the coefficient ring QH(CP ∞ ). Stabilization in this context is somewhat analogous to semi-classical approximation in physics. The "fully quantum objects" are the classes qc k ∈ ΩSU (n), and they are what's important in geometric applications, for example in Hofer geometry. We are still far from completely computing these classes, but as a corollary of the above we have the following. Theorem 1.2. The classes qc k on ΩSU (n) are algebraically independent and generate cohomology in the stable range 2k ≤ 2n − 2, with coefficients in QH(CP n−1 ). Since quantum classes on ΩSU (n) are pulled back from classes on ΩHam(CP n−1 ) as a simple topological corollary of Theorem 1.1 we obtain another proof of: [10]). The natural inclusion ΩSU (n) → ΩHam(CP n−1 ) is injective on rational homology in degree up to 2n − 2. Theorem 1.3 (Reznikov, Reznikov's argument is very different and more elementary in nature, he also proved a stronger topological claim, (he did not have conditions on degree) but of course his emphasis was different. 1.2. Applications to Hofer geometry. For a closed symplectic manifold (M, ω) recall that the (positive) Hofer length functional L + : LHam(M n , ω) → R is defined by (1.3) L + (γ) := 1 0 max(H γ t )dt, where H γ t is a time dependent generating Hamiltonian function for γ normalized by the condition M H γ t ω n = 0. Let j E : Ω E Ham(M, ω) → ΩHam(M, ω) denote the inclusion where Ω E Ham(M, ω) is the sub-level set with respect to the functional L + . Define (ρ, L + ) : H * (ΩHam(CP n−1 )) → R, to be the function (1.4) (ρ, L + )(a) = inf{E\| s.t. a ∈ image j E * ⊂ H * (ΩHam(CP n−1 ) )}. Let us also denote by i the natural map i : ΩSU (n) → ΩHam(CP n−1 ), and by i * L + the pullback of the function L + . We have an analogously defined function (ρ, i * L + ) : H * (ΩSU (n)) → R. Theorem 1.4. If a = 0 ∈ H 2k (ΩSU (n)) then (ρ, L + )(i * a) = 1, provided that 2 ≤ 2k ≤ 2n − 2, where the standard symplectic form on CP n−1 is normalized by the condition that the symplectic area of a complex line is 1. Moreover we have the following Hamiltonian rigidity phenomenon: (1.5) (ρ, i * L + )(a) = (ρ, L + )(i * a) = 1, for class a satisfying same conditions. What is already interesting is that (ρ, L + )(i * a) = 0, as Ham(CP n−1 ) is a very complicated infinite dimensional metric space, and sublevel sets Ω E Ham(CP n−1 ) may have interesting homology for arbitrarily small E. It's worth noting that I do not know if this theorem remain true for the full Hofer length functional, obtained by integrating the full oscillation max H γ t −min H γ t . Our argument does break down in this case, as it is not clear how to simultaneously bound both L + and L − . Excursion: On "quantization" of Mumford conjecture. The discussion in this section is strictly speaking outside the scope of methods and main setup of this paper. Nevertheless it may be helpful to some readers in order to put Theorem 1.1 into some further perspective. Let Σ be a closed Riemann surface of genus g, with a distinguished point x ∞ . Let I g,n be the set of isomorphism classes of tuples (E, α, j), where E is a holomorphic SL n (C)-bundle and α an identification of the fiber over x ∞ with C n and j a complex structure on Σ. It follows from [9, Section 8.11] that I g,n has a natural structure of an infinite dimensional manifold. Example 1.5. Take Σ = S 2 , the corresponding space I 0,n , is naturally diffeomorphic to ΩSU (n). This fact is related to the Grothendieck splitting theorem for holomorphic vector bundles on S 2 , or more appropriately in this context known as Birkhoff factorization, see [9, Section 8.10]. On I g,n there are natural characteristic cohomology classes qc k defined analogously to classes qc k on ΩSU (n). These classes are defined by counting certain holomorphic sections of the projectivization of the bundles (E, α, j), so that in particular we are now talking about genus g Gromov-Witten invariants. Remark 1.6. Note that the holomorphic structure will almost never be regular and must be perturbed within the class of π-compatible complex structures, see Definition 2.2. For computations such perturbations maybe unfeasible due to very complex and singular moduli spaces that arise, as apparent already in the proof of Proposition 2.4, in the genus 0 case. Instead it is preferable to work with virtual moduli cycle and virtual localization, for some naturally occurring circle actions, e.g. there is a natural Hamiltonian circle action on the Kahler manifold I 0,n = ΩSU (n) induced by the energy functional E : ΩSU (n) → R, which can be lifted to an action on the data for our classes (for properly chosen representatives f : B → ΩSU (n)). It can be shown that qc k stabilize to classes in the cohomology ring (1.6) H * ( lim g,n →∞ I g,n , QH(CP ∞ )). Stabilization in n is analogous to to what is done in this paper, while stabilization in g, uses semi-simplicity of the quantum homology algebra, as in Teleman [15]. The hypothesis is then that qc k are algebraically independent in this cohomology ring. To put this conjecture into some perspective, consider the embedding i : M g,1 ⊂ I g,n , where M g,1 is the moduli space of closed genus g Riemann surfaces with one marked point. The map i is defined by sending [Σ g , α, j] ∈ M g,1 to the class of the triple (Σ g × C n , α, j), with Σ g × C n endowed with a product holomorphic structure. On M g,1 there are Miller-Morita-Mumford characteristic classes c m k and Mumford's original conjecture was that they generate rational cohomology of the direct limit lim g →∞ M g,1 , where the limit is over natural (up to homotopy) inclusions M g,1 → M g+1,1 . This conjecture was proved by Madsen-Weiss as a consequence of a considerably stronger statement, [5]. On the other hand we have induced classes qc k ∈ H * (M g,1 , QH(CP n−1 )). Since i * is surjective on cohomology, as we have a continous projection pr : I g,n → M g,1 , with pr • i = id) a special case of the above conjecture is that these classes are algebraically independent in the cohomology ring H * ( lim g →∞ M g,1 , QH(CP ∞ )), For Miller-Morita-Mumford classes the corresponding statement is due to Morita. We may of course also conjecture that qc k generate H * (lim g →∞ M g,1 , QH(CP ∞ )), which is an interesting "quantum" analogue of the Mumford conjecture. Acknowledgements. I would like to thank Leonid Polterovich and Tel Aviv university for inviting me, and providing with a friendly atmosphere in which to undertake some thoughts which led to this article. In particular I am grateful to Leonid for compelling me to think about Gromov K-area. I also thank Dusa McDuff for helping me clarify some confusion in an earlier draft and Alexander Givental, Leonid and Dusa for comments on organization and content. Setup This section discusses all relevant constructions, and further outlines the main arguments. Quantum homology. In definition of quantum homology QH(M ), or various Floer homologies one often uses some kind of Novikov coefficients with which QH(M ) has special grading making quantum multiplication graded. This is often done even for monotone symplectic manifolds (M, ω), where it is certainly technically unnecessary, at least if one is not concerned with grading. Here we choose to work over C, which will also make definition of quantum classes more elegant and physical, as well as mathematically interesting. It also becomes more natural when we come to QH(CP ∞ ), as that no longer has any natural grading. Definition 2.1. For a symplectic manifold (M, ω) we set QH(M ) = H * (M, C) , which we think of as ungraded vector space, and hence drop the subscript * . 2.2. Quantum product. For integral generators a, b ∈ H * (M ), this is the product defined by (2.1) a * b = A∈H2(M) b A e −iω(A) , where b A is the homology class of the evaluation pseudocycle from the pointed moduli space, of J-holomorphic A-curves intersecting generic pseudocycles representing a, b, for a generic ω tamed J. This sum is finite in the monotone case: ω = kc 1 (T M ), with k > 0. The product is then extended to QH(M ) by linearity. For more technical details see [6]. Quantum classes. We now give a brief overview of the construction of classes qc k ∈ H k (ΩHam(M, ω), QH(M )), for a monotone symplectic manifold (M, ω), originally defined in [11]. The reader may note that this construction extends Seidel representation, [14]. Since our coefficients have no torsion these classes are specified by a map of Abelian groups Ψ : H * (ΩHam(M, ω), Q) → QH(M ), by setting qc k ([f ]) = Ψ([f ] ), for f : B → ΩHam(M, ω) a cycle. However, purely with this point of view we lose the extra structure of the Whitney sum formula. We now describe Ψ. By Smale's theorem rational homology is generated by cycles: f : B k → ΩHam(M, ω), where B is a closed oriented smooth manifold, which we may also assume to be smooth: the associated map f : B × S 1 → Ham(M, ω) is smooth). In fact, since ΩG has the homotopy type of a double loop space for any topological group G, by Milnor-Moore, Cartan-Serre [8], [1] the rational homology of ΩHam(M, ω) is freely generated via Pontryagin product by rational homotopy groups. In particular, the relations between cycles in homology can also be represented by smooth (in fact cylindrical) cobordisms. Given this we may construct a bundle p : P f → B, with (2.2) P f = B × M × D 2 0 B × M × D 2 ∞ / ∼, where (b, x, 1, θ) 0 ∼ (b, f b,θ (x), 1, θ) ∞ ,− → S 2 . Fix a family {j b,z }, b ∈ B, z ∈ S 2 of almost complex structures on M ֒→ P f → B × S 2 fiber-wise compatible with ω. Definition 2.2. A family {J b } is called π-compatible if: • The natural map π : (X b , J b ) → (S 2 , j) is J b holomorphic for each b. • J b preserves the vertical tangent bundle of M ֒→ X b → S 2 and restricts to the family {j b,z }. The importance of this condition is that it forces bubbling to happen in the fibers, where it is controlled by monotonicity of (M, ω). The map Ψ we now define measures part of the degree of quantum self intersection of a natural submanifold B × M ⊂ P f . The entire quantum self intersection is captured by the total quantum class of P f . We define Ψ as follows: (2.3) Ψ([B, f ]) = A∈j * (H sect 2 (X)) b A · e −iC( A) , Here, • H sect 2 (X) denotes the section homology classes of X. • C is the coupling class of Hamiltonian fibration (2.4) M ֒→ P f → B × S 2 , see [3, Section 3]. Restriction of C to the fibers X ⊂ P f is uniquely determined by the condition (2.5) i * (C) = [ω], M C n+1 = 0 ∈ H 2 (S 2 ). where i : M → X is the inclusion of fiber map, and the integral above denotes the integration along the fiber map for the fibration π : X → S 2 . • The map j * : H sect 2 (X) → H 2 (P f ) is induced by inclusion of fiber. • The coefficient b A ∈ H * (M ) is defined by duality: b A · M c = ev 0 · B×M [B] ⊗ c, where ev 0 : M(P h , A, {J b }) → B × M ev 0 (u, b) = (u(0), b) denotes the evaluation map from the space (2.6) M(P f , A, {J b }) of tuples (u, b), u is a J b -holomorphic section of X b in class A and · M , · B×M denote the intersection pairings. • The family {J b } is π-compatible in the sense above. The fact that Ψ is well defined with respect to various choices: the representative [f ], and the family {J b }, is described in more detail in [11], however this is a very standard argument in Gromov-Witten theory: a homotopy of this data gives rise to a cobordism of the above moduli spaces. Also the sum in (2.3), is actually finite. If T vert P f denotes the vertical tangent bundle of (2.4), then the natural restrictions on the dimension of the moduli space (2.7) 2n + 2k + 2 c 1 (T vert P f ), A , give rise to bounds on c 1 (T vert P f ), A , i.e. to degree d, where A = [S 2 ]+d[line] , as a class in X = M ×S 2 . Consequently only finitely many such classes can contribute. Bundles of the type P f above have a Whitney sum operation. Given P f1 , P f2 we get the bundle P f1 ⊕ P f2 ≡ P f2·f1 , where f 2 ·f 1 is the pointwise multiplication of the maps f 1 , f 2 : B → ΩHam(M, ω), using the natural topological group structure of ΩHam(M, ω). Geometrically this corresponds to doing connected sum on the fibers [11][Section 4.4] (which themselves are fibrations over S 2 ), and the set of (suitable) isomorphism classes of such bundles over B form an Abelian group P B , the group of homotopy classes of f : B → ΩHam(M, ω). For P f ∈ P B we set (2.8) qc k (P f ) = f * qc k . We may now state the properties satisfied by these classes, which we call axioms even though they do not characterize, these are verified in [11]. Quantum classes are a sequence of functions qc k : P B → H k (B, QH(M )) satisfying the following axioms: Axiom 1 (Naturality). For a map g : B 1 → B 2 : g * qc k (P 2 ) = qc k g * (P 2 ). Axiom 2 (Whitney sum formula). If P, P 1 , P 2 ∈ P B and P = P 1 ⊕ P 2 , then qc(P ) = qc(P 1 ) ∪ qc(P 2 ), where ∪ is the cup product of cohomology classes with coefficients in the quantum homology ring QH(M ) and qc(P ) is the total characteristic class Let us now specialize to M = CP n−1 with its natural symplectic form ω normalized by ω(A) = 1, for A the class of the line. And let us restrict our attention to the subgroup ΩSU (n) ⊂ ΩHam(CP n ). The map Ψ : ΩSU (n) → QH(CP n−1 ) will be denoted by Ψ n . Note that in this case (2.10) Ψ n ([pt]) = 1 = [CP n−1 ], since SU (n) is simply connected. We will proceed to induce Ψ on the limit ΩSU . This will allow us to arrive at a true normalization axiom, completing the axioms in that setting, and will also allow us to make contact with the splitting principle on BU . Let qc ∞ k ∈ H 2k (BU, QH(CP ∞ )) qc ∞ k ([f ]) = Ψ([f ]), for f : B 2k → BU a cycle. Theorem 2.6. Let K(B) denote the reduced K-theory group of B. The classes qc ∞ k ∈ H 2k (BU, QH(CP ∞ )) satisfy and are determined by the following axioms: Axiom 4 (Functoriality). For a map g : B 1 → B 2 , and P 2 ∈ K(B): g * qc ∞ k (P 2 ) = qc ∞ k (g * P 2 ). Axiom 5 (Whitney sum formula). If P, P 1 , P 2 ∈ K(B) and P = P 1 ⊕ P 2 , then qc ∞ (P ) = qc ∞ (P 1 ) ∪ qc ∞ (P 2 ), where ∪ is the cup product of cohomology classes with coefficients in the ring QH(CP ∞ ) and qc ∞ (P ) is the total characteristic class (2.12) qc ∞ (P ) = 1 + . . . + qc ∞ m (P ), where m is the dimension of B. Moreover, we will prove in Section 3 that they satisfy the following: Axiom 6 (Normalization). If f l : CP k → BU is the classifying map for the stabilized canonical line bundle, then (2.13) Ψ(f l ]) = [CP k−1 ]e i . Verification of the first two axioms is immediate from the corresponding axioms of the classes qc k ∈ H 2k (ΩSU (n), QH(CP n−1 )), using the well known fact that under the Bott isomorphism BU ≃ ΩSU , the group K(B) corresponds to the group of homotopy classes of maps f : B → ΩSU . Verification of the last axiom is done in Section 3. As we see the classes qc ∞ k formally have the same axioms as Chern classes, but they do not have the dimension property, i.e. qc k (P ) does not need to vanish if P is stabilization of rank r bundle with r < 2k: this already fails for the canonical line bundle. Proof of Theorem 1.1. To show generation, note that by the splitting principle, it is enough to show that qc classes generate cohomology of BT ⊂ BU , where T is the infinite torus. We need to show that for every f : B 2k → BT non vanishing in homology some quantum Chern number of P f is non-vanishing. That is we must show: (2.14) To show algebraic independence, note that SU has non-vanishing rational homotopy groups in each odd degree, indeed the rational homotopy type of SU is well known to be S 3 × S 5 × . . . × S 2k−1 . . . . Consequently, BU ≃ ΩSU has non-vanishing rational homotopy groups in each even degree, and by Milnor-Moore theorem these spherical generators do not vanish in homology. (In this case Milnor-Moore, Cartan-Serre theorem says that H * (ΩSU, Q) is freely generated by rational homotopy groups via Pontryagin product, see [8], [1]). For such a spherical homology class in degree 2k, clearly only qc k may be non-vanishing, and must be non-vanishing since qc k generate. Remark/Question 2.7. The space ΩSU is actually a homotopy ring space. On BU the ring multiplication corresponds to tensor product of vector bundles. In terms of bundles P f over B there is a kind of tensor product in addition to Whitney sum. The tensor product is induced by the Abelian multiplication map CP ∞ × CP ∞ → CP ∞ , which induces a multiplication on fibrations π : CP ∞ ֒→ X → S 2 . One then needs to check that P f1 ⊗ P f2 ∈ P B . It is then likely that this is the same as the ring coming from vector bundle tensor product. Or perhaps its a different ring structure? In any case, one would have to check that the proposed tensor product is well behaved for quantum classes, analogously to the case of Chern characters. If this ring is indeed different, then this is where "exoticity" of quantum classes may have an interesting expression. Proofs We are going to give two proofs of Proposition 2.4. The first one is more difficult, but less transcendental of the two, and we present here because it has the advantage of suggesting a route to computation of quantum classes in the non-stable range. Moreover, the energy flow picture in this argument is necessary for proof of Theorem 1.4. Unfortunately, this first proof requires some basics of the theory of loop groups. The second proof was suggested to me by Dusa McDuff, and basically just uses the index theorem and automatic transversality. For convenience, these will be presented independently of each other, so can be read in any order. Proof of Proposition 2.4. We will need the following. The proof of this fact uses cellular stratification induced by the energy functional E on ΩSU (n). For various details in the following the reader is referred to [9,Sections 8.8,8.9]. For us, the most important aspect of the proof of the proposition above is that there is a holomorphic cell decomposition of Kahler manifold ΩSU (n) up to dimension 2n − 2, with cells indexed by homomorphisms λ : S 1 → T with all weights either 1, −1, 0, where T ⊂ SU (n) is a fixed maximal torus. The closureC λ of each such cell C λ , is the closure of unstable manifold (for E) of a (certain) cycle in the E level set of λ. Let f λ : B 2k λ → ΩSU (n) be the compactified 2k-pseudocycle representingC λ . Each γ ∈C λ is a polynomial loop, i.e. extends to a holomorphic map γ C : C × → SL C (n). Consequently, X γ has a natural holomorphic structure, so that the complex structure on each fiber M z → X γ → S 2 is tamed by ω, and so we have a natural admissible family of complex structures on P f λ . In fact X γ is Kahler but this will not be important to us. What will be important is that X γ is biholomorphic to X γ∞ , where γ ∞ : S 1 → SU (n), γ ∞ ∈C λ is S 1 subgroup, which is a limit in forward time of the negative gradient flow trajectory of γ. In particular γ ∞ has all weights either 1, −1, 0. For γ = f λ (b) we call the corresponding γ ∞ (3.1) γ b ∞ . Aside from the condition on weights of the circle subgroups, here is another point where the stability condition 2k ≤ 2n−2 comes in. Let σ const denote some constant section of X ≃ CP n−1 × S 2 , which is our name for the topological model of the fibers of P f . For a class A = [σ const ] − d · [line] ∈ H 2 (X) the expected dimension of M(P f , A, {J b }) (see (2.6)) is 2n + 2k + 2c vert 1 ( A) ≤ 2n + 2n − 2 − 2d · n < 0 unless d ≤ 1, where c vert 1 is first Chern class of the vertical tangent bundle of M ֒→ P f λ → B λ × S 2 . Thus, A can contribute to Ψ(P f λ ) only if d = 1, (d = 0 can only contribute to c q 0 (P f ); which can be checked by analyzing the definition.) This fact will be crucial in the course of this proof. We set S = [σ const ] − A ∈ H 2 (X). Let us first understand moduli spaces of holomorphic S curves in X γ∞ , with γ ∞ an S 1 subgroup of the type above. Each X γ∞ is biholomorphic to S 3 × γ∞ CP n−1 i.e. the space of equivalence classes of tuples (3.2) [z 1 , z 2 , x], x ∈ CP n−1 , under the action of S 1 (3.3) e 2πit · [z 1 , z 2 , x] = [e −2πit z 1 , e −2πit z 2 , γ ∞ (t)x], using complex coordinates z 1 , z 2 on S 3 . Let H γ∞ be the normalized generating function for the action of γ ∞ on CP n−1 . Each x in the max level set F max of H γ∞ , gives rise to a section σ x = S 3 × γ∞ {x} of X γ∞ . It can be easily checked that [σ x ] = S. Moreover, by elementary energy considerations it can be shown that these are the only stable holomorphic S sections in X γ∞ , see discussion following Definition 3.3. in [12]. Consequently the moduli space M = M(P f λ , S, {J b }) is compact. It's restriction over open negative gradient trajectories R →C λ asymptotic to γ ∞ in forward time is identified with R × F max , where F max is as above. The regularized moduli space can be constructed from M together with kernel, cokernel data for the Cauchy Riemann operator for sections u ∈ M. This is well understood by know, see for example an algebro-geometric approach in [4]. It is however interesting that our moduli space is highly singular. What we need to show is that M and the local data for the Cauchy Riemann operator are identified with M s = M(P i•f λ , S, {J b }) and the corresponding local data for the Cauchy Riemann operator, where i • f λ : B λ → ΩSU (m). The fact that the moduli spaces M, M s are identified follows more or less immediately from the preceding discussion. The inclusion map i : ΩSU (n) → ΩSU (m) takes gradient trajectories in ΩSU (n) to gradient trajectories in ΩSU (m) and any circle subgroup is taken to a circle subgroup with m − n new 0-weights. There is a natural map j : P f → P i•f λ , and its clear that it identifies M with M s . We will denote j(X b ) by X s j(b) , and an element in M s identified to an element u = (σ x , b) ∈ M by j(u), where b ∈ B f λ . We show that the linearized Cauchy Riemann operators at u, j(u) have the same kernel and cokernel. Let V denote the infinite dimensional domain for the linearized CR operator D u at u. There is a subtlety here, since our target space P f λ is a smooth stratified space we actually have to work a strata at a time, but we suppress this. The domain for the CR operator D j(u) is the appropriate Sobolev completion of the space of C ∞ sections of the bundle j(u) * T vert X s j(b) , where j(u) * T vert X s j(b) ≃ S 3 × γ j(b) ∞ CP m−1 , is a holomorphic vector bundle, and ≃ is isomorphism of holomorphic vector bundles. Similarly, u * T vert X b ≃ S 3 × γ b ∞ CP n−1 . Consequently j(u) * T vert X s j(b) holomorphically splits into line bundles with Chern numbers either 0, −2, −1. All the Chern number 0 and −2 summands are identified with corresponding summands of u * T vert X b , and consequently the domain of D j(u) in comparison to D u is enlarged by the space of sections W , of sum of Chern number −1 holomorphic line bundles. But in our setting D j(u) coincides with Dolbeault operator and so we will have no "new kernel or cokernel" in comparison to D u . More precisely: D j(u) : W → Ω 0,1 (S 2 , j(u) * T vert X s j(b) )/Ω 0,1 (S 2 , u * T vert X) is an isomorphism, which concludes our argument. Second proof of Proposition 2.4. It is enough to verify the stabilization property for m = n + 1. We have to again make use of the homological stability condition, as in the first proof. Let f : B → ΩSU (n) be a map of a smooth, closed oriented manifold. For a class A = [σ const ] − d · [line] ∈ H 2 (X) the expected dimension of M(P f , A, {J b }) (see (2.6)) is 2n + 2k + 2c vert 1 ( A) ≤ 2n + 2n − 2 − 2d · n < 0 unless d ≤ 1,b ∈ M(P f ′ , S, {J f ′ ,b }) is O(−1). So we have an exact sequence (3.4) u * T vert P f → (j • u) * T vert P f ′ → O(−1). By construction of {J pert f ′ ,b } the real linear CR operator D i•u is compatible with this exact sequence. More explicitly, we have the real linear CR operator (3.5) Ω 0 (S 2 , O −1 ) → Ω 0,1 (S 2 , O(−1)) induced by D j•u b on Ω 0 (S 2 , (j • u) * T vert P f ′ /u * T vert P f ), with image Ω 0,1 (S 2 , (j • u) * T vert P f ′ /u * T vert P f ), since u * T vert P f ⊂ (j • u) * T vert P f ′ is J f ′ ,b invariant. Such an operator is surjective by the Riemann-Roch theorem, [6]. Consequently D j•u b is surjective. (Verification of Axiom 6). For the purpose of this computation we will change our perspective somewhat, since the moduli spaces at which we arrived in the first proof of Proposition 2.4 were very singular. (Although, as hinted in the Introduction we maybe able to also use virtual localization, and stay within the setup of that argument.) The reader may wonder if the perspective in the following argument could also be used to attack Proposition 2.4 as well, but this appears to be impossible as the picture below doesn't stabilize very nicely. Consider the path space from I to −I in SU (2n): Ω I,−I SU (2n). Fixing a path from I to −I, say along a minimal geodesic (3.6) p =         e πit . . . e πiθ e −πit . . . e −πit         , there is an obvious map m : Ω I,−I SU (2n) → ΩSU (2n). We will make use of the Bott map. The map i n B is the inclusion of the manifold of minimal geodesics in SU (2n) from I to −I, into Ω I,−I SU (2n). Any such geodesic is conjugate to p above and is determined by the choice of n complex dimensional e πiθ weight space in C 2n . The bound 2n is due to the fact that the index of a non-minimal closed geodesic from I to −I in SU (2n) is more than 2n. For more details on Bott map and this argument we refer the reader to the proof of Bott periodicity for the unitary group in [7]. Let f l : CP k → BU be as in wording of the axiom. For 2n sufficiently large the homotopy class of i B • f l is represented by i • m • i n B • f : CP k → BU, where f : CP k → Gr n (2n), is the map (3.7) f (l) = pl n−1 ⊕ l, where pl n−1 is a fixed complex n − 1 subspace in C 2n , l denotes a 1 complex dimensional subspace in the orthogonal C n+1 , and where i is just the inclusion ΩSU (2n) → ΩSU , however i is suppressed from now on. Therefore (3.8) i n B • f (l) =     e πit . . . e πit γ l ,     where the top n − 1 by n − 1 block is the diagonal matrix with entries e πit , and the bottom block γ l ∈ SU (n + 1) has e πit weight space determined by l ⊂ C n+1 . We can tautologically express m • i n B • f as the composition of the map g = i n B • f : CP n → ΩP SU (2n), and L : ΩP SU (2n) → ΩSU (2n), with L defined as follows: for a loop γ : S 1 → P SU (2n) pointwise multiply with the loop p : S 1 → P SU (2n) and lift to the universal cover SU (2n). Since γ · p is contractible the lift is a closed loop. By Corollary 2.3 (3.9) Ψ(m • i n B • f ) = Ψ(g) * Ψ(p), where * is the quantum product and we are in-distinguishing notation for the map p : S 1 → P SU (2n) and associated map p : [pt] → ΩP SU (2n). Note that the restriction of Ψ to H 0 (ΩHam(M, ω), Q) is the Seidel representation map S, [14]. Proof. Our bundle X p over S 2 is naturally isomorphic to S 3 × p CP 2n−1 , and so has a natural holomorphic structure. Further, we have natural holomorphic sections Similarly, the bundle P g has a natural admissible family {J b }, with all fibers (X b , J b ) biholomorphic to each other. Each X b is identified with a naturaly complex manifold S 3 × g b CP 2n−1 , where g b denotes the circle subgroup g(b) of P SU (2n), cf (3.3). The previous discussion can be applied to each X b . And it implies that with the cycle [CP k ] ⊗ [CP n−k ] ∈ H * (CP k × CP 2n−1 ). If we take a representative for [CP n−k ] not intersecting CP n−2 ⊂ CP 2n−1 corresponding under projectivization to the complex subspace pl n−1 (see (3.7)), then it is geometrically clear from construction that the two cycles intersect in a point by construction of the map g, and intersect transversally. It follows that Ψ(g) = [CP n−1+k ]e i/2 . Therefore, Ψ(i B • f l ) = [CP] n−1 e i/2 * [CP n−1+k ]e i/2 = [CP k−1 ]e i , since in this case there is only classical contribution to the quantum intersection product. σ x = S 3 × p {x}, where x ∈ F max ≃ CP n Proof of Theorem 1.4. Let 0 = a ∈ H 2k (ΩSU (n)), with 0 ≤ 2k ≤ 2n − 2. By Corollary 1.2 we have that (3.13) i qc αi βi , a = 0, for some α i , β i . Since these classes are pull-backs of the classes qc k ∈ H 2k (ΩHam(CP n−1 , ω)), the same holds for these latter cohomology classes and the cycle i * a. Proof. This is essentially [12, Lemma 3.2], and we reproduce it's proof here for convenience. The total space of P f is (3.14) P f = B × CP n−1 × D 2 0 B × CP n−1 × D 2 ∞ / ∼, where (b, x, 1, θ) 0 ∼ (b, f b,θ (x), 1, θ) ∞ , using the polar coordinates (r, 2πθ). The fiberwise family of Hamiltonian connections {A b } are induced by a family of certain closed forms { Ω b }, which we now describe, by declaring horizontal subspaces of A b to be Ω b -orthogonal to the vertical subspaces of π : F b → S 2 , where F b is the fiber of P f over b ∈ S k . The construction of this family mirrors the construction in Section 3.2 of [11]. First we define a family of forms { Ω b } on B × CP n−1 × D 2 ∞ . (3.15) Ω b \| D 2 ∞ (x, r, θ) = ω − d(η(r)H b θ (x)) ∧ dθ (2. 9 ) 9qc(P ) = qc 0 (P ) + . . . + qc m (P ), where m is the dimension of B. Axiom 3 (Partial normalization). qc 0 (P ), [pt] = 1 = [M ] ∈ QH(M ).Corollary 2.3. Ψ is a ring homomorphism from ΩHam(M, ω) with its Pontryagin product to QH(M ) with its quantum product. : CP n−1 → CP m−1 , j([z 0 , . . . , z n ]) = [z 0 , . . . , z n , 0, . . . , 0] be compatible inclusions. For a, b ∈ H * (CP n−1 ), of pure degreej * (a * b) = j * (a) * j * (b)if and only if a * b has a quantum correction, or if degree of deg a+deg b+2 ≤ 2(n−1). In particular the direct limit of abelian groups lim n QH(CP n−1 ) makes sense as a ring which we denote by QH(CP ∞ ), i.e. it's the natural formal quantum ring on H * (CP ∞ , C). Denote by QH(CP ∞ ) its unital completion, and 1 its unit, which we may informally view as [CP ∞ ]. This ring is just a free polynomical algebra over C on one generator.Here is the main step.Proposition 2.4. For a cycle f : B 2k → ΩSU (n) (2.11) Ψ m ([i • f ]) = j * Ψ n ([f ]), for 2k in stable range [2, 2n − 2].The above proposition is indeed fairly surprising. For even if one believes that the bundles induced by j • f , f have basically the "same twisting", the ambient spaces (fibrations, fibers) are certainly different and this same twisting can manifest itself in different looking elements of quantum homology. And in fact this different manifestation is certain to happen unless Ψ n ([f ]) is of the form c·e i ∈ QH * (CP n−1 ), i.e. unless the only contribution to Ψ n ([f ]) is coming from the section class A ∈ H 2 (CP n−1 × S 2 ) with C( A) = ω( A) = −1. For if its not, the above equality is impossible by dimension considerations, (this is a straightforward exercise involving definition of Ψ.) Corollary 2.5. There is an induced map Ψ : BU ≃ ΩSU → QH(CP ∞ ), induced by (2.11) by postulating that Ψ([pt]) = 1 (cf. (2.10)) and consequently cohomology classes (P f ), [B] = 0, for some β i , α i , s.t. i 2β i · α i = 2k. Clearly we may assume that B 2k = i [CP α(i) ] and f = f l 1 · . . . f l i · . . . , where i α(i) = k, and f l i : i [CP α(i) ] → BT is the classifying map for the pullback of the canonical line bundles on CP α(i) under the projection i [CP α(i) ] → CP α(i) . But then the claim follows by Axioms 6, and 5. There are natural maps i n B : ΩSU (n) → BU inducing isomorphism of homotopy groups up to dimension 2n − 2. Chern class of the vertical tangent bundle of M ֒→ P f → B × S 2 . Thus, A can contribute to Ψ(P f ) only if d = 1, (d = 0 can only contribute to c q 0 (P f ); which can be checked from the definition.) We set S = [σ const ] − [line]. Let j : ΩSU (n) → ΩSU (n + 1) be the inclusion map, and denote j • f by f ′ . Take a family of almost complex structures{J f,b } on P f for which the moduli space M(P f , S, {J f,b }) is regular. Extend {J f,b } to a family {J f ′ ,b} on P f ′ in any way. Consequently, for the families of almost complex structure {J f,b } on P f and {J f ′ ,b } on P f ′ the natural embedding of P f into P f ′ is holomorphic. The intersection number of a curve u ∈ M(P f ′ , S, {J f ′ ,b }) with P f ⊂ P f ′ , is c 1 of the normal bundle of CP n−1 inside CP n evaluated on −[line], i.e. -1. Consequently by positivity of intersections, (see[6]) for the family {J b } all the elements of the space M(P f ′ , S, {J b }) are contained inside the image of embedding of P f into P f ′ . We now show that {J f ′ ,b } is also regular. This will immediately yield our proposition. The pullback of the normal bundle to embedding, by u Theorem 3. 2 ( 2Bott). There is a natural inclusion m • i n B : Gr n (2n) → ΩSU (2n), inducing an isomorphism on homotopy (homology) groups in dimension up to 2n. p) = S(p) = [CP n−1 ]e i/2 . − 1 1the max level set of the generating function H p of the Hamiltonian action of p on CP 2n−1 . The class of such a section will be denoted [σ max ]. It can be easily checked that c vert 1 ([σ max ]) = −n and that C([σ max ]) = −1/2. By elementary consideration of energy it can be shown that the above are the only stable holomorphic [σ max ] class sections of X p ; see discussion following Definition 3.3,[12]. The corresponding moduli space is regular, (the weights of the holomorphic normal bundle to sections σ max are 0 and -1). Further, from dimension conditions it readily follows that there can be no other contributions to Ψ(p) and so Ψ(p) = [CP n−1 ]e i/2 . M = M(P g , [σ max ], {J b }) is compact and fibers over B = CP k with fiber CP n−1 . For A = [σ max ] − d[line] ∈ H 2 (X) The virtual dimension of M(P f , A, {J b }) is ( A) = 2n + (4n − 2) − 2n − d4n < 0, unless d = 0, i.e. A = [σ max ] is the only class that can contribute. Thus, we only need to understand the contribution from [σ max ] class. Once again M = M(P g , [σ max ], {J b }) is regular, i.e. the associated linearized Cauchy Riemann operator is onto for every (σ x , b) ∈ M, since by construction, the normal bundle N = S 3 × g b T x CP 2n−1 to a section u ∈ M is holomorphic with weights 0, −1. By definition Ψ(g) is determined by the intersection number of the evaluation cycle e 0 : M → CP k × CP 2n−1 , Lemma 3. 4 . 4For any representative f : B → ΩHam(CP n−1 , ω) for a,L + (i • f (b)) ≥ 1 for some b ∈ B. Here, H b θ is the generating Hamiltonian for f (b), normalized so that M H b θ ω n = 0, for all θ and the function η : [0, 1] → [0, 1] is a smooth function satisfying 0 ≤ η ′ (r), and η(r) = 1 if 1 − δ ≤ r ≤ 1,r 2 if r ≤ 1 − 2δ,for a small δ > 0. using the polar coordinates (r, 2πθ). With fiber modelled by a Hamiltonian fibration M ֒→ Xπ Of course such a section must exist by the discussion following (3.13). The theorem follows once we note that there is a representative f ′ : B → ΩHam(CP n−1 , ω) for such an a, in the form i • f , with f : B → ΩSU (n), s.t. the image of f ′ = i • f is contained in the sublevel set Ω 1 Ham(CP n−1 ω). Such a representative is readily found from the energy flow stratification picture of ΩSU (n), and we have already used it, see the proof of Proposition 2.4.It is not hard to check that the gluing relation ∼ pulls back the form Ω b \| D 2 ∞ to the form ω on the boundary CP n−1 × ∂D 2 0 , which we may then extend to ω on the whole of CP n−1 × D 2 0 . Let { Ω b } denote the resulting family on X b . The forms Ω b on F b restrict to ω on the fibers CP n−1 ֒→ F b → S 2 and the 2-form obtained by fiber-integration CP n−1 ( Ω b ) n+1 vanishes on S 2 . Such forms are called coupling forms, which is a notion due to Guillemin, Lerman and Sternberg[2]. 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A Pressley, G Segal, Loop groups. A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, (1986). Characteristic classes in symplectic topology. A G Reznikov, Selecta Math. (N.S.). 3Appendix D by L KatzarkovA. G. Reznikov, Characteristic classes in symplectic topology, Selecta Math. (N.S.), 3 (1997), pp. 601-642. Appendix D by L Katzarkov. Quantum characteristic classes and the Hofer metric. Y Savelyev, Geometry Topology. Y. Savelyev, Quantum characteristic classes and the Hofer metric, Geometry Topology, (2008). Virtual Morse theory on ΩHam(M, ω). J. Differ. Geom. 84, Virtual Morse theory on ΩHam(M, ω)., J. Differ. Geom., 84 (2010), pp. 409-425. arXiv:1104.5440On configuration spaces of stable maps. , On configuration spaces of stable maps, arXiv:1104.5440, (2011). 1 of symplectic automorphism groups and invertibles in quantum homology rings. P Seidel, Geom. Funct. Anal. 7P. Seidel, π 1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. 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[ "Unsupervised Synthesis of Anomalies in Videos: Transforming the Normal", "Unsupervised Synthesis of Anomalies in Videos: Transforming the Normal" ]	[ "Abhishek Joshi [email protected] \nDept. of Computer Science\nDept. of Computer Science\n\n", "Iit Kanpur \nDept. of Computer Science\nDept. of Computer Science\n\n", "Vinay P Namboodiri [email protected] \nDept. of Computer Science\nDept. of Computer Science\n\n", "Iit Kanpur \nDept. of Computer Science\nDept. of Computer Science\n\n" ]	[ "Dept. of Computer Science\nDept. of Computer Science\n", "Dept. of Computer Science\nDept. of Computer Science\n", "Dept. of Computer Science\nDept. of Computer Science\n", "Dept. of Computer Science\nDept. of Computer Science\n" ]	[]	Abnormal activity recognition requires detection of occurrence of anomalous events that suffer from a severe imbalance in data. In a video, normal is used to describe activities that conform to usual events while the irregular events which do not conform to the normal are referred to as abnormal. It is far more common to observe normal data than to obtain abnormal data in visual surveillance. In this paper, we propose an approach where we can obtain abnormal data by transforming normal data. This is a challenging task that is solved through a multistage pipeline approach. We utilize a number of techniques from unsupervised segmentation in order to synthesize new samples of data that are transformed from an existing set of normal examples. Further, this synthesis approach has useful applications as a data augmentation technique. An incrementally trained Bayesian convolutional neural network (CNN) is used to carefully select the set of abnormal samples that can be added. Finally through this synthesis approach we obtain a comparable set of abnormal samples that can be used for training the CNN for the classification of normal vs abnormal samples. We show that this method generalizes to multiple settings by evaluating it on two real world datasets and achieves improved performance over other probabilistic techniques that have been used in the past for this task.	10.1109/ijcnn.2019.8852035	[ "https://arxiv.org/pdf/1904.06633v1.pdf" ]	119,184,172	1904.06633	90bc452bd635aa0ddf224f6e1044ae77d53022b9	Unsupervised Synthesis of Anomalies in Videos: Transforming the Normal Abhishek Joshi [email protected] Dept. of Computer Science Dept. of Computer Science Iit Kanpur Dept. of Computer Science Dept. of Computer Science Vinay P Namboodiri [email protected] Dept. of Computer Science Dept. of Computer Science Iit Kanpur Dept. of Computer Science Dept. of Computer Science Unsupervised Synthesis of Anomalies in Videos: Transforming the Normal Abnormal activity recognition requires detection of occurrence of anomalous events that suffer from a severe imbalance in data. In a video, normal is used to describe activities that conform to usual events while the irregular events which do not conform to the normal are referred to as abnormal. It is far more common to observe normal data than to obtain abnormal data in visual surveillance. In this paper, we propose an approach where we can obtain abnormal data by transforming normal data. This is a challenging task that is solved through a multistage pipeline approach. We utilize a number of techniques from unsupervised segmentation in order to synthesize new samples of data that are transformed from an existing set of normal examples. Further, this synthesis approach has useful applications as a data augmentation technique. An incrementally trained Bayesian convolutional neural network (CNN) is used to carefully select the set of abnormal samples that can be added. Finally through this synthesis approach we obtain a comparable set of abnormal samples that can be used for training the CNN for the classification of normal vs abnormal samples. We show that this method generalizes to multiple settings by evaluating it on two real world datasets and achieves improved performance over other probabilistic techniques that have been used in the past for this task. I. INTRODUCTION Visual surveillance generates a huge amount of data. A single camera alone can generate terabytes of data over time. This data needs to be screened for abnormal events. This is usually done either through operators that keep active lookout on various screens or as a means of deterrence where the videos are analyzed after the effect if an untoward incidence (such as robbery) has occurred. Clearly, there is a need to automate this tedious process. Therefore, identifying abnormal activities automatically has been an important problem from the application perspective. The task however is challenging as abnormal activity is a highly class imbalance problem in a supervised setting. A number of problems have been solved by the community using deep learning techniques that have been trained using supervision. However, to adopt this approach a large set of labels need to be explicitly provided for abnormal activity. This is a challenge as usually the surveillance cameras typically capture the normal activity and abnormal occurrences are rare. A large number of these rare events would need to be Project webpage: https://abhjoshi8.github.io/VideoAnomalySynthesis (a) Traffic Junction Dataset [35]. The left image is normal while the right is an anomaly with a person crossing the road away from the zebra crossing (b) Traffic Junction Dataset [35]. The left image is normal while the right is an anomaly as the car enters the pedestrian area. (c) Highway Dataset [27]. The left image is normal while the right is an anomaly as a jaywalker crosses the road. explicitly annotated. Sometimes, an activity is unusual because there are no past occurrences and evidences of that activity. So, the data describing abnormal activities is scarce and we do not have much labeled data available for them. Hence, it would be desirable to have a video surveillance algorithm which leverages only data of normal activity to make a decision as to whether a particular activity is normal or abnormal. In this work, we follow this approach and transform the normal observations to automatically obtain abnormal observations. In this paper, we follow a synthesis based approach to obtain the supervision. There have been instances of interesting work that have used this approach based on synthesis. For instance, Rozantsev et al. [28] pursue the generation of synthetic data based on 3D models that are used for training object detectors. Very recent work pursued by Shrivastava et al. [29] obtains increased realism in generation of synthetic data using generative adversarial networks (GANs). However, the previous approaches obtained synthetic examples by modeling the 3D model and generating realistic renderings of the same. The challenge in our case is that we have samples of normal data but choose to not use any examples of abnormal data (as a few chosen abnormal examples would be limiting and not span the class of abnormal examples). In our approach, we adopt a different architecture where we use a synthesis technique that can generate large number of modified samples from normal samples to obtain abnormal samples. We then use a trained Bayesian network (compared also to non-Bayesian network) that assumes that the synthesized examples are fake and uses them to choose a set of examples that are abnormal and can be used. In Section II we discuss a number of approaches adopted in the literature. In Section III we describe the methodology followed by us i.e. the unsupervised setting for synthesizing anomaly and abnormal activity detection. In Section IV we provide experiments and compare our results with the unsupervised models that have been previously proposed in [35] and [27]. We provide analysis in Section V and finally conclude with a discussion in Section VI. II. RELATED WORK There are broadly two main approaches that have been followed to solve abnormal activity detection in videos. The first approach involves tracking objects in the video frames. The deviation in trajectory points leads to potential candidates for being abnormal [13] [17] [37]. Though these methods perform well, they are subject to trajectory abnormality constraints that may not be prevalent. For instance, due to occlusions there may be a number of truncated or abnormal trajectories obtained from normal sequences. Further, some abnormal sequences, due to short trajectories may appear normal. Moreover, a number of times, it is not the trajectory itself that is abnormal but the location where it occurs that makes it abnormal. For instance as illustrated in Figure 1(a), the act of crossing the road may appear normal but is abnormal, if not done on the pedestrian crossing. The second approach leverages feature descriptors to procure intrinsic patterns of the events, which are eventually used to model the behaviour. Although, absence of tracking information in the latter approach may lead to substantial loss but this approach shows a more convincing way to develop more generic and real-life models. We follow this broad category of approach for our task. Niebles et al. [24] apply topic models for the task of action recognition and classification by modeling the features in terms of visual words. In an unsupervised way, they use pLSA-LDA models to predict the actions. Sparse representation has been used in [20] [38] to learn the dictionary of normal behaviours. The behaviours which have large reconstruction errors are considered as anomalous behaviors during testing. Recently, a weakly supervised setting has been proposed in [33] which considers normal and anomalous videos as bags and video segments as instances in multiple instance learning (MIL) and predicts anomalous videos. Deep Learning approaches have lead to successes in many computer vision tasks [15] [10] including anomaly detection. Xu et al. [36] demonstrate the effectiveness of deep learning features through a multi-layer auto-encoder for feature learning. In the work [11], Hasan et al. propose a 3D convolutional auto-encoder to model normal frames, which have been further boosted in [6] [21] via both appearance and motion based model. Recently, Luo et al. [22] propose a temporally coherent sparse coding based method which maps to a stacked RNN framework. It is interesting to note that all these anomaly detection approaches are based on the reconstruction of normal training data with the assumption that abnormal events would correspond to larger reconstruction errors. Varadarajan et al. [35] adopt topic modelling for scene analysis and detection of abnormal events occurring in the video. The assumption is that in a domain, the set of usual events is fixed and can be extracted from the distribution of the visual words and the video clips in the domain. The idea is to build a model for the normal activities in the videos. A video clip having instance of abnormal events would then be expected to have a low likelihood over the model learnt from the normal events. Pathak et al. [27] propose a mechanism to extend the topic-based analysis of anomalous documents and is combined with a classifier based on spatio-temporal quantized words. As these are the main related approaches for our method, we benchmark our approach by comparing against these. Further, our approach uses a synthesis based transformation approach for generating abnormal samples that has to the best of our knowledge not been adopted for this problem. A. Background A crucial component in our model is the use of a Bayesian CNN for obtaining a discriminator. For the sake of completeness we present the related background for obtaining a Bayesian CNN. In probabilistic modeling, we infer a distribution over parameters w of a function y * = f (x * ; w), that is likely to generate our outputs. Also it is assumed that the training observations, input X = {x 1 , x 2 , ...x N } and their corresponding outputs Y = {y 1 , y 2 .....y N } are random variables from a distribution. The function f represents the network architecture, and w is the collection of the model parameters. Our objective is to model the uncertainty in prediction for classifying our synthesized abnormal samples. One of the popular ways to do this is the Bayesian approach. Given observation X, Y , we need to find the posterior distribution over space of functions, i.e., p(w\|X, Y ), which captures the most likely function parameters, given our observed data. The posterior can be modeled with the help some prior distribution p 0 (w) over the space of parameters w and with the likelihood function p(Y \|X, w). The likelihood function is the probability of the synthesized sample Y given the function parameter w. Then, we can predict the output of a new data point x * by integrating over all possible function parameters w. So the predictive distribution [9] [18] is given by the following equation. p(y * \|x * , X, Y ) = p(y * \|x * , w)p(w\|X, Y )dw(1) The posterior distribution p(w\|X, Y ) in eqn. 1 is intractable. To approximate the intractable posterior distribution p(w\|X, Y ), we need to define an approximating variational distribution q θ (w) parameterized by θ, whose structure is easy to evaluate. We thus minimize the KullbackLeibler(KL) divergence [16] between approximate posterior q θ (w) and the full posterior p(w\|X, Y ) w.r.t θ, which is denoted by KL(q θ (w)\|\|p(w\|X, Y )). Minimizing the KL divergence is equivalent to maximizing the log evidence lower bound [4] with respect to the variational parameters defining q θ (w), Loss = q θ (w) log p(Y \|X, w)dw−KL(q θ (w)\|\|p 0 (w)) (2) The posterior p(w\|X, Y ) in equation 1 is replaced with approximate posterior q θ (w). The integral of predictive distribution [9] is intractable for many models because it is integrated over all possible values of w. To approximate it, we could condition the model on a finite set of parameters w. So we approximate the integral with Monte Carlo integration. p(y * = c\|x * , X, Y ) = p(y * = c\|x * , w)p(w\|X, Y )dw ≈ p(y * = c\|x * , w)q θ (w)dw ≈ 1 M M m=1 p(y * = c\|x * ,ŵ m )(3) withŵ m ∼ q θ (w), which is the dropout distribution. III. METHODOLOGY Our aim is to synthesize frames in an unsupervised way, which appear to be abnormal leveraging solely the normal frames, to achieve detection of abnormal activities in a video. We discuss the intuition and techniques for the desired task in further subsections. As illustrated in Figure 2, our overall pipeline mainly consists of two components: Synthesis Block and Training Block. Given the normal videos frames as input, the task of Synthesis Block is to synthesize abnormal or fake frames by transforming the normal. This is achieved in several stages as shown in Figure 2. Firstly, through ViBe [1] technique we obtain foreground masks of moving objects. These noisy segments act as pseudo ground truths for training a fully convolutional network (FCN) [19] to obtain better foreground through semantic segmentation. Simultaneously, we keep track of the motion of foreground objects to obtain a representative map for overall motion in the video. Next, we randomly sample and cut a candidate foreground object and with the representative motion mask's assistance, the foreground object is maneuvered to a different place in the frame. In order to obtain a neat and indistinguishable fake frame, the original patch is algorithmically replaced with an appropriate patch from the prior frames. Subsection III-A explains the entire mechanism of the Synthesis Block in detail. The role of the Training Block is concisely that of a discriminator. Once fake frames have been synthesized, the challenge further lies in determining which ones among those are more likely to be considered abnormal. This block is responsible for selecting indistinguishable fake samples generated from the Synthesis Block, through incremental training to build a robust classifier. To serve the purpose, our idea is to incrementally train a Bayesian CNN, model the uncertainty and use it to classify abnormal frames. Based on the predictive posterior probabilities obtained from the classifier, we sample more fake abnormal ones and iteratively keep adding them to the training set. The methodology is further described in Section III-B A. Unsupervised Setting: Synthesizing Anomaly In order to obtain realistic anomalous frames we rely on the following principles, a) abnormal instances would be based on unusual location of entities (car on a pedestrian pathway, pedestrian in the middle of the road). b) realistic synthesis would require precise extraction through segmentation and realistic replacement of the extracted segment. We rely on motion cues for unsupervised segmentation of moving entities and further rely on this cue to obtain a motion map to evaluate possible placement and segment replacement cues. These are obtained without supervision. This is not unusual as extensive human vision studies show that motion plays an important role in the development of human visual system. For instance, it has been shown experimentally [25] that soon after gaining sight humans are better able to categorize objects that are seen to be in motion than those seen to be at rest. A similar observation has been studied for infants as well as described in [31]. 1) Unsupervised Motion Segmentation: In a real world scenario, almost all anomalous events occur due to objects in motion. For example, in a traffic road scene as shown in Figure 1, the abnormal events may include pedestrians crossing the road away from the zebra crossing or say, maybe a car entering pedestrian area. Since, the moving objects are responsible for such events, the idea is to focus primarily on the foreground and obtain segments of the moving objects. To serve the purpose of foreground extraction, we use ViBe technique as described in [1]. A similar method of foreground extraction has been adopted in [27], but for a very different purpose i.e. for formation of visual words for topic model. In addition to being faster, the advantage of ViBe is that it doesn't let the object in foreground fade away in the background quickly even when the object has stopped moving. This ensures that the objects are still highlighted as foreground in such events, for instance, when a pedestrian halts in the middle of the road for some time or an event where a car is being parked at a noparking zone. We further apply morphological transformations for denoising. Thus, we segment different blobs of foreground which we use for the purpose as described next. 2) Learning to Segment Through Motion and Noise: We follow an approach similar to Pathak et al. [26] to segment the objects. We use fully convolutional networks for semantic segmentation with default parameters [19], [34]. The foreground obtained from ViBe technique provide us with the pseudo ground truth for training. Moreover, the ViBe's output might not always lead to desired segments and is prone to noise. It is evident from extensive experiments in [26] that CNN is able to learn well even from noisy and often incorrect ground truth. Because of its finite capacity, a CNN will not be able to over-fit the noise. Instead, it closely learns the underlying correct segmentation leading to much smoother and visually more correct segmentation compared to the pseudo ground truth. 3) Sample Transformation: Once we segregate foreground from the background using ViBe motion segmentation, we obtain various blobs for each video frame. We find the (a) Traffic Junction Dataset [35] (b) Highway Dataset [27] rectangular connected blob of pixels using contour detection algorithm. These connected components represent the moving objects such as pedestrians and cars in video. There might be small groups of pixels as well in the frame which are not likely to be the actual objects of interest. We use area of those blobs as a parameter to eliminate noise, if any. Generally, anomalous events arise due to presence of objects or actions which we usually do not expect. In the Traffic Junction Dataset [35] , car entering pedestrian area, jay walking or people crossing the road away from the zebra crossing are among the unusual events as shown in Figure 1. This motivates our idea to crop the candidate object and place it to other parts of the same frame to make the frame appear abnormal. The object is confined within the rectangle which contains its background as well. If we crop the whole rectangular region, it would look unnatural when we place it on to other parts of the frame. Instead, we apply semantic segmentation [19] to the rectangular part using our trained model discussed earlier in Section III (A.1 and A.2) Thus, we now obtain a precise foreground object after segmentation. After obtaining the segment, it needs to be placed in other parts of the frame in an unsupervised way. We place objects only in those regions which have observed motion in the entire video. The intuition behind the algorithm is that it is highly unlikely for a car or a person to be present within a building's wall or flying in air. Again, we achieve this by leveraging ViBe motion segmentation. We pre-process the video by keeping track and storing all those pixel indexes which have witnessed motion as guided by ViBe. We achieve this task in real time. As shown in Figure 3, white pixels represent the regions where there has been foreground object motion at least once and the black pixels represent the regions that never observed any motion. We now have the set of candidate pixels (white pixels in Figure 3) on or in the vicinity of which the foreground objects shall be placed. Having cropped the object and placed its segmentation to another region in the frame, the next challenge lies in inpainting that particular missing area in the frame. The objective of region inpainting is to ensure that the synthesized frame appears visually natural. A basic heuristic has been proposed by Bertalmio et al. in [3] to provide an inpainting algorithm. This technique works well when the region to be inpainted is very small in size. However, as the missing region size is large, so is the case for size of real world objects, it fails to provide a smooth and clear inpainted region. We tackle this challenge of region inpainting with a simple approach. First, we exploit the fact that there are plenty of instances when no foreground object is present at a particular region of the frame in the entire video. Second, while the surveillance camera is static, the background remains almost unchanged throughout in the video. Our main idea here is to observe the missing rectangular region in previous video frames and choose the frame which has the least or ideally, no movement. We use ViBe for real time pixel level motion information. We cache a constant number of frames arriving earlier than the current frame and observe the ViBe pixels within the missing rectangular region's coordinates in those cached frames. Among these frames, the region having the least motion pixels count indicated by ViBe is chosen as the region that will replace the missing region in the current frame. This approach neatly replaces the missing region compared to the one inpainted using the algorithm in [3]. The qualitative results of our proposed algorithm for synthesizing abnormal frames are shown in Figure 4. B. Abnormal Activity Detection The sole purpose of generating abnormal frames, was to have a self-supervised learning mechanism to detect abnormal activities. We create our own dataset consisting of abnormal frames and normal frames as training samples. We can synthesize abnormal frames from each normal frame of the video. We train a Bayesian VGG-19 [30] for the two classes. We observe that while preparing the dataset, synthesis of abnormal frames does not always lead to a sample that actually appears to be abnormal. This is justifiable because the foreground object will be stochastically placed to other regions of a frame. For example, a car might be placed in the road itself, which is actually not abnormal. To overcome this problem, we introduce a Bayesian discriminator into the pipeline. This helps in statistically sampling a refined set of fake samples that can further be used to build a robust classifier. We interpret dropout as approximate Bayesian stochastic inference [8] over our model weights. Moreover, to impose Bernoulli distributions over model weights, we have adopted dropout [32]. Modeling Bernoulli distributions can be achieved by sampling multiple times (ten times, in our case) with dropping random units during inference. This sampling is possible because it is a Bayesian stochastic inference. These samples are considered as the Monte Carlo samples [14] from the posterior distribution over models. We experimented with various dropout ratios and use the following values for the same. For implementing Bayesian CNN, we use dropout ratio of [0.1, 0.1, 0.3, 0.4, 0.4] for each stack of convolutional layers respectively and 0.5 for FC layers. As the number of neurons increases in subsequent layers, we increased the dropout ratio for better generalization. The predicted posterior probability is obtained by applying the softmax function on the network output layer. We draw ten samples of each synthesized image I i using Monte-Carlo sampling from a distribution (this is predictive posterior distribution for the bayesian CNN). The predictive posterior probability signifies the probability of the sample belonging to the normal class. We calculate the mean µ i and σ i for image I i . p i = µ i − σ i 2(4) We consider p i as an upper bound on the predictive posterior probability as criteria to incrementally train the Bayesian CNN and use it to classify abnormal frames. As shown in Figure 7, we vary the predictive posterior probability p i and iteratively sample abnormal frames having probability ≤ p i . Our assumption here is that as there are large number of placements that are abnormal and the real samples are all normal, the model is able to distinguish the abnormal from the normal with some uncertainty. A few examples of the synthesized abnormal samples are provided in Figure 4. IV. EXPERIMENTS AND RESULTS We perform experiments on the Traffic Junction Dataset released in [35]. It consists of a video of 45 minutes recorded at a frame rate of 25 fps with frame size of 288 × 360. For comprehensive evaluation of the proposed algorithm on a real world surveillance video dataset, we perform experiments on the Highway Dataset released in [27]. The dataset consists of 6 and a half minute video having frame rate 25 fps with frame size of 288 × 360, illustrating the traffic scene of a highway in real world situation. Additionally, in Figure 5 we show that our Synthesis Block has application in data augmentation for thermal imagery as well. OSU Thermal Pedestrian Database [7] is a publicly available benchmark dataset which consists of only 284 images. Deep learning algorithms, however, don't perform well with such datasets having few data samples. So, our synthesis technique could help in training deep learning algorithms. Anomalous video frames were separated from the video for testing. From the remaining normal set of frames, 25% of the normal frames were also included in the test data along with the anomalous ones. For training the CNN, we used our synthesized dataset. Moreover, while adding the normal frames to the synthetic training set, we choose to skip frames by using frame rate of 5 fps instead of 25 fps. This would lower the number of similar frames in the train set, thereby, reducing redundancy. After having synthesized the abnormal frames, we now have the dataset consisting of two classes: normal and synthesized abnormal. We train a CNN with dropouts to have an approximate Bayesian stochastic inference as described in previous section. We train VGG-19 [30] on this dataset with image size 224 × 224. We use a weighted cross-entropy loss function for training in order explicitly weigh more penalties (five times more, in our case) for false negatives than false positives and SGD as the optimizer. We call this synthesized model. To further make our model robust, we obtain a refined dataset by removing the more probable normal fake examples. We apply softmax activation function on our trained model to obtain respective classification probabilities (predictive posterior) on the training synthesized abnormal samples. A low probability score for a sample frame to belong to the normal class could indicate a higher likeliness for it being abnormal. Thus, we refine our synthesized set of examples and iteratively train the Bayesian CNN on this set. Initially, we choose p i = 0 Figure 6. Furthermore, we experiment with a non-Bayesian setting, keeping our overall pipeline fixed except that we now do not adopt a Bayesian CNN and replace the discriminator with ResNet-50 [12] representing a non-Bayesian setting. The improved ResNet architecture results in an improvement of the area under the precision-recall curve (AUC) as shown in Table I. As can be seen from the quantitative results, we obtain consistent improvements in terms of AUC. The AUC being the standard metric shows improved performance which can be improved by further refining various blocks of the network. V. ANALYSIS In this section, we provide analysis for our proposed model from aspects such as distribution discrepancy, feature visualization and qualitative analysis. A. Distribution Discrepancy: proxy A-distance According to domain adaptation theory [2], A-distance as a measure of cross domain discrepancy, which, together with the source risk, will bound the target risk. The proxy A-distance is defined as d A = 2(1 − 2 ) where is the generalization error of a classifier(e.g. kernel SVM [5]) trained on the binary task of discriminating source and target. We can apply a similar concept to better analyze our results quantitatively. We use our model features and use it to calculate the proxy A-distance i.e. d A1 between two classes: our synthesized abnormal class and the ground truth abnormal class. Moreover, we compare that with the proxy A-distance i.e. d A2 between two classes: our synthesized abnormal class and the normal class. We obtain the values d A1 = 1.8589 and d A2 = 1.9782. Since, d A1 < d A2 . which suggests that our features can reduce the cross-domain gap more effectively. In other words, it gives an indication that features learned for our synthesized abnormal samples would be more closer to the real anomalous video frames. B. t-SNE Plot t-distributed Stochastic Neighbor Embedding (t-SNE) [23] is a technique to visualize high-dimensional data. It converts similarities between data points to joint probabilities and tries to minimize the Kullback-Leibler [16] divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data. In Figure 8(a), the red points correspond to our synthesized abnormal samples, while the green ones correspond to the normal frames in the video. It is clearly observed that the synthesized samples are mostly clustered towards the centre. In Figure 8(b), the red points correspond to our synthesized abnormal samples, while the blue ones correspond to the ground truth abnormal samples. The red points are spread across the blue points as shown in Figure 8(b). Through the t-SNE [23] visualizations in Figure 8, it can be observed that the feature representation of our synthesized abnormal samples makes it much closer to that of the actual ground truth samples distribution as shown in Figure 8(b) and far from the embeddings of the normal video frames since the points are mostly clustered at the centre as shown in Figure 8 C. Qualitative Analysis In the 44 minutes long video of the Traffic Junction Dataset [35], there is an instance shown in Figure 9 (a) where abnormality occurs in the form of a jay walker for a short period of 3 seconds. Our model is able to detect such an activity, which could be tiresome or could have gone undetected by a human operator for such surveillance videos. Also, the frame where a car enters the pedestrian area has also been detected as abnormal as shown in Figure 9 (b). However, the model fails to detect the frame in Figure 9 (c) as abnormal for which ground truth label is abnormal. This is a special case as it features a pedestrian crossing the road at the crossing. The context is that the pedestrian is doing so when the traffic signal is red. It is therefore an abnormal example. However, as this is a very subtle cue even for humans, it is very challenging to detect such cases. VI. CONCLUSION In this paper, we have presented a method that uses transformation of the normal for generating abnormal samples. These are then validated through a classifier to automatically obtain a set of abnormal examples. This is a novel approach as other approaches such as conditional GANs were also not suited for generating such abnormal examples, but our proposed pipeline is able to generate abnormal examples automatically without supervision. In future, we would be interested in considering spatio-temporal generation of such synthetic examples. We can also further consider other scenarios such as social interactions and imbalanced activity recognition cases where such approaches may be applicable. Fig. 1 . 1Sample Normal and Abnormal Frames from surveillance video datasets. Fig. 2 . 2Illustrative overview of the proposed approach. There are two main blocks, a Synthesis Block and a Training Block. Fig. 3 . 3Representative binary motion mask map for overall foreground motion in the surveillance video datasets. Fig. 4 . 4Sample frames from Traffic Junction and Highway Dataset annotated as normal (left) and our synthesized abnormal frames (right) from the normal. Fig. 5 . 5Our Synthesis Block technique has another application as a strategy for Data augmentation as well. Sample frames from OSU thermal imagery Dataset[7] (left) and synthesized frames (right) by our algorithm. Fig. 6 . 6Precision-Recall curves for anomaly detection in Traffic-Junction Dataset[35] using Bayesian CNN pipeline. Anomalous frames were considered as positive and the Normal frames as negative examples. Fig. 7 . 7Scores of AUC and Average Precision (AP) for Traffic-Junction Dataset [35] on incrementally training Bayesian CNN by varying the refined set of samples in the synthesized abnormal class. and vary it till p i = 0.5, thereby, adding more samples to incrementally train the CNN. Note that, at p i = 1 in Figure 7 is actually our initially proposed synthesized model. The experimental results on appropriately choosing the clean set of abnormal examples are shown in Figure 7. We adopt hold-out cross validation technique with validation set consisting of 30% of the dataset. The model with highest AUC performance on the validation set is chosen to report the results by evaluation on the test set. Additionally, we evaluate our other models obtained by varying the selected set of examples on the test set too. The results are consistent with the results in validation. Moreover, our proposed algorithm backed with experimental results suggests that actively choosing the training examples further assists in improving the results as shown in Fig. 8 . 8t-SNE visualizations for following distributions: (a) Green points represent Normal video frames distribution The centrally clustered red points represent our Synthesized frames distribution. (b) Blue points represent Ground truth abnormal distribution, while the red points represent our Synthesized distribution for Traffic-Junction Dataset[35]. 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[ "Helioseismic Constraints and Paradigm Shift in Solar Dynamo", "Helioseismic Constraints and Paradigm Shift in Solar Dynamo" ]	[ "Alexander G Kosovichev \nStanford University\n94305StanfordCAUSA\n", "Valery V Pipin \nStanford University\n94305StanfordCAUSA\n\nUCLA\n90095Los AngelesCAUSA\n\nInstitute of Solar-Terrestrial Physics\n664033IrkutskRussian Federation\n", "Junwei Zhao \nStanford University\n94305StanfordCAUSA\n" ]	[ "Stanford University\n94305StanfordCAUSA", "Stanford University\n94305StanfordCAUSA", "UCLA\n90095Los AngelesCAUSA", "Institute of Solar-Terrestrial Physics\n664033IrkutskRussian Federation", "Stanford University\n94305StanfordCAUSA" ]	[ "Progress in Physics of the Sun and Stars: A New Era in Helio-and Asteroseismology ASP Conference Series" ]	Helioseismology provides important constraints for the solar dynamo problem. However, the basic properties and even the depth of the dynamo process, which operates also in other stars, are unknown. Most of the dynamo models suggest that the toroidal magnetic field that emerges on the surface and forms sunspots is generated near the bottom of the convection zone, in the tachocline. However, there is a number of theoretical and observational problems with justifying the deep-seated dynamo models. This leads to the idea that the subsurface angular velocity shear may play an important role in the solar dynamo. Using helioseismology measurements of the internal rotation and meridional circulation, we investigate a mean-field MHD model of dynamo distributed in the bulk of the convection zone but shaped in a near-surface layer. We show that if the boundary conditions at the top of the dynamo region allow the large-scale toroidal magnetic fields to penetrate into the surface, then the dynamo wave propagates along the isosurface of angular velocity in the subsurface shear layer, forming the butterfly diagram in agreement with the Parker-Yoshimura rule and solar-cycle observations. Unlike the flux-transport dynamo models, this model does not depend on the transport of magnetic field by meridional circulation at the bottom of the convection zone, and works well when the meridional circulation forms two cells in radius, as recently indicated by deep-focus time-distance helioseismology analysis of the SDO/HMI and SOHO/MDI data. We compare the new dynamo model with various characteristics if the solar magnetic cycles, including the cycle asymmetry (Waldmeier's relations) and magnetic 'butterfly' diagrams.	null	[ "https://arxiv.org/pdf/1402.1901v1.pdf" ]	115,484,732	1402.1901	16eac914f357db80c800bc6250d704b61b190421	Helioseismic Constraints and Paradigm Shift in Solar Dynamo 9 Feb 2014 Alexander G Kosovichev Stanford University 94305StanfordCAUSA Valery V Pipin Stanford University 94305StanfordCAUSA UCLA 90095Los AngelesCAUSA Institute of Solar-Terrestrial Physics 664033IrkutskRussian Federation Junwei Zhao Stanford University 94305StanfordCAUSA Helioseismic Constraints and Paradigm Shift in Solar Dynamo Progress in Physics of the Sun and Stars: A New Era in Helio-and Asteroseismology ASP Conference Series 4799 Feb 2014 Helioseismology provides important constraints for the solar dynamo problem. However, the basic properties and even the depth of the dynamo process, which operates also in other stars, are unknown. Most of the dynamo models suggest that the toroidal magnetic field that emerges on the surface and forms sunspots is generated near the bottom of the convection zone, in the tachocline. However, there is a number of theoretical and observational problems with justifying the deep-seated dynamo models. This leads to the idea that the subsurface angular velocity shear may play an important role in the solar dynamo. Using helioseismology measurements of the internal rotation and meridional circulation, we investigate a mean-field MHD model of dynamo distributed in the bulk of the convection zone but shaped in a near-surface layer. We show that if the boundary conditions at the top of the dynamo region allow the large-scale toroidal magnetic fields to penetrate into the surface, then the dynamo wave propagates along the isosurface of angular velocity in the subsurface shear layer, forming the butterfly diagram in agreement with the Parker-Yoshimura rule and solar-cycle observations. Unlike the flux-transport dynamo models, this model does not depend on the transport of magnetic field by meridional circulation at the bottom of the convection zone, and works well when the meridional circulation forms two cells in radius, as recently indicated by deep-focus time-distance helioseismology analysis of the SDO/HMI and SOHO/MDI data. We compare the new dynamo model with various characteristics if the solar magnetic cycles, including the cycle asymmetry (Waldmeier's relations) and magnetic 'butterfly' diagrams. Basic Properties of Solar Magnetic Cycles The dynamo which operates in the solar convection zone and defines the properties of sunspot cycles remains enigmatic despite substantial observational and modeling efforts. In particular, the solar dynamo models must explain the magnetic "butterfly" diagram ( Fig. 1) and asymmetry of the sunspot cycles (Fig. 2a), so-called Waldmeier's effect (Waldmeier 1935). The magnetic butterfly diagram is obtained by stacking azimuthally averaged synoptic magnetic field maps, provided by the National Solar Observatory since 1976. The most prominent features of the butterfly diagram are migration of the sunspot formation zones towards the equator, migration of the following polarity flux towards the poles, polarity reversals of polar magnetic fields during sunspot maxima, and also polarity reversals of the toroidal magnetic field during the solar minima, which are observed as cyclic changes of polarity of leading and following sunspots in bipolar active 396 A. G. Kosovichev, V. V. Pipin, andJunwei Zhao 1980 1990 2000 2010 year -50 0 50 latitude Figure 1. Magnetic 'butterfly' diagram shows the evolution of the azymuthally averaged line-of-sight magnetic field a function of latitude and time. The range of magnetic field strength is from -10 G to +10 G. The synoptic magnetogram data are provided by the National Solar Observatory regions (the Hale's law). The Waldweier's effect (non-linear asymmetry of sunspot cycles) has the following three main characteristics: 1) the sunspot number growth time is shorter than the decay time (Fig. 2b); 2) the growth time of strong cycles is shorter than the growth time of weak cycles ( Fig. 2c-d); 3) the strong cycles are shorter than the weak cycles. The first property is often used for predicting the sunspot maximum from the growth rate at the beginning of a cycle, using the Waldmeier's "standard curves" (Fig. 2c). Thus, the dynamo theories have to explain both the Hale's law and the Waldmeier's effect. Bullard (1955) suggested that the sunspot pairs represent parts of subsurface toroidal magnetic rings, emerged from the depth comparable with the size of these pairs, i.e. 20 Mm, and that the toroidal magnetic field is produced from the poloidal field by the latitudinal differential rotation beneath the solar surface. Generation of the toroidal magnetic field through stretching of the poloidal field by the differential rotation is a common feature of most solar dynamo models. However, the models differ in the depth of the toroidal field generation, and also in the physics of the reverse process of the poloidal magnetic field generation from the toroidal field. Dynamo dilemma Two basic models of the solar dynamo operation have been developed. The first, so-called 'flux-transport' model (Babcock 1961;Leighton 1969) assumes that the poloidal magnetic field is produced by a combined action of the Coriolis force, turbulent diffusion and meridional circulation. In these models, the Coriolis force causes a tilt of emerging magnetic regions, relative to equator, which leads to preferential diffusion of magnetic flux of the trailing polarity and, subsequently, the polar field polarity reversals (Fig 3a). The sunspot butterfly diagram is explained by the equator-ward meridional flow at the bottom of the convection zone, slowly transporting the regenerated toroidal magnetic field from mid latitudes towards the equator (Wang et al. 1991). Among the well-known difficulties of such models is the requirement to generate coherent magnetic flux tubes at the bottom of the convection zone with the field strength of 6×10 4 −10 5 G, the magnetic energy density of which substantially exceeds the turbulent energy den-397 Figure 2. a) The sunspot number as a function of time illustrates the asymmetry of the solar cycles (the Waldmeier's effect); b) comparison of the sunspot number curves for four cycles (Bracewell 1988); c) a model of the Waldmeier relations (Bracewell 1988); d) dependence of the cycle amplitude from the growth time (Kitiashvili & Kosovichev 2009). sity. In addition, helioseismology revealed that speed of the meridional circulation substantially varies with the solar cycle, mostly due to large-scale converging flows around active regions. These flows may significantly affect the polarward diffusion process. In alternative models, initially suggested by Parker (1955), the poloidal field is generated by helical turbulence in the bulk of the convection zone, and is transported by turbulent diffusion in a form of 'dynamo waves', travelling along the isorotation surfaces (Fig. 3b, Yoshimura 1975). To explain the butterfly diagram this model requires that the rotation rate decreases towards the surface. However, measurements of the internal differential rotation by helioseismology showed that the rotation rate increases almost through the whole convection zone, except a shallow subsurface rotational shear layer, where the rotation sharply decreases (Fig. 4). This means that the dynamo waves in the deep convection travel poleward, contrary to the sunspot butterfly diagram. This model, like the flux-transport model, could not explain the Waldmeier's effect. Thus, both types of the solar dynamo models faced significant problems with explaining the solar magnetic cycles. Brandenburg (2005) suggested that the dynamo process can be distributed in the convection zone but 'shaped' in the subsurface layer, where the dynamo wave can migrate equatorward long the isorotation surfaces, according to the Parker-Yoshimura rule. This idea is supported the comparison of the magnetic flux rotation rate with the internal differential rotation (Benevolenskaya et al. 1999). This comparison showed that if the magnetic flux emerges radially and keep the rotation rate of its 'nest' then it most likely originates from the upper convection zone. If the magnetic flux tubes emerge from the base of the convection zone then the rotation rate of these flux tubes is generally slower than the observed rotation rate (Weber et al. 2012). Dynamo dilemma: a) illustration of the Babcock-Leighton dynamo model (Babcock 1961;Leighton 1969): the toroidal magnetic field producing sunspot regions is generated by the differential rotation in the convection zone while the poloidal magnetic field is produced near the surface by magnetic flux diffusion, the equator-ward migration of sunspot formation zones is provided by the equatorward meridional flow at the bottom of the convection zone (Wang et al. 1989) ; b) illustration of the propagation direction of dynamo waves along the isorotation surfaces in the Parker-Yoshimura model (Parker 1955;Yoshimura 1975), the equatorward migration of the sunspot zones requires a decrease of the internal differential rotation rate towards the surface. Solar Dynamo Modeling -Paradigm Shift To investigate the idea of subsurface-shear-shaped dynamo suggested by Brandenburg (2005), Pipin & Kosovichev (2011b) calculated a mean-field model, in which the dynamo effect is distributed in the bulk of the convection zone, and the toroidal magneticfield flux gets concentrated in regions of low turbulent diffusivity at the boundaries of the convection zone. They showed that if the conditions at the top of the convection zone such that the large-scale toroidal magnetic field penetrates close to the surface, then the butterfly diagram for the toroidal field in the upper convection zone is formed by the subsurface rotational shear layer, following the Parker-Yoshimura rule, and that this can explain the observed equator-ward migration of the sunspot formation zone during the solar cycles. In previous dynamo models such penetration of toroidal field was prevented because of an artificially high turbulent diffusivity or/and because of the boundary conditions representing potential (vacuum) magnetic field outside the convection zone. It was shown that changing just one of these assumptions results in significant changes of the solar dynamo properties. In particular, the potential field assumption can be lifted by adding electrical conductivity in the top boundary condition (Pipin & Kosovichev 2011b). Also, if the turbulent diffusivity is consistently calculated following the mixing-length theory the toroidal magnetic field penetrates into the subsurface shear layer even with the potential-field boundary condition (Pipin & Kosovichev 2011a). The dynamo model with the subsurface rotational shear is illustrated in Figures 5 and 6, which show the butterfly diagram for the toroidal magnetic field and the corresponding evolution of the radial component of the poloidal field at the solar surface, and also the distribution of the toroidal and poloidal magnetic fields in the convection zone at different phases of the dynamo cycle. The butterfly diagram and the phase rela- Figure 4. a) Solar differential rotation inferred by helioseismology (Schou et al. 1998); b) comparison of the internal rotation rate at different latitudes with the rotation rate of emerging magnetic flux at the beginning of the solar cycles ("new flux") and at their end ("old flux") (Benevolenskaya et al. 1999). tion between the toroidal and poloidal magnetic field correspond quite well to the solar observations (Fig. 1). These results mean that the Parker's dynamo wave model can be consistent with both the helioseismology inferences and magnetic field data. This brings new life to this model and represent a paradigm shift in solar dynamo modeling. In this model with the subsurface shear the meridional circulation affects the dynamics of magnetic field, but it no longer plays a key role in the dynamo process. The period of magnetic cycles is determined by the speed of the dynamo waves, which is controlled by turbulent magnetic diffusivity, and the strength of the sunspot cycles is determined by the turbulent kinetic helicity. Both these quantities can be consistently estimated from turbulence models based on the mixing length theory. The observed variations among the solar cycles, particularly, in the cycle maxima can be explained by long-term fluctuations in the kinetic helicity (Pipin et al. 2012). The new model can also explain the asymmetry of the sunspot cycles (the Waldmeier's effect), as a result of the magnetic helicity conservation which provides dynamic quenching for the poloidal field generation ('alpha'-effect) (Pipin & Kosovichev 2011a). Figure 7 shows the relationship between the rise and decay times for three different values of a parameter R ξ which controls the dissipation rate of magnetic helicity (this parameter is similar to an effective turbulent magnetic Reynolds number), and a histogram of the ratio of the rise and decay times obtained from the observations of the sunspot cycles (Fig. 2a) and from the model with R ξ = 100. Our dynamo model has several advantages compared to the flux-transport models. In particular, it can explain the relative stability of the duration of sunspot cycles compared to the flux-transport models, for which the cycle duration depends on the speed of the meridional flows, which can vary substantially during the cycles (Zhao & Kosovichev 2004). It does not require the formation of compact toroidal magnetic flux tubes with the field strength of ∼ 60 − 100 kG, which are problematic because the energy density of which exceeds the turbulent energy equipartition level, and also does require to explain coherent emergence of these flux tubes from the bottom of the convection zone. In our model, the magnetic field of sunspots and active regions can be provided by the magnetic field emerging from relatively shallow layers, similar to the process simulated numerically by Stein & Nordlund (2012). However, the tilt of bipolar active regions (the Joy's law) is not yet explained. New Helioseismology Observations and Constraints on Dynamo Models In addition to the differential rotation, the knowledge of the internal meridional flows is critical for developing the solar dynamo models. It was long assumed that the meridional flows are represented in the Northern and Southern hemispheres by circulation cells occupying the whole convection zone, and that at the flow is directed from the equator to the poles at the top of the convection zone and is reversed at the bottom. Since the flow speed is quite low (∼ 20 m/s) it is very difficult to measure these flows by helioseismology. Previous time-distance measurements by Giles et al. (1997) established that the polar-ward meridional flow extends into deeper convection zone, but were not able to detect the return meridional flow. The initial evidence that the return meridional flow can be shallow, starting at depth 20 Mm was obtained by Braun & Fan (1998). Later Mitra-Kraev & Thompson (2007) obtained an estimate of the depth of 401 Figure 7. Asymmetry of the sunspot cycles simulated in the dynamo model with the subsurface shear layer (Waldmeier's effect): a) the relationship between the rise and decay times for different values of a parameter R ξ that controls the dissipation rate of magnetic helicity; b) histogram of the asymmetry parameter (the ratio of the rise and decay times) obtained from observations of the sunspot cycles (Fig. 2a) and from the model with R ξ = 100. (Pipin & Kosovichev 2011a) the flow reversal at around 40 Mm. In both cases, the error estimate was comparable with the flow signal itself, and also no verification and testing of their techniques was done. The high-resolution helioseismology data from the Solar Dynamics Observatory (SDO) HMI instrument provides new opportunities for improving measurements of the meridional flow. Important advantages of the SDO/HMI data over the previous helioseismology observations are that the data have much higher resolution and almost uninterrupted, and also that the solar oscillations are observed not only in the Doppler shift, but also in other spectral parameters: continuum intensity, line width and line depth (Scherrer et al. 2012). Using these multi-parameter measurements Zhao et al. (2012) found previously unknown center-to-limb systematic shifts of the acoustic travel times. These systematic shifts are different for the different observables, and probably caused by a combination of the wave leakage and line formation effects in the solar atmosphere, due to changes of the effective observing height from the center to the limb. Near the limb the spectral line is formed higher in the atmosphere than near the center. This may cause systematic travel time shifts of the order of few seconds , which have to be taken into account in measurements of the meridional flows. The exact mechanism of these shifts is not understood, and Zhao et al. (2012) developed an empirical correction procedure. In this procedure the systematic variations were determined by measuring the travel-time variations along the equatorial regions during the periods when the solar rotation axis is perpendicular to the line of sight. After the substraction of the centerto-limb variations the measurements from the different observables gave very similar results. Inversion of the corrected travel times showed that the previous helioseismic measurements overestimated the meridional flow speed by about 10 m/sec. The corrected speed is more consistent with the surface flow speed obtained directly from the Doppler shift (e.g. Hathaway 2012). Recently, Zhao et al. (2013) used a deep-focus measurement scheme (Fig. 8a) Doppler-shift data covering its first 2-year period from 2010 May 1 through 2012 April 30, and calculated the time-distance cross-covariance functions for 60 measurement distances ranging from ∼ 2 to 44 degrees, covering almost the whole depth of the convection zone, according to the acoustic ray theory. The cross-correlation functions were averaged for the same latitudes, and then averaged again over one-month intervals. The acoustic travel times were determined by fitting the Gabor wavelet functions (Kosovichev & Duvall 1997), and corrected for the center-to-limb variation. Finally, for each distance the travel times were averaged again over the whole 2-year period, and used to determine the meridional flow speed by inversion in the acoustic ray-path approximation (Kosovichev 1996). Figure 8b shows the the North-South travel time differences as a function of the wave travel distance or the radius of the wave lower turning point (positive values are for the measurements in the northern hemisphere, and negative values are for the southern hemisphere), the solid curves are the measurement results (red curve is for the 403 SOHO/MDI instrument, black curve is for the SDO/HMI). In the asymptotic ray-path approximation, the sensitivity of acoustic travel times to the internal flows depends on the local sound speed and the angle between the ray path and the flow velocity. Therefore, most of the travel time sensitivity comes from the near-surface layers, where the sound-speed is low and from the region around the lower turning point of the acoustic ray paths because the waves travel along the flow. The sensitivity to the near-surface region is dominant, so that the travel times do not change sign when the waves travel through the deep regions of return flows. The return flow effects are reflected in the rate of decrease of the travel times with the depth of the wave turning point, or equivalently with the increase of the travel distance. The travel times from the SDO/HMI and also from the SOHO/MDI shown in Fig. 8b indicate a rapid decrease for the waves traveling into the deep convection zone, which is steeper than predicted for the standard single-cell models of the meridional circulation with the return flow near the bottom of the convection zone. Such rapid decrease indicates that the return flow is rather shallow. Also, quite unexpectedly, the travel times start rising for the acoustic waves traveling through the lower half of the convection zone, indicating the existence of deep poleward flows or a secondary circulation cell. The inversion results shown in Fig. 8c confirm this (Zhao et al. 2013). Thus, the new helioseismology measurements based on very long time series of observations of solar oscillations provide a strong evidence that the meridional circulation in the solar convection zone has a complicated structure and consists of at least two circulation cells stacked along the radius, as schematically illustrated in Figure 8d. Such doublecell meridional flow may be consistent the pole-ward migration of supergranulation, as recently suggested by Hathaway (2012). This result immediately puts in question the standard flux-transport dynamo models, which have to rely on the single-cell meridional flow to carry the magnetic flux towards the equator at the bottom of the convection zone in order to explain the sunspot butterfly diagram. This result also raises the question about how this type of meridional circulation can affect the distributed dynamo model with the subsurface shear layer. Dynamo model with double-cell meridional circulation To investigate the effects of the double-cell meridional circulation we calculated dynamo models with such circulation, using the mean-field magnetohydrodynamics theory approach which includes detailed modeling of the mean electromotive force and turbulent diffusion coefficient in the so-called "minimal tau-approximation" (e.g. Pipin 2008). The tau-approximation suggests that the second-order correlations do not vary significantly on the timescale τ c that corresponds to the typical turnover time of the convective flows. The theoretical calculations are done for the anelastic turbulent flows, and take into account the effects of density stratification, spatial inhomogeneity of the intensity of turbulent flows and inhomogeneity of the large-scale magnetic fields. The effects of the large-scale inhomogeneity of the turbulent flows and magnetic fields are calculated to the first order of the Taylor expansion in terms of the ratio typical spatial scales of turbulence and the mean quantities (for further details, see Pipin 2008). The meridional circulation is modeled in the form two spherical-shape circulation cells along the radius, occupying the whole convection zone (Pipin & Kosovichev 2013), as shown in Figure 8d. Results for this dynamo model are shown in Figures 9 and 10. Contrary to previous flux-transport dynamo models, which fail for such type of the meridional circulation (Jouve & Brun 2007), it is found that the dynamo model can robustly reproduce the basic properties of the solar magnetic cycles for a wide range of model parameters and the circulation speed. The best agreement with observations is achieved when the surface speed of meridional circulation is about 12 m/s. For this circulation speed the simulated sunspot activity shows the good synchronization with the polar magnetic fields. The toroidal magnetic field of the new cycle is generated near the bottom of the convection zone by the differential rotation. Simultaneously, in Fig. 9, we see a start of generation of the poloidal magnetic field (contour lines). The dynamo wave propagates by a turbulent diffusion process almost radially to the surface following the Parker-Yoshimura rule (Parker 1955;Yoshimura 1975). However, the propagation of the wave is inclined to the equator because of the anisotropy of the turbulent diffusion and turbulent transport effects. Near the surface the turbulent downward pumping and the subsurface rotational shear stop the radial propagation and deflect the dynamo wave toward the equator. The near-surface meridional circulation and the turbulent diffusion bring the decaying poloidal field to the poles. The meridional circulation modifies the propagation of the dynamo wave. It is found that the toroidal magnetic field is Figure 10. The magnetic "butterfly" diagrams at r = 0.82 R for: a) the dynamo model without the meridional circulation; b) for the model with the doublecell meridional circulation of U 0 = 12 m/s; c) tracking of the toroidal field maximum, which is an assumed latitudinal zone of sunspot formation; d) comparison of the toroidal field maxima for the models with different characteristic meridional flow speed with the observed speed of sunspot zone migration (Hathaway 2011). The toroidal field is shown by contours (plotted for ±100 G range), and the surface radial magnetic field is shown by background images. We draw these diagrams only for one hemisphere because the antisymmetric mode (dipole-like) is dominant (Pipin & Kosovichev 2013). transport models. Near the surface the poloidal field migrates towards the poles at high latitudes and towards the equator at low latitudes. Figure 10 shows the time-latitude "butterfly" diagrams of the toroidal (contours) and radial (background image) magnetic fields evolution in the upper part of the solar convection zone for the meridional flow speed U 0 = 0 and 12 m/s. In both cases (with and without the circulation) there is a qualitative agreement with observations. However, the maximum of the toroidal magnetic field migrates closer to the equator for the model with the circulation. Also, in this case the butterfly diagram wings are wider in latitude than in the case without circulation. Also it is found that the doublecell circulation reduces the latitudinal width of the polar branch for the radial magnetic field evolution and also reduces the overlap between the cycles. Conclusion The helioseismology discoveries of the rotational subsurface shear layer and the doublecell meridional circulation require to re-examine dynamo models of the solar activity cycles. They lead a new paradigm of the solar dynamo distributed in the convec-tion zone and equator-ward migrating dynamo waves in the sub-surface shear layer. The double-cell meridional circulation affects the dynamics of the large-scale magnetic fields, and puts additional constraints. In particular, if the observed equator-ward migration of the sunspot formation zone ('butterfly' diagram) is linked to migration of toroidal magnetic field in the convection zone then the magnetic field of sunspots may emerge from the depth of about 120 Mm. This is generally consistent with the observed rotation rate of surface magnetic fields of active regions and the internal differential rotation determined by helioseismology. Further helioseismology investigation will shed more light on the solar dynamo mechanism. Figure 3. Dynamo dilemma: a) illustration of the Babcock-Leighton dynamo model (Babcock 1961; Leighton 1969): the toroidal magnetic field producing sunspot regions is generated by the differential rotation in the convection zone while the poloidal magnetic field is produced near the surface by magnetic flux diffusion, the equator-ward migration of sunspot formation zones is provided by the equatorward meridional flow at the bottom of the convection zone (Wang et al. 1989) ; b) illustration of the propagation direction of dynamo waves along the isorotation surfaces in the Parker-Yoshimura model (Parker 1955; Yoshimura 1975), the equatorward migration of the sunspot zones requires a decrease of the internal differential rotation rate towards the surface. Figure 5 . 5Illustration of the dynamo model, which includes effects of the nearsurface rotational shear: the "butterfly" diagram of the toroidal magnetic field (color background) and the radial component of the poloidal field (contours) at the solar surface (r = 0.99 R ⊙ )(Pipin & Kosovichev 2011b). Figure 6 . 6Illustration of the dynamo model, which includes effects of the nearsurface rotational shear: the distribution of the toroidal (color) and poloidal magnetic field (contours) in the convection zone at different phases of the dynamo cycle(Pipin & Kosovichev 2011b). Figure 8 . 8to measure the meridional flow speed in the deep convection zone. They used the Helioseismic measurements of the meridional flow: a) the deep-focus time-distance helioseismology measurement scheme; b) variations of acoustic travel times, due to the meriodional flows, as a function of the wave travel distance or the radius of the wave lower turning point (positive values are for the measurements in the northern hemisphere, and negative values are for the southern hemisphere), the solid curves are the measurement results (red curve is for the SOHO/MDI instrument, black curve is for the SDO/HMI), the dashed curve shows the travel times calculated from the inversion results; c) the distribution of the meridional flow speed obtained by inversion of the SDO/HMI travel time measurements; d) schematic illustration of the double-cell meridional circulation(Zhao et al. 2013). Figure 9 . 9Evolution of the large-scale magnetic field inside the convection zone for the dynamo model with the meridional circulation speed U 0 = 12 m/s. 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[ "Algorithmic Mapping from Criticality to Self Organized Criticality Typeset using REVT E X 1", "Algorithmic Mapping from Criticality to Self Organized Criticality Typeset using REVT E X 1" ]	[ "F Bagnoli ", "P Palmerini ", "R Rechtman ", "\nDipartimento di Matematica Applicata\nDipartimento di Fisica\nUniversità di Firenze\nvia S. Marta3 I-50139FirenzeItaly\n", "\nCentro de Investigación en Energía\nUniversità di Firenze\nLargo E. Fermi, 2, I50125FirenzeItaly\n", "\nTemixco Mor\nUNAM\nApdo. Postal 3462850Mexico\n" ]	[ "Dipartimento di Matematica Applicata\nDipartimento di Fisica\nUniversità di Firenze\nvia S. Marta3 I-50139FirenzeItaly", "Centro de Investigación en Energía\nUniversità di Firenze\nLargo E. Fermi, 2, I50125FirenzeItaly", "Temixco Mor\nUNAM\nApdo. Postal 3462850Mexico" ]	[]	Probabilistic cellular automata are prototypes of non equilibrium critical phenomena. This class of models includes among others the directed percolation problem (Domany Kinzel model) and the dynamical Ising model. The critical properties of these models are usually obtained by fine-tuning one or more control parameters, as for instance the temperature. We present a method for the parallel evolution of the model for all the values of the control parameter, although its implementation is in general limited to a fixed number of values. This algorithm facilitates the sketching of phase diagrams and can be useful in deriving the critical properties of the model. Since the criticality here emerges from the asymptotic distribution of some quantities, without tuning of parameters, our method is a mapping from a probabilistic cellular automaton with critical behavior to a self organized critical model with the same critical properties. 64.60.Ht, 64.60.Lx, 05.50.+q Typeset using REVT E X	10.1103/physreve.55.3970	[ "https://arxiv.org/pdf/cond-mat/9605066v3.pdf" ]	18,062,728	cond-mat/9605066	1799d6aa67b3cfe08ffe70290a0df63878f03ac3	Algorithmic Mapping from Criticality to Self Organized Criticality Typeset using REVT E X 1 3 Dec 1996 F Bagnoli P Palmerini R Rechtman Dipartimento di Matematica Applicata Dipartimento di Fisica Università di Firenze via S. Marta3 I-50139FirenzeItaly Centro de Investigación en Energía Università di Firenze Largo E. Fermi, 2, I50125FirenzeItaly Temixco Mor UNAM Apdo. Postal 3462850Mexico Algorithmic Mapping from Criticality to Self Organized Criticality Typeset using REVT E X 1 3 Dec 1996(December 3, 2017) Probabilistic cellular automata are prototypes of non equilibrium critical phenomena. This class of models includes among others the directed percolation problem (Domany Kinzel model) and the dynamical Ising model. The critical properties of these models are usually obtained by fine-tuning one or more control parameters, as for instance the temperature. We present a method for the parallel evolution of the model for all the values of the control parameter, although its implementation is in general limited to a fixed number of values. This algorithm facilitates the sketching of phase diagrams and can be useful in deriving the critical properties of the model. Since the criticality here emerges from the asymptotic distribution of some quantities, without tuning of parameters, our method is a mapping from a probabilistic cellular automaton with critical behavior to a self organized critical model with the same critical properties. 64.60.Ht, 64.60.Lx, 05.50.+q Typeset using REVT E X I. INTRODUCTION Recently, several papers [1][2][3][4][5] have appeared discussing the relations between self organized criticality (SOC) [6] and usual critical phenomena. Some of them [5] stress the fact that one can reformulate classic critical systems (namely, directed percolation) in a way indistinguishable from SOC, while others [1] focus on the role of the control and order parameters. We started our investigation from the observation [7] that one can express the problem of directed site and bond percolation [8] in a form reminiscent of the invasion percolation process [9] or the Bak-Sneppen self organized model [10]. The advantage of this formulation is that the critical value of the percolation probability does not need to be adjusted carefully, but instead emerges from the probability distribution of a set of continuous variables, while the original model is defined in terms of Boolean variables. The directed percolation problem can be formulated in terms of probabilistic cellular automata (PCA) [11]. PCA are very general models that include for instance the kinetic Ising model [12]. In this paper we show how any critical PCA may be mapped into a SOC model. The mapping is presented constructively in Sec. II. It can also be considered as a multi-site coding technique [13], particularly adapted to probabilistic systems (where usual multisite performs badly). From a computational point of view, this algorithm allows a quick determination of phase diagrams and computation of critical properties. In Sec. III we apply this method to the study of the Domany-Kinzel model of directed percolation and to the two dimensional Ising model. On the other hand, the mapping implies that to each PCA corresponds a SOC model defined in a high or infinite dimensional space. This correspondence can give some insight in the nature of the SOC phase, as addressed in Sec. IV. We end with some conclusions and perspectives. II. THE FRAGMENT METHOD We deal with probabilistic cellular automata, i.e., discrete models defined on a lattice. Let us consider explicitly the one dimensional Boolean case. A configuration at time t + 1 is obtained from the configuration at time t by applying in parallel a probabilistic rule to each site. The rule is implemented on a computer by comparing (pseudo) random numbers with a certain number of fixed parameters (probabilities). One can think of PCA as the evolution of a deterministic discrete system on a random quenched field (the set of random numbers). For simplicity, we refer to the directed site percolation problem in 1+1 dimensions, where the higher p, the higher the probability of percolating. In this case one can visualize the random field as the height of a corrugated landscape, and p as the water level. There will be percolation if the water is able to percolate on the corrugated landscape, i.e., if the plane at height p is not completely blocked (in this directed model water is forbidden to backpercolate). For each value of p, we denote with a one the sites that are wet, and with a zero those that are dry. One can stack a set of planes, and let them evolve in parallel. We can read the state of a certain site for all values of p as a vector of ones and zeroes, each component being labeled by p. In the initial configuration sites are either wet or dry, independently of p. Thus, all vectors are either filled with ones or with zeroes. Going on with the percolation process, a component p of the vector at a certain site and time t + 1 will be wet if there is at least a wet component at the same height among its neighbors at time t, and if the height of the random field at that position is less than p. One can easily express this in computer language. Each component corresponds to a bit in a computer word. With words of n bits, p can assume the values 0/n, 1/n, . . . , i/n, . . . (n − 1)/n. The (bitwise) OR of the words in the neighborhood gives a one for all values of p for which there is at least one wet neighbor. Given a random number r in that site, all planes with p > r have the possibility of percolating. This is expressed in computer terms by taking a word R(r) filled with zeroes up to a fraction r of bits and then with ones, and performing the AND of R(r) with the previous word. Iterating this procedure, we get in the last line of the lattice (say at time T ) a set of partially filled words. If at time T a word has the bit number k equal to one, this means that for p = k/n, water would have percolated to that site (given the set of random numbers). The procedure can be generalized to words of arbitrary length. In the limit n → ∞, the Boolean vectors become the characteristic functions of subsets of the unit interval, that we call fragments. The manipulation of fragments is not limited to this bitwise implementation, as we shall see in the following. The fragment expressions do not depend explicitly on the control parameter p. The critical value of p and the critical scaling law of the order parameter are obtained a posteriori , from the distribution of fragments. Let us now formalize these concepts. For simplicity we refer to the Domany-Kinzel (DK) model [11], which is a simple one dimensional PCA. We denote with x t i = 0, 1 the state of a site i at time t, i = 0, . . . , L − 1, with L the size of the lattice. We shall simplify x ′ = x t+1 i , x ± = x t i±1 . All space index operations are modulo L (periodic boundary conditions). The evolution rule may be written as x ′ = [r < p](x − ⊕ x + ) ∨ [r < q]x − x +(1) where ⊕ represents the eXclusive OR operation (sum modulus two), ∨ the OR operation, and the multiplication (or ∧) stands for the AND operation, with the usual priority rules. The control parameters p and q are fixed, and r = r t i is a random number uniformly distributed between zero and one, and where [logical expression] is one if logical expression is true and zero otherwise [14,15]. In the case of directed site percolation p = q, Eq. (1) can be rewritten as x ′ (p) = [r < p](x − (p) ∨ x + (p)).(2) where we emphasize the dependence of x on p. The fragment approach consists in reading x t i (p) as the value of the characteristic function of the fragment X t i at p. The expression [r < p] is the characteristic function of a fragment R(r) = [r, 1). Equation (2) in terms of fragments is X ′ = R(r)(X − ∨ X + )(3) that does not depend on p. The Boolean functions AND, OR, XOR, NOT correspond to the set operations intersection, union, symmetric difference and complement, respectively. We shall use the same symbol for Boolean and set operations. The initial configuration is independent of p, this means that X 0 i are either the empty set or the unit interval, according with x 0 i . Applying the set operations we obtain the asymptotic fragments X ∞ i . For a given value of p, x t i (p) is one if the point p belongs to the fragment X t i and zero otherwise. Thus we can obtain the asymptotic value of x t i (p) from the asymptotic fragments X t i , that evolved without an explicit dependence on p. If some function of the x t i exhibits a phase transition in correspondence of a critical value p c , this behavior can be extracted from the asymptotic fragments. For instance, the density ρ ρ(p) = 1 L L i=1 x T i (p)(4) is proportional to the number of fragments the point p belongs to, ρ(p) = 1 L L i=1 [p ∈ X T i ].(5) The above procedure can be applied to all probabilistic cellular automata. The practical recipe for the implementation is PCA: Express the model as a probabilistic cellular automaton whose evolution rule only uses Boolean expressions, and convert the control parameters (p 1 , p 2 , . . . , p m ) to expressions like [r k < p k ] where p k appear alone on the right side. Fragments: Replace the variables x t i with fragments X t i ⊆ [0, 1) m , and substitute [r k < p k ] with R(r k ) ([r k > p k ] with its complement R(r k )). The initial configuration x 0 i is replaced by X 0 i = R(x 0 i ). Implementation: Implement the fragments as arrays of bits (the simplest approach) or as sparse vectors (see later) and iterate the rule. Criticality: The asymptotic distribution of fragments gives the critical properties (control parameters and exponents) of the original model. Let us illustrate separately each of the previous points. A. PCA The evolution rule is generally expressed by means of transition probabilities. Note, however, that the transition probabilities do not completely characterize the problem for the damage spreading transitions [16], since there are many ways of actually implementing the probabilistic choices in a computer code. The general approach for deriving a Boolean expression from transition probabilities is to write formally the future value of the dynamical variable (the spin) as a function of the spins in the neighborhood and of the transition probabilities converted to random Boolean variables, i.e., x ′ = f (x − , x + , . . . , [r < p 1 ], [r < p 2 ], . . .). Then there are several ways [18,19] of expressing a Boolean function using a set of standard Boolean operations like AND, OR, XOR, NOT. Clearly, one should spend some effort in reducing the length of the resulting expression. Sometimes (see the Ising model in Sec. III) one has to transform from [r < f (p)] to something like [f −1 (r) < p] (or more complex expressions). B. Fragments The method can be applied to any number of parameters. In the case of m parameters p 1 , . . . , p m the fragments are subsets of the m-dimensional unit hypercube. For instance, in the general DK model, Eq. (1), there are two control parameters (p and q) and the fragments are subset of the unit square. In this way it is possible to draw a sketch of a phase diagram, in just one simulation. However, if one is interested in crossing the critical surface along one line (see the computation of critical exponents in Sec. II D), one has to express the parameters as functions of a single variable, say s, and transform the expressions accordingly. For instance, the directed bond percolation problem corresponds to the curve q = p(2 − p), which can be expressed as p = s and q = s(2 − s). The corresponding expression is x ′ = [r < s](x − ⊕ x + ) ∨ [1 − √ 1 − r < s](x − x + )(6) which gives the following fragment expression X ′ = R(r)(X − ⊕ X + ) ∨ R(1 − √ 1 − r)(X − X + ).(7) C. Implementation The simplest way of implementing a fragment on a computer is by means of an array of n bits and using bitwise Boolean operations. Generally one uses computer words (32 or 64 bits) for efficiency, but it is possible to use multiple words to increase the sampling frequency of probability. The numerical advantage over other multi-spin approaches [13] is the use of just one random number for all the n simulations. Referring to a p-layer as a cut of the fragment space-time configuration with a given value of p, we see that the layers at different p are not independent, since they use the same random numbers. The influence of these correlations is discussed in Sec. II D. One can increase the sampling frequency around the region of interest (for instance the critical region) by appropriately defining the correspondence of the bits with the values of p. This affects the way R(r) is implemented. With the fragment method it is still possible to perform simulations starting from a single site (Grassberger method) keeping track of nonzero fragments. The method is powerful if one uses a small interval around the critical point, so that all clusters for various p are similar. When using two parameters (say p and q) one has to implement differently expressions like [r < p] (fill the unit square in the p direction from r to 1) from [r < q] (fill the unit square in the q direction from r to 1). The alternative approach in representing fragments consists in keeping track of the starting and ending points of all segments that form a one dimensional fragment. The approach is very similar to the treatment of sparse matrices, so we call it the sparse fragment method. The rules of combining sparse fragments are more complex than above. On the other hand in this way one has infinite precision, which comes useful in finding the critical behavior as explained in the Sec. II D. In general a fragment is formed by just one segment if the evolution rule can be expressed using only AND and OR, while for instance the XOR between two overlapping fragments causes holes. As an example, the site percolation rule Eq. (2) can be implemented as sparse fragments by considering the evolution of the lower extremum a = a t i of the segments [a, 1] as [20,1,5] a ′ = max(r, min(a − , a + )). Sometimes the problem can be reformulated without XOR operations. For instance, the bond percolation problem (6) can be rewritten as [5,20,21] x ′ = ([r − < p]x − ) ∨ ([r + < p]x + )(9) with two random numbers per site. The evolution of this rule can be easily implemented using sparse fragments. D. Criticality The critical properties of the original model are obtained from the asymptotic distribution of fragments. The fragment method introduces strong correlations among p-layers as also noted in Ref. [5]. One can exploit these correlations considering differences in the p direction. If the patterns for different p-layers have similar sizes, the fluctuations cancel out. This happens in general if the rule does not contain XOR (see Fig. 1). On the other hand, the XOR generally implies strong variations of clusters with p, so that the fluctuations can in principle be wider than uncorrelated simulations (see for instance Fig. 2). A powerful method for the computation of critical quantities exploits the scaling relation m(p, t) = α −β/ν m(α 1/ν (p − p c ) + p c , αt)(10) numerically solving it for the unknown β, ν and p c . This is an easy task for the sparse fragments approach, since one can obtain m(p, t) and m(p ′ , αt) with p ′ = α 1/ν (p − p c ) + p c for each value of ν and p c . For the bit approach one has to compute the value of the exponents and p c so to make all data collapse on a single (smooth) curve. This can be performed near p c approximating the curves with polynomials (or any other fitting function), and minimizing the χ 2 of the regression. III. APPLICATIONS In this section we show some results of the fragment method applied to classical problems: the determination of the phase diagram and critical properties of the Domany-Kinzel model, and the two-dimensional Ising model. The first example is the one-dimensional directed site percolation, i.e., the line q = p of the DK model, Eq. (2). A snapshot of part of the asymptotic fragment configuration is shown in Fig. 1, with the plot of the density ρ(p). As illustrated above, if one computes the XOR dilution (the line q = 0 of the DK model, Eq. (1)), the fragments decompose into several segments, as shown in the inset of Fig. 2. Correspondingly, the integrated density converges slowly to a smooth curve. The complete phase diagram of the DK model can be obtained in just one simulation by iterating two-dimensional fragments. The plot of the asymptotic density ρ(p, q) is shown in Fig. 3. It compares well with those obtained with other methods [11,16,17]. From the convergent behavior of the contour lines, the position (p = 1/2, q = 1) of the discontinuous transition for the density is clearly indicated. Near the corner (p = 1, q = 0) the surface becomes irregular: this is due to the prevalence of the XOR in eq. (1). One can also investigate the chaotic phase of the DK model by iterating two fragment configurations with the same random numbers. The Hamming distance between two replicas with a given value of p and q is the (p, q)-component of the density of the XOR between the two asymptotic fragment configuration. A plot of the resulting Hamming distance is shown in Fig. 4. Here one can notice a trace of the density phase boundary (near (p = 0.8, q = 0)), due to the critical slowing down. For the directed site percolation problem we found p c = 0.7055(4), β = 0.21065(5), ν = 1.7195(5) for a system of size 10 6 , in the interval 0.7 < p < 0.71 for different values of α = 1024, 2048, 4096, 8192. The agreement with previous measurements [8] is satisfactory. Moreover, we want to stress that these values were obtained with data coming from simulations of less than 20 minutes of CPU time on a 150 MHz PC running Linux [22]. As a second application, we consider now the kinetic version of an equilibrium system, the two dimensional Ising model with heat bath dynamics [23,24]. In Appendix A we show how to express the evolution equation of this model as a totalistic PCA, and how to translate its evolution in fragment language. In Fig. 5 we show the plot of the magnetization m(p) with respect to p = exp(−2J) for an Ising model with reduced interaction constant J. The transition is well characterized by plotting the second moment (standard deviation) of the magnetization as a function of p. We found p c = .172 ± 0.002 and β = .11 ± .002, in good agreement with the exact values p c = ( √ 2 − 1) 2 and β = 1/8. IV. CRITICALITY AND SELF-ORGANIZED CRITICALITY We have shown how the DK model and the Ising model, which are PCA with critical behavior, may be mapped into fragment models with no control parameter, that is models that show self organized criticality (SOC). It is evident that the fragment method may be applied to any critical PCA. This result is summarized in the diagram of Fig. 6. The state x t i (p) may be obtained by evolving the PCA with a given p (labeled by f p in the diagram) or by building the fragments X 0 i and evolving them with the fragment method (F in the diagram) that does not depend on p. Finally, by probing X t i with a p-layer, that is by checking if X t i extends down to p, we recover x t i . Although it is easier to think of one dimensional fragments, this result is valid for any number of control parameters. It is interesting to describe a "traditional" SOC model with the fragment language, trying to obtain the p-layer description that would make the SOC model correspond to a usual critical model. Let us discuss the one dimensional Bak-Snappen model [10] with nearest neighbors interactions. In this model one starts from an array of real numbers a i , i = 1, . . . , L uniformly distributed in the unit interval. One looks for the minimum of a i and replaces it and its nearest neighbors with newly generated random numbers, again uniformly distributed in the unit interval. The system auto-organizes so that the distribution of a i follows a powerlaw (with exponent 1), with a non-trivial avalanche distribution. We define now a parallel version of the previous model (which is not very efficient from a computational point of view). For the sake of simplicity, we divide the discussion in two parts: the research of the minimum and the actual evolution. Let us visualize the a i as the lower extremum of segments (fragments) X i , and cut the configuration with a line at height p. The minimum is localized at site k for which there is only one intersection. It can be expressed using a Boolean variable δ t i,k (a Kronecker δ) δ t i,k = p x t i (p)   j =i x t j (p)   ,(11) assuming that the minimum is unique in the continuous p limit. The fragments X i at and nearest to the minimum are replaced by segments of random length (R(r)) X t+1 i = X t i ⊕ (∆ t i−1,k ∨ ∆ t i,k ∨ ∆ t i+1,k )(R(r t i ) ⊕ X t i )(12) where the fragment ∆ t i,k is completely filled if δ t i,k = 1 and completely empty if ∆ t i,k = 0. The evolution can be expressed on a p-layer as x t+1 i = x t i ⊕ (δ t i−1,k ∨ δ t i,k ∨ δ t i+1,k )([r t i < p] ⊕ x t i ).(13) Similar but more complex expressions can be found for the invasion percolation process. One can see that in these "traditional" SOC models there are long range space interactions, and also interactions among p-layers (see Eq. (11)). We think that the second ingredient is the most important: if one knows how some quantity like the density varies with p it is not difficult to imagine a mechanism that automatically reaches the critical point. It is still to be proved that this is the actual mechanism of SOC. On the other hand, Eq. (8) shows that there exists space and p-local mechanisms that can be classified as SOC. V. CONCLUSIONS The fragment method can be considered both as a recipe for numerical studies of phase diagrams and as a mapping from criticality to self organized criticality. For what concerns the first topic, the possibility of having a sketch of the phase diagram without huge computation resources is useful in determining the position of the critical line. Numerical applications of the fragment method will be presented in future work. From the theoretical point of view, we think that the formalism presented in this work allows a clear characterization of the basic properties of self organized models, suggesting analogies between usual critical phenomena and self organized ones. ACKNOWLEDGMENTS We wish to acknowledge fruitful discussions with P. Grassberger, and R. Bulajich. Partial economic support from CNR (Italy), CONACYT (Mexico), Project DGAPA-UNAM IN103595, Centro Internacional de Ciencias A.C., the Programma Vigoni of the Conferenza permanente dei rettori delle università italiane and the workshop Chaos and Complexity at ISI-Villa Gualino under the CE contract n. ERBCHBGCT930295 is also acknowledged. APPENDIX: THE ISING MODEL Let us start by considering the one dimensional Ising model. Its reduced Hamiltonian can be written as H(x) = − J 2 L−1 i=0 σ i σ i+1 (A1) where σ i = 2x i − 1, and x i = 0, 1. We choose the heat bath dynamics, [23,24] for which the probability τ (x → y) of going from a configuration x to a configuration y that can differ from x in a certain number {i k } of sites is τ (x → y) = exp(−H(y)) ′ exp(−H(y ′ ))) (A2) where the sum in the denominator extends over all combinations of the differing sites y i k . This transition probability does not depend on x and satisfies the detailed balance principle. The configuration y is not limited to differ from x only at one site: the evolution can be applied in parallel changing all even (or odd) sites. Since the transition probabilities do not depend on the previous value of the cell, the space-time lattice decouples into two noninteracting sublattices: one with even sites at even times and odd site at odd times, and the complementary one. By considering only one sublattice, the neighborhood of the one dimensional Ising model is the same of the Domany-Kinzel model. The kinetic Ising model is just a totalistic cellular automaton (without adsorbing states). The local transition probabilities τ (x i−1 , x i+1 → y i ) can be computed from Eq. (A2) considering a difference in just one site. They are τ (0, 0 → 1) = p/(1 + p) τ (0, 1 → 1) = 1/2 τ (1, 0 → 1) = 1/2 τ (1, 1 → 1) = 1/(1 + p) with p = exp(−2J). It is convenient to introduce here the totalistic functions c k that take the value one if the sum of the variables in the neighborhood is k and zero otherwise. An efficent way of building these functions is described in [19]. For the Domany-Kinzel neighborhood they are c 0 = x − ∨ x + c 1 = x − ⊕ x + c 2 = x − x + . The evolution equation for the Ising cellular automaton is x ′ = [r < p/(1 + p)]c 0 ∨ [r < 1/2]c 1 ∨ [r < 1/(1 + p)]c 2 .(A3) Before using the fragment method one has to invert the [r < f (p)] ([r > f (p)]) expressions and substitute them with R(f −1 (r)) (R(f −1 (r))); p-independent expressions like [r < 1/2] trasform to R(0) or R(0) according with r. We leave them in the equations with the assumption that a true value means R(0) and a false value means R(0). We finally find that X ′ = R r 1 − r C 0 + [r < 1/2]C 1 + R 1 − r r C 2 . (A4) where now the C k are fragments. For the 2D square Ising model with heat bath dynamics, the Hamiltonian is H(x) = − J 2 L−1 i,j=0 σ i,j σ i+1,j + σ i,j σ i,j+1(A5) with σ i,j = 2x i,j − 1. The lattice decouples again in two noninteracting sublattices. Repeating the procedure as above, one has again a totalistic cellular automaton. Using the totalistic functions C k of four nearest neighbors, we find that X ′ = R r 1 − r C 0 ∨ R r 1 − r C 1 ∨ r < 1 2 C 2 ∨R 1 − r r C 3 ∨ R   1 − r r   C 4 (A6) where p = exp(−2J). The C k functions can be computed efficiently using the homogeneous polynomials D j [19] C 0 = C 1 ∨ C 2 ∨ C 3 ∨ C 4 C 1 = D 1 ⊕ D 3 C 2 = D 2 ⊕ D 3 C 3 = D 3 C 4 = D 4 where D 1 = X ++ ⊕ X +− ⊕ X −+ ⊕ X −− D 2 = X ++ X +− ⊕ (X ++ ⊕ X +− ) (X −+ ⊕ X −− ) ⊕ X −+ x −− D 3 = D 2 D 1 D 4 = X ++ X +− X −+ X −− . FIGURES FIG. 1. The density ρ vs. the control parameter p for the directed site percolation problem, Eq. (2). The inset shows asnapshot of the first 40 segments X t , Eq. (3) after t time steps.. One simulation with L = 320 and t = 1000. The resolution is 480 bits. FIG. 2. The density ρ vs. the control parameter p for the XOR dilution, Eq. (1) with q = 0. The inset shows a snapshot of the first 40 segments X t after t time steps.. One simulation with L = 320 and t = 1000. The resolution is 480 bits. Notice that the simulation reproduces well also the point p = 1, for which ρ = 0. FIG. 3 . 3The contour plot of ρ(p, q) of the Domany-Kinzel Model, Eq. (1), for a lattice of L = 2000 sites and t = 4000. The resolution is 128 × 128 bits. White corresponds to ρ = 0 and the contour lines are drawn at 0.1 intervals. FIG. 4. The contour plot of the asymptotic value of the distance between two replicas of the system evolving under the same realization of the noise. The parameters are those of the previous figure. FIG. 5. The magnetization m and the susceptibility V ar(m) (fluctuation of the magnetization) for a 2-dimensional Ising model of size L = 100 × 100, averaging over 10 samples every 1000 time steps after a transient of 10000 time steps. The resolution is 32 bits. FIG. 6. The diagram showing the mapping from criticality to self organized criticality. Figure 4 figure 6 ; Di Firenze, Drecam-Spec, Saclay, e-mail:[email protected] * * On leave from Facultad de Ciencias. Sur-Yvette Cedex, France; UNAM, Mexicoalso INFN and INFM sez. di Firenze; DRECAM-SPEC, CEA Saclay, 91191 Gif-Sur- Yvette Cedex, France; e-mail:[email protected] * * On leave from Facultad de Ciencias, UNAM, Mexico. . D Sornette, A Johansen, I Dornic, J. Phys. I (France). 5325D. Sornette, A. Johansen, and I. Dornic, J. Phys. I (France) 5, 325 (1995). . S Maslov, Y.-C Zhang, Phys. Rev. Let. 751550S. Maslov and Y.-C. Zhang, Phys. Rev. Let. 75, 1550 (1995). . M Paczuski, S Maslov, P Bak, Phys. Rev. E. 53414M. Paczuski, S. Maslov, and P. Bak, Phys. Rev. E 53, 414 (1996). . D Sornette, I Dornic, Phys. Rev. E. 543334D. Sornette and I. Dornic, Phys. Rev. E 54, 3334 (1996). . P Grassberger, Y.-C Zhang, Physica A. 224169P. Grassberger and Y.-C. Zhang, Physica A 224, 169 (1996). One can find an extended listing of articles on several aspects of SOC in the previuosly cited papers. P Bak, C Tang, K Wiesenfeld, Phys. Rev. Lett. 59381P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett 59, 381 (1987). One can find an extended listing of articles on several aspects of SOC in the previuosly cited papers. The original presentation occurred at ISI-Villa Gualino during the workshop. Chaos and Complexity. See also refs. 20,1,5The original presentation occurred at ISI-Villa Gualino during the workshop Chaos and Complexity (Torino, Italy 1994). See also refs. [20,1,5]. . J W Essam, A J Guttmann, K De'bell, J. Phys. A. 163815J.W.Essam, A.J. Guttmann, and K. de'Bell, J. Phys. A 16, 3815 (1983). . D Wilkinson, J F Willemsen, J. Phys. A: Math. Gen. 163365D. Wilkinson and J.F. Willemsen, J. Phys. A: Math. Gen. 16, 3365 (1983). . P Bak, K Sneppen, Phys. Rev. Lett. 714083P. Bak and K. Sneppen, Phys. Rev. Lett. 71, 4083 (1993). . E Domany, W Kinzel, Phys. Rev. Lett. 53229W. KinzelZ. Phys. BE. Domany and W. Kinzel, Phys. Rev. Lett. 53, 311 (1984). W. Kinzel, Z. Phys. B 58, 229 (1985). . A Georges, P Le Doussal, J. Stat. Phys. 541011A. Georges and P. Le Doussal, J. Stat. Phys. 54, 1011 (1989). . H J Herrmann, J. Stat. Phys. 2145H. J. Herrmann, J. Stat. Phys. 2, 145 (1991); . D Stauffer, J. Phys. A: Math. Gen. 24691D. Stauffer, J. Phys. A: Math. Gen. 24 L691 (1991). A Programming Language. K Iverson, WileyNew YorkK. Iverson, A Programming Language, (Wiley, New York, 1962). R L Graham, D E Knuth, O Patashnik, Concrete Mathematics. New YorkAddison WesleyR. L. Graham, D. E. Knuth, and O.Patashnik, Concrete Mathematics, (Addison Wesley, New York, 1989). . F Bagnoli, J. Stat. Phys. 85151F. Bagnoli, J. Stat. Phys. 85, 151 (1996). . F Bagnoli, R Bulaijch, R Livi, A Maritan, J. Phys. A:Math. Gen. 251071F. Bagnoli, R. Bulaijch, R. Livi, and A. Maritan, J. Phys. A:Math. Gen. 25, L1071 (1992). I Wegener, The Complexity of Boolean Functions. New YorkWileyI. Wegener, The Complexity of Boolean Functions (Wiley, New York, 1987). . F Bagnoli, Int. J. of Mod. Phys. C. 3307F. Bagnoli, Int. J. of Mod. Phys. C, 3, 307 (1992). . S Roux, A Hansen, E Guyon, J. Phys. 482125S. Roux, A. Hansen, and E. Guyon, J. Phys. (Paris) 48, 2125 (1987). . T Halpin-Healy, Y.-C Zhang, Phys. Rep. 254215T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. 254, 215 (1995). . A M Maritz, H L Herrmann, L De Arcangelis, J. Stat. Phys. 591043A. M. Maritz, H. L. Herrmann, and L. de Arcangelis, J. Stat. Phys. 59, 1043 (1990). . F Wang, M Suzuki, Physica A. 220534F. Wang and M. Suzuki, Physica A 220, 534 (1995).	[]
[ "Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions", "Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions" ]	[ "Markus Penz \nMax Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway\n", "Michael Ruggenthaler \nMax Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway\n", "Andre Laestadius \nMax Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway\n", "Erik I Tellgren \nMax Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway\n" ]	[ "Max Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway", "Max Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway", "Max Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway", "Max Planck Institute for the Structure and Dynamics of Matter\nDepartment of Chemistry\nHylleraas Centre for Quantum Molecular Sciences\nUniversity of Oslo\nHamburgGermany, Norway" ]	[]	The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions with Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown to converge to the correct ground-state density.The ubiquitous Kohn-Sham (KS) scheme [1] of ground-state density-functional theory (DFT) is the cornerstone of electronic structure calculations in quantum chemistry and solid-state physics. This scheme maps a complicated system of interacting electrons to an auxiliary, non-interacting Kohn-Sham system. Thereby it gives a set of coupled one-particle equations that need to be solved self consistently. Since a direct solution is unfeasible, practical approaches are variations of selfconsistent field methods taking the form of fixed-point iterations or of energy minimization algorithms[2][3][4][5][6][7]. To date, no iterative method has been rigorously shown to converge to the correct ground-state density. Convergence results for approximate schemes are available [8], but reliably achieving convergence in systems with small band gaps or for transition metals remains a hard practical challenge[9]. Hence, a method with mathematically guaranteed convergence is of central importance.An early insight is that iterations commonly fail unless oscillations between trial states are damped[2]. Work by Cancès and Le Bris[10,11]lead to the optimal damping algorithm (ODA) based on energy minimization by linesearch along a descent direction. Wagner et al.[12,13]presented a similar scheme and claimed to have proven convergence in the setting of exact DFT, while only the strict descent of energies was secured. This issue was addressed in Laestadius et al. [14] where a similar iterative scheme was proposed that proved a weak type of convergence after Moreau-Yosida (MY) regularization to ensure differentiability of the universal Lieb functional[15]. Such a regularized DFT has been previously suggested by Kvaal et al. [16]. Weak-type convergence here means that the energy converges to an upper bound for the exact energy. A rich study of possible strategies for self-consistent field iteration was recently put forward by Lammert[17]. Yet in all those works the question of a limit density and corresponding KS potential was still left open. On the other side, the result in Laestadius et al.[14]is fully ap-plicable to not only the setting of standard DFT, but to all DFT-flavors that fit into the given framework of reflexive Banach spaces. It has already been successfully applied to paramagnetic current DFT (CDFT)[18]. This general approach that reaches beyond standard DFT is followed also in this letter.	10.1103/physrevlett.123.037401	[ "https://arxiv.org/pdf/1903.09579v1.pdf" ]	85,459,163	1903.09579	b90ff3ec110d612add32ff070ebaacba9f873692	Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions 22 Mar 2019 Markus Penz Max Planck Institute for the Structure and Dynamics of Matter Department of Chemistry Hylleraas Centre for Quantum Molecular Sciences University of Oslo HamburgGermany, Norway Michael Ruggenthaler Max Planck Institute for the Structure and Dynamics of Matter Department of Chemistry Hylleraas Centre for Quantum Molecular Sciences University of Oslo HamburgGermany, Norway Andre Laestadius Max Planck Institute for the Structure and Dynamics of Matter Department of Chemistry Hylleraas Centre for Quantum Molecular Sciences University of Oslo HamburgGermany, Norway Erik I Tellgren Max Planck Institute for the Structure and Dynamics of Matter Department of Chemistry Hylleraas Centre for Quantum Molecular Sciences University of Oslo HamburgGermany, Norway Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions 22 Mar 2019 The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions with Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown to converge to the correct ground-state density.The ubiquitous Kohn-Sham (KS) scheme [1] of ground-state density-functional theory (DFT) is the cornerstone of electronic structure calculations in quantum chemistry and solid-state physics. This scheme maps a complicated system of interacting electrons to an auxiliary, non-interacting Kohn-Sham system. Thereby it gives a set of coupled one-particle equations that need to be solved self consistently. Since a direct solution is unfeasible, practical approaches are variations of selfconsistent field methods taking the form of fixed-point iterations or of energy minimization algorithms[2][3][4][5][6][7]. To date, no iterative method has been rigorously shown to converge to the correct ground-state density. Convergence results for approximate schemes are available [8], but reliably achieving convergence in systems with small band gaps or for transition metals remains a hard practical challenge[9]. Hence, a method with mathematically guaranteed convergence is of central importance.An early insight is that iterations commonly fail unless oscillations between trial states are damped[2]. Work by Cancès and Le Bris[10,11]lead to the optimal damping algorithm (ODA) based on energy minimization by linesearch along a descent direction. Wagner et al.[12,13]presented a similar scheme and claimed to have proven convergence in the setting of exact DFT, while only the strict descent of energies was secured. This issue was addressed in Laestadius et al. [14] where a similar iterative scheme was proposed that proved a weak type of convergence after Moreau-Yosida (MY) regularization to ensure differentiability of the universal Lieb functional[15]. Such a regularized DFT has been previously suggested by Kvaal et al. [16]. Weak-type convergence here means that the energy converges to an upper bound for the exact energy. A rich study of possible strategies for self-consistent field iteration was recently put forward by Lammert[17]. Yet in all those works the question of a limit density and corresponding KS potential was still left open. On the other side, the result in Laestadius et al.[14]is fully ap-plicable to not only the setting of standard DFT, but to all DFT-flavors that fit into the given framework of reflexive Banach spaces. It has already been successfully applied to paramagnetic current DFT (CDFT)[18]. This general approach that reaches beyond standard DFT is followed also in this letter. In what follows we give a fully rigorous proof of convergence for the exact KS iteration scheme in a finitedimensional state space. Due to the techniques involved, the new iteration scheme was baptized "MYKSODA" in Laestadius et al. [18]. The version given here includes a different, more conservative damping step that helps to prove convergence. To distinguish the two versions, we denote them "MYKSODA-S" for shorter, conservative steps, and "-L" for the original longer steps. The selection of the damping critically depends on MY regularization that makes the curvature of the universal Lieb functional bounded above. The convergence speed is increased by a larger regularization factor ε which corresponds to a smaller allowed curvature. The reason why a large ε is not desirable lies in numerical accuracy and the availability of good approximation functionals. We will now present the mathematical framework. For a much more detailed discussion of generalized KS iteration schemes in Banach spaces that can also be infinite dimensional we refer to Laestadius et al. [14]. The spaces for densities and potentials are chosen to be the Hilbert space X = X * = 2 (M ), M ∈ N, which corresponds to a finite one-particle basis, a lattice system with M sites, or many other possible settings. The reason for this dual choice of spaces is how densities and potentials couple in the energy expression. What is denoted R 3 vρ dx in standard DFT is v, ρ with ρ ∈ X, v ∈ X * in our generalized setting. For the internal energy (kinetic and interaction) of the full system a universal functionalF , like the one defined by constrained search [15,19] over all N -particle density matrices Γ that yield a given density ρ ∈ X, is introduced,F (ρ) = inf Γ →ρ {Tr((H kin + H int )Γ)} .(1) Here H kin stands for the kinetic energy contribution to the Hamiltonian and H int for interactions. Consequently the functionalF is defined on a setX ⊂ X of physical densities that come from an N -particle density matrix (ensemble N -representability). This setX will be assumed bounded in X. Since all physical densities are normalized in the 1 norm and all norms are equivalent in finite dimensions this follows naturally. It also holds for CDFT on a finite lattice, since the current density is bounded by the hopping parameter [20,Eq. (25)], and for one-body reduced density matrix functional theory (RDMFT) in finite basis sets, since the off-diagonal elements of the reduced density matrix are bounded by the diagonal ones that give the usual density [21,Eq. (3.49)]. On the other hand, elements in X will in general not constitute physical densities. In standard DFT this means that an arbitrary x ∈ X does not have to be normalized or even positive. Such an x ∈ X will thus be called a quasi-density. We keep the notation ρ to mark out physical densities. The total energy is the infimum of F (ρ) plus the the potential energy coming from a given external potential v ∈ X * , taken over all physical densities, E(v) = inf ρ∈X {F (ρ) + v, ρ }.(2) It is linked to a functional F on X by the Legendre-Fenchel transformation (convex conjugate). Then F can be transformed back to the same E as F (x) = sup v∈X * {E(v) − v, x },(3)E(v) = inf x∈X {F (x) + v, x }.(4) The functional F is by construction convex and lower semi-continuous [15,Th. 3.6] and has F (x) = +∞ whenever x is not in the domainX ofF . Minimizers of (2) are the ground-state densities, which establishes a link to the corresponding Schrödinger equation. They stay the same if one switches fromF to F and thus minimizers of (4) are always inX. Finding such minimizers of (4) is equivalent to determining the superdifferential of E, the set of all dual elements (quasi-densities) that yield a graph completely above E, ρ ∈ ∂E(v) ⊂X.(5) The MY regularization of the functional F on X is defined as F ε (x) = inf y∈X F (y) + 1 2ε x − y 2 .(6) The visual understanding of this is the following. As the vertex of the regularization parabola 1 2ε x 2 moves along the graph of F , the regularized F ε is given by the traced out lower envelope as visualized in Fig. 1. It also means the regularization puts an upper bound of ε −1 to the (positive) curvature of F ε . This will be an important ingredient in the convergence proof: A bound on the curvature means the convex functional F ε cannot change from falling to rising too quickly, yielding a secure bound on the possible step length for descent. The regularized F ε is then differentiable and even has a continuous gradient ∇F ε (Fréchet differentiability), something that will become important too in the convergence proof, if additionally X * is assumed uniformly convex [14,Th. 9]. Uniform convexity is trivially fulfilled in the cases of a finite dimensional, strictly convex Banach space X * or for any Hilbert space X = X * . With F ε we define the associated energy functional E ε (v) = inf x∈X {F ε (x) + v, x }.(7) The functional in (7) is not the MY regularization of E but the Legendre-Fenchel transformation of F ε . If z ∈ X is a minimizer in (7), called the ground-state quasidensity, then the gradient of F ε + v at z must be zero, ∇F ε (z) + v = 0.(8) Since the regularized functional is differentiable everywhere, the usual problem of (non-interacting) vrepresentability is avoided in the regularized setting. But E ε is still not differentiable, so we resort to the superdifferential like in (5). Since any such element z ∈ ∂E ε (v) automatically solves (8) it is the ground-state quasidensity of the regularized reference system for a potential v. Two important properties of E ε are [14, Th. 10 and Cor. 11] E(v) = E ε (v) + ε 2 v 2 ,(9)∂E(v) = ∂E ε (v) + εv ⊂X,(10) which relate the regularized system back to the unregularized one. Note that the ε in (10) takes a role comparable to that of permittivity, linking potentials to densities. Since E is already concave, the subtraction of a parabola in (9) makes E ε strongly concave. To set up a KS iteration scheme we define a reference system that is non-interacting bỹ F 0 (ρ) = inf Γ →ρ {Tr(H kin Γ)}(11) on the sameX ⊂ X and define E 0 , F 0 ε , and E 0 ε analogously. The analogue of (8) for F 0 ε at the same quasidensity z ∈ X is ∇F 0 ε (z) + v ε KS = 0,(12) and defines the KS potential v ε KS . Simply equating (8) and (12) gives ∇F ε (z) + v = ∇F 0 ε (z) + v ε KS ,(13) where the ground-state quasi-density z and the auxiliary KS potential v ε KS for the reference system are still unknown and neither F ε nor F 0 ε has a simple, explicit expression. The trick is to determine z and v ε KS in an iterative algorithm and to replace them by sequences x i → z, v i → v ε KS . The indicated convergence is our major concern in the following proof. We get an update rule for the potential sequence (v i ) i directly from (13), v i+1 = v + ∇F ε (x i ) − ∇F 0 ε (x i ),(14) and determine the next quasi-density by solving for the ground-state of the regularized reference system with v i+1 . This iteration has the stopping condition (14). Then x i is already the sought-after ground-state quasi-density z and thus also v i+1 = v ε KS is the respective KS potential yielding the same quasi-density for the reference system. v i+1 = −∇F 0 ε (x i ) which means v = −∇F ε (x i ) by The most important ingredient of practical KS calculations enters by giving suitable approximations for the expression ∇F ε −∇F 0 ε (Hartree-exchange-correlation (Hxc) potential including the correlated kinetic energy). For our purpose of showing convergence of exact KS theory we proceed as if this object is known. The KS algorithm is then the following: In step (a) get the new potential by (14) above. In step (b) solve the (simpler) ground-state problem for the reference system by choosing the next quasi-density from ∂E 0 ε (v i+1 ). From (10) it follows that ∂E 0 ε (v i+1 ) can be determined from the set of ground-state densities of the reference system, which just means solving the non-interacting Schrödinger equation. Finally, to be able to ensure a strictly descending energy and to show convergence of (x i ) i , (v i ) i , include a damping step (c) with an adaptively chosen step size. MYKSODA iteration scheme. AssumeX bounded and E 0 finite everywhere. For v ∈ X * fixed, set v 1 = v and select x 1 ∈ ∂E 0 ε (v). Iterate i = 1, 2, . . . according to: (a) Set v i+1 = v+∇F ε (x i )−∇F 0 ε (x i ) and stop if v i+1 = −∇F 0 ε (x i ) = v ε KS . (b) Select x i+1 ∈ ∂E 0 ε (v i+1 ) and get the iteration di- rection y i = (x i+1 − x i )/ x i+1 − x i . (c) Choose the step length τ i = −ε ∇F ε (x i )+v, y i > 0 and set x i+1 = x i + τ i y i . We prove below that this algorithm guarantees convergence to the correct KS potential, v i → v ε KS , and to the quasi-density, x i → z, of both, the full system with v and the reference system with v ε KS . The corresponding energy is then determined by E ε (v) = F ε (z) + v, z > −∞. These are still solutions of the regularized problem, but with (9) and (10) a transformation back to the unregularized setting is easily achieved. This, unlike the usually assumed unregularized KS iteration, gives different ground-state densities for the non-interacting and the interacting system, while circumventing all problems of differentiability and thus of v-representability. The assumption that E 0 is finite everywhere is trivially fulfilled in a finite-dimensional setting because E 0 is the outcome of a finite sum of numbers. It is still kept here to connect more closely to the setting of standard DFT and CDFT, where E 0 finite can be shown to hold even in the infinite-dimensional setting, see Lieb [15,Th Convergence proof. We refer to the the first part of the proof of Theorem 12 in Laestadius et al. [14] to show that the superdifferential ∂E 0 ε (v i+1 ) is everywhere non-empty because of E 0 finite, guaranteeing that the (regularized) ground-state problem in (b) always has at least one solution. The directional derivative of F ε +v at x i in direction x i+1 − x i can be rewritten by (a), [14,Lem. 4], we rewrite the right hand side of (15) with the help of (10), substitutingx ∇F ε (x i ) + v, x i+1 − x i = v i+1 + ∇F 0 ε (x i ), x i+1 − x i . (15) Realizing that x i+1 ∈ ∂E 0 ε (v i+1 ) from (b) and x i ∈ ∂E 0 ε (∇F 0 ε (x i )) from invertibilityi+1 = x i+1 + εv i+1 ∈ ∂E 0 (v i+1 ),(16)x i = x i + ε∇F 0 ε (x i ) ∈ ∂E 0 (∇F 0 ε (x i )),(17)which gives v i+1 + ∇F 0 ε (x i ),x i+1 −x i − ε v i+1 + ∇F 0 ε (x i ) 2 . (18) Now sincex i+1 ,x i are selected from the superdifferential of E 0 for the respective potentials v i+1 , ∇F 0 ε (x i ), the inner product is always smaller or equal to zero [14,Lem. 5]. This property is called monotonicity of ∂E 0 and directly follows from concavity of E 0 . What follows is strong monotonicity of E 0 ε , i.e., ∇F ε (x i ) + v, x i+1 − x i = v i+1 + ∇F 0 ε (x i ), x i+1 − x i ≤ −ε v i+1 + ∇F 0 ε (x i ) 2 = −ε ∇F ε (x i ) + v 2 .(19) The last line follows from (a) and is strictly smaller than zero if not ∇F ε (x i ) + v = 0, which means that we have already converged to the ground-state quasi-density. We thus infer that, unless converged, we always have a negative directional derivative of F ε + v at x i in the step direction y i which is parallel to x i+1 − x i , ∇F ε (x i ) + v, y i < 0.(20) Such a negative directional derivative means the left leg of the regularization parabola is aligned tangentially to the (differentiable) energy functional F ε + v, like it is depicted in Fig. 2. The next quasi-density step x i+1 = x i +τ i y i is then chosen at the vertex of this regularization parabola. This corresponds to a choice of step length τ i where the directional derivatives at x i in direction y i of the regularization parabola 1 2ε · −x i+1 2 and F ε + v are equal, which means x xi xi+1 x i+1 1 2ε x − xi+1 2 + mi Fε + v F + v ei mi ei+1∇F ε (x i ) + v, y i = − 1 ε x i+1 − x i , y i = − 1 ε x i+1 − x i = − τ i ε .(21) This construction yields a x i+1 = x i where the energy e i = F ε (x i ) + v, x i is always larger than the energy value m i at the vertex, see Fig. 2. Since the regularization parabola also lies fully above the energy functional F ε + v by construction, the energy e i+1 at x i+1 must obey e i+1 ≤ m i < e i . The strictly decreasing e i is now by definition bounded below by E ε (v) from (7) and thus converges. By determining e i − m i from the regularization parabola and then combining it with (21), τ 2 i 2ε = 1 2ε x i+1 − x i 2 = e i − m i ≤ e i − e i+1 → 0,(22) we can infer convergence of (x i ) i . Step (a) then defines an associated potential lim i→∞ v i+1 = v + ∇F 1 ε (x) − ∇F 0 ε (x),(23) since the gradients are both continuous. After having proved that the densities and potentials converge, it shall be demonstrated that they converge indeed to the expected ground-state quasi-density, z, and KS potential, v ε KS . We now come back to (19), where substituting x i+1 − x i = y i x i+1 − x i gives x i+1 − x i ∇F ε (x i ) + v, y i ≤ −ε ∇F ε (x i ) + v 2 ,(24) which together with (21) results in x i+1 − x i τ i ε ≥ ε ∇F ε (x i ) + v 2 .(25) We already know from the convergence of densities that (x i ) i is bounded, further x i+1 ∈ ∂E 0 ε (v i+1 ) = ∂E 0 (v i+1 ) − εv i+1(26) by (b) and (10). But ∂E 0 (v i+1 ) ⊂X, which is assumed to be bounded, and (v i ) i converges as well. Thus x i+1 − x i is bounded and since τ i → 0 it follows ∇F ε (x i ) + v → 0 and ∇F 0 ε (x i ) + v i+1 → 0. This in turn means v = −∇F ε (lim x i ) so lim x i = z is the ground- state quasi-density for the potential v in the full, regu- larized problem. Finally, lim v i+1 = − lim ∇F 0 ε (x i ) = −∇F 0 ε (z) = v ε KS is the KS potential. As noted above, the reference system reproduces the quasi-density z of the full system and they link back to the real densities by (10), Figure 3. Convergence of ∆Ei = ei − Eε(v) for a ring-lattice system with external potential v = cos(2θ) + 0.2 cos(θ) and different ε. The algorithm developed here is labeled "S" while the method "L" chooses the step length maximally. [14,18]. ρ = z + εv, ρ ε KS = z + εv ε KS ,(27) where typically ρ ε KS = ρ. Then v ε KS − v = ε −1 (ρ ε KS − ρ) is precisely the Hxc potential that depends on the regularization parameter ε here. This means every choice of ε defines a different reference system. A limit ε → 0 in the algorithm is unfeasible because of its relation to the step length. A simulation of two electrons in a discretized onedimensional quantum ring [22] allows us to illustrate the above method numerically. A previously reported implementation of MYKSODA-L in a CDFT setting [18] has been adapted to a pure DFT setting and extended by MYKSODA-S. A radius of R = 1 bohr, a uniform grid with 30 points, and the interaction energy H int = 3 1 + cos(θ 1 − θ 2 ) were used in the reported example. As expected, larger ε leads to faster convergence. Also, the more conservative steps taken by MYKSODA-S often lead to slower convergence in practice. Surprisingly, however, in some cases MYKSODA-S overtakes the less conservative MYKSODA-L. An example is shown in Fig. 3. Such a crossover is possible as the two algorithms follow different paths through the space of densities and potentials. Yet when the starting point is the same, the first step by MYKSODA-L always lowers the energy more than the first step by MYKSODA-S. Although it is a plausible conjecture that also MYKSODA-L, taking maximal steps, is guaranteed to converge, the present proof does not establish this. Maximal steps here means taking τ i maximally such that ∇F ε (x i+1 ) + v, x i+1 − x i ≤ 0 which means maximal decrease in energy in the direction chosen by step (b). In this letter we proved convergence of the exact, regularized Kohn-Sham iteration scheme with special adaptive damping. In short this means that KS-DFT is a veritable method to calculate the correct ground-state density. This strong statement holds for all flavors of DFT that are defined on a finite-dimensional density space X = 2 (M ) and have a linear coupling to external potentials of type v, ρ . This includes CDFT, where the potential v is a combination of scalar and vector potential, and the density ρ includes the paramagnetic current density. To allow for a combination of these different entities into one Banach space setting, the respective function spaces for one-particle densities and current densities have to fulfill a condition termed "compatibility" in Laestadius et al. [18]. A proof of MYKSODA convergence for infinite-dimensional Banach spaces X is feasible too, but turns out to be much more technical and will be presented elsewhere. The choice of step length implemented in (c) here is an essential part of the proof and similar choices could be of value in showing convergence of other ODA iteration schemes like for Hartree-Fock [10]. Next to this damping step the MY regularization is a vital part of the proof at hand, not only to have functional differentiability, but also for the strong monotonicity estimate needed to show convergence. How those findings can be transferred to realistic KS implementations will be the content of future research, but it is expected that they serve as useful guidelines for better convergence results. Figure 1 . 1Moreau-Yosida regularization Fε of an exemplary F , showing also the regulatization parabolas 1 2ε x − y 2 that trace out Fε. . 3.1(iii)] and Laestadius et al. [18, Lem. 20], respectively. Figure 2 . 2Illustration of one iteration step. Acknowledgments. We express our gratitude for fruitful discussion with Simen Kvaal. 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[ "Proton-Proton On Shell Optical Potential at High Energies and the Hollowness Effect", "Proton-Proton On Shell Optical Potential at High Energies and the Hollowness Effect" ]	[ "Enrique Ruiz Arriola ", "Wojciech Broniowski " ]	[]	[]	We analyze the usefulness of the optical potential as suggested by the double spectral Mandelstam representation at very high energies, such as in the proton-proton scattering at ISR and the LHC. Its particular meaning regarding the interpretation of the scattering data up to the maximum available measured energies is discussed. Our analysis reconstructs 3D dynamics from the effective transverse 2D impact parameter representation and suggests that besides the onset of gray nucleons at the LHC there appears an inelasticity depletion (hollowness) which precludes convolution models at the attometer scale.	10.1007/s00601-016-1095-z	[ "https://arxiv.org/pdf/1602.00288v2.pdf" ]	119,228,653	1602.00288	c70c8f434ef3b313e0ed2490a136dd469a5ebd23	Proton-Proton On Shell Optical Potential at High Energies and the Hollowness Effect March 23, 2016 Enrique Ruiz Arriola Wojciech Broniowski Proton-Proton On Shell Optical Potential at High Energies and the Hollowness Effect March 23, 2016Proceedings of Light Cone 2015 manuscript No. (will be inserted by the editor) We analyze the usefulness of the optical potential as suggested by the double spectral Mandelstam representation at very high energies, such as in the proton-proton scattering at ISR and the LHC. Its particular meaning regarding the interpretation of the scattering data up to the maximum available measured energies is discussed. Our analysis reconstructs 3D dynamics from the effective transverse 2D impact parameter representation and suggests that besides the onset of gray nucleons at the LHC there appears an inelasticity depletion (hollowness) which precludes convolution models at the attometer scale. Introduction The history of proton-proton scattering at high energies has been marked by continuous surprises; extrapolations have often been contradicted by actual measurements such as at ISR (see, e.g., [1; 2; 3; 4] for comprehensive accounts and references therein). The first run of the CERN-LHC on pp collisions has unveiled new features at CM energies about 7 TeV, measured by the TOTEM collaboration [5]. Contrary to naive expectations, probing strong interactions at such high energies, corresponding to a de Broglie wavelength of about half an attometer, becomes more intricate and may still be far from the asymptotics [6]. Regge phenomenolgy [7] motivated Barger and Phillips [8] to propose a parameterization which, when suitably extended, describes all ISR and the LHC data simultaneously [9; 10]. Dynamical calculations display the intricacy of non perturbative phenomena at these high energies from a fundamental viewpoint (see e.g. [11; 12]). In the present talk we return to a phenomenological level and unveil interesting features of pp scattering via the so-called on-shell optical potential model. The optical potential was first suggested to deal with inelastic neutron-nucleus scattering above the compound nucleus regime [13]. There, the concept of the black disk limit was first proposed and tested along with the observed Fraunhofer diffraction pattern [14], which also applies to the eikonal approximation [15]. The Serber model [16] was an incipient extension of the optical eikonal formalism to high energy particle physics. Based on a double spectral representation of the Mandelstam representation of the scattering amplitude, Cornwall and Ruderman delineated a definition of the optical potential directly rooted in field theory [17]. Field-theoretic discussions using the multichannel Bethe-Salpeter equation shed some further light [18; 19] (see, e.g., an early review on optical models [20]). The off-shell vs on-shell interplay was analyzed and an on-shell-type equation was proposed by Namyslowski [21]. The high energy grayness of the nucleon has been a matter of discussion since the 70's [22]. Amplitudes and parameterizations The invariant proton-proton elastic scattering differential cross section is given by dσ el dt = π p 2 dσ el dΩ = π p 2 \| f (s,t)\| 2 ,(1) where f (s,t) is the scattering amplitude having both the partial wave and the impact parameter expansions [23] f (s,t) = ∞ ∑ l=0 (2l + 1) f l (p)P l (cos θ ) = p 2 π d 2 b h(b, s) e iq·b = 2p 2 ∞ 0 bdbJ 0 (bq)h(b, s) ,(2) where s = 4(p 2 + M 2 ), p is the CM momentum, and t = −q 2 with q = 2p sin(θ /2) denoting the momentum transfer. In the eikonal approximation one has bp = l + 1/2 + O(s −1 ), hence h(b, s) = f l (p) + O(s −1 ) and P l (cos θ ) → J 0 (qb). The total, elastic, and total inelastic cross sections read, respectively [23] σ T = 4π p Im f (θ = 0) = 4p d 2 bImh(b, s) ,(3)σ el = dΩ \| f (s,t)\| 2 = 4p 2 d 2 b\|h(b, s)\| 2 ,(4)σ in ≡ σ T − σ el = d 2 bn in (b) .(5) Here, the transverse probability inelastic profile fulfills n in (b) ≤ 1 and is given by n in (b) = 4p Imh(b, s) − p\|h(b, s)\| 2 .(6) For our purposes we just need a working parameterization of the scattering amplitude. Here, and for definiteness we use the work in Ref. [9], and more specifically, their MBP2 form, which have been fitted separately for all known differential cross sections for √ s = 23.4, 30.5, 44.6, 52.8, 62.0, and 7000 GeV 1 and read f (s,t) = ∑ n c n F n (t)s α n (t) ,(7) where F n (t) are form factors, α n (t) = α n (0) + α n (0)t and c n are complex numbers which at variance with Regge theory [7], are assumed to be energy dependent. The fits produce χ 2 /d.o.f. ∼ 1.2 − 1.7 [9; 10]. The on-shell optical potential A general field theoretic approach requires solving a coupled channel Bethe-Salpeter equation involving all open channels, a most impractical a procedure, since their number becomes huge for the large energies at ISR or the LHC. Of course, a viable approach would be to determine the kernel, operating as a phenomenological optical potential, from the available NN scattering data. In the geometric picture, the diffraction pattern is manifest as a shadow of the inelastic scattering, such that the diffraction peak in the forward direction is due to a coherent interference. From a Quantum Mechanics point of view, the inelastic process can be interpreted as a leakage in the probability current. We propose a scheme below where a local and energy dependent potential can be directly computed from the data, unveiling the structure of the inelasticity hole. A standard tool for handling the two-body relativistic scattering is the Bethe-Salpeter equation (we use conventions of Ref. [24]), which in the operator form reads T = V +V G 0 T .(8) For the 2 → 2 sector with the kinematics (P/2 + k, P/2 − k) → (P/2 + p, P/2 − p) it can be written as T P (p, k) = V P (p, k) + i d 4 q (2π) 4 T P (q, k)S(q + )S(q − )V P (p, q) ,(9) where q ± = (P/2 ± q), P is the total momentum, T P (p, k) is the total scattering amplitude, and S(q ± ) denotes the nucleon propagator. The kernel V represents the irreducible four-point Green's function, and it is generically referred to as the potential. Equation (9) is a linear four-dimensional equation. It requires the off-shell behavior of the potential V P (k , k) and generally depends on the choice of the interpolating fields, although one expects the scattering amplitude for the on-shell particles to be independent of the field choice. 2 An approach which is manifestly independent of the off-shell ambiguities deduces an on-shell equation by separating explicitly those states which are on-shell and elastic from the rest [21] (for a similar and related ideas see, e.g., Ref. [24]). In the operator form, the final result can be written as (see also Ref. [17]) T el = W + T el G 0 T † el ,(10) where W is the on-shell optical potential and T el is the elastic on-shell scattering amplitude, T P,el (p, k) = T P (p, k)\| p 2 =k 2 =s/4−M 2 = −8π √ s f (s,t) .(11) When written out explicitly, the equation becomes T P,el (p, k) = W P (p, k) + i d 4 q (2π) 4 T P,el (q, k)S(q + )S(q − ) T P,el (q, p) * .(12) At the level of partial waves, we have a simplified form, which using p( s) = (s/4 − M 2 N ) 1 2 reads f l (s) = w l (s) + 1 π ∞ s 0 ds f l (s )p(s ) f l (s ) † s − s − i0 + ,(13) where s 0 = 4M 2 N . Note that only the on-shell amplitude enters here, whereas the equation is non-linear. As usual, the scattering amplitudes are defined as boundary values of analytic complex functions, such that f l (s) ≡ f l (s + i0 + ). Note that due to the Schwartz reflection principle f l (s) † ≡ f l (s − i0 + ), and thus the unitarity condition corresponds to a right-hand discontinuity cut 2iIm f l (s) = f l (s + i0 + ) − f l (s − i0), which reads Im f l (s) − p(s)\| f l (s)\| 2 = Imw l (s) at s > 4M 2 N and yields two contributions. The exchange, written as a left cut condition becomes Im f l (s) = Imw l (s) for s < 0, whereas the causality implies a dispersion relation in energy along the left and right cuts. Invoking the eikonal approximation, which works phenomenologically for √ s ≥ 23.5GeV, and using Eq. (5) we get w l (s)\| l+1/2=bp = n in (b)/4p + O(s −1 )(14) Solving the BS equation becomes very complicated and for a phenomenological kernel is not truly essential. Instead, we use a minimal relativistic approach [25] based on the squared mass operator [26] defined as M 2 = P µ P µ + V , where V represents the (invariant) interaction determined in the CM frame by matching to the non-relativistic limit with a local and energy-dependent phenomenological optical potential, V (r, s) = ReV (r, s) + iImV (r, s). This yields V = 8M N V (r, s); it could be obtained by fitting elastic scattering data [25]. After quantization we haveM 2 = 4(p 2 + M 2 N ) + 8M N V , withp = −i∇, such that the relativistic equation can be written asM 2 Ψ = 4(k 2 + M 2 N )Ψ , i.e., as a non-relativistic Schrödinger equation (−∇ 2 + M N V )Ψ = (s/4 − M 2 N )Ψ .(15) This equation incorporates the necessary physical ingredients which were also present in the BS equation: relativity and inelasticity. The optical potential V does not yet correspond to the on-shell one defined by Eq. (10). The argument given in Ref. [17] uses the optical theorem from the continuity equation, yielding σ T − σ el ≡ σ in = − M N p d 3 xImW (x, s) ,(16) where the on-shell optical potential is defined by its imaginary part ImW (x, s) ≡ ImV (x, s)\|Ψ (x)\| 2 and can be interpreted as the local density of inelasticity at a given CM energy √ s; it also becomes W = V + . . . perturbatively. Using ∑ ∞ l=0 (2l + 1) [ j l (pr)] 2 = 1, Eq. (16) becomes consistent with Eq. (14). We may further rewrite this in the impact parameter space by taking x = (b, z) and integrating over the longitudinal component. As a result we get Eq. (5), where the transverse probability profile function is given by n in (b) = − M N p ∞ −∞ dz ImW (x, s) = − M N p ∞ b 2rdr √ r 2 − b 2 ImW (r, s) .(17) In last step the spherical symmetry has been exploited. To determine W (r, s) we recognize this formula as an integral equation of the Abel type, hence it can be inverted using the standard method (see e.g. Ref. [27]) to give the on-shell optical potential directly from the inelasticity profile and hence from data, ImW (r, s) = 2p πM N ∞ r db n in (b) √ b 2 − r 2 .(18) This new formula is remarkable as it reconstructs the 3D on-shell dynamics from the effective transverse 2D impact parameter representation where the longitudinal physics has been integrated out. Numerical results and discussion For the MBP2 parameterization of Ref. [9] we obtain the inelastic profile function and its derivative for the measured and fitted energies √ s = 23.4, 30.5, 44.6, 52.8, 62.0, and 7000 GeV. The result for 14000 GeV is an extrapolation proposed in Ref. [9]. The amplitudes of the on-shell potentials depend strongly on the CM energy √ s, with a power-like behavior. Thus, in Fig. 1 we show the ratio normalized to the value at the origin, ImW (r, s)/ImW (0, s). As can be vividly seen, the lower energy values have a maximum at the origin, whereas the LHC pp data develop a dip in the origin, which suggests that the inelasticity becomes maximal at a finite value, around r = 1fm. The fact that the optical potential has its maximum away from the origin is most remarkable; a feature shared by the profile function n in (b), which shows that in an inelastic collision most damage is not necessarily produced by central collisions. The "hollowness" effect is less evident in 2D as the 3D hole is integrated over the longitudinal variables which effectively fill the 3D-hole. This goes beyond the idea that the protons become "gray" above 13TeV, as recently suggested by Dremin [28] (see also [29; 30]), rather than a black disk. Actually, this shows in particular that the hollowness effect cannot be reproduced by an intuitive folding structure. Indeed, for small r we get W (r) = d 3 yρ(y + r/2)ρ(y − r/2) = d 3 yρ(y) 2 − 1 4 d 3 y[r · ∇ρ(y)] 2 + . . .(19) showing that W (0) is a local maximum, in contrast to the phenomenological result, see Fig. 1. This conclusion also holds if the folding is made between wave functions with no extra weight. Finally, we note that the mean squared radius of ImW(r); r 2 = 3 2 b 2 displays a logarithmic growth with the energy. These surprising new high energy features as well as the fluctuations in n in (b) [31] become relevant in heavy ions collisions and will be addressed in more detail elsewhere. The energy interpolation of [9] suggests that the 3D depletion already happens at 0.5 − 1TeV, below LHC energies, generating a flattening of the 2D impact parameter dependence. Conclusions The on-shell optical potential is a meaningful concept under the most common and general assumption of the Mandelstam double spectral representation. We have shown that it is also a useful quantity when interpreting the proton-proton scattering data at very high energies; the shape of the inelasticity hole changes dramatically when going from ISR to the LHC. The shoulder-like form of the imaginary part of the on-shell potential resembles very much the traditional pattern found in the absorptive part of optical potential in the neutronnucleus reactions beyond the compound model regime [32]. For a heavy nucleus, the surface is much smaller than the volume and the shoulder merely shows that most inelastic processes occur at the surface. This can pictorially be imagined as derivatives of a Fermi-type distribution. In the case of the proton-proton scattering, extremely high energies seem necessary to resolve between the surface from the volume effects at the attometer scales. The puzzling hollowness effect sets in at 0.5 − 1TeV and awaits a dynamical explanation. 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[ "Devices with electrically tunable topological insulating phases", "Devices with electrically tunable topological insulating phases" ]	[ "Paolo Michetti \nElectronic\n\n", "Björn Trauzettel ", "\nInstitute of Theoretical Physics and Astrophysics\nUniversity of Würzburg\nD-97074WürzburgGermany\n" ]	[ "Electronic\n", "Institute of Theoretical Physics and Astrophysics\nUniversity of Würzburg\nD-97074WürzburgGermany" ]	[]	Solid-state topological insulating phases, characterized by spin-momentum locked edge modes, provide a powerful route for spin and charge manipulation in electronic devices. We propose to control charge and spin transport in the helical edge modes by electrically switching the topological insulating phase in a HgTe/CdTe double quantum well device. We introduce the concept of a topological field-effect-transistor and analyze possible applications to a spin battery, which also realizes a set up for an all-electrical investigation of the spin-polarization dynamics in metallic islands.The original prediction of the quantum spin Hall phase [1, 2] has generated a renewed interest in topological phases in solid state systems. In the following years, the realization of the topological insulator (TI) phase has been theoretically described and experimentally observed in two-dimensional (2D) HgTe/CdTe quantum wells (QWs)[3,4].1D helical edge modes are present at each boundary between a 2D TI and a normal insulator (NI), including the vacuum. Due to the topological protection against non-magnetic disorder[5,6]and the spin-filtered nature of such 1D channels, TI edge modes are extremely promising for spintronics. A crucial point, however, is to devise a reliable method allowing to control transport in these helical channels. Several authors have proposed quantum interference and the Aharanov-Bohm effect as a possible route to control charge and spin transport in 2D TIs[7][8][9][10][11][12][13][14]. However, the most straightforward route would be to change the TI into a NI and completely turn off the helical channel. Two recent proposals show that an electrically tunable TI phase can indeed be obtained in an InAs/GaSb type-II QW[15]and in a HgTe/CdTe double quantum well (DQW)[16].	10.1063/1.4792275	[ "https://arxiv.org/pdf/1301.1823v2.pdf" ]	118,486,563	1301.1823	0924ebaea44c212dd9e164f866b9a540097de24e	Devices with electrically tunable topological insulating phases 28 Feb 2013 Paolo Michetti Electronic Björn Trauzettel Institute of Theoretical Physics and Astrophysics University of Würzburg D-97074WürzburgGermany Devices with electrically tunable topological insulating phases 28 Feb 2013 Solid-state topological insulating phases, characterized by spin-momentum locked edge modes, provide a powerful route for spin and charge manipulation in electronic devices. We propose to control charge and spin transport in the helical edge modes by electrically switching the topological insulating phase in a HgTe/CdTe double quantum well device. We introduce the concept of a topological field-effect-transistor and analyze possible applications to a spin battery, which also realizes a set up for an all-electrical investigation of the spin-polarization dynamics in metallic islands.The original prediction of the quantum spin Hall phase [1, 2] has generated a renewed interest in topological phases in solid state systems. In the following years, the realization of the topological insulator (TI) phase has been theoretically described and experimentally observed in two-dimensional (2D) HgTe/CdTe quantum wells (QWs)[3,4].1D helical edge modes are present at each boundary between a 2D TI and a normal insulator (NI), including the vacuum. Due to the topological protection against non-magnetic disorder[5,6]and the spin-filtered nature of such 1D channels, TI edge modes are extremely promising for spintronics. A crucial point, however, is to devise a reliable method allowing to control transport in these helical channels. Several authors have proposed quantum interference and the Aharanov-Bohm effect as a possible route to control charge and spin transport in 2D TIs[7][8][9][10][11][12][13][14]. However, the most straightforward route would be to change the TI into a NI and completely turn off the helical channel. Two recent proposals show that an electrically tunable TI phase can indeed be obtained in an InAs/GaSb type-II QW[15]and in a HgTe/CdTe double quantum well (DQW)[16]. FIG. 1: (Color online) (a) Isometric sketch of a HgTe/CdTe DQW device with a back gate and two distinct top gates (left and right). In the ON state, top gates induce a gate-bias domain leading to a TI/NI interface (channel) where helical edge modes are found. Source (S) and drain (D) leads, placed along the interface between L and R top gates, collect charges from the edge modes. The lateral surface of the DQW is specifically treated to ensure negligible edge transport. (b) Schematic description of a TI/NI interface for direct gate polarization with indication of the helical spin transport of edge states. (c) Reverse gate polarization leading to opposite spin transport of the channel. HgTe DQWs are driven from NI to TI by the application of a inter-well potential bias \|V \| > V C , where V C is a critical value on the order of the gap of the individual QWs [16]. As shown in Fig. 1(a), a left (L) and right (R) top gate are employed to generate an inter-well potential bias domain (PBD), with V = V L and V = V R in the L and R region, respectively. The device is turned ON when the system realizes a TI/NI interface between the L and the R regions and helical edge modes run along the PBD line. Hence, we distinguish between a direct gate polarization accodingly to V L > V C > V R and Fig. 1(b), where spin up electrons run from source to drain, and a reverse gate polarization V L < V C < V R with opposite spin transport properties [ Fig. 1(c)]. The device is turned OFF when the L and R regions belong to the same topological class, i.e. V L , V R < V C (both NI) or V L , V R > V C (both TI). Source and drain electrodes, which could for example be obtained by diffusing metallic atoms in the DQW system, lay close to the orifice between L and R top gates in order to collect electrons from the helical channel. The design in Fig. 1(a) is studied to deal with a single helical channel (along the PBD), but helical edge states would also appear, if the DQW is in a TI phase, at the interface with the vacuum. We will assume the lateral surfaces of the DQW to be specifically treated in order to impede charge transport in such edge modes. Indeed numerical simulations on quantum spin Hall systems [17] suggest that transport of an helical edge mode at a physical edge can be suppressed by locally doping with magnetic impurities of random or in-plane magnetization. We will see that however such lateral edge modes, even when gapped out, play a relevant role in governing the spin properties of the system. The low-energy physics of a single HgTe/CdTe QW is known to be captured by the Bernevig-Hughes-Zhang (BHZ) model [3], which is, in first approximation, block-diagonal in the Kramers partner (spin) degree of freedom. For the spin up block the BHZ Hamiltonian reads [3] (1) allows for helical edge states at any TI/NI boundary [3]. h(k) = C − D k 2 I + M − B k 2 A (k x + ik y ) A (k x − ik y ) −M + B k 2 ,(1) The BHZ model has also been generalized to the case of a HgTe DQW [16], where the two QWs are separated by a thin spacing layer of thickness t allowing for tunneling. The Hamiltonian for the spin up block has then the following structure H DQW,↑ (k) = h(k) + V 2 h T (k) h T (k) h(k) − V 2 ,(2) where the tunneling Hamiltonian h T (k) is given by h T (k) = ∆ E1 α (k x + ik y ) α (k x − ik y ) ∆ H1 ,(3) omitting terms of order O(k 2 ). Numerical estimation of the tunneling parameters [16] indicates that ∆ E1 ≫ ∆ H1 so that ∆ H1 can be neglected. Moreover ∆ E1 and α can be taken constant as a function of k. Indicative estimates are reported in Table I for t ranging from 4 to 7 nm. For definiteness, we consider a structure of two equal QWs with M = 10 meV (QW thickness d ≈ 5.5 nm), with a thin inter-well spacing layer of thickness t = 5 nm, ensuring a significant amount of tunneling, and consider a PBD where V L = 30 meV and V R = 0 in the direct gate polarization [ Fig. 1(b)]. Fig. 2(a) shows the 2D bulk band structure of the system, where full lines refer to the L side and dashed lines to the R side. Note that the TI bandgap (L side) lays entirely in the NI bandgap (R side). The bandgap in the TI phase develops due to the inter-layer tunneling interaction and indeed, keeping fixed V = 30 meV, the TI gap is about E G ≈ 4, 2 and 1 meV for a spacing layer t = 5, 6, 7 nm, respectively. We solve now Eq. (2) in real space, imposing the continuity of the wave function and its normal derivative through the PBD line. When the device is ON, the system admits helical edge modes located at the TI/NI interface. The 1D dispersion curve of such helical edge states is shown in Fig. 2(b), together with the merging bulk bands. Let us now analyze ballistic transport in the helical edge states. Due to their spin-filtering nature, it is useful to define separate chemical potentials µ α,σ for the two spin species (σ =↑, ↓), with α = S, D for the source and drain contacts, respectively. Chemical potentials are kept in the DQW gap, so that at sufficiently low temperature no charge is present in the bulk of the L and R regions. For simplicity, we consider charge accumulation in the edge states (which are considerably extended) to have negligible effects. The unidirectional charge current for the spin species σ in the 1D helical channel is given by I (C) σ = −κ σ e h EG 0 dE f (E − µ Xσ ,σ ) ≈ −κ σ e h µ Xσ ,σ(4) with κ σ = 1 (−1) and X σ = S (D) for σ =↑ (↓). In the last passage we assumed k B T ≪ E G , µ Xσ ,σ and extended the integral to infinity. Note that µ is measured from the valence band edge of the TI region [Fig 2(b)]. The helical currents satisfy I Fig. 1(a) realizes the concept of a TI field effect transistor (FET), where charge transport in the 1D helical channel can be turned on and off by changing the topological phase of a part of the system. When the device is ON, the source-drain current in the 1D helical channel is (C) ↑ ∝ −µ S,↑ and I (C) ↓ ∝ µ D,↓ . The device inI (C) Q = I (C) ↑ + I (C) ↓ = e h [µ D,↓ − µ S,↑ ] ,(5) which gives I Q ≈ 0.1 µA for µ S = 1 meV and µ D = 3 meV (we assume spin-balanced leads with µ S,σ = µ S and µ D,σ = µ D ). This value does not significantly depend on temperature as long as k B T < E G , µ S , E G − µ D . We estimate the charge currents in the 2D bulk bands as I (X,±) Q = e αR 2 πL C dk k dE (X,±) k dk [f (E−µ D )−f (E−µ S )] ,(6) where X = R or L, ± refers to conduction and valence bands, the factor 2 accounts for the spin degeneracy, L C is the length of the channel and R the typical size of the contacts. α ≈ R 4πLc is the effective fraction of the electrons that has the right initial angle to propagate between the two contacts. We switch the device between the OFF state characterized by V L = V R = 0 and the ON state with V L = 0 and V R = 30 meV, compatible with the situation described in Fig. 2. When the FET is OFF the helical channel is absent, I Q + X,± I (X,±) Q . In Fig. 3, we show the current ratio I OFF /I ON as a function of temperature for different lateral size R of the contacts. It is clear that due to the narrow gap of the material small OFF/ON current ratios are only obtained at low temperature. We note however that the bulk bands responsible for the OFF current have a 2D character, while both the contacts and the helical channel are quasi-1D. Therefore the current ratio reduces for small R (or for long channels L C ) where transport in the 1D edge states becomes more efficient than that in the 2D bulk bands. A peculiarity of the topological FET described here is that a certain amount of disorder would be favorable for reducing the OFF/ON current ratio. This stems from the topological protection against non-magnetic disorder of the helical channel, which would be little affected by the presence of inhomogeneities, while transport in the bulk bands (and therefore I OFF ) would be suppressed . The spin current in the helical channel of Fig. 1 is given by I (C) s = − 2e I (C) ↑ − I (C) ↓ = 1 4π [µ S,↑ + µ D,↓ ] .(7) While the charge current Eq. (5) is evidently zero for the condition µ S,↑ = µ D,↓ , a net spin current would flow. This is a persistent pure equilibrium spin current, which does not lead to spin accumulation [19], because of the presence of helical edge states (even though they are gapped out) at the interface with the vacuum of the DQW. Indeed a conduction gap in the lateral helical edge modes is identical to a perfect backscattering process plus a spin-flip. Such backscattering processes lead to the following additional spin currents incoming on source and drain: I (S) s = 1 4π 2µ S,↓ , I (D) s = − 1 4π 2µ D,↑ ,(8) and, as expected, in equilibrium (µ S,σ = µ D,σ ) the total spin accumulation rate on the source (drain) due to helical edge modes I Fig. 4, realized with an electrically tunable TI material. We assume C 1 and C 2 to be mesoscopic metallic islands (MIs), while source and drain are macroscopic reservoirs with µ S,↑ = µ S,↓ = µ S and µ D,↑ = µ D,↓ = µ D . The system is described by three characteristic times: a spin-conserving relaxation time τ E , a current injection time τ I , and a spin relaxation time τ s [21]. We express the particle number of the σ spin species as n S,σ = N µ S,σ (and similarly n D,σ = N µ D,σ ), with N = mR 2 h the constant 2D density of states and R the typical dimension of the MIs. The current injection time is then defined as τ I = hN = mR 2 , which for m = 0.1m e and R ≈ 1 µm gives τ I ≈ 1 ns. For metallic islands of micrometer size generally τ E ≪ τ I and therefore we will assume intra-species relaxation to be instantaneous on the time scales of τ I . τ s is in the range 10 −9 − 10 −6 s [21]. If τ E ≪ τ I ≪ τ s the system is said to be in the slow spin relaxation limit. We set up a minimal model, which is similar in spirit to what was used in Refs. 21, 22, where the equilibration dynamics is described by master equations, and neglect for simplicity charging effects on the MIs. Hence, the charge and spin dynamics in the C 2 MI is governed by the following rate equation τ IμC2,σ = µ Xσ − µ C2,σ − τ I τ s [µ C2,σ − µ C2,−σ ] ,(9) while the equation for C 1 is obtained by replacing µ C2,σ ↔ µ C1,−σ . For spin-balanced leads with µ S = −µ D = δµ (for simplicity energy is now measured with respect to midgap) and the initially spin balanced MIs at µ C1,σ (t = 0) = µ C2,σ (t = 0) = µ 0 , Eq. (9) has the solution µ C2,σ (t) = κ σṼ + e − t τ I µ 0 − κ σṼ e − 2t τs and µ C1,σ (t) = µ C2,−σ (t), with V = δµ τs 2τI +τs . In the slow spin relaxation limit, µ C2,σ → κ σ δµ exponentially with characteristic time τ I . Opposite spin species accumulate on the two MIs with a spin-polarization factor P = \|Ṽ /E 0 \|, assuming the conduction band edge of the MIs to be E 0 < −\|δµ\|. Therefore C 1 and C 2 act like two opposite poles of a spin-battery. When a normal conducting circuit closes on such poles a pure spin current is supplied. Note however that dissipation occurs in the spin battery [ Fig. 4] because of a finite source-drain voltage and a finite charge current. The present configuration also realizes a set up where the dynamics of the spin densities on the MI can be studied by all-electrical means. In fact, by switching OFF the helical channel, once that the current is in a steady state, we store the spin densities on the two MIs, which then decays due to spin relaxation mechanisms. In order to measure the residual spin polarization µ C1,σ = −µ C2,σ = κ σ δµ(t) at the delay time t, we set µ S = µ D = 0 and then turn ON again the helical channels. The initial value of the source-drain charge current I = − e h 2δµ(t) allows to extract the residual spin polarization in the MIs. The time resolution is limited by the time it takes to switch the gate polarization (which does not involve any charging effects in the bulk). We neglected that in HgTe spin is not a properly conserved quantity but rather the total angular momentum is, see for example the discussion in Ref. 19. However, it can be shown that helical edge modes are spin polarized [9] (although spin is not perfectly aligned withẑ) and Kramers partners are characterized by opposite spin. Moreover the polarization axis is fixed as long as Rashba terms are negligible [23]. We also assumed that edge modes always ideally sink into the electric contacts, however due to a non perfect overlap of the edge mode profile with source and drain, they could partially avoid the contacts complicating the physical picture. For a quantitative purpose all these aspects should be taken into account together with the charging effects on the helical channel and, when considered, on the mesoscopic metallic islands. Many critical issues have still to be better understood before TIs can become a realistic solution for electronic devices. One of the limiting factors of HgTe DQWs comes from the relatively large extent (λ > 100 nm) of the edge states, which is due to the small band gap obtainable in that system. It is therefore a crucial challenge to search for materials featuring larger TI gaps and well-localized edge states, like ultra-thin Bi films [24,25] which has been proposed having λ < 10 nm, and devise a way to electrically tune the TI phase. Promising candidate materials include graphene doped with heavy adatoms [26], silicene [27,28], and 2D germanium [27]. written in the basis of the low energy subbands { \|E1, + , \|H1, + }. Conduction and valence bands of the TI for the spin up block are readily obtained by Eq. (1). Spin down eigenstates can then be obtained by applying the time reversal operatorT = −iK, withK the complex conjugation operator. We adopt A = 375 meV nm, B = −1.120 eV nm 2 and D = −730 meV nm 2 , which have been estimated by a comparison with the 8 × 8 Kane Hamiltonian[18], and assume C = 0 without loss of generality. The Dirac rest mass M depends on the QW thickness d. For d > 6.3 nm (M < 0) the system realizes a TI and the real space solution of Eq. FIG. 2 : 2(Color online) (a) 2D bulk dispersion curves for the left (L) and right (R) regions of the DQW device, which are in the TI and in the NI state, respectively. Bulk dispersion curves are degenerate in the spin degree of freedom. (b) Detail of the low energy spectrum with a plot of the 1D helical edge modes at the TI/NI boundary (the PBD line). Edge states in full thin lines and in dashed thin lines belong to opposite Kramers blocks (spin). We employ the following parameters: t = 5 nm, M = 10 meV, VL = 30 meV and VR = 0. . The ON current is instead I ON = I (C) FIG. 3 : 3(Color online) IOFF/ION current ratio as a function of temperature of the topological FET inFig. 1. The channel length is Lc = 1 µm, while we consider contacts of lateral size R = 0.2, 0.5, and 1 µm. noted in Ref. 19, 20, spin accumulation exploiting the TI edge modes requires an applied source-drain bias. Let us consider the setup shown in FIG. 4 : 4(Color online) Sketch of a spin battery realized with the exploitation of helical channels. Spin up (dashed line) and spin down (full line) spin filtered edge modes are shown at the boundary. TABLE I : ITunneling parameters α and ∆E1 for a HgTe/CdTe DQW, calculated at kx = ky = 0 for several values of the spacing layer thickness t[16]. AcknowledgmentsWe are grateful to G. Fiori and G. Iannaccone for helpful discussions and comments We would like to thank the Deutsche Forschungsgemeinschaft, the European Science Foundation, as well as the Helmoltz Foundation for financial support. . 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[ "Pregeometric First Order Yang-Mills Theory", "Pregeometric First Order Yang-Mills Theory" ]	[ "Priidik Gallagher \nInstitute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 150411TartuEstonia\n", "Tomi Koivisto \nInstitute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 150411TartuEstonia\n\nNational Institute of Chemical Physics and Biophysics\nRävala pst. 1010143TallinnEstonia\n\nUniversity of Helsinki and Helsinki Institute of Physics\nP.O. Box 64FI-00014HelsinkiFinland\n", "Luca Marzola \nNational Institute of Chemical Physics and Biophysics\nRävala pst. 1010143TallinnEstonia\n" ]	[ "Institute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 150411TartuEstonia", "Institute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nW. Ostwaldi 150411TartuEstonia", "National Institute of Chemical Physics and Biophysics\nRävala pst. 1010143TallinnEstonia", "University of Helsinki and Helsinki Institute of Physics\nP.O. Box 64FI-00014HelsinkiFinland", "National Institute of Chemical Physics and Biophysics\nRävala pst. 1010143TallinnEstonia" ]	[]	The standard description of particles and fundamental interactions is crucially based on a regular metric background. In the language of differential geometry, this dependence is encoded into the action via Hodge star dualization. As a result, the conventional forms of the scalar and Yang-Mills actions break down in a pregeometric regime where the metric is degenerate. This suggests the use of first order formalism, where the metric may emerge from more fundamental constituents and the theory can be consistently extended to the pregeometric phase. We systematically explore different realizations and interpretations of first order formalism, finding that a fundamental vector or spinor substructure brings about continuum magnetization and polarization as integration constants. This effect is analogous to the description of the cosmological dark sector in a recent self-dual formulation of gravity, and the similar form obtained for the first order Yang-Mills theory suggests new paths toward unification. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] Λ arise as integration constants[20]. This provides, to our knowledge, the unique candidate for a ΛCDM theory of cosmology. The topological origin of the CDM, due to the existence of a g µν = 0 phase, had been anticipated in the work of Bañados[21].	null	[ "https://arxiv.org/pdf/2202.05657v1.pdf" ]	246,823,261	2202.05657	f87620519aeb9f021401804a2d565be5086a5bda	Pregeometric First Order Yang-Mills Theory 11 Feb 2022 Priidik Gallagher Institute of Physics Laboratory of Theoretical Physics University of Tartu W. Ostwaldi 150411TartuEstonia Tomi Koivisto Institute of Physics Laboratory of Theoretical Physics University of Tartu W. Ostwaldi 150411TartuEstonia National Institute of Chemical Physics and Biophysics Rävala pst. 1010143TallinnEstonia University of Helsinki and Helsinki Institute of Physics P.O. Box 64FI-00014HelsinkiFinland Luca Marzola National Institute of Chemical Physics and Biophysics Rävala pst. 1010143TallinnEstonia Pregeometric First Order Yang-Mills Theory 11 Feb 2022* Electronic address: priidikgallagher@utee † Electronic address: tomikoivisto@utee ‡ Electronic address: lucamarzola@cernch The standard description of particles and fundamental interactions is crucially based on a regular metric background. In the language of differential geometry, this dependence is encoded into the action via Hodge star dualization. As a result, the conventional forms of the scalar and Yang-Mills actions break down in a pregeometric regime where the metric is degenerate. This suggests the use of first order formalism, where the metric may emerge from more fundamental constituents and the theory can be consistently extended to the pregeometric phase. We systematically explore different realizations and interpretations of first order formalism, finding that a fundamental vector or spinor substructure brings about continuum magnetization and polarization as integration constants. This effect is analogous to the description of the cosmological dark sector in a recent self-dual formulation of gravity, and the similar form obtained for the first order Yang-Mills theory suggests new paths toward unification. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] Λ arise as integration constants[20]. This provides, to our knowledge, the unique candidate for a ΛCDM theory of cosmology. The topological origin of the CDM, due to the existence of a g µν = 0 phase, had been anticipated in the work of Bañados[21]. I. INTRODUCTION The description of gravity in General Relativity (GR) is built on a 4-dimensional pseudo-Riemannian manifold supplying the fundamental field of interest: the metric. This describes distances and determines the curvature of spacetime through the Levi-Civita connection. The contemporary treatment of gauge fields can be taken to be just as geometric as that of gravity -Yang-Mills theory is then built on top of the background manifold by considering the dynamical connections and curvature of a G-bundle pertaining to the chosen symmetry group G. Scalar and fermionic fields can also be described with a similar apparatus, which highlights a difference in the treatment of fermions and bosons. In fact, whereas the Dirac action is inherently of the first order, the actions of bosonic fields are usually given in the second order form and rely on the metric to operate the required contractions. Thus, the formulation of Quantum Field Theory (QFT), and the Standard Model of particle physics in particular, presupposes a (constant, Minkowski) metric background [1], and in the context of GR this background is promoted to a dynamical field. Consequently, both frameworks completely break down in the hypothetical situation where the metric field becomes degenerate. This possibility was already considered by Einstein and Rosen, in their attempt of providing a geometrical description of elementary particles through "bridges" characterised by a vanishing metric determinant, g = 0 [2]. Later, such solutions and their topology have played an important part in attempts at quantum geometrodynamics [3,4]. In particular, it has been proposed that in quantum gravity, the ground state of the metric field should be g µν = 0 [5,6]. We point out a semantic inconsistency which occurs if the definition of a metric is taken to require an invertible, symmetric and covariant tensor. In the case that one insists upon a metric theory in this sense, the physical implication of an "ametric" phase, i.e. the loss of causal structure, then manifests in the inevitable nonlocality of the ultraviolet completion, e.g. in terms of infinite-derivative [7] or fake degrees of freedom [8]. A vanishing ground state for the metric was explicitly realised in a recent pregeometric gravity theory [9,10]. The related pregeometry programme proposes that the metric, or the (co)frame, is emergent and composed of other, potentially purely fermionic fields. The framework is in line with earlier studies such as that of Ref. [11], later invoked within unification [12], spinor gravity [13,14], analogue gravity [15], time-space asymmetry [16], lattice regularization [17] and cosmology [18,19]. By introducing the emergent coframe via the exterior covariant derivative of a Lorentz vector (or, a bispinor) in a Cartan-geometric language, Ref. [9] found that the consistent General-Relativistic solutions are immediately accompanied by dust interpretable as dark matter 1 . In the ground state, the gauge connection is arbitrary. To support a Minkowski background, the Lorentz vector spontaneously breaks the symmetry of the theory by acquiring a time-like expectation value and the gauge connection configuration has both torsion and curvature. Thus, in terms of the more primitive fields, there is a non-trivial structure underlying the Minkowski metric background. As we show in the present paper, the fact that the metric could be emergent and admit a singular phase forces to question the traditional descriptions of elementary fields. In fact, although the fermionic sector poses no problem, in a singular metric phase the conventional actions for scalar and Yang-Mills theory have to be abandoned due to the presence of a potentially singular inverse metric in the Hodge dualization. To overcome the problem, we systematically study the possibilities offered by the first order formalism, seeking forms for the bosonic actions which recover the usual equation of motions and are suitable for a possible pregeometric regime. Several approaches have been investigated before. For instance, Ref. [22], working in Euclidean signature, constructed matter actions invariant under O(5), independent of the metric and connection via introducing auxiliary fields, while Ref. [23] used a "preferred volume" formalism, without requiring general covariance, but only covariance under volumepreserving reparametrizations, likewise introducing an auxiliary field, which coincides with the inverse vierbein after symmetry breaking. Ref. [24] studied a Yang-Mills-Cartan action, where the gauge field included a Cartan index, and was the initial basis for developing the approach here. The Yang-Mills-Cartan action is included in their Cartan-unified theory, but separate de Sitter gauge invariance requires assuming that the contact vector is constant. Thus, the previous approaches have been based on 5-dimensional extensions of the 4-dimensional orthogonal symmetry. Starting from Lorentz symmetry, this article will instead discuss actions that essentially are alternative realizations of a first order Yang-Mills 1 In a unimodular version of this theory, both the cold dark matter (CDM) and the cosmological constant theory, that now with polynomially simple actions appear to be particularly applicable in the study of gravity. A similar approach to scalar field theory is possible as well, and is only briefly mentioned, but it deserves more future analysis. For terminological clarity, it may be useful to distinguish our approach from the premetric discourse in the Literature. The basic idea is the same: removing the metric from the fundamental equations of physics. The premetric program, put forward by Kottler in 1922, has developed into an axiomatic framework for analyzing and constructing the structure of a theory, beginning from the identification of the conserved quantities, and avoiding the reference to any metrical concepts as far as possible [25]. In principle, this framework allows to consider very general theories which do not even necessarily admit a Lagrangian formulation. In practice, however, the metric is often introduced at the stage when the theory is made predictive by postulating its constitutive law. Premetric electrodynamics has been very well studied, a classic textbook reference is [26], and a shorter overview is [27], for extension to gravity see [28,29]. Yet, an extension to Yang-Mills theory appears to be missing. The pregeometric theory that we pursue is, more particularly, a theory that is based on an invariant, polynomial action principle which remains well-defined when the inverse metric (that only emerges, potentially highly non-polynomially, as a composite of more primitive fields) does not exist, becomes degenerate or even singular. Thus, we could state that what we mean by a pregeometric theory is a premetric theory satisfying the two key axioms: 1) formulation as an action principle and 2) viability of the "ametric" phase. We shall make connection with the existing constructions of premetric electromagnetism, wherein the electromagnetic excitation is introduced in conjunction with the axiom of charge conservation and the form of the excitation is finally postulated (with or without invoking a metric) through the constitutive law, by instead promoting the excitation to a dynamical field. Thus, the constitutive law is the consequence of dynamics and only valid on-shell, and moreover, the usual relation between the excitation and the field strength may only emerge in the metric phase. We then proceed to explore the possibility of reducing, together with the metric field, the new dynamical excitation field into more primitive substructure. The article is structured as follows. In Section II we will briefly go over our conventions and set the formalism used to describe matter fields. The bulk of the article, in section III, is devoted to introducing the first order Yang-Mills actions and studying them. In particular, several interpretational questions will be looked over, and the vector substructure implications are investigated, producing analogous results to [9]. Section IV brings attention to the similarity of this first-order Yang-Mills theory and self-dual Palatini gravity and goes over some questions involving unification of gravity in first order formalism, before reaching the conclusion. II. CLASSICAL THEORY OF MATTER AND GRAVITY A. Some Mathematical Preliminaries Our conventions are (η ab ) = diag(−, +, +, +) and ǫ 0123 = −ǫ 0123 = −1. Lorentz indices are in Latin, with the spatial components capitalized, while Greek indices refer to the coordinate basis, or arbitrary basis in this section. We implicitly use natural units c = = 1, generally barring numerical coefficients if they do not modify the analysis nor the dynamics; coefficients and coupling constants are introduced in section IV for comparing GR with Yang-Mills theory. A few concepts that should be emphasized are more clear in arbitrary n dimensions, but will be restricted to 4 dimensions later on. Then, a general p-form ω = 1 p! ω µ 1 ...µp ϑ µ 1 ∧ . . . ∧ ϑ µp .(1) It is important to note that the Levi-Civita symbols ǫ µ 1 ...µn and ǫ µ 1 ...µn themselves are a premetric concept, arising from differential forms of maximal rank ϑ µ 1 ∧ . . . ∧ ϑ µn = ǫ µ 1 ...µn ϑ 0 ∧ . . . ∧ ϑ n−1 = ǫ µ 1 ...µnǫ(2) and their maximal interior product ǫ µ 1 ...µn = sgn(g) @ µn . . . @ µ 1ǫ ,(3) where the 1-forms ϑ µ i are an arbitrary cobasis with @ µ i its respective vector basis; note the addition of the sign is here only conventional, not strictly required for constructing the symbol. Without a metric, there is no immediate correspondence between the two symbols, which is sometimes notationally emphasized. The placement (or omission) of the sign of the metric determinant is the primary point of contention between various Levi-Civita symbol and tensor conventions. Here it is added to the symbol ǫ µ 1 ...µn , but alternatively it could instead be added to the symbol ǫ µ 1 ...µn or either of the Levi-Civita tensors. In practical terms, mainly the Levi-Civita tensors are used, but as can shortly be seen, in an orthonormal frame they coincide with the Levi-Civita symbols up to sign conventions. Furthermore, note that in the paper we further assume Lorentz symmetry as a starting point, therefore the symbols ǫ abcd and η ab are available as invariants of the symmetry, and in particular, η ab can be seen not as a field on spacetime, but simply as an invariant of Lorentz symmetry. Furthermore, the chain of interior productsǫ µ 1 = @ µ 1 ǫ,ǫ µ 1 µ 2 = @ µ 2 @ µ 1 ǫ, . . . yields a differential form basis equivalent to ϑ µ 1 , ϑ µ 1 ∧ ϑ µ 2 , . . .; see e.g. [26] for more discussion. Therefore it is also possible to expand differential forms as ω = 1 (n − p)! ω µ 1 ...µ n−pǫ µ 1 ...µ n−p .(4) Moving between a differential form description and the usual index formalism can be realized with the ⋄-dual density, which establishes a correspondence between p-forms and totally antisymmetric tensor densities of weight +1 and type (n − p, 0), i.e. (n − p) vectors. Generally in terms of basis vectors in n dimensions ⋄ (ǫ µ 1 ...µp ) = δ ν 1 ...νp µ 1 ...µp @ ν 1 ⊗ . . . ⊗ @ νp ,(5) thus for a general p-form ω the ⋄-dual tensor density ⋄ ω = 1 (n − p)! ω µ 1 ...µ n−p @ µ 1 ⊗ . . . ⊗ @ µ n−p ,(6) with the components ω µ 1 ...µ n−p = 1 p! ω µ n−p+1 ...µn ǫ µ 1 ...µn .(7) Note this duality between differential forms and tensor densities does not yet use the metric, as neither the expansion with respect to the Levi-Civita dual basisǫ µ 1 µ 2 ... nor the Levi-Civita symbols involve the metric, and is therefore viable in a premetric description. However, establishing the Hodge * -duality between differential forms of rank p and n − p does require the metric, see in components * ω = √ −g p!(n − p)! ω µ 1 ...µp ǫ µ 1 ...µpµ p+1 ...µn ϑ µ p+1 ∧ . . . ∧ ϑ µn ,(8) where the metric determinant appears explicitly and the inverse metric was used to raise indices. Likewise, the metric appears in the Levi-Civita tensors ε µ 1 ...µn = \|g\|ǫ µ 1 ...µn ,(9)ε µ 1 ...µn = 1 \|g\| ǫ µ 1 ...µn .(10) However, note the volume form in 4 dimensions Vol = 1 4! ε µνρσ dx µ ∧ dx ν ∧ dx ρ ∧ dx σ = 1 4! ǫ abcd e a ∧ e b ∧ e c ∧ e d ,(11) where in an orthonormal cobasis it loses explicit reference to the metric, as the determinant η = −1. B. Matter Actions in Differential Forms Equipped with the apparatus of differential geometry we review the standard actions of matter and gravity in the framework of Lorentz symmetry, paying special attention to the role of the metric. For a more thorough discussion of possible geometric descriptions of the background and their possible equivalence, we refer the reader to Ref. [30]. The simplest case of a massless Dirac fermion can be described in Riemann-Cartan geometry through the Dirac spinor action S ψ = − ψ γ ∧ * iDψ = d 4 x √ −gψγ µ iD µ ψ,(12) where γ = γ a e a is a 1-form and γ a are the Dirac gamma-matrices, while the exterior covariant derivative (barring possible gauge interactions) acts on the spinor 0-form as Dψ = dψ − i 2 ω ab − i 4 [γ a , γ b ] ψ.(13) The action can be straightforwardly extended to include a Dirac mass termψmψVol, possibly generated after the spontaneus breaking of gauge symmetries via the Higgs mechanism. The explicit mathematical construction of spinor theory on curved spacetime is lengthier. A simple exposition is that the 0-forms are spinor-valued in Minkowski space tangent to the background manifold 2 . Scalar field and Yang-Mills theories usually rely on second order actions, the kinetic terms being, respectively, S φ = − 1 2 dφ ∧ * dφ,(14)S A = − 1 2 Tr F ∧ * F .(15) Scalar fields φ are just functions, i.e. 0-forms on a manifold, while Yang-Mills fields A are the G-connection 1-forms of the corresponding gauge symmetry group G with Lie algebra g, and the respective curvature 2-form is given as F = dA + A ∧ A.(16) There are several ways to describe connections, for instance through a G-invariant horizontal distribution, as a family of 1-forms etc. Here, implicitly working with G-bundles E over M, and purely for a general description, we can assume that A is a section of End(E) valued in g. Therefore, locally A is a Lie algebra valued 1-form on the G-bundle. Finally, the Palatini gravity action is S e a ,ω ab = 1 2κ e a ∧ e b ∧ * R ab ,(17) where R ab is the curvature 2-form of the spin connection 1-form ω ab . The basic variables are the coframe e a and the spin connection ω ab , variation with respect to the first producing the Einstein equations, while the equations of motion of the latter fix the connection used in calculating curvature and used in the energy-momentum. In particular, when there is no contribution to spin density, the torsion vanishes and the connection reduces to Levi-Civita. As we can see, Hodge dualization appears in all the actions reported above, thereby apparently preventing us from immediately using them in a regime of the theory where the inverse metric, used in the dualization, is not available. Regarding this, we remark that the problem can actually be circumvented in pure spinor theories, as the action can be spelled out explicitly in an arbitrary orthonormal coframe S ψ = ǫ abcd e a ∧ e b ∧ e c ∧ψγ d iDψ,(18) without using the inverse metric nor the tetrad. On top of that, the Hodge star operator in the gravitational action is particularly amiable in self-dual formulation, discussed in section IV. The relevant formulation of scalar and Yang-Mills theory, suitable for a pregeometric regime independent of the metric, is not as well-known but can be resolved in a first order theory, as we will show below. III. THE YANG-MILLS KINETIC CYCLE A. An auxiliary 1-form To describe the dynamics of the G-connection gauge field A for a (generally non-Abelian) gauge group G, consider the action, for brevity neglecting overall coefficients, S = Tr G 1 2 ǫ abcd u a ∧ e b ∧ u c ∧ e d + η ab u a ∧ e b ∧ F ,(19) where u a is an auxiliary Lie algebra g valued 1-form, transforming as A → gAg −1 + gdg −1 ⇒ F → gF g −1 ,(20)u a → Ad g u a = gu a g −1 .(21) The action is thus by construction invariant under local gauge transformations, local Lorentz transformations and diffeomorphisms. Likewise, it is polynomial and only involves derivatives up to first order. Importantly, the action only makes use of the fundamental objects available: a coframe 3 e a , a 1-form u a , the vector potential A (contained in its field strength 2-form F ), and the invariants ǫ abcd and η ab of the Lorentz group. A second coframe is present in the conformal extension of the Lorentz symmetry [10], see also e.g. [32,33]. However, here we simply consider the (presumably broken) symmetry SO(1, 3) × G, and shall return to discuss paths to unification in section IV. The corresponding equations of motion of A and u a are respectively D(η ab u a ∧ e b ) = J,(22)ǫ abcd u b ∧ e c ∧ e d + e a ∧ F = 0.(23) Here J is the current 3-form, which is generally sourced by the matter terms in the total action 4 . The first equation is just the prototype inhomogeneous Yang-Mills equation, while the second equation, as we will show, enforces that on-shell η ab u a ∧ e b be related to the Hodge dual of gauge curvature. The classical equivalence of this formulation with the usual Yang-Mills theory can be shown by considering ⋄-dual densities of the differential forms, as defined in Eq. (6). In our case, the analysis will be done in a prototype orthonormal coframe e a , which is required to be a proper coframe in the non-singular phase det(η ab e a µ e b ν ) = 0. The differential forms u a and F can thus be expanded in the basis provided by e a and we can employ the Minkowski metric to raise and lower indices as necessary. In this case, in components ǫ abcd u b i ǫ icdk + 1 2 η ab F cd ǫ bcdk = 0,(24) which is 2u b i δ ik ab = 2(u k a − δ k a u i i ) = 1 2 η ab F cd ǫ bcdk .(25) Tracing over the indices a and k implies u i i = 0, and finally utilizing the Minkowksi metric to lower indices implies u (ab) = 0. The remaining antisymmetric part, u ab = u [ab] , can be reorganized to u ak = − 1 4 F ij ǫ ijak ,(26) which coincides with the Hodge star in an orthonormal coframe. As u a = u a k e k , in the nonsingular phase η ab u a ∧ e b = * F.(27) Substituting the above relation back into the prototype inhomogeneous equation produces the usual Yang-Mills equations. As DF = 0 is trivially satisfied, this first order formulation is classically equivalent to usual Yang-Mills theory by realization of a two-step "kinetic cycle". Likewise, the on-shell action neatly coincides with the usual Yang-Mills action, as multiplying Eq. (23) by u a and taking the trace immediately yields Tr ǫ abcd u a ∧ e b ∧ u c ∧ e d = Tr − η ab u a ∧ e b ∧ F ,(28) and the results follows through Eq. (27). The action in Eq. (19) can be considered to be pregeometric in the sense that it remains well defined, as do the corresponding equations of motion, even if the physical metric g µν = e a µ e b ν η ab is singular. In fact, neither the action nor the equations of motion depend on the inverse metric g µν = 4ǫ α 1 α 2 α 3 µ ǫ β 1 β 2 β 3 ν g α 1 β 1 g α 2 β 2 g α 3 β 3 ǫ α 1 α 2 α 3 α 4 ǫ β 1 β 2 β 3 β 4 g α 1 β 1 g α 2 β 2 g α 3 β 3 g α 4 β 4 .(29) The action is not completely premetric, however, as it still requires the Minkowski metric η ab . The general similarity of our approach to topological QFT, topological Yang-Mills theory, and BF theory has to be noted, for further inspiration. Heuristically, to emulate the Hodge dualization without having a regular metric at hand, it makes sense to begin with an expansion in the prototype orthonormal coframe, allowing for a possible singular phase. In order to recover the ordinary Yang-Mills theory, first setting aside possible topological terms, any prototype kinetic term X ∧ F will require X to become the dual field strength 2-form * F , which can easiest be done by introducing explicit Lorentz indices and 1-form substructure to X, the simplest substitution being X → η ab u a ∧ e b , as in Eq. (19). This can then be coupled with the Levi-Civita dualized basis as ǫ abcd u a ∧e b ∧u c ∧e d , similarly to how the Hodge dualization (8) connects differential form components to a dual basis. The procedure could be extended by introducing more auxiliary fields or Lagrange multipliers, thereby lengthening the cycle, but it does not appear helpful at present stage. Likewise, it is by no means the only method to construct a pregeometric Yang-Mills theory, as discussed earlier. Other approaches can be thought of too. For instance, introducing complex structure and instead considering the field strength F as a fundamental field, the Yang-Mills equations in 4 dimensions are equivalent to the system D ± F = ∓ i 2 J,(30) as * * F = −F in Minkowski signature. In particular, when considering U(1) theory, the covariant derivative is replaced by the exterior derivative, which then immediately implies the inhomogeneous Maxwell equation and that F is closed, therefore under suitable topological assumptions by de Rham's first theorem (cf. the Poincare lemma), F is also exact. In the non-Abelian case (30) does not work as neatly, but more crucially, this approach implicitly still requires the use of the Hodge dual in the definition of ± F . It is not clear whether it is possible to work around this. Initially, the proposed "kinetic cycle" was developed purely from the ideas in Ref. [24], but without introducing any Cartan "rolling" indices; note that the action (19) does not introduce more degrees of freedom than employing rolling indices would do. However, the action (19) essentially is a realization of first order Yang-Mills theory, applicable in arbitrary spacetime and in presence of metric singularities. The utility of first order (Abelian) Yang-Mills theory in degenerate spacetimes was already analyzed in Ref. [34] and earlier in context of degenerate tetrads discussed in Ref. [35], despite without the explicit continuation to non-Abelian theory. More expansive study of first order theory itself goes back to at least the 1970-s [36,37] in relation to strong coupling effects, but it has also appeared even longer ago, e.g. [38] in an appendix or see (the republication) [39] in remark and comparison of Maxwell electromagnetism with gravity, and is more generally related to the Ostrogradski procedure of lowering derivative order [40,41]. It has also been employed in relation to the Duffin-Kemmer formulation [42,43], or computation of loop effects [44] in emphasis of the simpler vertex structure. Proving the classical equivalence of first order and ordinary formulation of Yang-Mills theories is rather straightforward. Equivalence at the quantum level was recently studied via vacuum functionals [45], and earlier discussion can be found in e.g. [46], while other recent results include study into renormalization [47] and consistency conditions related to Green's functions and ultraviolet divergences [48]. Furthermore, as an aside to pregeometric deliberation, first order theory has been formulated as a deformation of topological BF theory [49]. In the electromagnetic U(1) theory, the 1-form u a can also be interpreted in terms of the electromagnetic excitation H, appearing from electric current conservation as dJ = 0 ⇒ J = dH. The excitation, both in the equations of motion and the action, appears on a premetric level, see Ref. [26] for details. In this light, rather than axiomatically defining correspondence between the excitation and dual field strength via the electromagnetic spacetime relation, in a first order theory this correspondence appears because of the specific form of the action and the excitation itself can be regarded as a fundamental field of the theory. In global formulations, u a is immediately reminiscent of an extra coframe field, albeit Yang-Mills charged. This can be further related to bimetric theory, see Ref. [50] for proposals, although for u a to be considered a proper coframe-like object, the implications of expansion w.r.t. u a require investigation 5 . Curiously enough, the 1-form u a allows to define a Yang-Mills derived pseudometric for any Yang-Mills theory g YM µν = Tr(u a µ u b ν η ab ) .(31) This is not necessarily canonical, however, as it is possible to derive similar structure from the interior product of the field strength 2-form F with respect to a vector basis, and likewise the interpretation of g YM µν is unclear. Quite interestingly, in D = 3 dimensions similar metric construction connects with gravity rather closely, see [53,54]. Finally, as expected, in a gravitational context the energy-momentum derived from e a also agrees with the usual Yang-Mills energy-momentum tensor. Variation w.r.t. the coframe e a , yields the canonical energy-momentum 3-form θ a := Tr G ǫ abcd e b ∧ u c ∧ u d − u a ∧ F .(32) One can also derive the energy-momentum from (14) and (15), and the equivalence is simplest seen via index calculations in the dual densities, investigating the component expression. Assuming u a is on-shell, so that Eq. (26) holds, and a regular metric phase, algebraic manipulation yields altogether Tr ε abcd u c i u d j ε bijk − u ab 1 2 F ij ε bijk = Tr − (F ai F ki − 1 4 δ k a F ij F ij ) .(33) That is T µν ∼ Tr F µρ F ν ρ − 1 4 g µν F ρσ F ρσ ,(34) therefore, up to conventions, the energy-momentum tensor agrees with that of the usual theory. 5 Interestingly, the action features precisely the partially massless interaction term of the Hassan-Rosen ghost-free bimetric theory [51]. Besides the partially massless term, there exist two other viable interactions [52]. It might be interesting to explore how including these terms would modify the first order Yang-Mills theory, and whether the bimetric modified gravity could perhaps be interpreted in this connection with particle physics. B. A Yang-Mills charged transformation Equivalently, it is possible to instead introduce a Lie algebra valued 0-form G ab with antisymmetric Lorentz indices, via the action S = Tr 1 24 G ab G cd η ac η bd ǫ ijkl e i ∧ e j ∧ e k ∧ e l + G ab η ac η bd e c ∧ e d ∧ F .(35) The resulting equations of motion w.r.t. A and G ab are respectively D(G ab e a ∧ e b ) = J,(36)1 12 G ab ε ijkl e i ∧ e j ∧ e k ∧ e l + e a ∧ e b ∧ F = 0.(37) The analysis mirrors that of the previous section, going to the dual basis yields G ab e a ∧ e b = * F,(38) and e.g. analyzing the energy-momentum would proceed in a similar way. Likewise the action (35) makes no reference to the inverse metric, thus allows for a singular phase and could be considered pregeometric. The main difference lies in the interpretation, discussed in section IV. Equivalently we can construct S = Tr 1 4 (G ab η ac η bd e a ∧ e b ) ∧ (G ij ǫ ijkl e k ∧ e l ) + (G ab η ac η bd e c ∧ e d ) ∧ F ,(39) where the coupling of G ab to the surface element e a ∧ e b is more explicit, similar to the appearance of u a ∧ e b in the 1-form approach. Either approach is classically equivalent to second order Yang-Mills theory, and there is evidence of quantum equivalence as well. Particularly in flat space, the differential form description can be reduced to (a variant of) first order Yang-Mills theory in the usual index formalism, for which the study of quantum properties and equivalence was discussed earlier, see e.g. [45]. The quantum properties of the first order formalism while remaining in curved spacetime require further investigation, however, and a deeper overview of applying the many possible quantization schemes promises to be insightful as well. C. Vector substructure In analogy with Ref. [9], we can produce the coframe from a single Yang-Mills charged vector φ a , such that u a = Dφ a = dφ a + ω a b φ b + [A, φ a ].(40) Therefore, rather than introducing a Lie-algebra valued 1-form u a , only a single Lorentz vector φ a is postulated. Explicitly, the action becomes S = Tr G 1 2 ǫ abcd Dφ a ∧ e b ∧ Dφ c ∧ e d + η ab Dφ a ∧ e b ∧ F ,(41) with the equations of motion for A and φ a respectively ε abcd [φ a , Dφ b ] ∧ e c ∧ e d + e a ∧ [φ a , F ] + D(Dφ a ∧ e a ) = J,(42)D ε abcd Dφ b ∧ e c ∧ e d + e a ∧ F = 0.(43) Further, note that it is possible to consider a single Dirac spinor ψ instead of a vector φ a , similarly to how various spinor-pregeometric approaches work with coframes. This is most Although the analysis is similar to earlier, the exterior covariant derivative yields extra effects, as was the case for the similar procedure in gravity. The second equation can be formally solved by introducing a Lie algebra valued 3-form X a such that DX a = 0; an integration constant, so to say. The formal solution ε abcd Dφ b ∧ e c ∧ e d + e a ∧ F = X a , DX a = 0,(44) can be contracted from the left or right by φ a . Then subtracting yields the commutator ε abcd [φ a , Dφ b ] ∧ e c ∧ e d + e a ∧ [φ a , F ] = [φ a , X a ].(45) 6 An attractive possibility would be to consider φ as a G-vector, andψ as its dual G-vector, so thatψγ a ψ would have its indices in the adjoint as desired. This would essentially realise the same trick internally as we are now performing externally, by considering the Lorentz vector φ a instead of the Lorentz adjoint G a b . The trick would considerably reduce the number of independent variational degrees of freedom. However, it remains to be investigated whether the gauge-invariant degrees of freedom in the (dual) field strength can be consistently encoded within one G-vector spinor (or whether we may would have to e.g. consider the ψ andχ =ψ as independent). Therefore the Yang-Mills equation prototype includes an arbitrary 3-form integration constant in the commutator, [φ a , X a ] + D(Dφ a ∧ e a ) = J. In case of Abelian groups [φ a , X a ] = 0, while otherwise this term is nontrivial. Establishing correspondence between Dφ a ∧e a and the dual field strength 2-form proceeds in analogy with the previous sections, but in the presence of the extra 3-form X a . For convenience, let Dφ a = u a . Investigating the dual density of Eq. (44) starts from ε abcd u b i ε icdk + 1 2 F cd ε a cdk = 1 3! X aicd ε icdk(47) and results in 2u ka = 1 2 F cd ε cdak − 1 3! X a icd ε icdk ,(48) so in global form u a ∧ e a = * F + 1 2 ( * X a ) ∧ e a .(49) Therefore, the equations of motion of the vector potential A are [φ a , X a ] + 1 2 D( * X a ∧ e a ) + D * F = J. The theory is equivalent to usual Yang-Mills theory when [φ a , X a ] + 1 2 D( * X a ∧ e a ) = 0. A particular solution is X a = 0 ⇒ DX a = 0, so a proper Yang-Mills limit exists. A solution for the φ a should generally exist, since (49) has the same number of equations as unknowns. Looking at this a bit more explicitly, in the geometric phase we can write the components of u ka in (48) in some coordinate system as u [µν] = −g α[µ ∇ ν] φ α . In the very simplest case of flat space g µν = η µν , ∇ µ = ∂ µ , Abelian group G = U(1) and setting X a = 0, the solution is simply that φ µ =Ã µ (up to the U(1) ambiguity φ µ → φ µ + ∂ µ ϕ), whereÃ µ is the electromagnetic gauge field corresponding to the dual field strength. In the generic case the solution for φ µ = e a µ φ a will be a nonlinear function of the gravitational fields, the gauge fields and the X a -field, but there is no obvious reason why such a solution shouldn't always exist. In general when X a = 0 the meaning of the additional terms is not particularly clear. In the case of Abelian U(1) theory, however, there is a simple interpretation in terms of vacuum magnetization and polarization. In usual electromagnetic theory, the current 3-form J splits into the contribution J mat from bound electric current inside matter and an external current J ext as J = J mat + J ext .(52) The total current is conserved, dJ = 0, and it is assumed there is no conversion between internal and external charges. Therefore it is meaningful to introduce a matter excitation H mat such that J mat = dH mat , see Ref. [26] for details. This excitation can then be split in terms of magnetization and polarization after proceeding with a 1+3 decomposition. Let spacetime have a local 1+3 foliation, parametrized by a monotonously increasing variable σ. Therefore topologically 7 let the differentiable manifold M = Σ × R. The vector field n corresponding to a congruence of observer worldlines is defined by n dσ = L n σ = −1.(53) Any p-form ω can be split into a component longitudinal to n by ⊥ ω = dσ ∧ ω ⊥ , ω ⊥ = n ω,(54) the remainder being the transverse component, ω = (1 − ⊥ )ω = n (dσ ∧ ω), n ω = 0.(55) Therefore ω = ⊥ ω + ω = dσ ∧ ω ⊥ + ω.(56) Applying this procedure to the internal excitation H mat = ⊥ H mat + H mat = −H mat ∧ dσ + D mat .(57) serves as the basis for defining the polarization 2-form P and magnetization 1-form M: D mat = −P,(58)H mat = M,(59) where the minus sign is convention. In the Abelian case, Eq. (51) reduces to d * F = J − 1 2 d( * X a ∧ e a ),(60) and we find 1 2 d( * X a ∧ e a ) is precisely of the current form J = dH, with the "vacuum excitation" H vac = 1 2 * X a ∧ e a .(61) This can then be split into the magnetization and polarization components, as M vac = 1 2 ( * X a (n)e a − e a (n) * X a ) ,(62)P vac = 1 2 e a ∧ * X a ,(63) where e a is the spatial coframe. In essence we have found that a suitable reformulation of electromagnetism allows for magnetization and polarization to appear simply as integration constants, similarly to how dark matter is described in the Khronon theory proposed in Ref. [9]. Defining analogues of magnetization and polarization for non-Abelian Yang-Mills theory is not conventional, however. By defining the dressed field strength, F = F − * H vac , we can rewrite equation (60) as d * F = J .(64) By straightforward manipulations, we can show that in the geometric phase the energymomentum tensor of the generic Yang-Mills theory becomes T µν = Tr 1 2 F µ α F να + F (µ α F ν)α − 1 4 g µν F αβ F αβ .(65) The vacuum excitation modifies the energy-momentum sources in an interesting way. It may break the conformal invariance of the gauge theory if T µ µ = − 1 2 Tr( * F µν H vac µν ) = 0. If the field strength vanishes, F µν = 0, the vacuum may still contain energy due to the H vac µν . On the other hand, the solution for the gauge field strength F µν = 0 is always available in the absence of material sources J µ = 0, and this solution has zero energy. In the next section III D we will see that when coupled to Khronon gravity, the X a -field can further generate a "hypermomentum" source. The resulting magnetization and polarization is not completely arbitrary, but is constrained by DX a = 0. It would be attractive to interpret this in terms of "covariantly closed" forms, but as in general D 2 = 0, this does not produce a proper cohomology theory. Rather, DX a = 0, and generally Dω = 0 for arbitrary forms ω, could be taken as the generalization of requiring X a or ω to be covariantly constant, compare with the exterior covariant derivative D on some vector bundle E mapping D : v ∈ TM → D v , such that for any section s ∈ Γ(E) and vector field v we have Ds(v) = D v s, among other axioms. Therefore the difficulty of solving DX a = 0, e.g. in terms of differential equations, should be of the same class as finding covariantly constant fields, possibly devolving into (numeric) integration in charts. The interior product yields the precise relation of X a to the vacuum excitation 2-form H vac , thus to the magnetization and polarization. Let @ a correspond to the vector basis dual to the coframe e a , that is e a (@ b ) = δ a b . Then directly H vac = 1 2 H ij e i ∧ e j = − 1 2 (@ a H vac ) ∧ e a = 1 2 * X a ∧ e a .(66) Therefore X a = − * (@ a H vac ).(67) In the regular metric phase, instead of X a , we could consider the Hodge dual Y a = * X a as the introduced fundamental quantity. The condition DX a = 0 reads in terms of the excitation as D( * (@ a H vac )) = 0, D. Pregeometric Yang-Mills theory and Khronon gravity The Cartan Khronon theory of gravity is based upon a new approach to the problem of time. Space and time emerge in a spontaneous symmetry breaking which might ultimately be connected to the collapse of the wavefunction. At the formal level, the key is the reduction of the coframe to the Cartan Khronon field τ a such that e a = Dτ a [9] (we will briefly discuss the spinor version of the formulation in Section IV). Then, a canonical clock field is built into the structure of the theory, and rather than introducing some σ by hand in the decomposition introduced above, we can identify σ = i η ab τ a τ b . The coupling of the pregeometric Yang-Mills theory to the Cartan Khronon gravity reveals further physical properties of the newly found 3-form X a . Consider the coupled theory S = Dτ a ∧ Dτ b + R ab + Tr Dφ a ∧ Dτ b 1 2 ε abcd Dφ c ∧ Dτ d + η ab F .(69) The equations of motion for the (Cartan) Khronon and the gravitational connection are, respectively, D + R a b ∧ Dτ b − θ a = 0 ,(70)1 2 D + Dτ [a ∧ Dτ b] = τ [a + R b] c ∧ Dτ c − τ [a θ b] − Tr φ [a X b] .(71) By integrating the first equation, we obtain the dark matter 3-form M a such that DM a = 0. Using this solution to simplify the second equation, we get + R a b ∧ Dτ b = θ a + M a ,(72)1 2 D + Dτ [a ∧ Dτ b] = τ [a M b] − Tr φ [a X b] .(73) In the second equation the LHS is self-dual, and thus must be the RHS. Assuming that this applies to each term separately, we see that whereas the dark matter 3-form satisfies Let us consider the situation that the Khronon and the iso-Khronon are aligned, i.e. φ a ∼ τ a (obviously, this implies the isotropy of φ a in the internal space). It immediately follows that − (X [a τ b] ) = 0. Then, one can deduce the two consequences of the conservation equation DX a = 0 (see [20]). Firstly, the 3-form X a is a purely spatial volume form. Secondly, its volume integral is a constant. However, in this case the physical effect of vacuum excitation vanishes, even though the 3-form X a may exist as a non-trivial 3-form. This is most easily seen in the time gauge τ a = τ δ a 0 , where we may write X 0 = X ⋆e 0 for some scalar X, and X I = 0, and obtain that dX 0 = 0. Notationwise, ⋆ is the internal dual, and the capital Latin letters are used for the spatial Lorentz indices. But firstly, we see that the time-like coframe e 0 is purely longitudinal e 0 = 0, since e 0 = Dτ 0 = dτ = −dσ. Secondly, the only non-vanishing component of * X a is * X 0 = * Xǫ IJK e I ∧ e J ∧ e K /6 = X( * ⋆ e 0 )/6 = Xe 0 is also longitudinal. So the H vac in (61) vanishes. Thus, the components of X a that satisfy the self-duality condition with respect to the Cartan Khronon τ a do not result in vacuum magnetization or polarization. In particular, if the iso-Khronon φ a picks up the time direction preferred by the symmetry-breaking field τ , the effect of X a is trivialised. When this is not the case, the theory predicts also novel gravitational effects due to the vacuum excitation. From (72) we see that the Yang-Mills fields contribute to the energymomentum and thus source gravity as usual. However, there is different kind of contribution in (73). Though it appears to be similar to the effect of dark matter, under closer inspection it turns out that this is not the case. Again, it is useful to adapt the system into the time gauge τ a = τ δ a 0 . In this gauge, the components of the self-dual curvature reduce to the triad of curvature two-forms R I . The independent components of the anti-self dual curvature are then encoded into the triad of torsion 2-forms T I . In the end, the field equations (72,73) can be re-expressed in the gauge-fixed form R I ∧ e I = −iθ 0 − iM 0 ,(74)R I ∧ dτ + iǫ I JK R J ∧ e K = −θ I ,(75)T I ∧ dτ − iǫ I JK T J ∧ e K = 2Tr φ [I X 0] .(76) The two first equations above are the energy and the momentum equations, respectively. As expected, the dark matter 3-form is associated with effectively pure energy density, and its effective pressure is identically zero. The new effect of the excitation 3-form X a is apparent in the last equation, where it appears as a source of torsion. Thus, the Yang-Mills vacuum excitation can generate nontrivial gravitational "hypermomentum" [56]. This effect disappears when the 3-form X a is aligned with the iso-Khronon such that φ a ∼ X a . The phenomenological implications of the Yang-Mills hypermomentum would be very interesting to explore, but we must leave that for future studies. E. Scalar fields The scalar field action in a singular metric regime runs into the same problem as the Yang-Mills action. Similarly, a first order theory can be defined for the field φ by introducing an auxiliary field G abc with totally antisymmetric indices, S = 1 4 G abc G abc + U(φ) ε ijkl e i ∧ e j ∧ e k ∧ e l + G abc e a ∧ e b ∧ e c ∧ dφ(77) again producing a 2-step kinetic cycle. Varying by φ and G abc produces respectively d(G abc e a ∧ e b ∧ e c ) + U ′ (φ)ε abcd e a ∧ e b ∧ e c ∧ e d = 0,(78)1 4 G abc ε ijkl e i ∧ e j ∧ e k ∧ e l + e a ∧ e b ∧ e c ∧ dφ = 0,(79) which is the prototype wave equation and the auxiliary equation. Everything goes by the usual procedure outlined earlier, enforcing * dφ = 1 3! d k φε kabc e a ∧ e b ∧ e c ,(80) and yielding the wave equation − d * dφ + U ′ (φ)ε abcd e a ∧ e b ∧ e c ∧ e d = 0.(81) Therefore all bosonic actions have a pregeometric first order formalism readily available. The interest is then of building a good theory of pregeometry. IV. PATH TO UNIFICATION First order formalism in gravity, that is the Palatini formulation in terms of (co)frames and connections, is well known and has been extensively studied, while the Yang-Mills analogue does not appear to be as popular. It is worth emphasizing how similar these theories appear to the exterior formulation of complex self-dual GR, while the remaining anti-self-dual component is appealing for unification attempts. Complex GR considers the complexified tensor bundle T C = r,s T r s (M) ⊗ C(82) over a real manifold M, see e.g. Ref. [57] for an overview. The structure group becomes SO(1, 3) C ∼ = SO(4) C , while the fields become complex-valued, i.e. sections of a complex bundle. In addition to investigating Hodge dualization and its eigenforms, we can define a dualization operation ⋆ in the complexified Lie algebra so(1, 3) C such that ⋆ ω ab = 1 2 ǫ ab cd σ cd ,(83) decomposing the Lie algebra into self-dual anti-self-dual components so(1, 3) C = so(1, 3) + C ⊕ so(1, 3) − C ,(84) such that so(1, 3) ± C = {ω ∈ so(1, 3)\| ⋆ ω = ±iω}.(85) The corresponding projector is P ± = 1 2 (1 ∓ i⋆).(86) Note so(1, 3) (±) C are simple Lie algebras, while so(1, 3) C is semi-simple. Significantly, the Palatini action S C = 1 2κ * (e a ∧ e b ) ∧ R ab(87) decomposes into S C = S + C + S − C = i 2κ e a ∧ e b ∧ + R ab − i 2κ e a ∧ e b ∧ − R ab(88) and it suffices to consider only one of the actions in the decomposition, as the stationary points of S C and S ± C lie over the same coframe fields. Working with the self-dual action is also the basis for Ashtekar's variables [58,59], establishing phase-space correspondence with SU(2) Yang-Mills theory. The natural continuation is with the cosmological constant Λ, which makes the similarity with Yang-Mills theory plain. The complete first order Λ-Einstein-Yang-Mills theory action is then, S = 1 2 Tr G κ −1 − 2Λǫ abcd e a ∧ e b ∧ e c ∧ e d + e a ∧ e b ∧ i + R ab + + 1 2 ǫ abcd u a ∧ e b ∧ u c ∧ e d + u a ∧ e b ∧ η ab F = = 1 2 Tr G (iκ −1 e a ∧ e b + u a ∧ e b ) ∧ + R ab + η ab F + ǫ abcd 2iΛe c ∧ e d + 1 2 u c ∧ e d(89) although the use of purely self-dual surface elements + (e a ∧e b ) in the GR action is conceivable as well. The Yang-Mills action can include a dimensionless constant, which is here set to unity. The many terms in the action can be grouped in various ways. For instance, in principle η ab could be put together with the vector potential A → η ab A, as to correspond to trace components in a connection-like 1-form ω ab = + ω ab + η ab A,(90) and furthermorẽ R ab = d( + ω ab + η ab A) + ( + ω ac + η ac A) ∧ ( + ω c b + δ c b A) = + R ab + η ab F.(91) When restricting to the case G = U(1), the gravitoelectromagnetic unification is achieved neatly, sinceR ab is precisely the (self-dual projection of the) Weyl extension of the Lorentz curvature. However, when considering the more complete unification along the lines e.g. SO(3, 13) → SO(1, 3) × SO(10) [60,61], which appears quite natural and attractive in this context [23], the generalization of the Weyl 1-form A into the adjoint of a non-Abelian algebra such as the SO(10) forces to rethink the most conventional GraviGUT schemes. The division into the surface element e a ∧ e b , resp. u a ∧ e b , and gauge curvature + R + F is clear, however. If Λ = 1 4 κ −1 , then the replacements iκ −1 e a ∧ e b → u a ∧ e b , + R ab → η ab F(92) would transform the gravitational action to a Yang-Mills one. It would be noteworthy to formulate this as a rigorous gauge principle, similar to how fermionic fields couple to gauge bosons, but this doesn't appear simple or unambiguous. The value of the cosmological constant is a problem, or alternatively a hint of the precise structure of the underlying theory. Generally it is expected to be of QFT origin, but the calculated value so far is sharply disconnected from measurement, see e.g. Ref. [62] for an overview. However, if it is to believed that gravity forms a unified theory with the Standard Model (or a suitable extension), then this analysis supports a fundamental origin for the cosmological constant, possibly related to symmetry breaking. The gauge group trace really only applies to the Yang-Mills term, but it is formally possible to introduce traceless Lorentz generators in the vector representation (J ab ) cd = i 2 (η ac η bd − η ad η bc ),(93) so that the Lorentz trace of the product Tr(J ab J uv ) = (J ab ) i j (J uv ) j i = δ [u a δ v] b ,(94) and the total action involves both traces, S = 1 2 Tr (e a ∧e b )(iκ −1 J ab +G ab )∧ + R ij J ij +F +(e c ∧e d )ǫ ijcd 2iΛJ ij + 1 4 G ij ,(95) implying that the proper path forward would be through some Lie algebra scheme, particularly when separating the (here) Λ = 1 8 κ −1 component. In this case, introducing an infinitesimal-like transformation k ij ab = iκ −1 δ a i δ b j + 1 4 G ij η ab + 1 4 η ij G ab ,(96) yields the action S = 1 2 Tr (e a ∧ e b )k ab ij ∧ + R ij + η ij F + k ij uv (e c ∧ e d )ǫ cduv ,(97) and the interpretation is that G ab is an infinitesimal-like surface excitation, or the surface element e a ∧ e b is transformed, such that the invariant part corresponds to gravity and the change to Yang-Mills theory. It is ambiguous which is the best interpretation for the auxiliary field, thus also leaving ambiguity in how exactly a unified formulation should arise. In the various formulations in this paper, the new field can be construed as either a 0-form or 1-form field and potentially physically meaningful or not; as a Yang-Mills charged coframe; as a linear transformation between Lorentz and Yang-Mills algebra; as Lagrange multiplier-like fields; as dynamical components of the dual field strength; as substructure of the electromagnetic excitation; as an additional set of vectors to the Khronon of Ref. [9]; as an infinitesimal Yang-Mills charged surface excitation. At the very least, in a Gravity -Standard Model unification, it can be assumed that the first order formulation represents the symmetry broken phase. In another approach, the 3 leftover anti-self-dual generators could be mapped to Yang-Mills generators [20]. For instance, it is possible to introduce a Yang-Mills charged "mixing matrix" z ab , in particular the complex setting of electromagnetic U(1) theory already appears in the complexified gravity formulation. So it could be defined A = z ab− ω ab , F = d(z ab− ω ab ).(98) In so simple a formulation, this is not a proper unified theory as commonly understood [60], i.e. roughly where the vacuum expectation value of a given order parameter selects commuting subgroups of a larger gauge group. Alternatively, if instead the target was SU(2) of the weak interaction, this could be adapted to some alternative of graviweak unification; a proper graviweak formalism is e.g. discussed in [63], considering SO(4) C and the soldering form as an order parameter. The theory of gravity is quite rich in similar theories. In addition to Palatini formalism, Plebanski formalism [64] S Plebanski = B ab ∧ R ab − 1 2 φ abcd B ab ∧ B cd(99) has to be noted as well. In comparison, the surface element is replaced with a single 2-form e a ∧ e b → B ab and fixed by essentially a Lagrange multiplier φ abcd = φ cdab = −φ bacd ,(100) which enforces that on-shell B ab agrees with + (e a ∧ e b ); see also Refs. [65,66] and the references therein. Plebanski theory was likewise utilized in unification [67], where the embedding into a larger gauge group and the addition of an extra term produces GR with a Yang-Mills action. In some spin foam approaches to quantum gravity, one considers only the first term in the Plebanski action (99), and may then at a suitable point impose the so called simplicity constraint that reduces the two-form to the exterior product of the tetrad one-forms [68]. While this establishes an appealing connection between gravity and topological QFT without local degrees of freedom, the tetrad is a quite complicated, mixed-index 16-component object that is completely alien to standard Yang-Mills theory. The insight of Ref. [9] that the tetrad should rather emerge as a covariant derivative of a (Lorentz-charged) Higgs-like scalar is not new, but goes back, via Akama [11] and others to the original generalization of F. Klein's geometrical framework to describe symmetries of physics byÉ. Cartan [69]. Since we can reduce the Lorentz-charged scalar to a Dirac bispinor, the possibility arises that the metric structure could be reduced to a property of matter fields. Further, the dynamics for this metric structure could arise solely from the quantum fluctuations of matter fields, as famously shown by Sakharov [70]. Perhaps all the fundamental interactions emerge in a similar fashion from a purely material origin? To recapitulate the above-described "series of further and yet further simplicity constraints" that connects the topological BF theory to what we may call the Dirac-Cartan Khronon gravity: B ab → e a ∧ e b → Dτ a ∧ Dτ b → D(ψγ a ψ) ∧ D(ψγ b ψ) .(101) Let us just in passing mention that the gravity theory [9] has the remarkable formulation, which is both quartic in the primordial spinor and quartic in the gauge-covariant derivative 8 S = i (ψγ a ψ)D + D + DD(ψγ a ψ) .(102) A partial integration and a couple of steps back in the chain (101) brings this into the familiar self-dual Palatini form. Throughout this article, we've had in mind that the more fundamental formulation of gravity should be considered in terms of the primordial ψ rather than e a . As we saw in Section III D, coupling to Khronon gravity can reveal further properties of the pregeometric Yang-Mills theory. Of course, any unification attempt with gravity and particle physics is constrained by various theorems, including the commonly cited Coleman-Mandula theorem [71], which implies that the symmetry group of the underlying QFT can only be the direct product of the Poincaré and an internal symmetry group. Although the assumptions of the Coleman-Mandula theorem appear natural, the common implication is not unavoidable. Ref. [67] avoids it as the Coleman-Mandula theorem requires the S-matrix symmetries include global Poincare invariance, while the proposal held no global symmetries; likewise in the broken phase of Ref. [63], the residual symmetry is precisely the required global Lorentz and local internal symmetry -the Coleman-Mandula theorem requires the existence of a Minkowski metric, while in the pregeometric regime with a vanishing soldering form, there is no explicit metric on the manifold. This also agrees with our discussion, implying that first order theory arises naturally in the broken phase. V. CONCLUSION The appearance of Hodge dualization in the actions describing matter and gravity can be avoided by using the first order formalism. For gravity, this proceeds from Palatini to a self-dual formulation. Spinor fields themselves require no inverse metric when explicitly 8 In our convention, the operator D acts only to the right. In an alternative convention, e.g. D(ψγ a ψ) in (101) would readψγ a ↔ Dψ. working with so(1, 3) indices. For the remaining bosonic scalar and Yang-Mills actions, the polynomial first order formulation then goes through a two-step process, such that the usual wave or Yang-Mills equations appear on-shell, while reference to Hodge dualization is in effect replaced with so(1, 3)-dualization, with no necessity of the inverse metric. The results obtained are consistent with those of the usual theories at a classical level. Consistency at the quantum level has been earlier investigated in usual index-notation approaches, and, depending on the precise formulation, is either immediately applicable or expected to hold barring the gravitational sector. Notably, we connect the fundamental axiom of charge conservation (and thus the appearance of the electromagnetic excitation) in the premetric programme to a fundamental field in the pregeometric programme. Generalizing previous Cartan-geometric results for dark matter and gravity, first order Yang-Mills formalism admits the description where a single Lorentz vector is introduced, rather than a 1-form. This applies to both Abelian and non-Abelian theories, requires introducing the least amount of extra degrees of freedom and produces interesting effects. In the U(1) theory of electromagnetism, the new terms can be interpreted as vacuum magnetization and polarization, while the generalization of such an interpretation to non-Abelian theory is not conventional. It is unlikely that more minimal schemes of this method exist, at least in 4 dimensions and excepting possible internal changes to the vector, as the method depends on coupling to Levi-Civita symbols via a Lorentz index. At the very least, the vector approach has a consistent phase with usual Yang-Mills theory, and deserves further investigation, e.g. in Cartan geometry. Comparing with first order formalism in the theory of gravity, there appears a strong case for some kind of dual or unified description of gravitation with gauge theories in complexified theory. However, the precise path remains yet ambiguous, not in small part due to the many possible interpretations of the auxiliary field. It is attractive to interpret it as some Yang-Mills charged coframe, but it could likewise be a transformation between Lie algebras or some excitation. Likewise in this duality, the value of the cosmological constant is curious. It remains to be seen whether this is coincidental or insightful, and what might resolve the problems. In the context of the results obtained, a natural direction appears to investigate Lie algebras and symmetry breaking, and perhaps dimension-dependence of this formulation. By the work of Ashtekar [58,59], we know that the Hamiltonian form of self-dual GR is closely related to Yang-Mills theory. Likewise we see that the Lagrangian of self-dual GR with the cosmological constant is very similar to that of first first-order Yang-Mills theory. Starting in a complexified first order theory seems promising, with the usual theories possibly only appearing in the end after symmetry breaking and applying suitable reality conditions. Let us wrap up the article. By construction, QFT is an effective framework that should robustly approximate physics up to a given energy scale [1]. A first order reformulation of the Standard Model could be a natural step towards a possibly more fundamental theory. Further, there are rather compelling arguments, beginning from the elementary, classical reasoning that is the basis of the premetric program, and extending to today's cutting-edge speculations about the nature of quantum gravity, that the metric tensor is an emergent field that may even vanish in its ground state. In this cross-lighting, it may seem surprising that a more systematic investigation of pregeometric first order Yang-Mills theory has not yet been undertaken in the Literature. Our basic finding is that the field excitation tensor H effectively becomes a fundamental field of the gauge theory, on the same footing as the connection A that gives rise to the field strength tensor F = DA. We considered several formulations of this principle, suggesting several new directions to pursue, but they can be all classified according to which kind of field is considered to be the variational degree of freedom. • The standard formulation imposes H = * F without dynamical variation. In the premetric language, this is the axiom of constitutive law. • The coframe variation considers u a , and results in H ≈ u a ∧ e a . Whilst perhaps uneconomical, such theories suggest interesting connections to bimetric gravity on one side, and to geometric formulations of QCD on the other. • The group element G ab as a variational degree of freedom results in H ≈ G ab e a ∧ e b . This approach allows the interpretation of the unified theory (97) as a surface excitation of a topological action. • The vector substructure φ a results in a relation H + X ≈ Dφ a ∧e a , where X is an extra 2-form, that (at least in the electromagnetic case) allows the interpretation in terms of vacuum magnetization and polarization, as well as its surprising analogy with the cosmological dark matter. • The spinor substructure ψ is an alternative to the vector substructure, based on that φ a ≈ψγ a ψ. This is the approach we intend to study at more depth in the future. The unification of the primordial spinor gravity (102) and the pregeometric Yang-Mills theory reduced to a spinor substructure might be a step towards the lower-level QFT that we have been seeking. In a complementary approach, ascending from the first principles towards the higher level of a dynamical QFT, progress is being made indeed (rather than e.g. qubits) in terms of fermions [72]. Each brick bridging the gap between these levels is paving the way for a new paradigm. Most of the various descriptions are equivalent, emphasizing different properties, and it is possible to consistently work in local charts if necessary. Proceeding forward, variation with respect to φ produces the Klein-Gordon equation, while the variation with respect to A yields the inhomogeneous Yang-Mills equation D * F = J. The homogeneous equation is trivial as the Bianchi identity DF = 0 is satisfied by construction of the curvature 2-form. clear in the commutative case of U(1) theory, where we could consider a substitution of the type φ a →ψγ a ψ, with the objects in the adjoint representation being invariant under transformations. The non-Abelian case, however, requires more structure than a single spinor ψ, cf. extra Yang-Mills indices 6 . which is rather a co-covariant constancy condition, if a covariant codifferential δ D = * D * were to be introduced. The interpretation of terms in the non-Abelian case remains unclear, however. From the above we see that though the dressed field strength F = F − * H vac satisfies (64) (in the premetric context called the first fundamental equation), only the contracted field strength F a = @ a F − * (@ a H vac ) would satisfy the adapted Bianchi identity (respectively, the second fundamental equation), DF a = − # T a F = 0 in the absence of torsion (c.f. Lemma 1 of Ref.[55]). the self-duality condition − (τ [a M b] ) = 0 wrt the Khronon τ a , the vacuum excitation 3-form satisfies the self-duality condition Tr − (φ [a X b] ) = 0 wrt the iso-Khronon φ a . This justifies the interpretation of the cosmological dark matter as the gravitational analogy to the vacuum magnetization/polarization in the internal gauge theory. We refer the reader to e.g. Ref.[31] for a discussion pertaining to spin geometry and the construction of spinor bundles. Notably, it is not possible to define such structure on completely arbitrary manifolds, as generally there can be topological obstructions. Pregeometry could liberate from such obstructions. (One can comb a hairy ball if it is allowed to have a bald spot.) Note in particular that the (prototype) coframe e a used here need not be regular everywhere and is only required to allow a proper expansion with respect to it in the nonsingular phase. In this sense, the coframe 1-forms produce the geometric structure of interest, rather than just being part of the description of spacetime. For conciseness we will refer to e a just as a coframe rather than a pseudo-coframe. When a metric is available, J is usually taken to be the Hodge dual of a current 1-form, so J = * j. 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[ "Polarization dependence and symmetry analysis in indirect K-edge RIXS", "Polarization dependence and symmetry analysis in indirect K-edge RIXS" ]	[ "G Chabot-Couture \nDepartment of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA\n", "J N Hancock \nDépartement de Physique de la Matière Condensée\nUniversité de Genève\nCH-1211GenèveSwitzerland\n", "P K Mang \nDepartment of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA\n", "D M Casa \nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n", "T Gog \nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n", "M Greven \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n" ]	[ "Department of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA", "Département de Physique de la Matière Condensée\nUniversité de Genève\nCH-1211GenèveSwitzerland", "Department of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA", "School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA" ]	[]	We present a study of the charge-transfer excitations in undoped Nd 2 CuO 4 using resonant inelastic X-ray scattering (RIXS) at the Cu K-edge. At the Brillouin zone center, azimuthal scans that rotate the incidentphoton polarization within the CuO 2 planes reveal weak fourfold oscillations. A comparison of spectra taken in different Brillouin zones reveals a spectral weight decrease at high energy loss from forward-to back-scattering. We show that these are scattered-photon polarization effects related to the properties of the observed electronic excitations. Each of the two effects constitutes about 10% of the inelastic signal while the '4p-as-spectator' approximation describes the remaining 80%. Raman selection rules can accurately model our data, and we conclude that the observed polarization-dependent RIXS features correspond to E g and B 1g charge-transfer excitations to non-bonding oxygen 2p bands, above 2.5 eV energy-loss, and to an E g d → d excitation at 1.65 eV.	10.1103/physrevb.82.035113	[ "https://arxiv.org/pdf/1003.5348v1.pdf" ]	55,089,253	1003.5348	f9e871568817ea924cac3e0d13cd5110349d217b	Polarization dependence and symmetry analysis in indirect K-edge RIXS 28 Mar 2010 G Chabot-Couture Department of Applied Physics Stanford University 94305StanfordCaliforniaUSA J N Hancock Département de Physique de la Matière Condensée Université de Genève CH-1211GenèveSwitzerland P K Mang Department of Applied Physics Stanford University 94305StanfordCaliforniaUSA D M Casa Advanced Photon Source Argonne National Laboratory 60439ArgonneIllinoisUSA T Gog Advanced Photon Source Argonne National Laboratory 60439ArgonneIllinoisUSA M Greven School of Physics and Astronomy University of Minnesota 55455MinneapolisMinnesotaUSA Polarization dependence and symmetry analysis in indirect K-edge RIXS 28 Mar 2010(Dated: March 30, 2010)arXiv:1003.5348v1 [cond-mat.supr-con] We present a study of the charge-transfer excitations in undoped Nd 2 CuO 4 using resonant inelastic X-ray scattering (RIXS) at the Cu K-edge. At the Brillouin zone center, azimuthal scans that rotate the incidentphoton polarization within the CuO 2 planes reveal weak fourfold oscillations. A comparison of spectra taken in different Brillouin zones reveals a spectral weight decrease at high energy loss from forward-to back-scattering. We show that these are scattered-photon polarization effects related to the properties of the observed electronic excitations. Each of the two effects constitutes about 10% of the inelastic signal while the '4p-as-spectator' approximation describes the remaining 80%. Raman selection rules can accurately model our data, and we conclude that the observed polarization-dependent RIXS features correspond to E g and B 1g charge-transfer excitations to non-bonding oxygen 2p bands, above 2.5 eV energy-loss, and to an E g d → d excitation at 1.65 eV. Raman scattering and optical spectroscopy have enabled tremendous contributions to the study of condensed matter systems. Both probes use ∼ 1 eV light and are limited to essentially zero momentum transfer. In order to investigate the charge response of a material throughout the Brillouin zone, photons in the X-ray regime must be used. X-ray Raman scattering, more commonly referred to as resonant inelastic X-ray scattering (RIXS), allows the measurement of the momentum dependence of charge excitations. Even though it has successfully been used to study the physics of a wide array of systems, [1][2][3][4][5] this resonant technique is relatively new and there is still much to learn about the details of its cross section. One of the strengths of conventional Raman scattering derives from the fact that the technique allows the determination of the symmetry of excitations by selecting the polarization of the incident and scattered photons. Only polarization dependent soft (or direct 6 ) X-ray RIXS studies, at the O K-edge or the Cu L-and M-edges of the cuprates, [7][8][9][10][11] have shown that photon polarization can be used to select different electronic excitations. In these cases, the photon polarization effects are understood to come from the direct interaction of the photoelectron, excited into the valence system, with the valence electrons. This is why this type of RIXS is commonly referred to as direct. 6 On the other hand, tuning the incident energy to the Cu K-edge excites the photoelectron into the Cu 4p band approximately 10-20 eV above the 3d valence levels. The relatively large separation in energy from the valence levels as well as the large spatial extent of the 4p orbital are believed to prevent the photoelectron from interacting with the valence system, and this type of RIXS is in turn referred to as indirect. 6 Accordingly, theoretical models of this type of RIXS accordingly do not include possible interactions of the 4p photoelectron with the valence system, [12][13][14][15] which is commonly referred to as the '4p-as-spectator' approximation. RIXS investigations of the photon polarization effects have so far concluded that the incident polarization does not affect the probed valence band excitations, but only determines their specific resonance energies based on the crystal field levels of the 4p photoelectron, 14,16,17 in accordance with the 4p-as-spectator approximation. Nonetheless, there still exists no quantitative experimental evidence for an incident-photon polarization independence. Furthermore, the scattered-photon polarization effects at the transition metal K-edge have received very little attention, and it remains unclear whether they are present and can be used to learn about the symmetry of charge-transfer excitations in transition metal oxides. This paper is separated in three parts. In Section I, we discuss the photon-polarization dependence expected within the 4p-as-spectator approximation. In Sec. II, we study the photon-polarization and scattering-geometry dependence of RIXS at the Cu K-edge of Nd 2 CuO 4 and observe scatteredphoton polarization effects beyond the 4p-as-spectator approximation. In Sec. II A we present data obtained upon rotating the incident-photon polarization within the CuO 2 planes while in Sec. II B we present a comparison of zone-center spectra taken in different Brillouin zones. The results are discussed in Sec. III. The normalization procedure used to correct for sample self-absorption and compare RIXS signal across different scattering geometries is described in Appendix A. I. POLARIZATION DEPENDENCE OF THE RIXS CROSS SECTION Before presenting our experimental observations of photon polarization effects in RIXS, it is instructive to discuss what is expected according the current understanding of the indirect RIXS cross section. At the Cu K-edge, the RIXS process starts with the resonant absorption of an X-ray photon which creates a 1s core hole and a 4p photoelectron on the Cu site. During the lifetime of the core hole, before it recombines with the photoelectron and an X-ray photon is emitted, the core-hole interacts strongly with the valence system to create electronic excitations. The 4p photoelectron, on the other hand, is believed to be only a spectator and to evolve without interacting with the valence system. In order to determine the cross-sectional implications of this 4p-as-spectator approximation, we consider a simple model where the photoelectron's energy eigenstates are dictated by the crystal-field: E 4p x,y = ∆ and E 4p z = 0. While this approach neglects 4p band effects, in practice, the RIXS intensity is often confined within narrow and well-separated intervals in incident energy which should be approximated well by this two-level model. Since the initial and final states of the 4p photoelectron are determined by the initial (ǫ i ) and scattered (ǫ f ) photon polarizations, the transition amplitude (I 4p ) of the photoelectron is determined by its evolution within the crystal-field during the lifetime of the core hole ( /Γ) and contains polarization dependent factors that modulate the intensity of the excitations created by the core-hole: I 4p = ǫ f 1 E i − H − iΓ \|ǫ i 2 = ǫ x f ǫ x i + ǫ y f ǫ y i E i − ∆ − iΓ + ǫ z f ǫ z i E i − iΓ 2(1) In the case where ∆ ≫ Γ, the cross section simplifies to a polarizer: only the polarization components along the resonating 4p crystal-field level contribute to the inelastic signal: ǫ x f ǫ x i + ǫ y f ǫ y i 2 for an in-plane resonance (E i = ∆) and \|ǫ z f ǫ z i \| 2 for an out-of-plane resonance (E i = 0). In their study of CuO, Döring et al. 18 pointed out that, within the 4p-as-spectator approximation, the electric dipole absorption-emission matrix element is equivalent to the resonant elastic X-ray scattering cross section described in detail by Hannon et al. 19 . Furthermore, they successfully apply this approximation to describe the scattering angle dependence of the intensity of the 5.4 eV local charge-transfer excitation in horizontal scattering geometry. Equation 1 is a simplification of the formula presented in Ref. 19 but it succinctly captures the important effect of scattering geometry on the RIXS signal. Including different valence states, like the well-and poorlyscreened intermediate states, would improve it. In general, Eq. 1 shows that the inelastic intensity is maximized when both the incident-and scattered-photon polarizations are parallel to each other and point along a crystalfield eigenstate. These conditions can naturally be fulfilled in vertical-scattering geometry where both polarizations are perpendicular to the scattering plane (σ polarized). 2,17 For horizontal scattering geometry, with both polarizations within the scattering plane (π polarized), these conditions can only be approached for forward and backward scattering. 20,21 On the other hand, in horizontal-scattering geometry the elastic intensity can be minimized independently of the inelastic intensity. This can be an advantage since it allows the suppression of the elastic 'tail' due to the non-zero energy resolution, leading to a higher signal-to-background ratio. At a scattering angle (2θ) of 90 • , the non-resonant elastic contribution is zero because the polarization factor of Thompson scattering (ǫ f · ǫ i ) is zero. To have non-zero inelastic intensity, the non-degenerate 4p eigenstates can be used as cross-polarizers. By polarizing the incident photon between 4p crystal-field eigenstates of different energies, e.g., ǫ i \|\| x+ z, the inelastic signal will be dominated by the excitations cre-ated by the incident photon polarization component along the resonating 4p crystal-field eigenstate. Since this resonating component is not perpendicular to the scattered-photon polarization, the resonant inelastic signal will be detectable while the non-resonant elastic signal will be zero. Note that this effectively rotates the photon polarization 90 • . This should not be mistaken for a scattered-photon polarization effect where the 4p photoelectron scatters during the RIXS process. This type of scattering geometry is used extensively. 22,23 Based solely on the 4p polarization effects, the maximum RIXS intensity in this horizontal scattering geometry is observed when both incident and scattered photon polarization are at 45 • from the resonantly excited 4p eigenstate. In the tetragonal crystal-field symmetry of cuprates for example, Q \|\| c and 2θ = 90 • allows maximum RIXS intensity for both in-plane and out-of-plane resonances while minimizing the non-resonant contribution to the elastic line. As is shown in Appendix A, self-absorption effects will move this maximum of RIXS intensity towards a more grazing incidence angle (keeping 2θ = 90 • ) when the scattering surface is perpendicular to the [0 0 L] direction. II. PHOTON POLARIZATION EFFECTS IN NCO For our study of the scattering-geometry and photonpolarization dependence of RIXS, we chose Nd 2 CuO 4 (NCO), the tetragonal Mott-insulating parent compound of the electron-doped high-temperature superconductor Nd 2−x Ce x CuO 4 . A single crystal was prepared as described previously 24 and studied in its as-grown state. A larger piece from the same growth was measured with neutron scattering and found to have a Néel temperature T N ∼ 270 K. 24 The RIXS data were collected in vertical scattering geometry with the X-ray spectrometer on beamline 9-ID-B, at the Advanced Photon Source, and using the 2 m arm configuration. The energy resolution (FWHM) was 0.32 eV (Sec. II A) and 0.25 eV (Sec. II B). The 'elastic tails' due to the elastic peak and non-zero energy resolution are subtracted from the inelastic spectra above 1.5 eV energy loss. This is accomplished by (i) using the elastic peak to establish zero energy transfer, (ii) fitting energygain data (typically up to 3 eV) to the heuristic modified Lorentzian form 1/(1 + \|w\| α ) where w is energy-loss (with typical values of α in the range 1-2), (iii) and then subtracting the result of the fit from the energy-loss part of the spectrum. A. Azimuthal (ψ-) scans In order to study the effects of photon polarization independently of the momentum transfer, we perform azimuthal scans, which rotate the incident-photon polarization within the CuO 2 planes, while keeping the energy-and momentumtransfer constant (the rotation axis is parallel to Q. 25 ) This scattering geometry is illustrated in Fig. 1a, with ω = 0, and in Fig. 1b. The azimuthal angle ψ is the scanned variable. of the scattering geometry used throughout this paper. The polarization of the incident beam (σ) and the two possible polarization conditions for the scattered beam (σ and π) are emphasized in both panels. Note that ψ-scans are collected at ω = 0 only (b) inset: two-dimensional (2D) Brillouin zone: the red line shows the full region of integration arising from the momentum resolution whereas the blue line shows its half width. (c) Energy scans taken at the 2D zone center with incident energy E i =8997 eV (circles) and 9001 eV (squares). The superposed colored circles and square show where the ψ-scans of corresponding color and energy are measured in (d) are measured. The charge-transfer gap measured with optical conductivity is indicated for comparison. 26 (d) Azimuthal scans taken in a full circle around the 2D zone center at 1.625, 2, 3.5, 5 eV energy-loss (for E i = 8997 eV) and at 6 eV energy-loss for (E i =9001 eV) for Q = (0 0 9.1) and symmetrized with respect to the 90 • rotations and mirror planes of the underlying tetragonal structure. (e) Energy scans with elastic tail subtracted (see text) taken at Q = (0 0.5 11.1) for different values of the azimuthal angle (ψ). The scans indicated by circles were taken at E i = 8997 eV whereas those indicated by squares were taken at E i = 9001 eV. Energy scans with ψ = 45 • are corrected for changes in self-absorption (+5.2%) and incident polarization (+0.8%) compared to ψ = 10.7 • . (f) Fourier components of the ψ-scan at 3.5 eV energy-loss with amplitude given in percent of 0-fold (DC) component (not shown). The 4-fold component is well above the statistical noise level (gray region). There is also a 2-fold component present in the raw data. Figure 1c presents characteristic RIXS spectra taken at two different incident energies. These line scans show electronic excitations in the 1-10 eV energy-loss range associated with the electronic structure of the strongly-correlated CuO 2 plane, and they indicate the energies where ψ-scans are made. At the incident energy E i = 8997 eV, the chosen values span from 1.625 eV, just above the optical charge-transfer gap, 26 to the maximum of the inelastic intensity at 5 eV. At E i = 9001 eV, we study the molecular orbital excitation 27 by measuring at 6 eV. The raw azimuthal scans are symmetrized to be consistent with the underlying tetragonal symmetry of the CuO 2 planes. This symmetrization consists of folding back the data within the 0-45 • arc and then averaging it, the results are shown in Fig. 1d. This procedure averages out all non-4-fold components exactly which, in practical terms, filters out experimental noise and leaves only the intrinsic electronic properties. For E i = 8997 eV, the resulting amplitudes of the 4fold oscillations are −0.8 ± 3.5% (at 1.625 eV), −0.2 ± 0.9% (2 eV), 3.2 ± 0.9% (3.5 eV), and 3.2 ± 0.9% (5 eV), while for E i = 9001 eV, we find 1.6 ± 0.5% (6 eV). Only the last three 4-fold oscillations, albeit small, are statistically significant (better than 3σ) and inconsistent with being statistical noise. Fourier analysis of the ψ-scan at 3.5 eV (E i = 8997 eV) is presented in Fig. 1f, where the amplitude is in percent of the 0-fold (DC) component. Of the four physical components (0-,1-,2-,4-fold), only the 0-, 2-, and 4-fold components are above the statistical noise level. 28 The 2-fold component is systematically observed in all ψ-scans, but filtered out by the symmetrization procedure. optical gap 3.5eV a b k in k sc k in k sc pl an ar c Q 2θ ε in ε sc ε in ε sc 0 0 Q x y Q Intensity (count/sec.) 0 (a) (b) (c) (f) (e) 90 180 270 360 (deg) Azimuthal Angle E i =8997eV E i =9001eV 2 1 0 6eV 1.6eVIntensity (count/sec.) 2 1 =10.7 o =45 o =10.7 o =45 o ( 0 ( Energy loss (eV) ( 0 ( 0 2 1 0 10 8 6 4 2 0 E i =9001eV E i =8997eV We can rule out the extrinsic effects of the experimental configuration details as the origin of the observed 4-fold oscillations. Extrinsic effects due to sample self-absorption and the X-ray beam 'footprint' are expected to be 1-and 2-fold respectively. In order to minimize these effects, we aligned the scattering surface normal along the ψ-scan rotation axis (i.e., ω = 0 in Fig. 1a) and we made sure that the X-ray beam spot was contained by the sample surface. After this procedure, no measurable 1-fold component was observed, but a systematic 2-fold component of 2% remained. This 2-fold component is observed both within our RIXS signal and with a fluorescence monitor placed below the analyzer. Its strong dependence on the X-ray beam position within the scattering surface confirms that it is a footprint effect. Furthermore, extrinsic effects should be independent of both the selected intermediate state and the final electronic state that is probed and, as a result, be equally prominent for all energy-loss and incident-photon energies. This is inconsistent with the variability of the 4-fold oscillation amplitude we observe. The rotation of the anisotropic momentum-resolution-ellipsoid in reciprocal space (see Fig.1b), might create an artificial 4-fold oscillation pattern in the ψ-scans, but a calculation based on the ellipsoid's shape and on the anisotropy of the momentum dependent inelastic signal gives an upper bound on this effect to less than 0.05%, much smaller than the observed oscillation amplitudes. We can furthermore rule out that the 4-fold oscillations result from the resonant nature of RIXS, i.e., from the details of the resonantly excited intermediate state. In our experiment, the intermediate state consists of a 1s core hole and an in-plane 4p photoelectron created by the absorption of the incident photon. A local distortion of the lattice could introduce a 2-fold oscillation by splitting the 4p x and 4p y levels, as in the case of the manganites, 25 but the crystal structure of Nd 2 CuO 4 is tetragonal and the 4p x and 4p y levels are de-generate. The 4p states mix non-locally with the 3d x 2 −y 2 state, which introduces a 4-fold component to the intermediate state. On the other hand, two factors make this effect negligible: quadrupole transitions are typically two orders of magnitude weaker than dipole transitions and the 3d-4p mixing is weak because of small overlap integrals and a large energy separation between the two bands. In addition, an intermediatestate effect should be independent of energy-loss value and be seen equally in all measured spectra which is inconsistent with what we observe. We conclude that the 4-fold oscillation is a property of the electronic excitations in the final state. That is, as we vary the energy-loss value, we probe varying admixtures of final states with different symmetries. Since the incident-photon polarization is kept within the CuO 2 planes during the azimuthal rotations, the 4p-asspectator approximation predicts the RIXS signal to be independent of ψ. While this accurately describes approximately 95% of the inelastic signal, the observed 4-fold oscillations with peak-to-peak amplitudes up to 6.4% are evidence of excitations created by the interaction of the 4p photoelectron with the valence system. Because the 4p photoelectron can transfer angular momentum, these excitations can be of a different nature than those created by the 1s core-hole. In indirect K-edge RIXS, this result constitutes the first evidence of scattered photon polarization effects beyond the 4p-as-spectator approximation. In Fig. 1e, we extend our analysis beyond the zone center. This figure shows the ψ-angle dependence of the inelastic spectra at (0 π) at E i = 8997 eV and 9001 eV. The molecular orbital excitation at 5.6 eV measured for E i = 9001 eV has a weak ψ angle dependence: it is approximately 6% weaker for ψ = 45 • than for ψ = 10.7 • . There might also be a ψdependent feature for E i = 8997 eV at approximately 3.8 eV. B. Scattering-geometry dependence Scattering-geometry dependence of the RIXS signal can have three different origins: sample self-absorption, incident and scattered photon polarization effects, and momentumtransfer effects. Since all three generally affect the signal simultaneously, it is difficult to separate their contributions. In Sec. II A, we used a particular scattering geometry that allows for the polarization degrees of freedom to be varied independently of the momentum transfer while minimizing selfabsorption effects. Alternatively, the scattering geometry dependence can be investigated by measuring the Brillouin zone dependence of the inelastic signal. Recently, Kim et al. 17 studied the Brillouin-zone dependence of charge-transfer excitations at the Cu K-edge of the Mott insulator La 2 CuO 4 . Based on remarkable agreement between inelastic spectra taken at high symmetry points of different Brillouin zones, they concluded that the RIXS signal is the same in all Brillouin zones. However, a closer look at the experimental spectra shows an interesting energy-lossdependent difference of approximately 10% between spectra measured in different Brillouin zones. In this Section, we study this subtle effect further by measuring the scattering-geometry dependence of the related Mott insulator Nd 2 CuO 4 . Inelastic line scans (energy-gain side subtracted) taken at the zone center of six different zones are shown in Fig. 2a. The proper normalization of each spectrum is important as it allows the comparison of relative intensities even between widely different scattering geometries. In order to separate extrinsic effects from intrinsic features, we compare four different normalization techniques. The first (FM) consists of using the fluorescence signal as a monitor. The second (FM+SA) adds the self-absorption correction described in Appendix A. The third (SW1) and fourth (SW2) both use the fluorescence signal as a monitor and further normalize the spectra by the integrated inelastic spectral weight in a fixed energy-loss range, between 1.4-2.9 eV and 1.9-2.4 eV respectively. The resulting normalization factors are compiled in Table I where the 'raw' fluorescence monitor factors are separated from the different corrections using either selfabsorption or integrated spectral weight. As expected, (0 0 L) scattering geometries (where ω = 0) all have approximately the same normalization factor, except for (0 0 7.1), where the X-ray footprint starts to be limited by the scattering surface size. The (H 0 L) grazing-incidence normal-emission geometries (where ω < 0) suffer less self-absorption and have lower normalization factors accordingly. Based on the tabulated values, the fluorescence monitor provides the largest contribution to the normalization (≈ 20%), which validates its use as a first-order self-absorption correction. However, the additional self-absorption correction described in Appendix A is not negligible (≈ 5%) and should be used. We analyze the scattering geometry dependence by dividing each of the six energy spectra into energy-loss bins, each bin half the size of the experimental resolution (0.125 eV). Within each bin, the scattering-geometry dependence of the intensity is fitted to a linear form (y = σ 1 + σ 2 x), where the dependent variable (x) is either L, \|Q\| 2 , or cos 2 (θ −ω) (see Fig. 1a for the definition of θ and ω). Each fit is performed using the four different normalization procedures described above. The quality of each linear fit is characterized by a reduced chi-squareχ 2 (the total chi-square divided by the number of degrees of freedom d). The chi-square values and degrees of freedom can be summed across the energy bins to compose a collective reduced chi-squareχ 2 = i χ 2 i / i d i (i: bin index) which characterizes the fit functions' ability to represent the observed Brillouin-zone dependence across the entire data set. The goodness-of-fit indicator (G) is also calculated. 29 The values of these two indicators, for each combination of fit function and normalization procedure, are compiled in Table II. The cos 2 (θ − ω) linear dependence provides the best fit to our data as it robustly yields the lowest χ 2 and the highest G values, independently of the normalization procedure. The L and \|Q\| 2 fits are poorer, no matter what normalization procedure is used. Examples of the linear cos 2 (θ − ω) fits are presented in Fig. 2b-c and compare different normalization procedures. The energy-loss values chosen for this comparison span the spectral range of our data and correspond to the thick and gray vertical lines in Fig. 2a,d-e. In Fig. 2d-e, the resulting fit parameters (intercept σ 1 and slope σ 2 ) for each energy bin are compared for three of the four different normalization procedures. The choice of normalization procedure affects the extracted amount of Brillouin zone dependence. For example, the FM+SA correction in Fig. 2b,d adequately includes the effects of scattered-photon polarization: the self-absorption is larger for normal emission (in-plane scattered photon polarization, cos 2 (θ − ω) ≈ 0) than for grazing emission (mix of inplane and out-of-plane photon polarization, cos 2 (θ − ω) ≈ 1). This difference is apparent form the polarization dependent X-ray absorption curves in Fig. 4c. The SW2 normalization yields the lowest χ 2 and largest G which suggests that the 1.9-2.4 eV energy-loss region is Brillouin-zone independent. On the other hand, this normalization produces unphysically high values of σ 2 (Ref. 30 ) which suggests that the 1-3 eV energy-loss region in σ 2 is artificially reduced by a source of error beyond the selfabsorption correction. cos 2 (θ − ω) Q 2 L While a priori not unphysical, 31 the negative offset in σ 2 below 3 eV (using FM+SA normalization) is probably an artifact the elastic-line subtraction. In our subtraction of the energy-gain side, we assume that the elastic line is symmetric. While this approximation is in principle valid, weak anisotropic elastic signal could 'leak' into our RIXS spectra (due to the non-zero energy resolution) and introduce an artificial Brillouin-zone dependence of the signal at low energy loss. Errors in the fitted elastic line position can also introduce a weak Brillouin zone dependence. While this anisotropy is limited to low energy loss, an error in fitted position would be proportional to the RIXS spectrum's energy-loss slope and create artifacts at both high and low energy-loss. We emphasize that whereas the broad negative offset observed in σ 2 below 3 eV (within FM+SA) could be explained by a Brillouin-zone-varying asymmetry in the elastic line, neither of the above effects can create sharp features like the one observed at 1.65 eV. Finally, a slight crystal misalignment could account for the overall larger intensity measured between 3-5 eV at (0 0 11.1), since the zone center spectrum at 8997 eV is a local minimum of inelastic intensity. On the other hand, the different spectral shape observed for (2 0 11) and (3 0 12) around 4 eV is not explicable by a crystal misalignment and could be evidence of a Brillouin-zone dependence not captured by the analysis presented in Fig. 2. While the many sources of error make the quantitative comparison of the spectra difficult, the quantities σ 1 and σ 2 exhibit robust features. The Brillouin-zone independent part of the RIXS spectrum (σ 1 ) consists of a broad feature centered at 4.5 eV, a shoulder at 2 eV (and possibly another at 1.4 eV), and a (linearly-extrapolated) onset of 0.8 eV. On the other hand, the Brillouin zone dependent part (σ 2 ) consists of a broad feature, centered around 5 eV with an onset around 2.5 eV, and its most interesting feature is a resolution-limited peak at 1.65 eV. Because the spectra cannot be scaled to collapse onto one common curve, no matter what normalization is used, the Brillouin-zone dependent part (σ 2 ) cannot be spurious. The energy-loss dependence also rules out the resonant crosssection as the origin of the Brillouin-zone dependence, leaving only the properties of the measured electronic excitations to explain the effect. For the employed scattering geometry, the 4p-asspectator approximation predicts no photon-polarizationbased Brillouin-zone dependence of the inelastic signal. In contrast, the observed Brillouin-zone dependence is best fit by the cos 2 (θ − ω) form which implies that the effect is photonpolarization-based and not momentum-based. As such, this observation implies that the 4p photoelectron interacts with the valence system during the RIXS process. We note that the integration of the FM+SA normalized σ 2 spectral weight above 2.5 eV 32 sums to 15 ± 2% of the integrated σ 1 spectral weight. This departure from the 4pas-spectator approximation is larger than the ψ-scan peak-topeak amplitude variations (∼ 6%), but it is of the same order of magnitude, which suggests a similar mechanism for both effects. III. DISCUSSION The analysis of the incident-and scattered-photon polarization dependence of the cross section is key to the study of the symmetry of electronic excitations in Raman scattering. 33 Within (direct) soft RIXS, this has been used to distinguish the Zhang-Rice singlet (ZRS) excitation from local d → d excitations. 7,34 On the other hand, for (indirect) hard RIXS, it is unknown what excitation symmetries are measured and what their relative strengths are. It is furthermore unknown if RIXS obeys selection rules linking the incident-and scatteredphoton polarizations and the underlying excitation symmetries. While theoretical treatments have made assumptions about what types of excitations are measurable, they have not discussed selection rules explicitly. Treatments using jointdensity-of-states-type cross sections limit the scattering from the core-hole to interband transitions between bands of the same point-group symmetry at Q = 0. 35,36 Furthermore, calculated RIXS spectra using one-band Hubbard models are automatically limited to states of x 2 − y 2 local symmetry (the ZRS combination of oxygen orbitals is an x 2 − y 2 combination of the O p σ orbitals), so that the symmetry of the chargetransfer excitations is limited to A 1g at Q = 0. 37 On the other hand, non-A 1g transitions have been suggested to explain new features in measured RIXS spectra: charge-transfers to nonbonding bands 4 and local d → d excitations 38 are both exam-ples of such non-A 1g transitions. While it remains unclear if the polarization-based Raman selection rules can be applied to interpret indirect RIXS spectra at the Brillouin zone center (zero reduced q), we test their applicability by comparing their predictions with our data. From the definition of the symmetry channels allowed by the tetragonal (D 4h ) crystal structure of Nd 2 CuO 4 (A 1g , A 2g , B 1g , B 2g , and E g ) and the evolution of the incident-and scatteredphoton polarizations with the azimuthal angle (ψ) and the angular difference (θ − ω), we can write the photon-polarizationbased Raman selection rules, at Q = 0, as a function of the scattering power within each allowed symmetry channel: I Inel. ∝ σ(A 1g ) + σ(A 2g ) + σ(B 1g ) + σ(B 2g ) + σ(E g ) − σ(B 1g ) − σ(A 2g ) cos 2 (θ − ω) + σ(B 1g ) − σ(B 2g ) cos 2 (2ψ) cos 2 (θ − ω)(2) Since the scattered-photon polarization is not analyzed, we must include both the σ and π channels (polarization perpendicular and parallel to the scattering plane, respectively), as shown in Fig. 1a,b. Note that these rules inform us about the polarization dependence of these symmetry channels, but not about their relative size. At fixed θ and for ω = 0, Eq. 2 can be rewritten as σ 0 + σ 4 cos 2 (2ψ) which is precisely the functional form of the 4-fold oscillations presented in Sec. II A. With a peak-to-peak 4-fold amplitude equal to σ(B 1g ) − σ(B 2g ) cos 2 (θ − ω), these azimuthal oscillations can be interpreted in terms of the B 1g and B 2g symmetry channels. While we cannot determine the B 1g and B 2g amplitudes independently, our data suggest the presence of B 1g -type electronic excitations at 3.5 and 5 eV energy loss for E i = 8997 eV and at 6 eV for E i = 9001 eV. Correcting the 4-fold peak-to-peak amplitude to account for the cos 2 (θ − ω) dependence, we obtain an adjusted B 1g − B 2g amplitude of +8.7% (σ 4 /σ 0 ) between 3.5 − 5eV for E i =8997 eV. These adjusted data are shown in Fig. 3. For ψ = 0, Eq. 2 reduces to a cross section of the form σ 1 + σ 2 cos 2 (θ − ω), where σ 1 = A 1g + A 2g + B 1g + B 2g and σ 2 = E g − A 2g − B 2g , which is precisely the function that best fits the Brillouin-zone dependence presented in Sec. II B. This suggests that the decrease in spectral weight toward backscattering observed here for Nd 2 CuO 4 and by Kim et al. 17 for La 2 CuO 4 can be interpreted in terms of E g excitations at high energy-loss. These symmetry assignments for σ 1 and σ 2 are reproduced in Fig. 3. In tetragonal symmetry, the incident-and scattered-photon polarizations of an E g excitation correspond to 4p crystal-field eigenstates with different energies (e.g., x→z or z→x), unlike the A 1g , A 2g , B 1g , B 2g symmetry channels where the incident and outgoing 4p crystal-field eigenstates are degenerate (and planar). This creates a unique resonance profile 17,18,23,39 for E g excitations which an increase or decrease of intensity compared to the other symmetry channels. For example, in the scattering geometry used here, at the incident energy E i = 8997 eV, E g and B 1g excitations have the same incident-photon polarization resonance but their scattered-photon polarization resonances differ. For B 1g excitations, the scattered-photon polarization is ǫ f = x and the FIG. 3. Symmetry interpretation of the photon-polarization dependent and independent RIXS spectral weight for tetragonal Nd 2 CuO 4 . While composed of many different symmetry channels, the blue spectrum is most likely dominated by the A 1g symmetry channel. See text for details. final energy resonance is at E f = 8997 eV whereas for E g excitations, the scattered-photon polarization is ǫ f = z and there are scattered-photon resonances at E f = 8985, 8993, and 8997 eV according to the XAS data in Fig. 4. Around 4 eV and 12 eV energy-loss, the scattered photon is tuned to an out-of-plane resonance and E g excitations will be enhanced compared to B 1g excitations. In this case, assuming that both symmetry channels have a comparable density of states, we estimate that E g excitations are comparatively enhanced by a factor of 2-5. If we correct for this enhancement, the observed 15% Brillouin-zone dependence becomes a 3-8% effect, more comparable to the observed 8.7% azimuthal-angle dependence. For the scattering geometry used here, the core-hole is expected to create the majority of the inelastic signal because it has the strongest effect on the valence system. We suggest that the Brillouin-zone-independent and ψ-angle independent spectral weight is created by core-hole scattering and consists of A 1g symmetry. We note, however, that this contribution can be reduced to zero in certain scattering geometries because its cross-section follows the 4p-as-spectator approximation discussed in Sec. I. While we have shown that the Raman selection rules can accurately model the observed photon-polarization effects and suggest symmetry assignments for different inelastic features, these assignments should be supported by a theoretical understanding of the excitations. The molecular orbital (MO) excitation at the poorly-screened state is understood to be of A 1g symmetry, which is consistent with its weak ψ-angle dependence at (0 π) (see Fig. 1e). While the MO excitation does show a weak 4-fold oscillation at the zone center, a comparison of the (0 π) and (0 0) spectra suggests that the 4-fold oscillation might instead be a property of a momentum-dependent shoulder that disappears away from (0 0). In principle, d → d excitations provide a testing ground for the validity of conventional Raman selection rules within RIXS, and such excitations have been well studied with soft X-ray RIXS at the Cu L- [8][9][10]40,41 and M-edges, 11 with optical absorption, 42 with large-shift Raman scattering, 43 and with different theoretical methods. 44,45 Although for Nd 2 CuO 4 only the A 2g (d x 2 −y 2 → d xy ) Cu crystal-field excitation has been observed around 1.4 eV, 43 the crystal-field excitations of tetragonal Sr 2 CuO 2 Cl 2 have been extensively studied 10,11,42,44 and their energies should be similar to those of Nd 2 CuO 4 . For the latter, Raman selection rules suggest that the resolution-limited feature we observe at 1.65 eV (Fig. 2) is of E g symmetry. The E g crystal-field excitation (d x 2 −y 2 → d xz ) in Sr 2 CuO 2 Cl 2 has an energy of 1.7 eV which, supports this symmetry assignment and the applicability of the Raman selection rules. A complete determination of the d → d excitations in tetragonal Nd 2 CuO 4 should be possible with the experimental methods described in this paper, i.e., without analyzing the scattered-photon polarization, since only three types of d → d excitations are possible (the B 2g symmetry change does not exist within the Cu 3d orbitals). For example, with better energy resolution, the A 2g excitation should be observable at 1.4 eV with a Brillouin-zone dependence study, whereas the B 1g (d x 2 −y 2 → d 3z 2 −r 2 ) excitation should be observable with a ψ-angle dependence study. Charge-transfers from the non-bonding oxygen bands to the upper Hubbard band, which have been used to explain the multiplet of inelastic features between 2 and 6 eV, 4 are expected to be of non-A 1g symmetry. As studied with ARPES, [46][47][48] O 2p non-bonding bands have approximately 1.5 eV more binding energy than the ZRS band. Accordingly, the RIXS charge transfer to such bands should start at an energy 1.5 eV above the overall onset of RIXS excitations. In Nd 2 CuO 4 , the onset of excitations is approximately 0.8 eV. On the other hand, the 4-fold azimuthal-scan oscillations start between 2 and 3.5 eV, and the broad feature in the Brillouinzone dependent part starts at around 2.5 eV. Both have onsets approximately 1.5 eV above 0.8 eV and could be interpreted as B 1g and E g charge-transfers to oxygen non-bonding bands, respectively. In their EELS study of Sr 2 CuO 2 Cl 2 , Moskvin et al. 49 identify many charge-transfer excitations, three of which have A 1g symmetry and should be RIXS active. The two strongest excitations are around 8 eV, but do not have obvious RIXS equivalents. The third A 1g excitation at 2 eV, just above the optical gap, is identified as the Zhang-Rice singlet. This may correspond to the 2 eV feature in RIXS, which is apparent as a weak shoulder in Fig. 2d-e and is seen as a clear peak in La 2 CuO 4 and Sr 2 CuO 2 Cl 2 . This suggests that the 2 eV feature in RIXS is of A 1g symmetry. An exact diagonalization calculation Tohyama 50 of charge-transfer excitations within a one-band Hubbard model of the CuO 2 plane agree quite well with our data and the A 1g symmetry assignment of the 2 eV feature. The calculation shows a peak at 2 eV that is exclusively of A 1g symmetry, and a continuum of excitations between 2 − 3.5 eV that is of predominantly B 1g character, in good agreement with the symmetry interpretation of our data. The relative intensity of these two features disagrees with our observations, but quantitative agreement is not expected, since the resonant and non-resonant cross-sections are very differ-ent. In order to better understand the symmetry selectivity of the RIXS cross section, it is worthwhile to compare it to wellestablished probes. On the other hand, symmetry-selectivity differs from probe to probe, which renders direct comparisons hazardous. For example, in crystals with an inversion center, optical conductivity measures a current-current correlation function only sensitive to ungerade excitations, while electron-energy-loss-spectroscopy (EELS) and IXS measure a density-density correlation function that is sensitive to both ungerade and gerade excitations. Furthermore, the symmetrysensitivity of these probes has a strong and non-periodic Qdependence 51,52 in contrast to the nearly Brillouin-zone independent RIXS spectra. IV. CONCLUSION By studying the photon-polarization dependence of indirect RIXS at the Cu K-edge of the tetragonal Mott insulator Nd 2 CuO 4 , we uncover anomalous excitations of two different types, which is evidence of scattered-photon polarization effects. While the majority (80%) of the inelastic signal is describable by the 4p-as-spectator approximation, the anomalous remainder constitutes the first evidence of RIXS excitations created by the interaction of the 4p photoelectron with the valence system. The successful modeling of the observed azimuthal-scan 4fold patterns and of the Brillouin-zone dependence by photonpolarization-based Raman selection rules suggests that these rules can be used to interpret zone-center RIXS spectra. Using these rules, we tentatively assign the sharp peak at 1.65 eV in the Brillouin-zone-dependent spectral weight to an E g d → d excitation and the broad features above 2.5 eV in both the Brillouin-zone-dependent and azimuthal-angle-dependent spectral weight to E g and B 1g charge-transfers to non-bonding oxygen bands. Establishing photon-polarization-based methods to characterize the electronic excitations' symmetry is a pivotal challenge for RIXS. Such methods should facilitate the interpretation of experimental spectra and help provide a better understanding of the underlying physics of the cuprates and other transition metal oxides. While a complete scatteredphoton polarization analysis is currently prohibited by low count rates, ongoing instrumentation development and future increases in photon flux should soon make the full polarization analysis of the inelastic signal possible. We would like to acknowledge valuable conversations with J. van den Brink, T.P. Devereaux, and K. Ishii. This work was supported by the DOE under Contract No. DE-AC02-76SF00515 and by the NSF under Grant No. DMR-0705086. Appendix A: RIXS Normalization with a fluorescence monitor Aside from effects intrinsic to the cross section, the RIXS scattering intensity is also modulated by the sample selfabsorption, an extrinsic effect that depends on the scattering geometry. In X-ray scattering, the absorption processes which determine the X-ray absorption length of a crystal are dominated by Auger emission of electrons and core-level fluorescence lines. Together, these processes' cross-sections dwarf the RIXS cross-section. As a result, when the X-ray beam travels through the sample, before and after the RIXS event, its intensity is attenuated. This attenuation has a strong dependence on the scattering geometry and determines the illuminated sample volume. The RIXS signal is proportional to the number of Cu atoms resonantly excited in this volume. This attenuation is referred to as self-absorption, and correcting for it is common practice for other spectroscopies scattering probes (for example, in EELS, 53 in EXAFS, 54 and in direct RIXS 55 for example) but has not yet become standard for indirect RIXS. To calculate this effect, we must know the X-ray absorption length and the scattering geometry. In reflection geometry, the RIXS intensity given by Eq. A1. It includes the incident beam intensity I 0 , the intrinsic RIXS scattering amplitude per Cu atom F(E i , ∆E), the incident beam cross-sectional area B a , the density of Cu atoms ρ Cu , the absorption coefficient for the incident (scattered) photons µ i (µ f ) with polarization ǫ i (ǫ f ) and energy E i (E f ), as well as the scattering angles θ i and θ f given relative to the sample surface, as shown in Fig. 4b. The maximum scattering intensity is obtained for a photon with grazing incidence which is emitted perpendicular to the sample surface (an example is shown in Fig. 4b). In turn, the minimum intensity is observed for normal incidence and grazing emission angle. The reduction in RIXS signal from having a footprint (the beam spot on the sample) larger than the scattering surface is not included but is an important effect at grazing incidence. I = (I 0 ρ Cu B a ) S E i , ǫ i , E f , ǫ f F E i , E i − E f = I 0 ρ Cu B a µ(ǫ i , E i ) + µ(ǫ f , E f ) sin(θ i ) sin(θ f ) F E i , E i − E f (A1) In an ideal experiment, this formula could be used directly to normalize the RIXS spectra taken in different scattering geometries. In practice though, the footprint is highly dependent on the scattering surface and can affect the RIXS signal in non-trivial ways. To counteract this difficulty, we use a fluorescence detector put at a known position close to the analyzer crystal, as presented in Fig. 4a. After measuring the position of the sample surface relative to the crystal axes and tuning the fluorescence detector to an emission line (for example Cu K α1 ), we can calculate the scattering geometry dependence of both the fluorescence monitor (FM) signal (S FM ) and of the RIXS signal (S RIXS ). In order to correct for the selfabsorption effect, which changes the intensity of the RIXS signal based on scattering geometry, we normalize with the following ratio: I C RIXS = I RIXS I FM × S FM S RIXS (A2) The measured fluorescence and RIXS intensities are I FM and I RIXS . Simply dividing the RIXS signal by the FM signal is a first-order correction that can partially account for changes in the X-ray footprint on the sample. However, the exact correction must include variations in X-ray self-absorption based on the scattering geometry, the photon's polarization and energy all of which require calculating the self-absorption factors S FM and S RIXS . Note that the S FM calculation includes the difference in location between the spectrometer's analyzer crystal and the fluorescence monitor, as shown in Fig. 4a. In practice, the further away the FM is placed from the analyzer crystal, the more difficult it will be to accurately account for positional differences in the calculation of the self-absorption correction. This can become a large source of systematic error in the normalization procedure, although this is not the case in the present experiment. The fluorescence monitor signal described above is collected with a solid-state detector (Amptek). The signal is fed into a scaler and only photons within a fixed energy range are counted, in our case, in a 1 keV range around the strong Cu K α1 emission line at 8.1 keV. An absorption curve for Nd 2 CuO 4 with in-plane incident-photon polarization collected with this fluorescence monitor is presented in Fig. 4d. The non-zero signal below the absorption edge is parasitic but is not the result of dark current within the detector. Instead, it is composed of two signals that leak into the energy integration window: elastic scattering at 9 keV and neodymium L-edge fluorescence at 7 keV. Within our theoretical calculation, this parasitic component is tentatively compensated for by modeling it as 70% elastic signal and 30% neodymium fluorescence. Since the parasitic component does not vary congruently with the measured emission line, it constitutes a source of error and should in general be minimized. In order to calculate the polarization dependent X-ray absorption coefficients (µ) in the Cu K-edge region, we collected X-ray absorption data by partial fluorescence yield at the Cu K α1 emission line with an X-ray spectrometer. 57 (1.2 eV resolution) The effect of self-absorption, typically important close to absorption edges, was calculated as described by Carboni et al. 58 , but was not significant here. In order to complete the construction of the absorption coefficients, the curves were then scaled to match the tabulated values for X-ray absorption in Nd 2 CuO 4 above and below the Cu K-edge and following Ref. 56 . The results are shown in Fig. 4c. While both the normalization procedure above the edge and the self-absorption correction of the fluorescence signal are sources of error, a conservative estimation of their effects is included in the normalization factor's error but, in our case, they are negligible. This type of correction includes absorption effects due to changes of the scattered-photon energy that simply dividing by the fluorescence signal cannot account for. For example, based on Fig. 4c, an incident in-plane photon with an energy of 8997 eV would see more absorption on its way into the sample than the scattered photon on its way out of the sample with 10 eV energy-loss (E f = 8987 eV). Within an inelastic spectrum taken in these conditions and corrected by the fluorescence signal alone, the high-energy-loss response would be artificially larger than that at low energy, based solely on this change in the XAS coefficient. Finally, the scattered photon can in principle have σ or π polarization which affects the calculated self-absorption correction. In vertical scattering geometry, at the in-plane 4p resonance, and within the bounds of the 4p-as-spectator approximation, the scattered photon must be σ polarized and we calculate the self-absorption correction based on this prescription. As seen in Sec. II B, there is evidence that this approximation only describes part of the RIXS signal, with the complementary contribution in the π polarization channel. This latter polarization channel, when isolated, can be normalized by modifying the calculated scattered photon X-ray absorption coefficient (µ f ) accordingly and multiplying the σ → π scattering intensity by the ratio of the σ → π and σ → σ calculated self-absorption corrections: f σ→π RIXS = S σ→π S σ→σ (A3) On the other hand, within the measured spectral range, the maximum correction is only 4% of the Brillouin-zone dependent signal and does not noticeably affect the normalization outlined above. FIG. 1. (a) Side and (b) top view of the scattering geometry used throughout this paper. The polarization of the incident beam (σ) and the two possible polarization conditions for the scattered beam (σ and π) are emphasized in both panels. Note that ψ-scans are collected at ω = 0 only (b) inset: two-dimensional (2D) Brillouin zone: the red line shows the full region of integration arising from the momentum resolution whereas the blue line shows its half width. (c) Energy scans taken at the 2D zone center with incident energy E i =8997 eV (circles) and 9001 eV (squares). The superposed colored circles and square show where the ψ-scans of corresponding color and energy are measured in (d) are measured. The charge-transfer gap measured with optical conductivity is indicated for comparison. 26 (d) Azimuthal scans taken in a full circle around the 2D zone center at 1.625, 2, 3.5, 5 eV energy-loss (for E i = 8997 eV) and at 6 eV energy-loss for (E i =9001 eV) for Q = (0 0 9.1) and symmetrized with respect to the 90 • rotations and mirror planes of the underlying tetragonal structure. (e) Energy scans with elastic tail subtracted (see text) taken at Q = (0 0.5 11.1) for different values of the azimuthal angle (ψ). The scans indicated by circles were taken at E i = 8997 eV whereas those indicated by squares were taken at E i = 9001 eV. Energy scans with ψ = 45 • are corrected for changes in self-absorption (+5.2%) and incident polarization (+0.8%) compared to ψ = 10.7 • . (f) Fourier components of the ψ-scan at 3.5 eV energy-loss with amplitude given in percent of 0-fold (DC) component (not shown). The 4-fold component is well above the statistical noise level (gray region). There is also a 2-fold component present in the raw data. FIG . 2. (a) Two-dimensional zone-center spectra collected for a wide range of scattering geometries. Data sets are separated vertically by 0.3 units. The solid lines through the data are smooth interpolations and they are also overlaid at the bottom of the figure to highlight the evolution of the scattering geometry data. The data sets are scaled using the FM+SA method (see text for details). (b-c) Fits of the Brillouin-zone dependence to the linear form y = σ 1 + σ 2 cos 2 (θ − ω) at 1.65, 3, 4, and 6.5 eV, comparing the FM+SA normalization procedure with (b) FM and (c) SW2 normalizations. (d-e) Fitted coefficients σ 1 and σ 2 as a function of energy-loss, comparing the FM+SA normalization procedure with (d) FM and (e) SW2 normalizations. FIG. 4 . 4(a) Position of the fluorescence monitor relative to the analyzer crystal (b) Self-absorption parameters in reflection geometry. (c) X-ray absorption spectroscopy (XAS) data (by partial fluorescence yield of the Cu K α1 emission line) in the Cu K-edge region. The black lines correspond to the calculated X-ray absorption coefficients above and below the Cu K-edge for Nd 2 CuO 4 . The measured XAS curves are scaled to follow the published curves by Tranquada et al.56 . (d) Nd 2 CuO 4 absorption curve at the Cu K-edge measured with the fluorescence monitor as described in the text. The black arrows indicate incident energies of 8997 eV and 9001 eV. TABLE I. Normalization factors for six different scattering geometries. The raw fluorescence monitor (FM) normalization factors are compared to different correction methods (see text for details).Raw Correction factor Q FM FM FM+SA SW1 SW2 (0 0 7.1) 1.185 1 0.960 1.081 1.091 (0 0 9.1) 0.995 1 0.984 1.065 1.060 (0 0 11.1) 1.000 1 1.000 1.000 1.000 (0 0 15.1) 0.930 1 1.023 1.065 1.003 (2 0 11) 0.966 1 1.043 1.072 0.967 (3 0 12) 0.774 1 1.049 1.077 1.017 TABLE II . 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Casa, and T. Gog, Phys. Rev. B, 70, 205128 (2004). While the unphysical 9-fold component is above noise level, its amplitude is 1.4σ and is not statistically inconsistent with being zero. While the unphysical 9-fold component is above noise level, its amplitude is 1.4σ and is not statistically inconsistent with being zero. It depends on the number of degrees of freedom and the χ 2 such that, given a chi-square distribution, G is the probability that a randomly obtained χ 2 value be bigger than the one resulting from the fit. Typically, a fit with G above 10% is judged good. However, this value is not hard-set and the distribution of errors or their systematics should be taken into account and can make smaller values of G acceptable. The goodness-of-fit indicator (G) is another metric to quantify the quality of a fitThe goodness-of-fit indicator (G) is another metric to quantify the quality of a fit. It depends on the number of degrees of freedom and the χ 2 such that, given a chi-square distribution, G is the prob- ability that a randomly obtained χ 2 value be bigger than the one resulting from the fit. Typically, a fit with G above 10% is judged good. However, this value is not hard-set and the distribution of errors or their systematics should be taken into account and can make smaller values of G acceptable. The ratio of the integrated spectral weight in the σ 2 and σ 1 channels for SW2 normalization is 39% compared to 15% for FM+SA. The ratio of the integrated spectral weight in the σ 2 and σ 1 chan- nels for SW2 normalization is 39% compared to 15% for FM+SA. They correspond to an increase of the RIXS spectral weight towards backscattering. They correspond to an increase of the RIXS spectral weight to- wards backscattering. We do not include the negative region at low energy-loss since it is most likely a systematical error introduced by the elastic line subtraction. 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[ "The Progenitors of Planetary Nebulae in Dwarf Irregular Galaxies", "The Progenitors of Planetary Nebulae in Dwarf Irregular Galaxies" ]	[ "Michael G Richer [email protected] ", "Marshall L Mccall [email protected] ", "\nDepartment of Physics & Astronomy\nOAN\nInstituto de Astronomía\nUniversidad Nacional Autónoma de México\nP.O. Box 43902792143San DiegoCA\n", "\nYork University\n4700 Keele StreetM3JTorontoOntarioCanada\n" ]	[ "Department of Physics & Astronomy\nOAN\nInstituto de Astronomía\nUniversidad Nacional Autónoma de México\nP.O. Box 43902792143San DiegoCA", "York University\n4700 Keele StreetM3JTorontoOntarioCanada" ]	[]	We present chemical abundances for planetary nebulae and H II regions in the Local Group dwarf irregular galaxy NGC 6822 based upon spectroscopy obtained at the Canada-France-Hawaii Telescope using the Multi-Object Spectrograph. From these and similar data compiled from the literature for planetary nebulae in the Magellanic Clouds, Sextans A, Sextans B, and Leo A, we consider the origin and evolution of the stellar progenitors of bright planetary nebulae in dwarf irregular galaxies. On average, the oxygen abundance observed in the bright planetary nebulae in these galaxies coincides with that measured in the interstellar medium, indicating that, in general, the bright planetary nebulae in dwarf irregulars descend primarily, though not exclusively, from stars formed in the relatively recent past. We also find that the ratio of neon to oxygen abundances in these bright planetary nebulae is identical to that measured in the interstellar medium, indicating that neither abundance is significantly altered as a result of the evolution of their stellar progenitors. We do find two planetary nebulae, that in Sextans A and S33 in NGC 6822, where oxygen appears to have been dredged up, but these are the exception rather than the rule. In fact, we find that even nitrogen is not always dredged up, so it appears that the dredge-up of oxygen is uncommon for the abundance range of the sample.	10.1086/511410	[ "https://arxiv.org/pdf/astro-ph/0611833v1.pdf" ]	18,425,788	astro-ph/0611833	87f43805c67e1a83b727df82136f7be4653b31d8	The Progenitors of Planetary Nebulae in Dwarf Irregular Galaxies 28 Nov 2006 Michael G Richer [email protected] Marshall L Mccall [email protected] Department of Physics & Astronomy OAN Instituto de Astronomía Universidad Nacional Autónoma de México P.O. Box 43902792143San DiegoCA York University 4700 Keele StreetM3JTorontoOntarioCanada The Progenitors of Planetary Nebulae in Dwarf Irregular Galaxies 28 Nov 2006Subject headings: galaxies: abundances-galaxies: individual (NGC 6822)-galaxies: irregular-ISM: abundances-ISM: planetary nebulae (general)-stars: evolution We present chemical abundances for planetary nebulae and H II regions in the Local Group dwarf irregular galaxy NGC 6822 based upon spectroscopy obtained at the Canada-France-Hawaii Telescope using the Multi-Object Spectrograph. From these and similar data compiled from the literature for planetary nebulae in the Magellanic Clouds, Sextans A, Sextans B, and Leo A, we consider the origin and evolution of the stellar progenitors of bright planetary nebulae in dwarf irregular galaxies. On average, the oxygen abundance observed in the bright planetary nebulae in these galaxies coincides with that measured in the interstellar medium, indicating that, in general, the bright planetary nebulae in dwarf irregulars descend primarily, though not exclusively, from stars formed in the relatively recent past. We also find that the ratio of neon to oxygen abundances in these bright planetary nebulae is identical to that measured in the interstellar medium, indicating that neither abundance is significantly altered as a result of the evolution of their stellar progenitors. We do find two planetary nebulae, that in Sextans A and S33 in NGC 6822, where oxygen appears to have been dredged up, but these are the exception rather than the rule. In fact, we find that even nitrogen is not always dredged up, so it appears that the dredge-up of oxygen is uncommon for the abundance range of the sample. Introduction It is well-known that the stellar progenitors of planetary nebulae modify their initial chemical composition through the course of their evolution. Typically, the matter that is returned to the interstellar medium via the nebular shell is enriched in helium and often in nitrogen, carbon, and sprocess elements (e.g., Forestini & Charbonnel 1997;Marigo 2001). Recently, theoretical work has suggested that oxygen is also produced as a result of dredge-up in the progenitors of planetary nebulae from very low metallicities to those as high as that of the LMC (e.g., Marigo 2001;Herwig 2004). These models indicate that the dredge-up of oxygen should be a relatively common result of the third dredge-up that accompanies thermal pulses on the asymptotic giant branch, at least at low metallicities (e.g., Z < 0.25Z ⊙ ), especially for planetary nebulae derived from progenitors in the 2 − 3 M ⊙ range. However, clear examples of its occurrence are rare. Péquignot et al. (2000) found evidence of oxygen dredge-up in the planetary nebula He 2-436 in the Sagittarius dwarf spheroidal from a detailed comparison with Wray 16-423, another planetary nebula in the same galaxy. In principle, planetary nebulae in dwarf irregulars allow the hypothesis that oxygen is dredged up to be tested rather carefully, since the limiting composition of the stellar progenitors is known, namely that of the interstellar medium. At least in the Magellanic Clouds, Sextans B, and Leo A, bright planetary nebulae have oxygen abundances similar to those found in the interstellar medium (Richer 1993;Magrini et al. 2005;Kniazev et al. 2005;van Zee et al. 2006). Preliminary results indicate that this situation also holds in NGC 6822 and NGC 3109 (Knaizev et al. 2006;Leisy et al. 2006). The evidence currently available also indicates that the progenitors of these planetary nebulae generally did not modify their initial oxygen abundances significantly ( §3; . Consequently, the high oxygen abundance in bright planetary nebulae in dwarf irregulars implies that their stellar progenitors are derived from relatively recent star formation. If so, some of the progenitors of these planetary nebulae may have been relatively massive, in which case the chances of observing the dredge-up of nitrogen and, perhaps, oxygen should be high. Here, we present an analysis of nitrogen and oxygen production by considering the chemical compositions of the bright planetary nebulae in the Magellanic Clouds, Sextans A and B, NGC 6822, and Leo A. In Section 2, we present our observations of planetary nebulae and H II regions in NGC 6822 as well as the reduction and analysis of these data. In Section 3, we compile similar data for the planetary nebulae in the Magellanic Clouds, Sextans A and B, and Leo A, and demonstrate that the abundances in planetary nebulae and H II regions in all of these galaxies are generally similar. In Section 4, we consider the nucleosynthetic production in the progenitors of bright planetary nebulae, finding that oxygen production is rare and that even nitrogen production is not as common as supposed. We present our conclusions in Section 5. Observations and Analysis Observations and Data Reductions Our observations of NGC 6822 were obtained at the Canada-France-Hawaii Telescope using the Multi-Object Spectrograph (MOS;Le Fèvre et al. 1994) on UT 17-19 August 1998. The MOS is a multi-object, imaging, grism spectrograph that uses focal plane masks constructed from previously acquired images. The object slits were chosen to be 1 ′′ wide, but of varying lengths to accommodate the objects of interest. In all cases, however, the slit lengths exceeded 12 ′′ . We used a 600 l mm −1 grism that gave a dispersion of 105Å mm −1 and a central wavelength of 4950Å. The detector was the STIS2 2048 × 2048 CCD. The pixel size was 21 microns, yielding a spatial scale at the detector of 0. ′′ 43 pix −1 . The spectral coverage varied depending upon the object's position within the field of view, but usually covered the 3700-6700Å interval. Likewise, the dispersion was typically 2.1Å pix −1 , but varied depending upon the object's position within the field of view (from 1.9Å pix −1 to 2.2Å pix −1 ). Images in the light of [O III]λ5007 and the continuum at 5500Å were acquired on UT 17 August 1998 and used to select objects for observation. Spectra were obtained the following two nights. A total of five exposures of 1800 seconds each were obtained. The photometric calibration was achieved using observations of BD+17 • 4708, BD+25 • 3941, BD+26 • 2606, BD+33 • 2642, and HD 19445. Spectra of the HgNeAr lamps were used to calibrate in wavelength. Pixel-to-pixel variations were removed using spectra of the halogen lamp taken through the masks created for the standard stars and NGC 6822. The spectra were reduced using the specred package of the Image Reduction and Analysis Facility 2 (IRAF). The procedure for data reduction followed approximately that outlined in Massey et al. (1992). The mean of the overscan section was subtracted from all images. A zerocorrection image was constructed from overscansubtracted bias images and was subtracted from all images. The fit1d task was used to fit the columns of the overscan-and zero-correctionsubtracted flat field images with a many-piece linear spline to remove the shape of the halogen lamp to form normalized flat field images. Images of the standard stars and NGC 6822 were divided by the appropriate normalized flat field image. The spectra of the planetary nebulae and H II regions in NGC 6822 were then extracted and calibrated in wavelength using the spectra of the arc lamps. Next, the spectra of the standard stars were used to calibrate in flux. Finally, the fluxand wavelength-calibrated spectra were averaged to produce the final spectra. Table 1 presents the raw line intensities normalized to Hβ and their uncertainties (1 σ) for each object, measured on a scale where the Hβ line has an intensity of 100. The line intensities were measured using the software described in McCall et al. (1985). This software simultaneously fits a sampled Gaussian profile to the emission line(s) and a straight line to the continuum. The quoted errors include the uncertainties from the fit to the line itself, the fit to the reference line (Hβ), and the noise in the continuum for both lines. For those lines where no uncertainty is quoted, the intensity given is an upper limit (2 σ). Table 1 also includes the reddening determined from Hα/Hβ and Hγ/Hβ, assuming intrinsic ratios appropriate for the temperature and density observed. The temperature and density finally adopted for each object is given in Table 2 (see §2.4 for details). Normally, both estimates of the reddening agree within errors. The Fitzpatrick (1999) reddening law was used, parametrized with a total-to-selective extinction of 3.041. This parametrization delivers a true ratio of total-to-selective extinction of 3.07 when integrated over the spectrum of Vega (McCall 2004), which is the average value for the diffuse component of the interstellar medium of the Milky Way (McCall & Armour 2000). The uncertainties quoted for reddenings, temperatures, and densities are derived from the maximum and minimum line ratios allowed considering the uncertainties in the line intensities. Line Intensities The line intensities were corrected for reddening according to log F (λ) F (Hβ) = log I(λ) I(Hβ) −0.4E(B−V )(A 1 (λ)−A 1 (Hβ)) (1) where F (λ)/F (Hβ) and I(λ)/I(Hβ) are the observed and reddening-corrected line intensity ratios, respectively, E(B −V ) is the reddening determined from the F (Hα)/F (Hβ) ratio, when available, and A 1 (λ) is the extinction in magnitudes for E(B − V ) = 1 mag, i.e., A 1 (λ) = A(λ)/E(B − V ), A(λ) being the reddening law (Fitzpatrick 1999 Object Identification and Classification Object names are taken from Hodge (1977), Killen & Dufour (1982), Hodge et al. (1988), and Leisy et al. (2005a). Only the object denoted pn-020 is not included among the sources found in previous studies. This new object is superposed on the H II region Ho15. Below, we argue that this object is a planetary nebula, so we denote it as pn-020, extending the naming scheme of Leisy et al. (2005a). We classified the objects as planetary nebulae if He II λ4686 was present (a sufficient, though not necessary, criterion), the object was pointlike, and no continuum was observed. Based upon these criteria, S33, pn-010, pn-012, S16, and pn-017 appear to be planetary nebulae, in agreement with previous studies (Dufour & Talent 1980;Richer & McCall 1995;Leisy et al. 2005a). At the distance of NGC 6822 (0.5 Mpc; Mateo 1998), a planetary nebula with a typical radius of 0.1 pc should be unresolved. KD29, Ho06, Ho10, and S10 are not point sources and therefore H II regions. The foregoing leaves only pn-019 and pn-020 unclassified. Both of these objects are point sources in our images, but neither present He II λ4686 emission and both also present a continuum. It is tempting to automatically disqualify any object with an observable continuum as a planetary nebula, but an obvious problem with this approach is that there are an abundance of bright stars in star-forming galaxies that chance may throw along the line of sight. The case of pn-019 is the simpler of the two. It is evident from the on-and off-band images that there is a star about 0. ′′ 9 to the south that is also visible on the finding chart presented by Leisy et al. (2005a). In all likelihood, it is this object that contributes the continuum emission. In the case of pn-020, onand off-band images as well as a section of the twodimensional spectrum are presented in Fig. 1. In the two-dimensional spectrum, the line emission is displaced from the continuum by about 0. ′′ 5 to the east. A second, fainter continuum source appears about 1. ′′ 5 farther to the east of the pn-020. Careful inspection of the on-and off-band images reveals these two objects to be stars. Consequently, we conclude that pn-020 is not a source of continuum emission. The emission line spectrum further argues that this object is not an H II region or an unresolved nova or supernova remnant: the observed [O III]λ5007/Hβ intensity ratio (Table 1) is nearly double the highest value observed in any H II region in NGC 6822 (Pagel et al. 1980;Peimbert et al. 2005;Lee et al. 2006, this study) and the [S II]λλ6716,6731 lines in this object are weaker than those in the four H II regions observed here. To summarize, we find that pn-019 and pn-020 both appear to be planetary nebulae. Fig. 1 serves to locate pn-020. Leisy et al. (2005a) present finding charts for the other planetary nebulae. We conclude that our sample includes seven planetary nebulae and four H II regions. Table 2 presents the electron temperatures and densities as well as the ionic and elemental abundances derived for each object. The atomic data employed for ions of N, O, and Ne are listed in Table 3. For H • and He + , the emissivities of Storey & Hummer (1995) were used. For He • , we used an extended list of the emissivities from Porter et al. (2005) that was kindly provided by R. Porter. Physical Conditions and Chemical Abundances The ionic abundances were derived using the SNAP software package (Krawchuk et al. 1997). The elemental abundances were calculated from the ionic abundances using the ionization correction factors (ICFs) proposed by Kingsburgh & Barlow (1994). The uncertainties quoted for all quantities account for the uncertainties in the line intensities involved in their derivation, including the uncertainties in the reddening. However, the quoted uncertainties in the elemental abundances do not include the uncertainties in the ICFs. Note that no ICFs are involved in the N/O and Ne/O abundance ratios that are tabulated in Table 2. In the calculation of the elemental abundances, we made an effort to adopt a scheme that treated all objects as homogeneously as possible. Occasionally, this will result in less than optimal elemental abundances for some objects, but has the advantage of uniformity, i.e., the drawbacks are common to all objects. For example, it is feasible to use singlet lines of He I to compute He + /H in some objects, but not in all. To avoid the risk of introducing spurious differences, we chose to use common lines for all objects. The one exception to this rule is the O + /H ionic abundance. When available, the ionic abundance based upon [O II]λ3727 was used in favor of that based upon [O II]λλ7319,7331. The electron temperature adopted is always based upon the [O III]λλ4363/5007 ratio, the upper limit being used when [O III]λ4363 was not detected. Likewise, the only electron densities adopted were based upon the [S II]λλ6716,6731 lines, substituting values of 100 cm −3 and 2000 cm −3 for H II regions and planetary nebulae, respectively, when the density could not be derived. The density adopted for planetary nebulae was chosen based upon the results of Riesgo & López (2006). Oxygen abundances for the planetary nebulae S16 and S33 were previously presented by Dufour & Talent (1980) and Richer & McCall (1995). Within uncertainties, the oxygen abundances found here agree with their previous values. We note that the electron temperatures we find for the planetary nebulae are generally higher than those found for the H II regions, as expected. Our oxygen abundances for the H II regions KD29, Ho10, and Ho06 are similar to those derived previously for other H II regions in NGC 6822 by Pagel et al. (1980), Peimbert et al. (2005), and Lee et al. (2006) (Table 1), but it is quite low, so there is reason to believe that the oxygen abundance we derive is not unreasonably low. For S10, we have a reasonable detection of [O III]λ4363, so its low oxygen abundance is somewhat surprising. Had we adopted lower temperatures for the O + zone, as done in Lee et al. (2006), we would find slightly higher abundances for all objects except Ho06 (its electron temperature would hardly change in the Lee et al. (2006) scheme). The reverse would be true had we adopted higher temperatures in the O + zone as Peimbert et al. (2005) did. Lee et al. (2006) find a scatter in their measured oxygen abundances of about ±0.1 dex. We find about double this scatter, driven by the low abundance for S10. The helium abundances we derive for the H II regions in our sample are similar to those obtained by Pagel et al. (1980) and Peimbert et al. (2005). Chemical Similarity between Planetary Nebulae and H II Regions We now consider the data available in the literature for bright planetary nebulae in dwarf irregular galaxies. By bright planetary nebulae, we adopt all planetary nebulae within 2 mag of the brightest in each galaxy. Selecting planetary nebulae based upon high [O III]λ5007 luminosity affects the resulting sample in two known ways (and perhaps others so far unidentified). First, high [O III]λ5007 luminosity will favor the most oxygen-rich planetary nebulae in each galaxy, at least for oxygen abundances below that of the ISM in the SMC Richer & McCall 1995). As we argue below, this should favor objects resulting from recent star formation. Second, high [O III]λ5007 luminosity will also favor planetary nebulae early in their evolution (e.g., Jacoby 1989; Stasińska et al. 1998). In Table 4, we compare the average oxygen abundances in the bright planetary nebulae in dwarf irregulars with the corresponding average oxygen abundances in their H II regions. For all of the planetary nebulae in all of the dwarf irregular galaxies, we have re-computed their chemical abundances from the line intensities given in the data sources, following the procedure outlined above for the planetary nebulae in NGC 6822. For the H II regions in the Magellanic Clouds, Sextans A and B, and Leo A, we have adopted the abundances given in the original studies. Table 4 graphically. In all of the galaxies considered, the mean oxygen abundance in planetary nebulae and H II regions is the same, within uncertainties, except in Sextans A. If Leo A is excluded, the data suggest a trend. However, the significance of the slope of a line fit to the data (excluding Leo A) is then critically dependent upon the inclusion of Sextans A, whose "population" of bright planetary nebulae contains a single member. If Sextans A is excluded, an F-test indicates that the slope of the resulting relation is not statistically significant. If Sextans A is included, the slope of the line fit to the data is similar, but its statistical significance improves notably, reducing the probability of obtaining the relation by chance to 6%. Independent of the existence of any trend, Fig. 2 indicates that the dredge-up of oxygen is uncommon among the progenitors of the bright planetary nebulae, since the mean abundance found for bright planetary nebulae agrees with that in the interstellar medium, at least for dwarf irregular galaxies with metallicities similar to or greater than that of the SMC. Figure 3 illustrates the excellent correlation between neon and oxygen abundances in bright planetary nebulae in dwarf irregular galaxies. No error bars are drawn for the planetary nebulae in the Magellanic Clouds since this information is not generally available from the original sources. However, it is likely that the uncertainties are similar to those for the other planetary nebulae. In this figure, we also plot the relationship between these abundances found by Izotov et al. (2006) in emission line galaxies, where the relation is set by the nucleosynthetic yields of type II supernovae. The agreement between the two relations is excellent. A linear least squares fit to the planetary nebula data yields 12 + log(Ne/H) = (1.043 ± 0.075)X − 1.04 ± 0.61 (2) where X = 12 + log(O/H). Within uncertainties, the slope and intercept coincide with the relation found by Izotov et al. (2006). Since the stellar progenitors of planetary nebulae are not expected to modify their neon abundance during their evolution (Marigo et al. 2003), the good correlation found between the oxygen and neon abundances indicates that the progenitors of bright planetary nebulae in dwarf irregulars generally do not modify either abundance. This correlation is the basis for supposing that the most oxygen-rich planetary nebulae in dwarf irregulars are the result of recent star formation rather than dredge-up. Nucleosynthesis in the Progenitors of Planetary Nebulae Returning to Fig. 2, population synthesis models indicate that the oxygen abundance in the planetary nebula population is always expected to be below that observed in the interstellar medium , though those models assume that the progenitors of planetary nebulae do not dredge up oxygen. As the oxygen abundance decreases, these models indicate that the difference in abundance between the planetary nebulae and the interstellar medium should decrease. The sense of any possible trend in Fig. 2 is therefore in agreement with these expectations. The novelty is that Fig. 2 indicates that the bright planetary nebulae may in fact have oxygen abundances exceeding those in the interstellar medium at low oxygen abundances. In the context of population synthesis models, an oxygen abundance in planetary nebulae exceeding that found in the interstellar medium requires that the progenitors of the planetary nebulae commonly dredge up oxygen. The planetary nebula in Sextans A has a strong bearing on whether oxygen is judged to be dredged up at low metallicity. In Fig. 5, we present the N/Ne abundance ratio as a function of the Ne/O abundance ratio for the planetary nebulae in our sample. We adopt these abundance ratios since neon is neither expected nor observed to change significantly as a result of the prior evolution of the progenitors of planetary nebulae (Marigo et al. 2003). We also plot the Ne/O ratio found at an oxygen abundance of 12 + log(O/H) = 8.0 dex (Izotov et al. 2006). If oxygen is dredged up, one expects that Ne/O should be low. N/Ne will depend upon which processes have dredged up nitrogen (Marigo 2001). Low Ne/O is found for the planetary nebula in Sextans A and for S33 in NGC 6822. Thus, oxygen is dredged up on occasion. While the above examples support the hypothesis that oxygen can occasionally be dredged up, they do not address the principal concern, which is whether oxygen is routinely dredged up at low metallicity. The planetary nebula in Leo A provides useful guidance. As van Zee et al. (2006) argue, this object may not be the result of recent star formation in Leo A, but may have been formed some time in the past. The oxygen abundance in this object is the same as that in the interstellar medium. Furthermore, the nebula falls on the neon-oxygen relation for star-forming galaxies. It follows that the progenitor of this planetary nebula did not substantially modify its initial store of oxygen. If one accepts that the planetary nebula in Leo A is not derived from a recently formed, and therefore relatively massive, stellar progenitor, the comparison of this object with the planetary nebula in Sextans A allows the possibility that only massive progenitors of planetary nebulae dredge up oxygen, provided, of course, that the planetary nebula in Sextans A descends from a massive progenitor. If the planetary nebula in Leo A is the result of recent star formation, one must conclude that oxygen is only dredged up under some fraction of circumstances, even at very low metallicity. The prevalence of oxygen self-enrichment should become clearer in the near future. Three groups have undertaken spectroscopy of planetary nebulae in NGC 3109 (Leisy et al. 2005b;Knaizev et al. 2006;Peña et al. 2006). Given NGC 3109's oxygen abundance of 12 + log(O/H) ∼ 7.8 dex (Lee et al. 2003), its population of planetary nebulae should help clarify whether the trend in Fig. 2 Lee et al. 2003). It would also be useful to confirm that planetary nebulae have systematically lower abundances than the interstellar medium at oxygen abundances exceeding 12 + log(O/H) = 8 dex. Spectroscopy of the planetary nebulae in IC 10 (12 + log(O/H) ∼ 8.2 dex; Lee et al. 2003) or in the disks of M31, M33, or other nearby spirals would be useful in this regard. (Planetary nebulae in the disk of the Milky Way are less desirable since their absolute luminosities are much more difficult to ascertain.) Figure 4 emphasizes the difficulty of interpreting nitrogen abundances. It is apparent that N/O does not vary in a uniform way as a function of oxygen abundance. In the LMC, the planetary nebulae with the highest oxygen abundances have the lowest N/O ratios. As the oxygen abundance decreases, the N/O ratio increases, indicating that the progenitors of these objects enriched themselves more efficiently in nitrogen. In contrast, the planetary nebulae in the SMC are all approximately uniformly enriched in N compared to the ISM by ∆ log(N/O) ∼ 0.7 dex. For the other galaxies, no clear trends are evident, though it is noteworthy that the bright planetary in most of them achieve approximately the same limiting nitrogen enrichment relative to oxygen, log(N/O) ∼ 0.5 dex (cf. Fig. 6). Perhaps, the simplest conclusion is that the progenitors of all planetary nebulae enrich themselves in nitrogen to some arbitrary extent, depending upon parameters other than initial mass, as appears to occur in the progenitors of planetary nebulae in systems without star formation ). An important question is whether nitrogen enrichment occurs as a result of the consumption of carbon or of oxygen also. (The conversion of carbon to nitrogen should occur in any case.) That the neon and oxygen abundances for the planetary nebulae in dwarf irregulars generally follow the trend found in ELGs argues that the nitrogen enrichment comes at the expense of carbon. (The same conclusion is drawn from plotting the Ne/O abundance ratio as a function of oxygen abundance.) The N/O ratio may reflect the chemical evolution of the host galaxies as well as the nucleosynthesis within the progenitors of these planetary nebulae. Many of the planetary nebulae in Sextans B and NGC 6822 are examples of the interplay of these influences since they generally have both oxygen abundances and N/O ratios that agree with the value found in the interstellar medium. Either their progenitors did not dredge up nitrogen, which is surprising if they were rel-atively massive, or these galaxies have evolved chemically very little in the recent past, allowing the progenitors of their bright planetary nebulae to be less massive. The latter option, however, is somewhat at odds with the expectation that bright planetary nebulae should arise from progenitors of order 2 M ⊙ (e.g., Marigo et al. 2004). The lack of nitrogen production is surprising, especially if the stellar progenitors are massive, since both the first and second dredge-up as well as hot bottom burning should all result in nitrogen production (e.g., Forestini & Charbonnel 1997). At any rate, if nitrogen, which should be more easily produced, is not always dredged up in these progenitors of planetary nebulae, oxygen should be dredged up even less frequently. Finally, in Fig. 6, we plot the helium abundances for the planetary nebulae in our sample as a function of log(N/O). There is little correlation between helium abundance and log(N/O). Comparing the helium abundances in planetary nebulae and H II regions in NGC 6822 (Table 2), the helium enrichment in planetary nebulae spans the range from zero to a doubling of the original helium content. Conclusions We have obtained spectroscopy of a sample of seven planetary nebulae and four H II regions in the Local Group dwarf irregular galaxy NGC 6822. From these data, we calculate their chemical composition and find that the planetary nebulae in NGC 6822 have oxygen and neon abundances very similar to those in the interstellar medium, from which we infer that these planetary nebulae arise preferentially, though not exclusively, as a result of recent star formation in NGC 6822. The difference between the mean oxygen abundance in bright planetary nebulae and the mean for HII regions is consistent with zero regardless of the level of enrichment, except perhaps in Sextans A, but the result for this galaxy is based upon a single planetary nebula. Very generally, we find that bright planetary nebulae in dwarf irregulars have oxygen and neon abundances similar to those found in the interstellar medium in star-forming galaxies. Since the latter abundance ratio is set by the nucleosynthetic yields of type II supernovae, and since the progenitors of planetary nebulae are not expected to modify their neon abundances, it is unlikely that the progenitors of the vast majority of the bright planetary nebulae in our sample modified their initial oxygen or neon abundances. Two planetary nebulae in our combined sample, that in Sextans A and S33 in NGC 6822, do present abundance ratios indicative of the dredgeup of oxygen. While it remains to be shown that oxygen is routinely dredged up, there is a slight suggestion of a trend wherein this might happen at low metallicity under some circumstances. On the other hand, many of the progenitors of the bright planetary nebulae in Sextans B and NGC 6822 did not dredge up nitrogen, implying that the dredge-up of nitrogen might be a less common process than has been thought. We thank the time allocation committee of the Canada-France-Hawaii Telescope for granting us the opportunity to observe. We thank R. This 2-column preprint was prepared with the AAS L A T E X macros v5.2. -We present the correlation of neon and oxygen abundances in bright planetary nebulae in the sample of dwarf irregular galaxies for which these data exist. In this and subsequent plots, no uncertainties are shown for the planetary nebulae in the LMC and SMC since this information is not available from the original sources, though it is likely that their uncertainties are similar to those for the objects in other galaxies. The heavy dashed line is the relation between neon and oxygen abundances in H II regions in emission line galaxies (ELGs) from Izotov et al. (2006), while the thin dashed lines indicate the scatter about this relation. The generally excellent agreement between the planetary nebulae and the ELGs indicates that the stellar progenitors of most bright planetary nebulae in dwarf irregulars do not significantly modify either of these abundances. -Here, we plot the helium abundance as a function of the N/O abundance ratio for bright planetary nebulae in dwarf irregular galaxies. The relation is surprisingly flat. It is also perhaps noteworthy that a similar limiting nitrogen enrichment, log(N/O) ∼ 0.5 dex, is achieved in most galaxies, independently of the helium abundance. Figure 2 2Figure 2 presents the information in Table 4 graphically. In all of the galaxies considered, the mean oxygen abundance in planetary nebulae and H II regions is the same, within uncertainties, except in Sextans A. If Leo A is excluded, the data suggest a trend. However, the significance of the slope of a line fit to the data (excluding Leo A) is then critically dependent upon the inclusion of Sextans A, whose "population" of bright planetary nebulae contains a single member. If Sextans A is excluded, an F-test indicates that the slope of the resulting relation is not statistically significant. If Sextans A is included, the slope of the line fit to the data is similar, but its statistical significance improves notably, reducing the probability of obtaining the relation by chance to 6%. Independent of the existence of any trend, Fig. 2 indicates that the dredge-up of oxygen is uncommon among the progenitors of the bright planetary nebulae, since the mean abundance found for bright planetary nebulae agrees with that in the interstellar medium, at least for dwarf irregular galaxies with metallicities similar to or greater than that of the SMC. Figure 3 illustrates the excellent correlation between neon and oxygen abundances in bright planetary nebulae in dwarf irregular galaxies. No error bars are drawn for the planetary nebulae in the Magellanic Clouds since this information is not generally available from the original sources. However, it is likely that the uncertainties are similar to those for the other planetary nebulae. In this figure, we also plot the relationship between these abundances found by Izotov et al. (2006) in emission line galaxies, where the relation is set by the nucleosynthetic yields of type II supernovae. The agreement between the two relations is excellent. A linear least squares fit to the planetary nebula data yields Figure 4 4plots the N/O abundance ratio as a function of the oxygen abundance. In this figure, both the N/O ratio and the oxygen abundance in the planetary nebulae are calculated relative to the values observed in the ISM (H II regions). This figure emphasizes that most of the bright planetary nebulae in these galaxies have oxygen abundances close to the ISM values. It also emphasizes how little N/O varies from its ISM values in some of these galaxies, particularly NGC 6822 and Sextans B. Porter for providing us with a more extensive list of He • emissivities. MGR acknowledges financial support from CONACyT through grant 43121 and from UNAM-DGAPA via grants IN112103, IN108406-2, and IN108506-2. MLM thanks the Natural Sciences and Engineering Research Council of Canada for its continuing support. Canada-France-Hawaii Telescope ( Fig. 2 . 2-We compare the mean oxygen abundance in bright planetary nebulae and H II regions in dwarf irregular galaxies. The vertical error bars indicate the uncertainty in the ratio of abundances. The horizontal bar on each point indicates the abundance range spanned by the planetary nebulae in each galaxy. No range is shown for Leo A or Sextans A since only one planetary nebula is known in each of these galaxies. The horizontal dashed line indicates agreement between oxygen abundances in planetary nebulae and H II regions. Generally, the oxygen abundances in bright planetary nebulae are very similar to those in H II regions. The open squares connected by a line are the predictions for star-forming galaxies fromRicher et al. (1997). Fig. 4 .Fig. 5 . 45-We plot the N/O abundance ratio as a function of the oxygen abundance for bright planetary nebulae in dwarf irregular galaxies. Both log(N/O) and log(O/H) are computed with respect to the values found in the ISM for the host galaxies in order to emphasize that most bright planetary nebulae have oxygen abundances similar to those in the ISM and that sometimes even nitrogen is enriched very little. -Here, we plot the N/Ne abundance ratio as a function of the Ne/O abundance ratio for bright planetary nebulae in dwarf irregular galaxies. Objects that have dredged up oxygen should have low values of Ne/O, such as occurs for the planetary nebula in Sextans A and S33 in NGC 6822, while N/Ne may vary according to which dredge-up episodes have occurred. The vertical dashed line is the value of the Ne/O ratio in star-forming galaxies for 12 + log(O/H) = 8.0 dex(Izotov et al. 2006). for the objects in which they detect [O III]λ4363. For KD29, we only have an upper limit to [O III]λ4363 is significant. Likewise, spectroscopy of the planetary nebulae recently reported in IC 1613 and WLM would be helpful (both galaxies have 12 + log(O/H) ∼ 7.7 dex; Table 1 1Observed Line Intensities a for PNe b and H II regions b in NGC 6822wavelength ion A1(λ) c pn-020 S33 pn-010 pn-012 S16 pn-017 3727 [O II] 4.52 < 51 28.4 ± 7.2 318 ± 51 92 ± 42 22 3868.76 [Ne III] 4.39 41 ± 11 31.7 ± 2.4 52 ± 14 50 ± 18 46.5 ± 6.5 < 47 3889.049 H I 4.37 9.8 ± 2.0 3967.47 [Ne III] 4.30 10.6 ± 2.2 8.5 ± 5.8 49 ± 15 3970.072 H I 4.29 8.9 ± 2.2 9.3 ± 5.9 4101.765 H I 4.18 21.3 ± 1.8 15.0 ± 3.2 < 34 4340.495 H I 3.97 < 11 43.6 ± 1.4 34.2 ± 7.8 49.2 ± 8.7 34.9 ± 3.6 37.5 ± 9.2 4363.21 [O III] 3.95 < 11 29.1 ± 1.2 20.2 ± 6.3 10.9 ± 6.8 14.0 ± 2.9 23.4 ± 8.3 4387.929 He I 3.93 2.2 ± 0.9 4471.477 He I 3.86 3.43 ± 0.97 4685.75 He II 3.66 < 8 56.7 ± 1.3 35.4 ± 6.5 < 13 20.9 ± 2.2 41.4 ± 9.1 4711.34 [Ar IV] 3.63 0.4 ± 1.1 4713.375 He I 3.63 1.2 ± 1.2 4740.2 [Ar IV] 3.62 3.76 ± 0.81 4861.332 H I 3.49 100.0 ± 4.9 100.0 ± 1.0 100.0 ± 4.7 100.0 ± 6.2 100.0 ± 2.1 100.0 ± 7.6 4958.92 [O III] 3.39 323 ± 19 380.5 ± 7.2 265 ± 15 257 ± 18 424 ± 14 239 ± 22 5006.85 [O III] 3.34 1009 ± 51 1178.0 ± 14 787 ± 38 789 ± 50 1310 ± 30 747 ± 59 5200 [N I] 3.19 9.5 ± 1.1 5412 He II 3.02 6.33 ± 0.82 5754.57 [N II] 2.75 30.3 ± 1.4 5875.666 He I 2.65 33.6 ± 5.3 13.9 ± 1.2 16.0 ± 5.7 17.6 ± 6.7 24.8 ± 2.6 < 6 6300.32 [O I] 2.40 40.1 ± 1.5 6312.06 [S III] 2.39 1.5 ± 1.0 6363.81 [O I] 2.36 15.3 ± 1.3 6548.06 [N II] 2.26 243.3 ± 5.3 12.0 ± 6.0 22.5 ± 8.3 20.5 ± 9.1 31.3 ± 9.7 6562.817 H I 2.26 1097 ± 55 477.4 ± 7.0 455 ± 23 545 ± 36 703 ± 19 420 ± 34 6583.39 [N II] 2.25 < 12 720.0 ± 9.2 34.4 ± 7.1 12.5 ± 7.8 36.2 ± 9.4 60 ± 11 6678.149 He I 2.20 12.5 ± 3.5 3.66 ± 0.97 6.4 ± 5.3 < 9 14.0 ± 3.6 6716.42 [S II] 2.17 4.4 ± 3.2 4.39 ± 0.99 77.0 ± 9.0 < 9 13.4 ± 3.5 40.2 ± 9.5 6730.78 [S II] 2.16 20.2 ± 4.3 5.8 ± 1.1 61.8 ± 8.4 < 9 3.5 ± 2.6 19.8 ± 7.4 7065.179 He I 2.01 46.3 ± 8.7 8.6 ± 1.5 26.9 ± 3.4 7135.8 [Ar III] 1.98 57.3 ± 9.2 7.9 ± 1.5 17.9 ± 3.1 7319.92 [O II] 1.90 27.8 ± 8.0 < 26 13.7 ± 3.3 < 25 7330.19 [O II] 1.90 9.6 ± 6.1 < 26 12.8 ± 3.3 < 25 E(B − V )Hα (mag) 1.21 ± 0.05 0.49 ± 0.02 0.45 ± 0.06 0.59 ± 0.08 0.82 ± 0.03 0.38 ± 0.08 E(B − V )Hγ (mag) 0.19 ± 0.07 0.74 ± 0.53 −0.09 ± 0.41 0.68 ± 0.23 0.53 ± 0.57 Table 2 2Chemical abundances for PNe a and H II regions a in NGC 6822quantity line(s) used pn-020 S33 pn-010 pn-012 S16 pn-017 Te 4363/5007 < 15790 19479 +835 −1253 19754 +5220 −4183 14930 +5456 −4877 14019 +1552 −1441 21711 +8775 −5866 Ne 6716/6731 2000 2183 13563 −1784 186 685 +84 −88 2000 2000 He 0 /H 5876 0.080 +0.010 −0.011 0.053 +0.018 −0.014 0.087 +0.026 −0.029 0.075 +0.029 −0.029 0.083 +0.010 −0.009 0.024 +0.005 −0.006 He + /H 4686 < 0.009 0.055 +0.002 −0.002 0.034 +0.007 −0.007 < 0.012 0.020 +0.003 −0.002 0.039 +0.013 −0.010 10 6 N + /H 6584 0.217E +0.028 −0.025 20.1 +6.0 −1.8 0.97 +0.71 −0.40 0.5 +1.3 −0.4 1.28 0.69 −0.49 1.6 +1.4 −0.7 10 5 O + /H 3727 < 1.6 0.25 +0.66 −0.13 1.9 +2.6 −1.0 1.6 +9.0 −1.2 0.67 +0.33 −2.0 10 5 O + /H 7325 0.62 +0.53 −0.66 < 3.8 1.3 +1.9 −1.3 < 1 10 5 O 2+ /H 5007 8.0 +1.1 −1.0 6.4 +1.2 −0.6 4.2 +3.3 −1.6 8.0 +18 −4.1 14.7 +5.5 −3.7 3.3 +3.6 −1.6 10 6 Ne 2+ /H 3869 26 +12 −9 6.5 +1.8 −1.1 10 +14 −6 23 +105 −17 30 +20 −12 6.8 +8.9 −3.5 ICF(N) 14.88 41.97 3.95 6.61 26.53 8.31 ICF(O) 1.07 1.61 1.25 1.10 1.16 1.91 ICF(Ne) 1.15 1.67 1.82 1.32 1.21 2.49 He/H 0.088 +0.010 −0.30 quantity line(s) used pn-019 S10 KD29 Ho10 Ho06 Te 4363/5007 15735 +1499 −1419 14774 +2279 −2108 < 13872 12935 +4179 −4465 10596 +1549 −1638 Ne 6716/6731 100 188 +143 −108 100 629 +1162 −511 100 He 0 /H 5876 0.174 +0.011 −0.011 0.048 +0.004 −0.005 0.075 +0.009 −0.009 0.067 +0.020 −0.017 0.078 +0.002 −0.005 He + /H 4686 < 0.002 10 6 N + /H 6584 0.23 +0.22 −0.16 1.30 +0.80 −0.49 2.06 +0.14 −0.14 1.1 +3.3 −0.7 10 5 O + /H 3727 < 6 5.85 +0.34 −0.34 6.1 +9.2 −3.1 10 5 O + /H 7325 2.1 +4.0 −2.2 < 1.6 < 2 10 5 O 2+ /H 5007 4.3 +1.3 −0.9 3.0 +1.7 −0.9 1.34 +0.03 −0.03 6.5 +21 −3.4 12.3 +9.9 −4.3 10 6 Ne 2+ /H 3869 6.4 +4.1 −2.6 4.0 +4.4 −2.1 ICF(N) 69.53 2.48 1.23 4.25 3.00 ICF(O) 1 1 1 1 1 ICF(Ne) 1.01 1.68 5.38 1.31 1.50 He/H 0.176 +0.011 −0.011 0.049 +0.004 −0.005 0.075 +0.009 −0.009 0.067 +0.020 −0.017 0.078 +0.002 −0.005 Table 3 3Atomic data usedion Transition Probabilities Collision Strengths N + Wiesse et al. (1996) Lennon & Burke (1994) O + Wiesse et al. (1996) Pradhan (1976) McLaughlin & Bell (1993) O 2+ Wiesse et al. (1996) Lennon & Burke (1994) Ne 2+ Mendoza & Zeippen (1982) Butler & Zeippen (1994) Kauffman & Sugar (1986) Table 4 4Oxygen Abundances in H II regions and PNe in Dwarf IrregularsFig. 1.-From left to right, we present the on-band, off-band, and the two-dimensional spectrum of pn-020 (the wavelength interval including Hβ at bottom and [O III]λλ4959,5007 at top; from UT 19 August 1998).In the images, north is up and east is to the left. In the spectrum, the spatial axis has east to the left. The vertical and horizontal lines in the on-band image locate pn-020. Its coordinates are α = 19:45:11.5 δ = -14:48:54 (J2000, uncertainty ±1 ′′ ). The black bar at the bottom of the on-band image is 30 ′′ long. The spectrum clearly demonstrates that the planetary nebula is offset to the east from the faint star visible in the off-band image. Likewise, the spectrum emphasizes the dramatically different line intensity ratios compared to the background H II region Ho15Galaxy 12 + log(O/H)H II 12 + log(O/H)PN ∆ log(O/H) a Data b Leo A 7.38 ± 0.10 7.34 ± 0.10 −0.04 ± 0.14 1 Sextans A 7.60 ± 0.20 7.84 ± 0.11 +0.29 ± 0.23 2,3 SMC 8.03 ± 0.10 8.13 ± 0.06 +0.10 ± 0.12 4,5 NGC 6822 8.10 ± 0.10 8.01 ± 0.14 −0.09 ± 0.17 6,7,8 Sextans B 8.12 ± 0.12 7.95 ± 0.16 −0.17 ± 0.20 2,3 LMC 8.35 ± 0.06 8.24 ± 0.10 −0.11 ± 0.12 4,5 Visiting Astronomer, Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, le Centre National de la Recherche Scientifique de France, and the University of Hawaii IRAF is distributed by the National Optical Astronomical Observatories, which is operated by the Associated Universities for Research in Astronomy, Inc., under contract to the National Science Foundation. + log(N/H) 7.20 +0.42 −0.31 6.51 +0.26 −0.16 6.40 +0.03 −0.03 6.67 +1.30 −0.28 12 + log(O/H) 7.64 +0.12 −0.09 7.71 +0.37 −0.20 7.86 +0.02 −0.02 7.93 +1.08 −0.17 8.27 +0.32 −0.12 12 + log(Ne/H) 6.81 +0.28 −0.18 6.82 +0.48 −0.23 log(N/O) −0.44 +0.42 −0.31 −1.20 +0.89 −0.48 −1.45 +0.04 −0.04 −1.26 +1.30 −0.28 log(Ne/O) −0.83 +0.30 −0.20 −0.88 +0.54 −0.26 a pn-020, S33, pn-010, pn-012, S16, pn-017, and pn-019 are planetary nebulae while S10, KD29, Ho10, and Ho06 are H II regions. 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[ "On the Use of L-functionals in Regression Models", "On the Use of L-functionals in Regression Models" ]	[ "Ola Hössjer ", "Måns Karlsson " ]	[]	[]	In this paper we survey and unify a large class or L-functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L-moments (Sittinen, 1969), L-skewness, and Lkurtosis (Hosking, 1990) and introduce order numbers for a large class of L-functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2) respectively and describe how a family of L-functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follow a transformed linear model, with a link function that determines the appropriate class of L-functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a unified asymptotic theory of estimates of L-functionals, and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.	null	[ "https://arxiv.org/pdf/2204.13552v1.pdf" ]	248,427,235	2204.13552	56d05488e655bf24e3d41be95ab32c8bffae9447	On the Use of L-functionals in Regression Models Ola Hössjer Måns Karlsson On the Use of L-functionals in Regression Models Bird phenologyCoefficient of determinationL-functionalsL- statisticsOrder numbersOrthogonal series expansionQuantile functionQuantile regression In this paper we survey and unify a large class or L-functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L-moments (Sittinen, 1969), L-skewness, and Lkurtosis (Hosking, 1990) and introduce order numbers for a large class of L-functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2) respectively and describe how a family of L-functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follow a transformed linear model, with a link function that determines the appropriate class of L-functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a unified asymptotic theory of estimates of L-functionals, and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion. properties are conveniently represented in terms of L-functionals of the empirical distribution formed by the sample, as summarized in Chapter 8 of Serfling (1980). Many authors have proposed extensions of L-statistics for regression models. In a pioneering article Koenker and Bassett (1978) introduced regression quantiles. These nonparametric estimators make few assumptions on the conditional distribution of the response variable. They naturally extend order statistics to linear regression models, and have been applied to many fields of science (Koenker and Hallock, 2001). Koenker and Portnoy (1989) introduced linear combinations of regression quantiles and showed that much of the computational and asymptotic theory of L-functionals of order statistics for i.i.d. models extend to the regression framework. The monograph of Koenker (2005) summarizes these and a number of other aspects of regression quantiles, including nonlinear regression. Other contributions include parametric versions of regression quantiles (Gilchrist, 2000, Frumento and Bottai, 2016, and the use of L-functionals in survival analysis (Frumento and Bottai, 2017). In this paper we review, unify, and extend the use of L-functionals for regression models. We consider four classes of L-functionals or ratios of L-functionals that represent measures of location, scale, skewness, and heavytailedness (kurtosis). Then we introduce order numbers for a large class of L-functionals through orthogonal series expansions of quantile functions (Takemura, 1983, Okagbue et al., 2019, and motivate why location, scale, skewness, and heavytailedness functionals have order numbers 1, 2, (3,2), and (4,2) respectively. In this context we describe how a given reference distribution gives rise to a whole collection of L-functionals. If Legendre polynomials and a uniform reference distribution is used, the resulting class of L-functionals of order 1, 2, . . . corresponds to L-moments (Sillito, 1969), whereas the ratios of L-functionals of order (2,1), (3,2), and (4,2) agree with the L-coefficient of variation, the L-skewness, and the L-kurtosis (Hosking, 1990(Hosking, , 1992(Hosking, , 2006 up to normalizing constants. This Legendre class of L-functionals is a natural choice for distributions with bounded support, but Hermite polynomials (with a Gaussian reference distribution) or Laguerre polynomials (with an exponential reference distribution) might be preferable for data whose support is on the real line and on the positive real line respectively, in particular if the distribution of the response is close to the reference distribution. We will apply the framework of L-functionals to a wide range of transformed linear regression models, with linear models a special case. We argue that the transformation (or link function) will determine the appropriate type of L-functionals to use. We also demonstrate how parametric and nonparametric methods of estimating conditional L-functionals can be put into a unified framework, for response variables with or without censoring and truncation. In more detail, the paper is organized as follows. In Section 2 we introduce L-functionals for models without covariates, with particular emphasis on functionals that quantify location, scale, skewness or heavytailedness, and how their order numbers can be assessed. Then in Section 3 we generalize the framework of Section 2 to linear and transformed linear regression models, for models with our without censoring or truncation. Section 4 contains numerical examples, in Section 5 we analyze a data set with migration times of birds and introduce a novel version of the coefficient of determination, whereas Section 6 provides a summarizing discussion. 2 L-functionals without covariates Definition of L-functionals Let F Y (y) = F (y) = P (Y ≤ y) be the unknown distribution function of a random variable Y . Suppose we want to infer a certain functional θ = T (F ) of F , using a data set Y 1 , . . . , Y n of independent and identically distributed random variables with P (Y i ≤ y) = F (y), for i = 1, . . . , n. In this paper we will focus on L-functionals, i.e. linear combinations θ = T (F ) = 1 0 Q(p)dG(p) (1) of quantiles Q(p) = Q Y (p) = F −1 (p) = inf{y; F (y) ≥ p}(2) of F , using some weight function G = G + − G − that corresponds to a signed measure. When θ is a measure of location of F , the weight function is often a positive measure (G = G + ), but this is not the case for measures of scale, skewness and kurtosis (cf. Section 2.3). Following Serfling (1980) we consider measures dG(p) = g(p)dp + M m=1 g m δ πm (p) on [0, 1] that split into one absolutely continuous part g(p)dp and another finite sum of point masses g m δ πm at π m with weights g m for m = 1, . . . , M , with 0 ≤ π 1 < . . . π M ≤ 1. This allows us to work with all the common types of distributions, i.e. continuous, discrete and mixtures, within the same framework. The L-functional is robust if extreme quantiles of F are excluded, i.e. if the total variation measure \|G\| = G + + G − satisfies \|G\|([0, π) ∪ (1 − π, 1]) = 0 for some sufficiently small 0 < π ≤ 0.5. Then the breakdown point (Rousseeuw and Leroy, 1987) of T is at least π. A wide class of estimatorsθ = T (F ) = 1 0Q (p)dG(p),(4) of θ are obtained by pluggingF , an estimate of F , into (1), withQ =F −1 the corresponding estimate of Q. This estimate of F could be nonparametric, i.e. the empirical distribution function F (y) = 1 n n i=1 1(Y i ≤ y)(5) formed by the sample. Thenθ = n i=1 w i Y (i)(6) is an L-statistic, i.e. a linear combination of the order statistics Y (1) ≤ . . . ≤ Y (n) , with w i = i/n (i−1)/n dG(p) the weight assigned to the ith order statistic. It is also possible to insert a parametric estimator F (y) = F (y;ψ) of F into (4). In this case F (y) = F (y; ψ) is fully determined by a finitedimensional parameter ψ = (ψ 1 , . . . , ψ r ), of whichψ = (ψ 1 , . . . ,ψ r ) is an estimate. If the density function F (y; ψ) = f (y; ψ) is tractable, this is typically the maximum likelihood estimator of ψ. For some distributional families, such as the Generalized Lambda Distributions (Karian and Dudewicz, 2000) or mixtures of quantile functions (Karvanen, 2006, Karvanen andNuutinen, 2008), the quantile function Q(p; ψ) = F −1 (p; ψ) has a more explicit form. It might then be more tractable to estimate ψ by fitting some of the order statistics Y (i) to Q (Gilchrist, 2007) or some empirical L-functionals to the corresponding population-based L-functionals (Karvanen, 2006). Asymptotics In order to study the large sample behavior ofθ =θ n as n → ∞, we introduce S, the space of real-valued and integrable functions on (0, 1), equipped with a seminorm Q S = 1 0 \|Q(p)\|\|g(p)\|dp + M m=1 \|g m \|\|Q(π m )\|. where g(p) and g 1 , . . . , g M refer to the absolutely continuous part and the point masses of the weight measure G in (3) respectively. We may also regard (8) as a norm of the restriction of Q to supp(G). If F 1 and F 2 are two distributions with quantile functions Q 1 and Q 2 , it follows that \|T (F 2 ) − T (F 1 )\| ≤ 1 0 \|Q 2 (p) − Q 1 (p)\|d\|G\|(p) = 1 0 \|Q 2 (p) − Q 1 (p)\|\|g(p)\|dp + M m=1 \|g m \|\|Q 2 (π m ) − Q 1 (π m )\| = Q 2 − Q 1 S .(9) This implies in particular that T (F ) is a continuous functional with respect to the distance measure d(F 2 , F 1 ) = Q 2 −Q 1 S introduced by · S . In particular, when M = 0 and g(p) ≡ 1, this distance measure equals the Wasserstein metric of order 1 (Olkin and Pukelsheim, 1982). The asymptotic properties ofθ n are determined by the weight function G and the large sample behavior of the rescaled quantile process Z n (p) = √ n[Q n (p) − Q(p)], 0 < p < 1,(10) which is a random element of S. We will assume weak convergence (Billingsley, 1999) Z n L −→ Z as n → ∞, with respect to the topology introduced by the norm in S, i.e. E[h(Z n )] → E[h (Z)] for all bounded and continuous functions h : S → R. It is further assumed that the limit in (11) is a Gaussian process with mean function E[Z(p)] = 0 and covariance function Cov[Z(p), Z(s)] = R(p, s) for p, s ∈ supp(G). Notice in particular that for discrete weight measures (g ≡ 0), only the last term on the right hand side of (8) is present. Then (11) corresponds to weak convergence of finite-dimensional distributions of Z n towards Z at π 1 , . . . , π m . This follows from the fact that Z n and Z, restricted to supp(G) = {π 1 , . . . , π M }, represent weak convergence of random vectors of dimension M when g ≡ 0. On the other hand, (11) represents functional weak convergence on an infinite index set p ∈ supp(G) when g = 0, due to the first term of (8). It is also possible to define Q S = Q ∞ = sup 0<p<1 \|Q(p)\|(12) as the supremum of Q on (0, 1). It is easy to see that T (F ) is continuous with this choice of norm, since \|T (F 2 ) − T (F 1 )\| ≤ \|G\|(0, 1) Q 2 − Q 1 S . Note that Q S in (12) does not involve the weight measure G. This is advantageous when simultaneous weak convergence of several L-functionals, based on different weight functions, is of interest. On the other hand, the advantage of (8) is that this (semi)norm exists for a larger class of functions Q. It also gives rise to a weaker topology on S, so that (11) requires less. Indeed, when (8) is used, weak convergence of Z n need only be established on the index set supp(G). Since (9) implies thatT : S → R, defined byT (Q) = 1 0 Q(p)dG(p), is a continuous functional, it follows from the Continuous Mapping Theorem that √ n(θ n − θ) =T (Z n ) L −→T (Z) ∼ N (0, Σ)(13) as n → ∞, where the asymptotic covariance matrix of the limiting normal distribution satisfies Σ = 1 0 1 0 R(p, s)dG(p)dG(s).(14) The expression for the covariance function R depends on which estimator of F that is used in (4). The nonparametric estimate (5) corresponds to R(p, s) = min(p, s) − ps f (Q(p))f (Q(s)) ,(15) where f (y) = F (y) is the density function of Y . When (15) is inserted into (14), we get a well known expression for the asymptotic variance of L-statistics, cf. Mosteller (1946), Bennett (1952), Jung (1955), Chernoff et al. (1967), and Moore (1968). In the parametric case we have that R(p, s) = dQ(p; ψ) dψ V dQ(s; ψ) dψ T ,(16) providedψ =ψ n is asymptotically normal with covariance matrix V , i.e. √ n(ψ n − ψ) L −→ N (0, V )(17) as n → ∞. In particular, V equals the inverse of the Fisher information matrix of ψ, whenψ n is the ML-estimator of ψ. Examples of L-functionals The choice of weight function G in (4) will determine the type of L-functional. In the present paper we will mainly focus on four types of statistical functionals, presented in Oja (1981), namely for location, scale, skewness and kurtosis: Example 1 (Measures of location) A measure of location (T = T loc ) is equivariant with respect to linear transformations of data, i.e. T loc (F aY +b ) = aT loc (F Y ) + b(18) for any real valued a and b. Since Q aY +b = aQ Y + b, it follows that (18) holds for all L-functionals with a weight function satisfying 1 0 dG(p) = 1.(19) For location functionals we will also require that G is a positive measure, so that G − = 0.(20) The two regularity conditions (19)-(20) imply that T loc preserves stochastic ordering of distribution functions. By this we mean that T loc (F 1 ) ≤ T loc (F 2 ) whenever F 2 is stochastically larger than F 1 , that is, when the quantile functions of the two distributions satisfy Q 1 (p) ≤ Q 2 (p) for all 0 < p < 1. Examples of location functionals with a discrete weight function (g ≡ 0) include quantiles T loc (F ) = Q(π)(21) for some fixed 0 < π < 1 and for distributions F with bounded support the midrange T loc (F ) = 1 2 [Q(0) + Q(1)] .(22) Examples of location functionals with an absolutely continuous weight density include smoothed quantiles T loc (F ) = 1 0 1 b K p − π b Q(p)dp,(23) where K is the smoothing probability density and 0 < b < min(π, 1 − π) the bandwidth (Parzen, 1979, Sheather andMarron, 1990), and compound expectations (CEs) T loc (F ) = 1 π 1 − π 0 π1 π0 Q −1 (p)dp(24) that average quantiles of F between π 0 and π 1 for some appropriately chosen 0 ≤ π 0 < π 1 ≤ 1. Notice in particular that (24) is a special case of (23) that corresponds to a rectangular kernel K(x) = 1(\|x\| ≤ 1)/2, π = (π 1 + π 2 )/2 and b = (π 1 − π 0 )/2. A centralized measure of location is one whose weight function is symmetric around p = 0.5 and satisfies (19). If F Y has a symmetric distribution around µ, a centralized measure of location will equal the center of symmetry (T loc (F Y ) = µ). These types of weight functions include the median (π = 0.5 in (21)), the midrange (22), and the trimmed mean (π 0 + π 1 = 1 in (24), see for instance Tukey and McLaughlin (1963), Bickel (1965), and Stigler (1977)). When π 0 = 0 and π 1 = 1, the trimmed mean simplifies to the expected value T loc (F Y ) = 1 0 Q Y (p)dp = E(Y ).(25) A location-scale family F (y) = F 0 y − µ σ(26) corresponds to a parametric family (7) with ψ = (µ, σ) and F 0 known. It is then of interest to find the optimal weight function G of location for the Lstatistic (6). This weight function minimizes the asymptotic variance (14)-(15) among all L-functionals T loc that estimate µ in (26). It is well known (Chernoff et al., 1967, Chapter 8 of Serfling, 1980 that if the density f 0 = F 0 is twice differentiable, the asymptotically optimal L-estimator of µ (when σ is known or when f 0 is symmetric so that σ is Fisher orthogonal to µ) corresponds to an absolutely continuous weight measure with density g(p) = − (log f 0 ) (F −1 0 (p)) I(f 0 ) ,(27) where I(f 0 ) = f 0 (y) 2 /f 0 (y)dy is the Fisher information. In particular, the expected value (25) with weight function g(p) ≡ 1 is optimal when F ∼ N (µ, σ 2 ), whereas the midrange (22) is optimal for a uniform distribution, as can be seen by approximating this distribution by a smooth F 0 . 2 Example 2 (Measures of scale) A measure of scale (T = T scale ) is nonnegative and satisfies T scale (F aY +b ) = \|a\|T scale (F Y )(28) for all real-valued a and b. In the context of L-functionals, (28) is satisfied for weight functions that take on positive as well as negative values, in such a way that the conditions dG is skew-symmetric around p = 0.5, and G + = G restricted to (0.5, 1], G − = G restricted to [0, 0.5)(30) are fulfilled. In order to see that (29)-(30) correspond to a measure of scale, it is instructive to insert these conditions into (1). This makes it possible to rewrite the scale functional as T scale (F ) = 1 0 (Q(p) − Q(0.5))dG(p) = 1 0 \|Q(p) − Q(0.5)\|d\|G\|(p) = 1 0.5 [Q(p) − Q(1 − p)] dG + (p).(31) Notice in particular that (31) has an intuitive interpretation as a linear combination of interquantile ranges Q(p) − Q(1 − p). Bickel and Lehmann (1976) and Oja (1981) defined a spread-ordering among distributions, where F 2 is said to be at least as spread out as F 1 if the difference Q 2 (p) − Q 1 (p) between the quantile functions of F 2 and F 1 is non-decreasing. It follows from (31) that scale functionals preserve spread ordering, i.e. T scale (F 1 ) ≤ T scale (F 2 ). Since the unit of T scale is somewhat arbitrary, some reference distribution F 0 is typically chosen to have scale 1, i.e. T scale (F 0 ) = 1 0 Q 0 (p)dG(p) = 1.(32) If this reference distribution is a standard normal N (0, 1), it follows that T scale (F Y ) = σ for Y ∼ N (µ, σ 2 ). The simplest scale functional that satisfies (29)- (30) and (32), is the one for which G + is a point measure at π for some 0.5 < π ≤ 1. Then T scale (F ) = Q(π) − Q(1 − π) Q 0 (π) − Q 0 (1 − π)(33) equals the standardized interquartile range of F when π = 0.75 and the standardized range when π = 1 and F has bounded support. The standardized Gini's mean difference T scale (F ) = KE\|Y 1 − Y 2 \| = 1 0.5 [Q(p) − Q(1 − p)] (p − 0.5)dp 1 0.5 [Q 0 (p) − Q 0 (1 − p)] (p − 0.5)dp(34) corresponds to having dG + (p) = K(p − 0.5)1(0.5 ≤ p ≤ 1)dp in (31), with K chosen so that (32) holds. A third measure of scale T scale (F ) = 1 0.5 [Q(p) − Q(1 − p)] dp 1 0.5 [Q 0 (p) − Q 0 (1 − p)] dp(35) has a constant density dG + (p) = K1(0.5 ≤ p ≤ 1)dp on (0.5, 1), with K > 0 a constant chosen so that (32) holds. It is sometimes of interest to choose the weight function G so that the asymptotic variance Σ in (14) is minimized. It can be shown (Chernoff et al., 1967, Chapter 8 of Serfling, 1980) that for the location-scale family (26), an Lfunctional with absolutely continuous weight function g(p) = K (log f 0 ) F −1 0 (p) + F −1 0 (p)(log f 0 ) F −1 0 (p)(36) corresponds to an asymptotically optimal estimator of the scale parameter σ whenever the location parameter µ is known or orthogonal to σ, and the density function f 0 is twice differentiable, with K chosen so that (32) holds for some appropriately chosen reference distribution. In particular, K = 1 yields a consistent estimator of the scale parameter σ in (26). For instance, g(p) = F −1 0 (p) is optimal for the normal distribution (F 0 ∼ N (0, 1)). A robustified, skewsymmetric and trimmed version of this weight function has been studied by Welsh and Morrison (1990). More generally, it follows from (36) that the optimal g is skew-symmetric around 0.5 whenever f 0 is symmetric around 0, but not when symmetry of f 0 fails. The advantage of having a skew-symmetry requirement (29) on the weight function is the intuitive interpretation (31) of T scale (F ) in terms of a linear combination of interquantile ranges. Koenker and Zhou (1994) define scale functionals more generally by dropping the skew-symmetry condition and only requiring that the weight function satisfies G((0, p]) < 0 for all 0 < p < 1 and G([0, 1]) = 0. An example of such a scale functional is presented in Section 2.4. 2 Example 3 (Measures of skewness) The traditional measure T skew (F Y ) = E (Y − E(Y )) 3 {E [(Y − E(Y )) 2 ]} 3/2(37) of skewness compares the left and right tails of F Y . It is not robust, since the third moment of Y must be finite. Here we will analyze versions of skewness that are more robust than (37), and defined as the ratio T skew (F ) = T uskew (F ) T scale (F )(38) of two L-functionals. The numerator of (38) corresponds to an unstandardized measure of skewness, which transforms as T uskew (F aY +b ) = aT uskew (F Y )(39) under linear mappings, for all real-valued a and b. The denominator of (38) is another L-functional that measures scale. It follows from (28) and (39) that the standardized skewness satisfies T skew (F aY +b ) = sgn(a)T skew (F Y )(40) for all non-negative a, with sgn(a) = a/\|a\|. We will assume here that T skew is chosen so that T skew (F 0 ) = 1 holds for some reference distribution F 0 whose right tail is heavier than the left tail (e.g. an exponential distribution). A class of weight functions whose unstandardized skewness functional transform linearly, as in (39), are those that satisfy 1 0 dG(p) = 0, dG symmetric around p = 0.5,(42) and supp(G − ) ⊆ [π, 1 − π], supp(G + ) ⊆ [0, π] ∪ [1 − π, 1],(43) for some 0 ≤ π ≤ 0.5. Indeed, inserting (42)-(43) into (1), we find that T uskew (F ) = 1 0.5 [Q(p) + Q(1 − p) − 2Q(0.5)] dG(p) = 1 1−π [Q(p) + Q(1 − p) − 2Q(0.5)] dG + (p) − 1−π 0.5 [Q(p) + Q(1 − p) − 2Q(0.5)] dG − (p).(44) In particular, if F Y is symmetric around its center of symmetry µ, if follows that T uskew (F ) = 0. Since Q(p) + Q(1 − p) − 2Q(0.5) measures skewness for each quantile 0.5 < p < 1, (44) is a functional that compares skewness of the tails of F with the skewness of the central part of F . Of particular interest is the case when G − is a point measure at 0.5. Then (44) simplifies to T uskew (F ) = 1 0.5 [Q(p) + Q(1 − p) − 2Q(0.5)] dG + (p).(45) If the positive part of the weight function is chosen as dG + (p) = Kδ 1−π (p) + Kδ π (p) for some 0 < π < 0.5 and K > 0, and if the interquantile range in (33) is used for standardization, one obtains the standardized measure T skew (F ) = K · Q(1 − π) + Q(π) − 2Q(0.5) Q(1 − π) − Q(π)(46) of skewness. It was introduced by Galton (1883) and Bowley (1920) for π = 0.25 and K = 1, and for arbitrary π by Hinkley (1975). Here we will rather define K so that (41) holds for some appropriate reference distribution F 0 . An alternative to (46) is to choose an unstandardized skewness functional (45) for which G + has a constant density in the numerator of (38), and then use a scale measure with a constant weight density g + for quantiles above 0.5, in the denominator of (38). The corresponding standardized measure T skew (F ) = K · 1 0.5 [Q(p) + Q(1 − p) − 2Q(0.5)] dp 1 0.5 [Q(1 − p) − Q(p)] dp(47) of skewness with K = 1 was proposed by Groeneveld and Meeden (1984). Here we will rather choose K in order for (41) to hold. Notice that the skewness measure (47) puts higher weights on the tails of the distribution, compared to (46). See also Kim and White (1994) for an overview of different robust measures of skewness. Groeneveld and Meeden (1984) argued that a reasonable skewness measure should satisfy (40) and in addition preserve the partial skewness-ordering among distributions, due to van Zwet (1964). By this we mean that T skew (F 1 ) ≤ T skew (F 2 ) whenever F 2 is at least as skewed to the right as F 1 , i.e. if x → F −1 2 (F 1 (x)) is a convex function. Groeneveld and Meeden verified that (46) and (47) preserve this skewness-ordering among distributions. This and other partial skewness-orderings are discussed by Oja (1981), McGillivray (1986), and Garcia et al. (2018). 2 Example 4 (Measures of heavytailedness) There is no universal agreement whether the (excess) kurtosis T (F Y ) = E[(Y − E(Y )) 4 ] {E [(Y − E(Y )) 2 ]} 2 − 3(48) quantifies peakness versus tails of F Y or modality versus bimodality. Following Chissom (1970) and Oja (1981), we will regard kurtosis as a measure of heavytailedness. In order to find more robust measures of heavytailedness, we will consider functionals T heavy (F ) = T uheavy (F ) T scale (F )(49) that are defined as the ratio of two L-functionals. The functional T uheavy in the numerator of (49) corresponds to an unstandardized measure of heavytailedness, and it transforms as T uheavy (F aY +b ) = \|a\|T uheavy (F Y )(50) under linear mappings, for all real-valued a and b, whereas the functional in the denominator of (49) corresponds to a measure of scale. It follows from (28) and (50) that T heavy is invariant with respect to linear transformations, i.e. T heavy (F aY +b ) = T heavy (F Y ). Notice that (50) is identical to the corresponding relation (28) for scale functionals. But whereas T scale is always non-negative, we want T heavy to be positive for heavy-tailed distributions and negative for light-tailed distributions. In order to accomplish this we choose the weight function in (1), for T uheavy (F ), as dG skew-symmetric around p = 0.5, and G + = G restricted to [π, 0.5) ∪ (1 − π, 1], G − = G restricted to [0, π) ∪ (0.5, 1 − π](53) for some 0 < π < 0.5. In addition, we require that the unstandardized and standardized measures of heavytailedness satisfy T uheavy (F 0 ) = 1 0 Q 0 (p)dG(p) = 0, T heavy (F 1 ) = 1 0 Q 1 (p)dG(p)/T scale (F 1 ) = 1(54) for two distributions F 0 and F 1 with quantile functions Q 0 and Q 1 . When there is no restriction on the range of Y , F 0 is typically a normal distribution, whereas F 1 is a symmetric and moderately light tailed distribution, such as the Laplace distribution. In order to motivate that (52) leads to a measure of heavytailedness, we insert this equation into (1) and notice that T uheavy (F ) = 1 1−π [Q(p) − Q(1 − p)] dG + (p) − 1−π 0.5 [Q(p) − Q(1 − p)] dG − (p) (55) equals the difference between the weighted interquantile differences Q(p) − Q(1 − p) in the tails and in the central part of the distribution, respectively. Several robust measures of kurtosis fit into our framework. As a first example, Moors (1988) introduced T heavy (F ) = K Q(0.875) − Q(0.625) + Q(0.375) − Q(0.125) Q(0.75) − Q(0.25) − 1.23 ,(56) with K = 1, where the unstandardized kurtosis in the numerator has a weight function G such that the restrictions of G + and G − to (0.5, 1) have a one point distribution at 0.875 and a two point distribution at 0.625 and 0.75 respectively. The scale measure in the denominator of (56), on the other hand, corresponds to an unstandardized interquartile range. Finally, the constant 1.23 is chosen so that the upper part of (54) holds when F 0 ∼ N (0, 1). Here we will additionally choose K > 0 in (56) so that the lower part of (54) is satisfied for some appropriately chosen reference distribution F 1 . Second, the tail ratio of Gilchrist (2000) can be normalized as T heavy (F ) = K 1 Q(0.9) − Q(0.1) Q(0.75) − Q(0.25) − K 0 ,(57) where K 0 and K 1 are chosen so that (54) holds. A third class of kurtosis measures T heavy (F ) = K 1 1 1−π0 [Q(p) − Q(1 − p)] dp 1 1−π1 [Q(p) − Q(1 − p)] dp − K 0(58) was introduced by Hogg (1972Hogg ( ,1974 for some conveniently selected 0 < π 0 < π 1 ≤ 0.5, and with K 0 chosen so that the upper part of (54) holds when F 0 ∼ N (0, 1). Hogg conducted simulations for which π 0 = 0.05, π 1 = 0.5, and K 0 = 2.59 gave satisfactory results. Whereas Hogg used K 1 = 1, we will rather choose K 1 so that the lower part of (54) is satisfied for some appropriate reference distribution F 1 . See also Kim and White (1994) for an overview of different robust measures of kurtosis. A partial kurtosis-ordering of symmetric distributions (van Zwet, 1964, Oja, 1981 states that F 2 has more kurtosis than F 1 if x → F −1 2 (F 1 (x)) is a concave (convex) to the left (right) of the point of symmetry. It can be shown that both (56) and (58) preserve this kurtosis-ordering among symmetric distributions. Sometimes it is only one tail of F that is of interest. It is possible then to split the weight function of an unstandardized heavytailedness functional as dG(p) = 1(p < 0.5)dG(p) + 1(p > 0.5)dG(p) =: dG left (p) + dG right (p), where only the low quantiles are included in G left in order to study the left tail of F , whereas only the high quantiles are used in G right to study the right tail of F . It follows from (52) that the weight measure for the right tail satisfies G + right = G restricted to (1 − π, 1], G − right = G restricted to (0.5, 1 − π].(59) The corresponding functional T uheavyr (F ) = 1 1−π Q(p)G + (p) − 1−π 0.5 Q(p)dG − (p)(60) can be viewed as a restriction of (55) to an interval (0.5, 1) of quantiles, and it is standardized as T heavyr (F ) = T uheavyr (F ) T scale (F ) . Suppose, for instance, that e Y has a heavy tail to the right, in the sense that P (e Y ≥ y) = y −α L(y) for some tail parameter α > 0, and with a function L(y) that varies slowly as y → ∞, i.e. L(ty)/L(y) → 1 as y → ∞ for all t > 0. Then Hill's estimator of α (Hill 1975, Haeusler andTeugels, 1985) corresponds to an estimatorθ = T uheavyr (F ) in (6) with weight function g + right (p) = 1(1 − π < p < 1)/(1 − π), dG − right (p) = δ 1−π (p).(61) In order forθ =θ n to be a consistent estimator of α, it is either required that L(·) is constant for large enough y, or that π = π n → 0 as n → ∞ at a rate depending on how much L(·) varies for large y. 2 The order of L-functionals In this section we introduce order numbers for a certain subclass of L-functionals. Since each L-functional is determined by its weight measure G, this amounts to introducing order numbers for a subclass of weight measures. We will state two mandatory conditions on G and a third optional symmetry condition, in order for it to be of order m ∈ {1, 2, . . .}. First, there has to exist a disjoint decomposition of [0, 1] into m intervals I 1 < I 2 < . . . < I m such that G − (I k ) = 0 and G + (I k ) > 0, if m − k is even, G − (I k ) > 0 and G + (I k ) = 0, if m − k is odd.(62) Second, 1 0 dG(p) = 1, if G has order m = 1, 0, if G has order m > 1.(63) Third, dG is symmetric (skew-symmetric) around p = 0.5 if m is odd (even). (64) When all three conditions (62)-(64) hold, we refer to G as a symmetric (skewsymmetric) weight measure of order m. It may be verified from Section 2.3 that location functionals have order 1, scale functionals order 2, unstandardized skewness functionals order 3, and unstandardized heavytailedness functionals order 4. Write T m (F ) = 1 0 Q(p)dG m (p)(65) for an L-functional whose weight function G = G m is of order m, and T ml (F ) = T m (F ) T l (F )(66) for a functional that is the ratio of two L-functionals of order m and l respectively. We will refer to (m, l) as the order number of T ml . Consequently, skewness functionals have order (3, 2) and heavytailedness functionals order (4, 2). It is possible to obtain collections {T m (F )} ∞ m=1 of L-functionals from orthogonal series expansions of the quantile function Q = F −1 . Each such collection makes use of a reference distribution F 0 with density function f 0 (y) = F 0 (y) on its support [a, b], where −∞ ≤ a < b ≤ ∞. Introduce the scalar product f, g = b a f (y)g(y)f 0 (y)dy for functions that are square integrable with respect to f 0 (y)dy, and suppose there exists an orthonormal system of polyno- mials {P k } ∞ k=0 of degrees k = 0, 1, 2, . . . such that b a P k (y)P l (y)f 0 (y)dy = 1(k = l).(67) Then define the absolutely continuous weight densities (dG m (p) = g m (p)dp) g m (p) = P m−1 (Q 0 (p)) (68) for m = 1, 2, . . ., where Q 0 = F −1 0 is the quantile function of F 0 . It follows from (67)-(68) that {g m } forms an orthonormal system of basis functions on [0, 1], i.e. 1 0 g m (p)g l (p)dp = 1(m = l). Equations (65) and (69) imply that {T m (F )} ∞ m=1 are the coefficients of an orthonormal series expansion of Q. Indeed, from Theorem 2.3 of Takemura (1983) we find that if F has a finite second moment Q(p) = ∞ m=1 T m (F )g m (p),(70) for all continuity points p of Q(·), strictly between 0 and 1. This is to say that {T m (F )} ∞ m=1 will quantify all aspects of the quantile function Q of F . Takemura (1983) used (70) in order to define goodness-of-fit tests of a locationscale family (26), where F 0 in equation (26) is also the reference distribution of the series expansion. If we restrict ourselves to functionals up to order 4, it is clear that {T m (F )} 4 m=1 carries the same information about F as the four types of functionals of Section 2.3, i.e. T 1 (F ), T 2 (F ), T 32 (F ), and T 42 (F ). In order to verify that (65) and (68) for m = 1, 2, . . . define a valid collection of L-functionals, the first condition (62) is equivalent to each polynomial P k (y) having k distinct zeros with a leading positive coefficient of y k . The second condition (63) follows by choosing l = 1 in (69), since g 1 (p) = 1. The third symmetry condition (64) holds whenever F 0 is symmetric, and if P k (y) is an even (odd) function of y when k is even (odd). (64) holds for all T m , and otherwise it is asymmetric. For a symmetric collection of functionals {T m } ∞ m=1 we introduce a symmetric reference distribution F 0 with mean 0 and variance 1 that satisfies Symmetric and asymmetric collections of L-functionals A collection {T m } ∞ m=1 of L-functionals is symmetric ifT m (F 0 ) = 1, m = 2, 0, m = 2.(71) For a symmetric collection of L-functionals, we also introduce one distribution F 1 that is moderately skewed to the right, and another symmetric distribution F 2 , which is more heavytailed than F 0 , such that T 32 (F 1 ) = 1, T 42 (F 2 ) = 1.(72) Let us verify that a symmetric polynomial collection (69) of L-functionals satisfies (71) and (72). Starting with (71), recall that F 0 is assumed to be symmetric with expected value 0 and variance 1. Applying (69) with m, l ∈ {1, 2} we find that g 1 (p) = 1 and g 2 (p) = Q 0 (p). Then a second application of (69) with variable m and l = 2 implies (71). It is also possible to relax the orthonormality condition (69) and adjust g 3 (·) and g 4 (·) by multiplicative constants in such a way that (72) holds for some appropriately chosen distributions F 1 and F 2 . When F is a life length distribution we typically choose a non-symmetric ref- erence distribution F 0 , supported on [a, b) = [0, ∞) , and use a non-symmetric collection of L-functionals. Then we replace condition (71) by the milder requirement T 2 (F 0 ) = 1.(73) Since skewness and kurtosis are somewhat less natural concepts for life lengths, we will not always impose condition (72) in this context. For instance, we are typically more interested in quantifying how thick the right tail of F is (as in (60)-(61)), and this requires combined information from all of {T m (F )} 4 m=1 . Examples of polynomial collections of L-functionals In this subsection we give three collections {T m } ∞ m=1 of L-functionals based on (65) and (68). Example 5 (Legendre collection of L-functionals) This collection of symmetric L-functionals has absolutely continuous weight functions g m (p) = √ 2m − 1L m−1 (2p − 1),(74) where L m−1 is the Legendre polynomial of order m − 1 on [−1, 1]. Thus we have that g 1 (p) = 1, (25) and T 2 (F ) is proportional to Gini's mean difference (34). g 2 (p) = √ 3(2p − 1), g 3 (p) = √ 5[3(2p − 1) 2 − 1]/2, g 4 (p) = √ 7[5(2p − 1) 3 − 3(2p − 1)]/2, so that T 1 (F ) = E(Y ) equals the mean The fact that g m gives rise to an L-functional of order m follows from the general construction in (68), with a uniform reference distribution F 0 ∼ U (− √ 3, √ 3). Notice in particular that this symmetric distribution has first two moments E(Y ) = 0 and E(Y 2 ) = 1, as required above (71). Moreover, since (65) and (69) that (71) holds for all m. Equation (69) follows from well known orthogonality properties of Legendre polynomials, whereas (62) is referred to as the interlacing property of Legendre polynomials. It is possible to adjust g 3 and g 4 by multiplicative constants so that (72) holds for some appropriate reference distributions, for instance the beta distributions F 1 ∼ B(2, 1) and F 2 ∼ B(0.5, 0.5). Q 0 (p) = √ 3(2p − 1) = g 2 (p) it follows from It turns out that the Legendre system of L-functionals is equivalent to the L-moments γ m (F ) = T m (F )/ √ 2m − 1 of F or order 1, 2, . . ., introduced by Sillito (1969). Likewise, T 32 (F ) and T 42 (F ) are equivalent to the L-skewness τ 3 = γ 3 /γ 2 and L-kurtosis τ 4 = λ 4 /λ 2 of Hosking (1990). Another closely related concept is the family of probability weighted moments Greenwood et al. (1979) µ qrs (F ) = E [Y q F (Y ) r (1 − F (Y )) s ] of, where F = F Y is the distribution function of Y . In fact, it can be seen that each µ 1rs (F ) is a linear combination of Legendre functionals {T m (F )} r+s+1 m=1 of F up to order r + s + 1. We argue that the Legendre system (74) of L-functionals is appropriate for bounded random variables Y ∈ [a, b], and Hosking (1990) proves that {T m (F )} ∞ m=1 exist for distributions F with a finite mean. In spite of this, each T m is non-robust with a breakdown point of 0. It is possible though to define a robustified collection {T π m } ∞ m=1 of L-functionals, for each 0 < π < 0.5, with weight functions g π m (p) = √ 2m − 1 1 − 2π L m−1 2p − 1 1 − 2π 1(π < p < 1 − π)(75) and breakdown point π. Notice in particular that g 1 equals the trimmed mean, i.e. π 0 = π and π 1 = 1 − π in (24). In analogy with (70), one finds that {T π m (F )} ∞ m=1 provides information about the conditional distribution of F for all quantiles between π and 1 − π, since Q(p) 1 − 2π = ∞ m=1 T π m (F )g π m (p), at all continuity point π < p < 1 − π of Q(·). In order for F π 0 to serve as a reference (71) for {T π m } ∞ m=1 , it is required that Y \| Q π 0 (π) < Y < Q π 0 (1 − π) ∼ U (− √ 3, √ 3) whenever Y ∼ F π 0 and Q π 0 = (F π 0 ) −1 . More generally, we argue that the robustified Legendre collection of L-functionals is appropriate whenever Y \| Q(π) < Y < Q(1 − π) is bounded within some finite interval [a, b]. For instance, if F Y \|Q(π)<Y <Q(1−π) ∼ U (a, b) is uniform, then T π 1 (F Y ) = (a + b)/2 and T π 2 (F Y ) = (a − b)/(2 √ 3) equal the mean and standard deviation of this uniform distribution, whereas T π m (F Y ) = 0 for m ≥ 2. Other ways of robustifing the Legendre system of L-functionals have been proposed by Mudholkar and Huston (1988) and Elamir and Seheult (2003). 2 Example 6 (Hermite collection of L-functionals) In this example we introduce a collection of symmetric L-functionals for which F 0 ∼ N (0, 1) is a reference distribution. The weight densities g m (p) = 1 (m − 1)! H m−1 [Q 0 (p)](76) are defined in terms of the quantile function Q 0 = F −1 0 and the probabilistic Hermite polynomials H k (y) = (−1) k e y 2 /2 d k e −y 2 /2 dy k , −∞ < y < ∞,(77) of order k = 0, 1, 2, . . .. Inserting (77) into (76), we find that the first four weight densities have the form g 1 (p) = 1, g 2 (p) = Q 0 (p), g 3 (p) = [Q 0 (p) 2 − 1]/ √ 2, g 4 (p) = [Q 0 (p) 3 − 3Q 0 (p)]/ √ 6.(78) Recall from (27) and (36) that g 1 and g 2 correspond to optimal L-functionals of location and scale for F 0 . It follows from well known orthogonality properties of Hermite polynomials that the weight functions in (76) constitute an orthonormal system (69) on [0, 1], and the series expansion (70) of Q can be interpreted as a robust Cornish-Fisher expansion (Fisher and Cornish, 1960). In particular, since g 2 (p) = Q 0 (p), it follows from (65) and (71) that F 0 indeed is a reference distribution for the Hermite class of L-functionals. As in Example 5, one may multiply g 3 and g 4 by constants so that (72) holds for some appropriate reference distributions, for instance a non-central t-distribution F 1 and central t-distribution F 2 , with appropriate parameters. Robustified versions T π m (F ) of the Hermite functionals are constructed in the same way as in Example 5, with weight functions g π m (p) = 1 (1 − 2π) (m − 1)! H m−1 [Q 0 ( p 1 − 2π )]1(π < p < 1 − π). The reference (71) of this system is the improper mixture distribution F π 0 ∼ πδ −∞ + (1 − 2π)N (0, 1) + πδ ∞ , with probabilities π at minus and plus infinity. We argue that the (robustified) Hermite system of L-functionals is appropriate whenever Y ∈ R, that is, when there are no upper or lower bound restrictions on Y . 2 Example 7 (Laguerre collection of L-functionals) In this example we consider an asymmetric collection of L-functionals, which is of interest when Y ≥ 0, for instance a lifetime. This class of L-functionals will have F 0 ∼ Exp(1) as a reference distribution. Let La k (y) be the Laguerre polynomial of degree k = 0, 1, 2, . . .. These polynomials form an orthonormal system on [0, ∞) with respect to the density measure dF 0 (y) = e −y dy, i.e. The weight function of T m is defined as g m (p) = (−1) m−1 La m−1 (Q 0 (p)) = (−1) m−1 La m−1 log (1 − p) −1 ,(80) for m = 1, 2, . . .. The factor (−1) m−1 of (80) ensures that all weight functions g m (p) will have a leading positive coefficient of Q m−1 0 (p), as required by (62). It can be seen that the first four weight functions are g 1 (p) = 1, g 2 (p) = Q 0 (p) − 1, g 3 (p) = Q 2 0 (p)/2 − 2Q 0 (p) + 1, g 4 (p) = [Q 3 0 (p) − 9Q 2 0 (p) + 18Q 0 (p) − 6]/6. It follows from (67) and (80) that {g m } ∞ m=1 forms an orthonormal system (69) of weight functions on [0, 1], with T m (F 0 ) = 1 0 Q 0 (p)g m (p)dp = 1 0 (g 1 (p) + g 2 (p))g m (p)dp = 1, m = 1, 2, 0, m ≥ 3, in agreement with (73). Hence, according to this definition T 32 (F 0 ) = 0 although F 0 is skewed to the right. The rationale is that most lifetime distributions are skewed in this direction. Using F 0 as a yardstick we may therefore interpret T 32 (F ) > 0 as F being more skewed to the right than F 0 . When Y ≥ 0 it is typically only the heaviness of the right tail of F = F Y that is of interest. A functional for right-heavytailedness is T heavyr (F ) = T 4,right (F ) T 2 (F ) , where T 4,right (F ) has weight function g 4,right (p) = g 4 (p)1(1 − π < p < 1), and 0 < π < 1 is the smallest positive integer satisfying 1 1−π Q 0 (p)g 4 (p)dp = 0. This definitions guarantees that T heavyr (F 0 ) = 0, so that F 0 is a reference for right-heavytailedness. 2 L-functionals for regression models In this section we regard Y as the outcome variable of a regression model with a vector x = (x 1 , . . . , x q ) T of covariates. Let F Y \|x (y) = P (Y ≤ y \| x), −∞ < y < ∞,(81) be the conditional distribution function of Y given x. By inverting this function we obtain the p:th conditional quantile (CQ) Q(p \| x) = F −1 Y \|x (p) = inf{y; F Y \|x (y) ≥ p}.(82) Each functional (1) gives rise to a linear combination θ(x) = T (F Y \|x ) = 1 0 Q(p \| x)dG(p)(83) of CQs. We will refer to (83) as a conditional L-functional. The weight functions of Section 2 give rise to different conditional L-functionals that correspond to location, scale, unstandardized skewness or unstandardized heavytailedness measures of F Y \|x . For each 0 < p < 1, letQ(p \| x) be an estimator of the conditional quantile (82), based on a sample of size n. The corresponding estimatorθ (x) = 1 0Q (p \| x)dG(p)(84) of θ(x) is a conditional L-statistic that reduces to (4) for a model without any covariates. In order to study standardized measures of skewness and heavytailedness of F Y \|x , we need to consider ratios of two conditional L-functionals T 1 (F Y \|x ) and T 2 (F Y \|x ) that involve two different weight functions G 1 and G 2 . We will therefore study quantities θ(x) = 1 0 Q(p \| x)dG 1 (p) 1 0 Q(p \| x)dG 2 (p) ,(85) and their estimatorsθ (x) = 1 0Q (p \| x)dG 1 (p) 1 0Q (p \| x)dG 2 (p) .(86) For a model with covariates, the definitions of (84) and (86) will depend on whether the response variable Y is censored/truncated or not, and on the type of regression model that is used. In the following subsections we will consider several such models. Data without censoring/truncation In this subsection we assume there is no censoring or truncation, so that the response variable Y is observed. In more detail, suppose that a sample of independent random vectors (x 1 , Y 1 ), . . . , (x n , Y n ) is available, with the same conditional distribution F Yi\|xi = F Y \|xi of the response variable as in (81). In order to estimate θ(x) in (83) or (85) we have to make some smoothness assumptions on x → θ(x), which in turn requires smoothness of the conditional quantiles x → Q(p \| x). The most general approach is to estimate Q(p \| x) (and hence also θ(x)) by some nonparametric method, for instance local polynomial regression (Chauduri, 1991), smoothing splines , regression splines (He and Shi, 1994), piecewise polynomial regression tree methods (Chauduri and Loh, 2002), or a semiparametric linear model with varying coefficients (Kim, 2007). In the following two subsections we will rather concentrate on two fully parametric models for the relation between x and Q(p \| x); linear models and transformed linear models. Linear models Models and estimators When the outcome variable Y ∈ (−∞, ∞) is unbounded, it is natural to use a linear model Q(p \| x) = x T β(p),(87) so that each conditional quantile (82) is a linear function of the covariates, with β(p) = (β 1 (p), . . . , β q (p)) T a vector of regression parameters. The simplest special case of (87) is the homoscedastic linear model, with F Y \|x (y) = F 0 (y − x T b),(88)for some vector b = (b 1 , . . . , b q ) T . If the first regression component is an intercept (x = (1, x 2 , . . . , x q ) T ) and F 0 has median 0, it follows that β(p) = b + [Q 0 (p) − Q 0 (0.5)] e 1 , where e 1 = (1, 0, . . . , 0) T and Q 0 is the inverse of F 0 . In particular, the framework of Section 2 is a special case of (88) with q = 1 and b 1 the median of F Y . It is also possible to incorporate heteroscedastic regression models of type F Y \|x (y) = F 0 y − x T b x T c (89) into (87), with c = (c 1 , . . . , c q ) T , for all x such that x T c > 0. Again, it follows that β(p) = b + [Q 0 (p) − Q 0 (0.5)] c if F 0 has median 0. The first L-based inference methods of regression focused on estimating b for the linear and homoscedastic model (88), based on some preliminary estimate (Bickel, 1973, Ruppert and Carroll, 1980, Welsh, 1987. Gutenbrunner and Jurečková (1992) introduced another approach based on regression rankscores. Here we will focus on procedures that estimate the conditional quantile (87) by first finding an estimator of β(p). The most general such estimator is a solution of the minimization problem β(p) = (β 1 (p), . . . ,β q (p)) T = arg min b∈R q n i=1 ρ p (Y i − x T i b),(90) where ρ p (y) = [p − 1(y < 0)]y is the so called check function. This estimator is nonparametric in the sense that it makes few assumptions about the conditional distribution of Y \|x, apart from its linear dependency on x in (87). It is usually referred to as a regression quantile and for the model of Section 2, with no explanatory variables, it simplifies to the sample quantile Y ([np]) . Koenker and Bassett (1978) introduced (90) for the homoscedastic regression model (88), and later it was extended by Koenker and Bassett (1982) and Koenker and Zhao (1994) to the heteroscedastic model (89). Efron (1991) proposed a slightly different nonparametric estimate of β(p), by first minimizing an asymmetric squared loss function of the residuals Y i − x T i b. Frumento and Bottai (2016) modeled β(p) = β(p; ψ) = r k=1 ψ k β k (p)(91) parametrically in terms of a q × r matrix ψ = (ψ 1 , . . . , ψ r ), where β 1 (p), . . . , β r (p) are known functions of p. This includes, for instance, the homoscedastic and heteroscedastic regression models (88) and (89), when F 0 is regarded as known. Both of these models have β 1 (p) ≡ 1, β 2 (p) = Q 0 (p) − Q 0 (0.5), and ψ 1 = b, whereas ψ 2 = e 1 for the homoscedastic and ψ 2 = c for the heteroscedastic model. The model in (91) gives rise to a parametric regression quantile estimator, wherê β(p) = β(p;ψ) = r k=1ψ k β k (p),(92) andψ k is an estimator of ψ k . Frumento and Bottai (2016) used ψ = arg min ψ n i=1 1 0 ρ p (Y i − x T i β(p; ψ))dp.(93) The conditional L-functional (83) of a linear model has a very tractable form. Recall first of all from Example 6 that the Hermite system (76) of weight functions G = G m is a natural choice of L-functionals of order m = 1, 2, . . . for a linear model, whenever the range of the outcome variable is unbounded. Moreover, since the conditional quantile (87) is a linear function of x, this linearity is preserved for conditional L-functionals (83). Indeed, inserting (87) into (83), we find that θ(x) = x T 1 0 β(p)dG(p) =: x T B,(94) where B = (B 1 , . . . , B q ) T and B j = 1 0 β j (p)dG(p). In order to estimate the conditional L-functional θ(x), the chosen estimatorβ(p) (for instance the nonparametric (90) or the parametric (92)) is first plugged into (87) and then into (84). This gives an estimated conditional quantileQ(p \| x) = x Tβ (p) and an estimatê θ(x) = x TB (95) of θ(x), whereB = (B 1 , . . . ,B q ) T = 1 0β (p)dG(p) is a linear combination of all β(p). In the context of regression quantiles, the estimatorB was first proposed by Koenker and Portnoy (1987) for the homoscedastic model (88), and then extended to heteroscedastic models (89) by Koenker and Zhao (1994). García-Pareja and Bottai (2018) studied (94) for weight functions that correspond to a uniform distribution on [π 0 , π 1 ] for some 0 ≤ π 0 < π 1 ≤ 1. They referred to θ(x) as a conditional compound expectation, and it generalizes the compound expectation (24) of the location model. Asymptotics In order to study the large sample asymptotics ofθ(x) =θ n (x), we follow García-Pareja and Bottai (2018) and view β = {β(p); 0 < p < 1} as an element of the q-dimensional product space S q . In our setting the norm β S q = q j=1 β j S of this space generalizes (8) from q = 1 to q ≥ 1. For q-dimensional vectors B ∈ R q we introduce the L 1 -norm \|B\| = q j=1 \|B j \|. Moreover, if W, W 1 , . . . , W n , . . . are random variables of a metric space W, equipped with norm \| · \|, we say that the sequence W n converges in probability towards W as n → ∞, i.e. W n p −→ W , if P (\|W n − W \| > ε) → 0 for each ε > 0. Equipped with these preliminaries, we have the following: Proposition 1 (Consistency of (95).) Supposeβ =β n ∈ S q is a consistent estimator of β, i.e.β n p −→ β as n → ∞. ThenB =B n p −→ B and the estimated conditional L-functional in (95) is consistent, i.e.θ n (x) p −→ θ(x) as n → ∞. Proof. In analogy with (9) we have the inequalities \|B n − B\| ≤ q j=1 1 0 \|β j (p) − β j (p)\|d\|G\|(p) = q j=1 1 0 \|β j (p) − β j (p)\|\|g(p)\|dp + M m=1 \|g m \|\|β j (π m ) − β j (π m )\| = β n − β S q(96) By assumption, P ( β n − β S q > ε) → 0 as n → ∞ for each ε > 0. In conjunction with (96) As a next step, in order to establish asymptotic normality ofθ(x) =θ n (x) as n → ∞ we first generalize (10) from the location model (q = 1, x i ≡ 1) and introduce the rescaled process Z n (p) = √ n[β n (p) − β(p)], 0 < p < 1(97) ofβ n (p). Viewing Z n = {Z n (p); 0 < p < 1} as a random element of S q , we will assume weak convergence Z n L −→ Z as n → ∞(98) with respect to the topology in S q introduced by · S q . The limit in (98) is assumed to be a Gaussian process Z = {Z(p); 0 < p < 1} with mean E(Z(p)) = (0, . . . , 0) T for all 0 < p < 1 and covariance function R(p, s) = Cov(Z(p), Z(s)) for all 0 < p, s < 1. In the context of regression quantiles (90), in order to find the covariance function R(p, s), it is helpful to rewrite (97) as Z n (p) = 1 √ n D −1 n (p) n i=1 ρ p (Y i − x T i β(p)) + o p (1), where D n (p) = n i=1 fi(p)xix T i n f i (p) = dF Y \|x i (y) dy y=Q(p\|xi) ρ p (y) = p − 1(y < 0)(99) and o p (1) is an asymptotically negligible remainder term, i.e. o p (1) p −→ 0 as n → ∞, uniformly for p ∈ supp(G). Then assume there exist positive definite matrices A and D(p) such that (i) lim n→∞ n i=1 x i x T i /n = A, (ii) lim n→∞ D n (p) = D(p), (iii) max i=1,...,n \|x i \|/ √ n → 0.(100) The regularity conditions in (100) imply that the Gaussian limit process has a covariance function R(p, s) = [min(p, s) − ps]D −1 (p)AD −1 (s)(101) for regression quantiles, see for instance Chapter 4 of Koenker (2005). This covariance function simplifies to R(p, s) = min(p, s) − ps f 0 (Q 0 (p))f 0 (Q 0 (s)) A −1(102) for the homoscedastic regression model (88), cf. Koenker and Portnoy (1987). We notice that (102) generalizes the asymptotic covariance function (15) of the normalized quantile process (10) for data without covariates (x i = 1 and A = 1). For the parametric estimator (92), in order to find the asymptotic covariance function of {β(p); 0 < p < 1}, we first need to generalize (17) and establish asymptotic normality ofψ =ψ n . To this end, it is convenient to introduce vec(ψ) = (ψ T 1 , . . . ,ψ T r ) T , the column vector of length rq in which the columns ofψ are stacked on top of each other. It is shown in Frumento and Bottai (2016), under regularity conditions similar to (100), that √ n(vec(ψ n ) − vec(ψ)) L −→ N (0, V ) = N   0,    V 11 . . . V 1r . . . . . . . . . V r1 . . . V rr       ,(103) where V is a square matrix of order rq, and V kl is a square matrix of order q that corresponds to the asymptotic covariance matrix betweenψ k andψ l . The exact form of V can be found in Frumento and Bottai (2016). It follows from (92) and (103) that weak convergence (98) holds with asymptotic covariance function R(p, s) = r k,l=1 β k (p)β l (s)V kl(104) of the limit process Z. Notice also that this covariance function is a special case of (16) when q = 1 and x i ≡ 1, with dQ(p; ψ)/dψ = (β 1 (p), . . . , β r (p)). Equipped with these preliminaries, the following result provides asymptotic normality ofB n andθ n (x): Proposition 2 (Asymptotic normality of (95)) Suppose (98) holds for some Gaussian limit process Z whose covariance function R(p, s) is the asymptotic covariance function of {β n (p); 0 < p < 1}. Suppose further that the function g in (3) is bounded by g ∞ < ∞. Then the estimatorB =B n of B is asymptotically normal as n → ∞, in the sense that √ n(B n − B) L −→ N (0, Σ),(105) where the mean vector of the q-dimensional limiting normal distribution is 0 = (0, . . . , 0) T and the covariance matrix equals Σ = 1 0 1 0 R(p, s)dG(p)dG(s).(106) Moreover, the estimatorθ(x) =θ n (x) of the conditional L-functional in (95) is also asymptotically normal, with √ n[θ n (x) − θ(x)] L −→ N (0, x T Σx)(107) as n → ∞. Proof. Introduce the functional h : S q → R q by means of h(z) = 1 0 z(p)dG(p). From the proof of Proposition 1, and the fact that g ∞ < ∞, we know that h is a continuous functional. Since Propositions 1-2 are applicable whenever the goal is to estimate conditional L-functionals that correspond to measures of location, scale, unstandardized skewness or unstandardized heavytailedness of F Y \|x , defined as in (83). However, in order to study standardized measures of skewness and heavytailedness of F Y \|x we consider quantities (85) that are defined as a ratio of two conditional L-functionals T 1 (F Y \|x ) and T 2 (F Y \|x ), with different weight functions G 1 and G 2 . For the linear model (87) we find that θ(x) = x T B 1 x T B 2 ,(108) with B k = 1 0 β(p)dG k (p) for k = 1, 2. The corresponding estimator θ(x) = x TB 1 x TB 2 ,(109) is defined analogously withB k = 1 0β (p)dG k (p) for k = 1, 2. The following proposition shows thatB 1 =B 1 n andB 2 =B 2 n are jointly asymptotically normal, and as a consequence, thatθ(x) =θ n (x) in (109) is asymptotically normal as well. Proposition 3 (Consistency and asymptotic normality of (109)) Suppose the regularity conditions of Propositions 1 and 2 hold. ThenB where 0 = (0, . . . , 0) T is a 2q-dimensional vector of zeros, Σ kl = 1 0 1 0 R(p, s)dG k (p)dG l (s)(111) for k, l = 1, 2, and R(p, s) is the asymptotic covariance function of {β n (p); 0 < p < 1}. Moreover, the estimatorθ n (x) in (109) of the ratio θ(x) of the two conditional L-functionals in (108), is also asymptotically normal as n → ∞, in the sense that √ n[θ n (x) − θ(x)] L −→ N 0, Σ † Σ † = x T Σ 11 x (x T B 2 ) 2 − 2x T B 1 · x T Σ 12 x (x T B 2 ) 3 + (x T B 1 ) 2 · x T Σ 22 x (x T B 2 ) 4 .(112) Proof. Consistency ofB 1 n andB 2 n follows as in the proof of Proposition 1. In order to verify (110) we first look at linear combinations αB 1 n + βB 2 n = 1 0β (p)(αdG 1 (p) + βdG 2 (p)) =: 1 0β (p)dG(p) ofB 1 n andB 2 n . Then we apply Proposition 2 with G = αdG 1 + βdG 2 in order to deduce √ n[αB 1 n + βB 2 n − (αB 1 + βB 2 )] L −→ N (0, α 2 Σ 11 + 2αβΣ 12 + β 2 Σ 22 ) as n → ∞ for any real-valued α and β. Then (110) follows from the Cramér-Wold device. Next we consider the estimatorθ n (x) of θ(x). Consistencyθ n (x) p −→ θ(x) follows as in the proof of Proposition 1, from the consistency of (B (112), it is convenient to denote the numerators and denominators of (108) and (109) as N/D andN /D respectively. Notice first that √ n[(N ,D) − (N, D)] L −→ N (0, 0), x T Σ 11 x x T Σ 12 x x T Σ 21 x x T Σ 22 x as n → ∞, using first the Cramér-Wold device then same argument as in the proof of (109). Then (112) follows after a first order two-dimensional Taylor expansionθ n (x) ≈ θ(x) + 1 D (N − N ) − N D 2 (D − D) Transformed linear models Suppose the response variable Y ∈ [a, b] is constrained to lie in an interval with end points −∞ ≤ a < b ≤ ∞. If at least one of these two end points is finite, the conditional quantile (87) of the linear model may fall outside [a, b] for some covariate vectors x. In order to avoid this, it is common to introduce a known and strictly increasing link function h : [a, b] → R and assume that regression data (x, h(Y )) for the transformed outcome variable follows the linear model of Section 3.1.1. This is analogous to link functions of Generalized Linear Models (McCullagh and Nelder, 1989), although here we focus on transformations of quantiles rather than of expected values. Since quantiles are preserved by monotone transformations it follows from (87) that the conditional quantiles satisfy Q(p \| x) = h −1 x T β(p)(113) for all 0 < p < 1. If our objective is to estimate a conditional location, conditional scale, conditional unstandardized skewness or conditional unstandardized kurtosis of Y , we use the conditional L-functional in (83). For the transformed linear model (113), this functional θ(x) = 1 0 h −1 (x T β(p))dG(p)(114) is no longer a linear function of x, as in (94). In order to estimate θ(x) from data, we proceed as in Section 3.1.1. We first estimate the regression parameter β(p) of the transformed data set by some method. One possibility is to estimate β(p) nonparametrically with regression quantileŝ β(p) = arg min b∈R q n i=1 ρ p (h(Y i ) − x T i b),(115) in analogy with (90). When p = 0.5, we notice that (115) is a special case of L 1 -estimation for nonlinear regression models (Oberhofer, 1982). Alternatively we estimate β(p) parametrically aŝ β(p) = arg min ψ n i=1 1 0 ρ s (h(Y i ) − x T i β(s; ψ))ds    β 1 (p) . . . β r (p)    ,(116) in analogy with (92)-(93). As we will see below, for some models it is more convenient to estimate β(p) parametrically by maximum likelihood rather than using (116). By plugging a nonparametric or parametric estimate of β(p) into (114), we finally obtain an estimator θ(x) = 1 0 h −1 (x Tβ (p))dG(p)(117) of θ(x). For some of the examples below, the link function h is unknown, and then we have to estimate h(·) (parametrically or nonparametrically) as well as β(p). This gives rise to an estimator θ(x) = 1 0ĥ −1 (x Tβ (p))dG(p)(118) of θ(x), whereĥ(·) is a monotone estimate of h(·). We will now give several examples of link functions for which the corresponding transformed response variables follow a linear model, as described in (113). Example 8 (Logit transformations.) Suppose the outcome variables is bounded, so that −∞ < a < b < ∞ are both finite. Liu et al. (2009) and Bottai et al. (2010) used model (113) with a logit transformation h(y) = logit y − a b − a = log y − a b − y for such data. 2 Since Y is bounded in Example 8, it is possible to use L-functionals in (117)-(118) with a weight density dG(p) = g m (p)dp chosen from the Legendre system (74) of Example 5. The remaining examples of this section concern outcome variables such as life lengths, where Y is constrained to be positive (a = 0, b = ∞). Recall from Example 7 that it is possible then choose the weight density dG(p) = g m (p)dp in (117)-(118) from the asymmetric collection (80) of basis functions, with an exponential reference distribution. The construction in Example 7 can be modified though so that some other lifetime distribution is used as reference. Example 9 (Logarithmic transformations.) When Y is non-negative (a = 0, b = ∞) it is common to use the logarithmic link h(y) = log(y). The Accelerated Failure Time (AFT) model (Kalbfleich and Prentice, 2002) is often used when Y > 0 is a lifetime, and it corresponds to having a parametric location-scale regression model (88) for the transformed data {(x i , log(Y i ))} n i=1 . In more detail, log [Q(p \| x)] = µ + q k=2 x k b k−1 + σQ 0 (p) = x T β(p),(120) where Q 0 is the inverse of some reference distribution F 0 of the log lifetime. Here x = (1, x 2 , . . . , x q ) T is the covariate vector with an added intercept, β(p) = (µ + σQ 0 (p))e 1 + (0, b T ) T , and b = (b 1 , . . . , b q−1 ) T contains the effect parameters of the q−1 covariates. When F 0 is known, for instance a logistic distribution, equation (120) corresponds to a parametric model (116) with r = 2, ψ 1 = (µ, b 1 , . . . , b q−1 ) T , ψ 2 = (σ, 0, . . . , 0) T , β 1 (p) ≡ 1 and β 2 (p) = Q 0 (p). When computingβ(p) it is possible though to use the maximum likelihood estimates of the q + 1 nonzero model parameters µ, σ, b 1 , . . . , b q−1 of ψ, rather than (116). García-Pareja et al. (2019) used this approach, with a piecewise constant weight function (24), to estimate the conditional compound expectation θ(x) of an AFT model. When F 0 is left unspecified it is also possible to estimate µ and b nonparametrically according to (115) (Ying et al., 1995). 2 Example 10 (Power transformations.) Assume as in Example 9 that Y is positive, i.e. a = 0 and b = ∞. Mu and He (2007) estimated conditional quantiles (114) with class of power link functions (Box and Cox, 1964), i.e. h γ (y) = (y γ − 1)/γ, γ = 0, log(y), γ = 0. It was assumed in Mu and He (2007) that not only the regression parameter vector β(p), but also the parameter γ = γ(p) of the power transformation (121), were unknown. For this reason their estimate included a combination of a CUSUM procedure and regression quantiles (115). The resulting estimated link functionĥ(y; p) = hγ (p) (y) is then inserted into (118) in order to estimate θ(x). 2 Example 11 (Log cumulative baseline hazard transformations.) As in the previous two examples, consider a positive response variable Y , so that a = 0 and b = ∞. The Cox regression model (Cox, 1972) expresses the hazard function λ(y \| x 2 , . . . , x q ) = λ bl (y) exp( q k=2 x k b k−1 )(122) of the lifetime Y as a product of a baseline hazard λ bl and a term that involves the covariates x 2 , . . . , x q and a regression vector b = (b 1 , . . . , b q−1 ) T . Equivalently, (122) can be rewritten in terms of the cumulative hazard function as Λ(y \| x 2 , . . . , x q ) = y 0 λ(z \| x 2 , . . . , x q )dz = Λ bl (y) exp( q k=2 x k b k−1 ),(123) where Λ bl (y) = y 0 λ bl (z)dz is the corresponding cumulative baseline hazard. It is well known (Doksum and Gasko, 1990, Koenker and Geling, 2001, Portnoy, 2003, García-Pareja et al., 2019) that (123) can be rewritten as a transformed linear model (113), with a link function h(y) = log [Λ bl (y)] y=Q(p\|x) = log log(1 − p) −1 − q k=2 x k b k−1 = x T β(p) (124) that is the logarithm of the cumulative baseline hazard, x = (1, x 2 , . . . , x q ) T is a covariate vector with added intercept, and the regression vector is β(p) = log log(1 − p) −1 e 1 −(0, b T ) T . This is an instance of a parametric model (116) for which r = 2, ψ 1 = e 1 is known, ψ 2 = (0, b T ) T , β 1 (p) = log log(1 − p) −1 is a quantile of a Gumbel distribution and β 2 (p) ≡ −1. García-Pareja et al. For a Cox model it is traditional to estimate b directly by partial likelihood rather than using the nonparametric or parametric estimators (115) and (116). Typically the baseline hazard is estimated nonparametrically (see for instance Kalbfleish and Prentice, 2002). This corresponds to a nonparametric estimatê h(y) = log[Λ bl (y)](125) of the link function h, which is inserted into (118) in order to estimate θ(x). On the other hand, if the baseline distribution F bl (y) = 1 − exp(−Λ bl (y)) = 1 − exp(−ay b ) is Weibull, with a, b > 0 known, then (124) is equivalent to an AFT model with a known logarithmic link function (119), and with Q 0 (p) a quantile of a Gumbel distribution in (120). Gelfand et al. (2000) proposed a larger parametric model for the baseline distribution F bl ; a mixture of Weibull distributions. In our context this corresponds to having a finite number of unknown parameters γ of the link function (124), and a parametric estimate (125) of this link function based onΛ bl (y) = Λ bl (y;γ). Finally, Royston and Parmar (2002) assumed a version log[Λ(y \| x 2 , . . . , x q )] = s(log(y); γ) + q k=2 x k b k−1(126) of (123) where log[Λ bl (y)] is replaced by a cubic spline function s(·; γ) of log(y) that is parametrized by γ. It can be seen that this corresponds to replacing the link function in (124) by h(y; γ) = s(log(y); γ). This gives rise to an estimated link functionĥ (y) = s(log(y);γ), which is inserted into (118) in order to estimate θ(x). 2 Example 12 (Log baseline odds transformations.) As in the previous example, let x 2 , . . . , x q represent q − 1 covariates. Bennett (1983) introduced a model for which the proportional odds of death satisfies Γ(y \| x 2 , . . . , x q ) = F Y \|x2,...,xq (y) 1 − F Y \|x2,...,xq (y) = Γ bl (y) exp( q k=2 x k b k−1 ),(128) for some effect parameters b = (b 1 , . . . , b q−1 ) T and baseline odds function Γ bl (y). Doksum and Gasko (1990) showed that this model can be rewritten as a transformed linear model (113), with link function h(y) = log [Γ bl (y)] y=Q(p\|x) = log[ p 1 − p ] − q k=2 x k b k−1 = x T β(p).(129) The vector x = (1, x 2 , . . . , x q ) T contains an intercept and covariates, whereas β(p) = log[p/(1 − p)]e 1 − (0, b T ) T includes the effect parameters b of the covariates and an intercept parameter log[p/(1 − p)] that is a quantile of a standard logistic distribution. This corresponds to a parametric model (116) with r = 2, a known ψ 1 = e 1 , and unknown ψ 2 = (0, b T ) T , β 1 (p) = log[p/(1 − p)], and β 2 (p) ≡ −1. If the baseline distribution F bl (y) = Γ bl (y)/(1 + Γ bl (y)) = [1 + exp(−(log(y) − a)/b)] −1 is log-logistic, for some known a and b > 0, it can be seen that the proportional odds model (129) is equivalent to an AFT model (120) with a logarithmic link function, where Q 0 (p) is the quantile of a logistic distribution. Royston and Parmar (2002) studied a version log[Γ(y \| x 2 , . . . , x q )] = s(log(y); γ) + q k=2 x k b k−1(130) of (128) where the log baseline odds log[Γ bl (y)] is replaced by a cubic spline function s(·; γ) of log(y). It can be seen that this corresponds to replacing the link function in (129) by h(y) = s(log(y); γ). The corresponding estimate (127) is then inserted into (118) in order to estimate θ(x). 2 Example 13 (Log power transformations.) Younes and Lachin (1997) considered a class of models which includes proportional hazards and proportional odds as special cases. In more detail, they assumed that the logarithm of a Box-Cox transformation (121) of the survival function 1 − F Y , conditionally on covariates x 2 , . . . , x q , satisfies log (1 − F Y \|x2,...,xq (y)) −γ − 1 γ = log (1 − F bl (y)) −γ − 1 γ + q k=2 x k b k−1 for some γ > 0, where γ = 1 corresponds to the proportional odds model (128) and γ → 0 to the proportional hazards model (123). This is a transformed linear model with link function h(y) = log (1 − F bl (y)) −γ − 1 γ {y = Q(p \| x)} = log (1 − p) −γ − 1 γ − q k=2 x k b k−1 = x T β(p),(131) covariate vector x = (1, x 2 , . . . , x q ) T , and regression parameter vector β(p) = log{[(1 − p) −γ − 1]/γ}e 1 − (0, b T ) T . 2 In order to study the asymptotic properties of the estimatorθ(x) in (117) of the conditional L-functional θ(x) in (114), we will assume that the link function h is continuously differentiable, with a strictly positive derivative. It is helpful to approximate (117) by a first order Taylor expansion θ(x) ≈ θ(x) + x T 1 0 β (p) − β(p) dG x (p),(132) where dG x (p) = 1 h [h −1 (x T β(p))] · dG(p)(133) can be thought of as an effective weight function, which determines how much different quantiles contribute to the estimation error of θ(x). The following result is a corollary of Propositions 1-2: (117) is then a consistent estimator of θ(x), so thatθ n (x) Corollary 1 (Consistency and asymptotic normality of (117).) Suppose that the regularity conditions of Propositions 1-2 hold for the transformed regression model (x, h(Y )) and that h is a known link function such that h (y) ≥ c x > 0 for all y ∈ [a x , b x ], where (a x +ε, b x −ε) includes the set h −1 {x T β(p); p ∈ supp(G)}) for some ε > 0. The conditional L-statisticθ n (x) inp −→ θ(x) as n → ∞. It is asymptotically normal as well, i.e. √ n[θ n (x) − θ(x)] L −→ N (0, x T Σ x x)(134) as n → ∞, with a covariance matrix Σ x = 1 0 1 0 R(p, s)dG x (p)dG x (s)(135) that involves the effective weight function G x in (133) and R(p, s), the asymptotic covariance matrix of {β n (p); 0 < p < 1}. Proof. The result can be derived similarly as in the proofs of Propositions 1-2, using the Taylor expansion (132). Indeed, the regularity conditions on h imply that the remainder term of this Taylor expansion is asymptotically negligible, both for the consistency and the asymptotic normality proofs. 2 When our objective is to estimate conditional skewness or kurtosis we focus on quantities θ(x), defined as the ratio (85) of two conditional L-functionals T 1 (F Y \|x ) and T 2 (F Y \|x ) with different weight functions G 1 and G 2 . For the transformed linear model (113) we find that θ(x) = 1 0 h −1 (x T β(p))dG 1 (p) 1 0 h −1 (x T β(p))dG 2 (p) .(136) We estimate θ(x) by plugging an appropriate estimator of β(p) into (114), i.e. θ(x) = 1 0 h −1 (x Tβ (p))dG 1 (p) 1 0 h −1 (x Tβ (p))dG 2 (p) .(137) When the link function h is unknown, as in (118) it is possible to define a version ofθ(x) where h is replaced by a parametric or nonparametric estimatê h in the numerator and denominator of (137). The following result is essentially a consequence of Proposition 3 and Corollary 1: Corollary 2 (Consistency and asymptotic normality of (137).) Suppose that the regularity conditions of Proposition 3 and Corollary 1 hold. The quantityθ n (x) in (137) is then a consistent estimator of θ(x) in (136). It is also asymptotically normal, in the sense that √ n[θ n (x)−θ(x)] L −→ N (0, Σ ) Σ = x T Σ 11 x x T 2 (F Y \|x ) − 2T 1 (F Y \|x ) · x T Σ 12 x x (T 2 (F Y \|x )) 3 + (T 1 (F Y \|x )) 2 · x T Σ 22 x x (T 2 (F Y \|x )) 4 (138) as n → ∞, where Σ kl x = 1 0 1 0 R(p, s)dG k x (p)dG l x (s),(139) G 1 x and G 2 x are defined as in (133), with G 1 and G 2 in place of G, and R(p, s) is the asymptotic covariance matrix of {β n (p); 0 < p < 1}. The consistency and asymptotic normality ofθ n (x) in Corollaries 1-2 is a unified result for nonparametric or parametric estimates (115)-(116) of β = {β(p); 0 < p < 1}. It is only required that the link function h is known, and thatβ = {β(p); 0 < p < 1} is a consistent and asymptotically normal estimator of β, whose covariance function R(p, s) appears in (135) and (139). In the nonparametric case, R(p, s) is defined as in (101), provided the density in the expression for D n (p) is changed to f i (p) = dF h(Y )\|xi (v)/dv v=h(Q(p\|xi)) . In the parametric case, R(p, s) is defined as in (104). See also Newey and McFadden (1994), Ying et al. (1995), Chen et al. (2003), Mu and He (2007), and references therein, for a discussion on when the estimator (118) of θ(x) with estimated link function is asymptotically equivalent to the corresponding estimator (117) where h is known. Censored and truncated data Assume there exists a collection {(x i , Y i )} n i=1 of i.i.d. random vectors, and that F Y \|x follows the transformed linear model (113). The objective is to estimate the L-functional (114) or the ratio (136) of L-functionals, when some data is lost due to censoring or truncation. This boils down to finding an estimatorβ(p) of β(p) for all 0 < p < 1, and then plugging this estimator into (117), (118) or (137). Notice that Corollaries 1-2 apply to censored and truncated data as well, if consistency and asymptotic normality is established for {β(p); 0 < p < 1}, with some limiting covariance function {R(p, s); 0 < p, s < 1}. Censoring When a distorted versionỸ of the outcome variable Y is observed, it is often the case that a censoring variable C causes this distortion. The two most common types of censoring arẽ Y = max(Y, C), left-censoring, min(Y, C), right-censoring. Without loss of generality we restrict ourselves to right-censoring. To this end, assume that {(x i , Y i , C i )} n i=1 are i.i.d. copies of covariate vectors, response and censoring variables, such that Y i and C i are conditionally independent given x i , and withỸ i = min(Y i , C i ) the right-censored version of Y i . Let us first assume that all censoring variables C i are observed, whether Y i is censored or not. This corresponds to having a data set consisting of the observations (x i ,Ỹ i , C i ) for i = 1, . . . , n. In this case the conditional quantilẽ Q(p \| x, C) = F −1 Y \|x (p) = min(x T β(p), C)(140) of censored data is an explicit function of β(p). This gives rise to the estimatê β(p) = arg min b n i=1 ρ p [h(Ỹ i ) − min(x T i b, C i )] of β(p) due to Powell (1986). For the general right-censoring problem only {(x i ,Ỹ i , ∆ i )} n i=1 is observed, where ∆ i = 1(Y i =Ỹ i ) indicates whether observation i has been censored or not. The relation between the quantile functions of censored and uncensored response variables is then somewhat more complicated than (140). Because of conditional independence of Y and C given x, it follows that FỸ \|x (y) = 1 − (1 − F Y \|x (y))(1 − F C\|x (y)),(141) and consequently the quantile functionQ(· \| x) of the right censored response variable is related to the quantile function Q(· \| x) of the uncensored response asQ (π(x, p) \| x) = Q(p \| x),(142) where π(x, p) \| x))) tells which quantile of the censored observation an uncensored p-quantile relates to. If F C\|x would be known for all x, then based on (142) and the fact that quantiles are preserved under the monotone transformation h, the estimator = 1 − (1 − p)(1 − F C\|x (Q(pβ(p) = arg min b n i=1 ρ π(xi,p) (h(Ỹ i ) − x T i b)(143) of β(p) due to Lindgren (1997) could be used. When the censoring distribution is unknown, a nonparametric Kaplan-Meier estimatorF C\|x was employed by Lindgren (1997) in order to replace π(x i , p) in (90) by an estimatorπ(x i , p). Another consequence of (141) is that the random variable 1(h(Ỹ ) − x T β(p) ≥ 0) 1 − F C\|x (x T β(p)) − (1 − p) has zero expectation. This motivated Ying et al. (1995) and Leng and Tong (2013) to propose and study the score-based estimator Leng and Tong (2013) proved that this estimator is asymptotically normal with a √ n-rate of convergence. As a drawback, (144) does not account for which observations that have been censored (∆ i = 0) or not. In order to use this information, Wang and Wang (2009) β(p) = arg min b n i=1 x i 1 −ω i (b) 1 −F C\|xi (x T i b) − (1 − p) (144) of β(p), withω i (b) = 1(h(Ỹ i ) ≤ x T i b).introduced β(p) = arg min b n i=1 x i p −ω i (b) + (1 − ∆ i )ω i (b)(1 − p) 1 −F Y \|xi (Ỹ i ) ,(145) whereF Y \|x is a local Kaplan-Meier estimate of F Y \|x . This estimator can be motivated by noticing that each term of (145) has zero expectation when b = β(p), andF Y \|xi is replaced by the true but unknown F Y \|xi . Other estimators of β(p), for censored observations, have been proposed by Yang (1999), Portnoy (2003), Neocleous et al. (2004), and Peng and Huang (2008). Truncation Truncation means that some observations (x, Y ) are lost, depending on the value of some other truncation random variable L. The two most common types of truncation are that (x, Y ) is lost when Y < L, left-truncation, Y > L, right-truncation. We will assume that left-truncation occurs together with right-censoring, so that the observed data set is Frumento and Bottai (2017) generalized (145) and presented an estimator Frumento and Bottai (2017) gave conditions under which this estimator of β(p) is consistent and asymptotically normal. {(x i ,Ỹ i , ∆ i , L i ); i = 1, . . . , n, Y i > L i }.β(p) = arg min b n i=1 x i ω i (b) − ω i (b)(1 − p) 1 −F Y \|xi (L i ) −ω i (b) + (1 − ∆ i )ω i (b)(1 − p) 1 −F Y \|xi (Ỹ i ) ,(146)of β(p), where ω i (b) = 1(h(L i ) ≤ x T i b). Numerical examples In this section we will analyze numerical properties of L-functionals without covariates, as described in Section 2. In more detail we consider the four (standardized) L-functionals T 1 (F ), T 2 (F ), T 32 (F ), and T 42 (F ) of a target distribution F . First, in Section 4.1, we quantify how well these functionals approximate selected, well known distributions' quantile functions using the polynomial series of Examples 5-7. For the Legendre system (74), this has previously been done by Hosking (1990Hosking ( , 1992, Karvanen (2006Karvanen ( , 2008 and Elamir and Seheult (2003). We will find that typically, matching the "essential support" (or form) of a linearly standardized version of the target distribution F (i.e. a region harbouring most of the probability mass of a standardized version of F ) with the support (or form) of the polynomial system's reference distribution F 0 , produces the most accurate approximations. In Section 4.2, we present graphs showing the change in the L-functionals T 2 , T 32 and T 42 for Beta distributions F ∼ B(ψ 1 , ψ 2 ), when the two shape parameters ψ 1 and ψ 2 are varied in such a way that the expected value T 1 (F ) = ψ 1 /(ψ 1 + ψ 2 ) in (25) is held constant. All computations were performed in R (R Core Team, 2021). Quantile function approximation errors In this subsection we will investigate the suitability of the three orthogonal polynomial series of Examples 5-7 for approximating the quantile function Q(p) = F −1 (p), using m 0 terms. For instance, m 0 = 4 terms is the approximation using only location, scale, skewness and kurtosis. This will be done for distributions F with various types of essential support (that is, a set that supports close to 1 of the probability mass of F ). The three types of essential support are interval support [a, b], half-infinite support of life-time distributions [0, ∞) and doubly infinite support on the real line R. As a rule of thumb we hypothesize that the polynomial series whose reference distribution F 0 has a support of the same type as the essential support of the distribution F whose quantile function we try to approximate, will be the most suitable. Thus, the Legendre polynomials should suit distributions with bounded essential support, Laguerre polynomials should be appropriate for distributions with half infinite essential support, whereas Hermite polynomials are preferable for distributions with unbounded essential support, to the left and right. The approximated quantile function is denoted Q appr (p) = m0 m=1 T m (F )g m (p). Note that, because of the orthogonality property (69)-(70), the Integrated Squared Error (ISE) of the approximation error Q − Q appr , satisfies 1 0 (Q(p) − Q appr (p)) 2 dp = 1 0 ∞ m=1 T m (F )g m (p) − m0 m=1 T m (F )g m (p) 2 dp = 1 0 ∞ m=m0+1 T m (F )g m (p) 2 dp, = ∞ m=m0+1 T 2 m (F ),(147) meaning that the approximation error amounts to what cannot be summarized about the distribution from the first m 0 L-moments. To compute the ISE, we evaluate 1 0 Q(p) − m0 m=1 1 0 Q(ρ)g m (ρ) dρ g m (p) 2 dp(148) numerically. This makes it possible to compute how large fraction ∆ m0 = 1 − ISE 1 0 Q 2 (p) dp(149) of the variation of Y that is explained by the first m 0 L-moments. In Table 1 we have computed ∆ 4 for selected distributions, using the abovementioned three polynomial series. Sometimes the numerical procedures fail to evaluate either integral of (148) when F has heavy tails. Most commonly, m = 4 causes problems for the Hermite polynomials, as can be seen from (76), (78), and (148), and less commonly for the higher order L-functionals using the Laguerre polynomials (cf. (80) and (148)). Shortening the inner and outer integration intervals of (148) to ( , 1 − ) would mitigate these problems. Therefore, in order to reduce the numerical problems associated with the approximation error (148) we construct a new series g m of orthonormal polynomials to replace g m in Q appr . We do this by using the Gram-Schmidt process to orthonormalize the base functions g 1 (p), . . . , g 4 (p) with an inner product 1− g k (p)g l (p) dp, k, l ∈ {1, . . . , 4}. Thus, we can approximate (148) using g m (p) for any chosen and distribution. In particular, = 0 corresponds to using g 0 m = g m in Q appr . Notice that g m is conceptually different from g π m used in the robustified functionals in Examples 5, 6 and 7. Since the numerical issues occur when p approaches 0 and 1, we cannot mitigate it with g π m , since it still utilizes the whole support of the chosen polynomial series. (149). The first column denotes the distribution, the next three specify which polynomial series was used to approximate the quantile function, and the subcolumns specify the values of ∆ 4 and for each approximation. The best approximation is highlighted. Exp(ψ) refers to an exponential distribution with scale parameter ψ, so that E F (Y ) = ψ, Γ(ψ 1 , ψ 2 ) is a gamma distribution with shape parameter ψ 1 and scale parameter ψ 2 , so that E F (Y ) = ψ 1 ψ 2 and Γ(1, ψ) = Exp(ψ). Wei(ψ 1 , ψ 2 ) corresponds to a Weibull distribution with shape parameter ψ 1 and scale parameter ψ 2 , so that Wei(1, ψ) = Exp(ψ). Notice how it is not the true support of the distribution that reveals which polynomial series that best approximates the quantile function, but rather the essential support. For instance, a Wei(3, 1) distribution has support on [0, ∞), but the Hermite polynomials give the best approximation since the left and right tails are light, similarly to a normal distribution. Figure 1: Quantile function Q(p), and the associated approximations Q appr (p), for any of the polynomial systems of Section 2.4 that is computable, for each chosen . The most suitable polynomial series for approximating Q, measured by ∆ 4 , is the Legendre system for (a), the Hermite system for (b) and the Laguerre system for (c) and (d). For (a) to (c), the reference distribution Q 0 of the best performing polynomial series equals Q, see Examples 5-7. For (d), notice how the quantile function Q three times crosses the approximating Q appr when using Laguerre polynomials. In Figure 1 we showcase four distributional approximations, where (a)-(c) are approximations using = 0 and (d) is the same approximation as in (c), but using = 10 −5 . A comparison between c) and d) reveals the effects of extreme quantiles on Q appr . Plotting the change in scale, shape and location for beta distributions with fixed location Using the canonical parametrization B(ψ 1 , ψ 2 ) of the beta distribution we generated a grid of parameter values for five fixed choices (0.5, 0.6, 0.7, 0.8, 0.9) of the expected value T 1 (F ) = E F (Y ) = ψ 1 /(ψ 1 + ψ 2 ). For each value pair (ψ 1 , ψ 2 ) we computed the three (standardized) L-moments T 2 (F ), T 32 (F ), and T 42 (F ) according to (65) and (66), using all three polynomial series of Section 2.4.2. As T 2 (F ) → 0, it can be seen from properties of the beta distribution that the standardized version of F will converge to a normal distribution, and consequently T 32 (F ) → T 32 (N(·, ·)). Since the Hermite and Legendre polynomials generate symmetric collections of L-functionals, it follows from (71) of Section 2.4.1 that T 32 (N(·, ·)) = 0, whereas T 32 (N(·, ·)) = −0.340 for Laguerre polynomials. Similarly, T 42 → T 42 (N(·, ·)), where T 42 (N(·, ·)) = 0 for Hermite polynomials, whereas T 42 (N(·, ·)) = 0.187 for Legendre polynomials and T 42 (N(·, ·)) = 0.201 for Laguerre polynomials. As T 2 (F ) increases, F will converge to a Bernoulli Be(T 1 (F ))-distribution, so that in the limit F ({0}) = 1 − T 1 (F ) and F ({1}) = T 1 (F ). In Figures 2-4 we have plotted T 32 (F ) and T 42 (F ) as functions of T 2 (F ) for each expected value and polynomial series. It can be seen, for instance, that for the Hermite and Legendre systems, standardized skewness is zero (negative) for a beta distribution with T 1 (F ) = 0.5 (T 1 (F ) > 0.5). On the other hand, for the Laguerre system, skewness is negative for all beta distributions with T 1 (F ) ≥ 0.5. The reason is that skewness of the Laguerre system is quantified in relation to the asymmetric exponential reference distribution. ) and T 42 (N(·, ·)). The colored dots illustrate the values of T 32 (Be(T 1 (F ))) and T 42 (Be(T 1 (F ))) for each of the five fixed values of T 1 (F ). Notice that the deviations from the expected Bernoulli limits increase as the expected value increases. This is due to an increasing probability mass in the extreme tails, outside of ( , 1 − ), which is missed by our adapted orthogonal weight functions g m , to a higher extent the larger T 1 (F ) is. Bird migration timing analysis As an application of the regression methodology of Section 3, we consider bird migration timing analysis, often called phenological analysis. We will use Lfunctionals and quantile regression to analyze changes over time in location, scale, skewness and kurtosis of bird migration timing distributions. We will also present the varying effects of covariates across the quantiles, and highlight reflections on how the covariates influence the values of the L-functionals. Our approach enables analysis of a larger number of distributional aspects than available with canonical methods, such as the one in Lehikoinen et.al. (2019). The data The data set we will analyze is collected by the Falsterbo Bird Observatory and concerns the species Common Redstart (Phoenicurus phoenicurus). This species was selected since it is a bird where the plumages of juveniles, adults, females and males are all distinct. Thus, reliable information on sex and age is available for use as covariates in the analysis. Birds were captured and ringed by the bird observatory under similar schemes each year during the period 1980-2019, although we will use data from the years 1982-2019, since the identification of juvenile females was not performed the first two years. We will only use data on newly ringed birds. Recaptures between years are rare, but ideally these data should be incorporated as well. Due to the extent of the ringing effort each year, it is safe to assume that the whole migration period was covered by the annual sampling window. The covariates and response variable of the data set are summarized in Table 2. Exploratory visualizations Selecting all the observations from 2010, and splitting the data set into four subsets -one for each combination of age and sex -allows us to compute the first four L-moments for each subset this particular year. In Figure 5, the empirical conditional quantile function for each subpopulation x of year 2010, age, and sex is presented, along with the Legendre-based approximationQ appr (·\|x) of each empirical conditional quantile function. The function in (151) is the inverse of the conditional empirical response distributionF Q(p; x) =F −1 (p; x) = inf{y;F (y\|x) ≥ p}(151)(y\|x) = n i=1 1(Y i ≤ y, x i = x) n i=1 1(x i = x) ,(152) for each covariate vector x that appears in the data set. Although the plot in Figure 5 contains data from just one of the years under study, it is possible to trace differences in location, and to some extent differences in scale and skewness, between the subpopulations. Quantile regression Fitting quantile regression models to the Common Redstart data makes it possible to find out whether covariates have different magnitudes of effect for different quantiles. Moreover, we can estimate the first four conditional L-functionals from (84), and the standardized conditional L-functionals from (86). In particular, we will study how these estimated conditional L-functionals change with covariates. Model setup We fitted models using two approaches: an identity link and a logit link. As presented in Section 3.1.2, quantiles are preserved under monotone transfor-mations. The response variable is Julian day and thus the response values will be located within an interval [a, b], where the sampling window has end points a = 80 and b = 162. As presented in Example 8 the logit-link function is a monotone transformation. In accordance with Section 3.1.2, we fit the model logit Q (p \| x) − a b − a = x T β(p),(153) with q = 4 parameters in β(p) (corresponding to an intercept and an effect parameter for each one of the three covariates of Table 2). Then we findβ(p) nonparametrically through (115), i.e. β(p) = arg min b∈R q n i=1 ρ p logit Y i − a b − a − x T i b .(154) For comparison with (153)-(154), we will also fit a linear quantile regression model (87). All implementations were made in R (R Core Team, 2021). We chose to implement the objective function ourselves and used optim to find estimates, but it is also possible to use software packages readily available for many types of quantile regression, such as qgam (Fasiolo et al., 2017) and quantreg (Koenker, 2020). Model selection When fitting the quantile regression model we may choose any link function that is a monotone transformation. Moreover, when having fitted the model we may choose between the three sets of weight functions g m (p) presented in Examples 5-7 and then compute the associated four conditional (standardized) L-functionalsθ(x). An ideal model selection method should yield the combination of link function and weight functions that gives the best overall approximation ofQ(· \| x) for all x. It should also be useful for selecting which covariates to use in the model. To this end, we modify (148) to create R 2 m0 = 1 − X 1 0 Q (p \| x) −Q appr (p \| x) 2 dp dF X (x) 1 0 (Q(p) − EF (Y )) 2 dp ,(155) whereF X gives equal weight to all covariates vectors that appear in the data set (regardless of the number of observations with this covariate vector), Q(· \| x) =F −1 (· \| x) (156) Q(p) =F −1 (p) = inf{y;F (y) ≥ p}(157) andF (y) = XF (y \| x) dF X (x).(158) Thus, EF (Y ) = T 1 (F ) = T 1 (F )g 1 (p), and note that this is not the sample mean, but rather a weighted sample mean, since we weight the unique covariate vectors x (not the observations) equally. For the particular analysis of the Common Redstart data, this has the effect of weighting data from each year equally, meaning we do not let years with a larger number of registered birds have a larger influence. Since the number of birds can vary a lot between years, this is deemed advantageous for the purpose of the analysis. As with (148), the m 0 index specifies the degree ofQ appr . With m 0 = 1, this reduces to a weighted version of the classical R 2 , and for any m 0 ≥ 1 we get a measure of how much variation in the response is captured by the first m 0 conditional L-moments. Similarly as in Hössjer (2008), any R 2 m0 with m 0 ≥ 2 includes variation in the response not explained be the covariates. In particular, if R 2 m0 increases significantly when m 0 gets larger, this indicates that the corresponding weight functions g m0 (p) capture an essential part of the variation in the conditional response distributions. We fitted models for the identity link and logit link functions and computed θ(x) for all three polynomial weight function sets. When computing the integrals in (84) and (86) we interpolated the grid of β j (p)-estimates using the cubic splines of Forsythe (1977). The resulting R 2 m0 values are presented in Table 3. Observe that R 2 m0 for m 0 > 1 not necessarily is an increasing function of the number of covariates included in the model. This is partly due to the fact that all covariate vectors are weighted equally in the definition of R 2 m0 , regardless of their number of observations. Covariate and L-functional estimates In this section we will first illustrate how the q = 4 parameter estimates of (154) change as a function of the quantile p (cf. Figure 6) over a grid of 100 values in (0, 1) for the case of an identity link function. Next we plot the change in the location, scale, standardized skewness and standardized kurtosis over time, for each subpopulation, i.e. each combination of age and sex. These plots are shown in Figure 7. All estimates include approximate 95% bootstrapped pointwise confidence intervals, represented by ribbons. The negative estimates of the effect of year in Figure 6d, and the negative slopes over time of the expected value of conditional response distribution, in Figure 7a, both demonstrate that birds in recent years tend to arrive earlier. We also use the resampling results to present tables over the Mahalanobis distance between different pairs of covariates' L-functionals. For this we need some additional notation. For any subset I ⊂ {1, 2, 32, 42} of order numbers of location, scale, standardized skewness and standardized kurtosis, we introduce the collection θ I (x) = (T m (F Y \|x ); m ∈ I) of conditional (standardized) L-moments for covariate vector x. The corresponding vector of estimated conditional L-moments is denoted Table 3: R 2 m0 -values for two choices of covariates, the identity and logit link functions and the three polynomial systems of weight functions defined in Examples 5-7. (A weighted version of) the classical coefficient of determination, R 2 1 , reflects that more of the variation in the response is explained when age and sex are added as covariates, compared to having only year as covariate. Notice though that R 2 m0 for m 0 > 1 is higher for a model with year as the only covariate compared to using all three covariates. The reason is that the year only model has an increased number of observations in each covariate specific subset, which leads to a smootherQ(p\|x) with smaller jumps at each discontinuity point. This, in turn, leads to a closer fit ofQ appr (p\|x). θ I (x) = (T m (F Y \|x ); m ∈ I) .(159) Recall that each x corresponds to a subpopulation (a combination of age group and sex) at a specific time point. Assume that the data set is resampled B times, and let F * b Y \|x be the b:th resampled conditional response distribution (b = 1, . . . , B) for covariate vector x. The corresponding resampled vector of conditional L-moments is θ * b I (x) = (T m (F * b Y \|x ); m ∈ I) .(160) These B vectors will be scattered around (159), with an estimated covariance matrixΣ I (x) = 1 B B b=1 (θ * b I (x) −θ I (x))(θ * b I (x) −θ I (x)) . In particular, the Mahalanobis distance between (160) and the center point (159) of the distribution is M b I (x) = (θ * b I (x) −θ I (x)) TΣ I (x) −1 (θ * b I (x) −θ I (x)).(161) This distance can be interpreted as how many standard deviations away from the center pointθ I (x) the point θ * b I (x) is. We would however like to measure the distance between the center of the resampled clouds for different pairs x 1 and x 2 of covariate vectors, rather than each resampled point's distance to its center. The Mahalanobis distance between x 1 and x 2 , for the collection I of conditional L-moments, is M I (x 1 , x 2 ) = (θ I (x 1 ) −θ I (x)) T [0.5(Σ I (x 1 ) +Σ I (x 2 ))] −1 (θ I (x 1 ) −θ I (x)). (162) In Tables 4 and 5 we present values of the modified Mahalanobis distances (162) for different combinations of I, x 1 , and x 2 . From these tables it can be seen that the location functional is most important for distinguishing the arrival distributions between subpopulations (Table 4) and years (Table 5). Although the scale, standardized skewness and standardized heavytailedness functionals are less important, they still help to discriminate even more between the arrival time distributions of these groups. Discussion In this paper we developed a general theory of L-functionals of the response variable distribution of regression models. Based on orthogonal series expansions of the quantile functions of these distributions we generalized the concept of L-moments and identified collections {T m (F )} 4 m=1 of L-functionals that correspond to measures of location, scale, unstandardized skewness and unstandardized heavytailedness of the response. Different collections of L-functionals were introduced, depending on whether the domain of the response variable is bounded, or unbounded in one or two directions. Figure 6: Plots of parameter estimates for the regression model with the best fit, using the identity link. The black lines represent the q = 4 estimateŝ β 1 (p),β 2 (p),β 3 (p),β 4 (p), where p takes values on a grid of 100 equispaced points in (0, 1). The grey ribbon of each subplot consists of approximate 95% pointwise bootstrapped confidence interval for {β k (p); 0 < p < 1}. Note that in (b), the confidence interval is very broad for the lowest quantiles, which is due to there being extremely few juvenile birds among the early arrivers. In (d), the effect of year quickly approaches 0 in the upper quantiles. This might be due to an increased chance of catching stray birds instead of migrating birds at that time of the year. Lastly, smoothingβ(p) seems like a good idea, given the wigglyness of the estimates. (95) and (109), using the Legendre weight functions g m (p). A subplot illustrates a specific conditional L-functional. It contains four curves, each one of which corresponds to a fixed combination of age and sex, whereas year varies along the horizontal axis. For the location plot (a), all polynomial series would give the same result (the conditional mean), since the first polynomial in each series is 1. For the scale plot (b) the lines correspond to Gini's mean difference times a proportionality constant (cf. Example 5). The standardized skewness (c) and standardized kurtosis (d) are measured relative to a uniform distribution, since we use weight functions based on Legendre polynomials. The important information obtained from the figure is qualitative; how the various L-functionals change over time, rather than their actual values. From (c) we notice that the arrival distribution shifts from right-skewed towards symmetric over the study period, whereas from (d) we observe that the kurtosis increases. The conclusion would be that the left tail slowly becomes as thick as the right tail, meaning that early arrivers become more frequent over time. A number of generalizations of our work is possible. The first extension is to study more systematically the standardized L-functionals of skewness, heavytailedness, ... T m2 (F ) = T m (F )/T 2 (F ) when m ≥ 3. In particular, it is of interest to know which types of stochastic orderings between response distributions (Oja, 1981) these functionals preserve. Second, Takemura (1983) defined orthogonal series expansions of quantile functions for arbitrary reference distributions F 0 for which T m (F 0 ) vanish when m ≥ 3. In Examples 5-7 we considered expansions for uniform, normal, and exponential distributions. It is also possible to define a system (68) of L-functionals for Weibull, log-logistic, and other reference distributions F 0 that are of interest in survival analysis. Third, a general way of robustifying a collection {T m (F )} of L-functionals is to introduce a weight function w that downweights contributions from the lower and upper tails of F . The weight density g m (p) = w(p)P m−1 (Q 0 (p)) of order m is a generalization of (68), using a variable rather than a constant weight function w(p) ≡ 1. For instance, the trimmed L-moments of Elamir and Seheult (2003) correspond to a weight function w(p) = (1 − p) t p t for some positive integer t. Notice however that in general the orthogonality property (69) is lost for such a system of weight functions. Fourth, recursive estimation of quantiles (Stephanou et al., 2017) could be extended to online estimation of Lfunctionals. Fifth, regression quantiles have been used for time series Zu, 2008, White et al., 2008) in order to estimate quantiles and robust measures of skewness/kurtosis of predictive distributions. It is of interest to analyze (ratios of) L-functionals of such predictive distributions. Sixth, measures of location, scale, skewness and heavytailedness of multivariate distributions can be defined in terms of multivariate L-statistics (Liu, 1990, Liu et al. 1999, Zuo et al., 2004, and Dang et al., 2009. In this context it is of interest to study different polynomial systems of L-functionals and their order numbers. La k (y)La l (y)e −y dy = 1(k = l). it follows thatB n p −→ B. Finally, sinceθ n (x) = h(B n ) and θ(x) = h(B) for the continuous function h : R q → R, defined by h(B) = x T B, consistencyθ n (x) p −→ θ(x) follows by the Continuous Mapping Theorem. 2 √ n(B n −B) = h(Z n ) and h(Z) ∼ N (0, Σ), equation (105) is a consequence of the Continuous Mapping Theorem. Then (107) follows from (105) by a second application of the Continuous Mapping Theorem since √ n[θ n (x) − θ(x)] = √ nx T (B n − B) = h[ √ n(B n − B)], using the function h : R q → R defined by h(B) = x T B. 2 n are consistent and (jointly) asymptotically normal estimators of B 1 and B 2 as n → ∞, in the sense that ) − (B 1 , B 2 )] L −→ N 0, Σ 11 Σ 12 Σ 21 Σ 22 , 1 , B 2 ) and the Continuous Mapping Theorem. In order to verify asymptotic normality ofθ n (x) =N /D =: h(N ,D) around the point (N, D), noticing that θ(x) = h(N, D). 2 estimate the conditional compound expectation θ(x) of a Cox model based on the piecewise linear weight function (24). ) X ∼ Exp(1) with = 10 −5 . Figure 2 : 2Plots of standardized skewness T 32 (F ) (a) and standardized kurtosis T 42 (F ) (b) versus scale T 2 (F ) for various beta distributions F ∼ B(ψ 1 , ψ 2 ) using the Legendre polynomials and = 0. The black dots correspond to values of T 32 (N(·, ·)) and T 42(N(·, ·)). The colored dots illustrate the values of T 32 (Be(T 1 (F ))) and T 42 (Be(T 1 (F ))) for each of the five fixed values of T 1 (F ). Notice that close to the Bernoulli limit, because of the numerical approximations involved in computing the L-functionals, both T 32 and T 42 show slight deviations from the ideal value for T 1 (F ) ∈ {0.8, 0.9}. Figure 3 : 3Plots of standardized skewness T 32 (F ) (a) and standardized kurtosis T 42 (F ) (b) versus scale T 2 (F ) for various beta distributions F ∼ B(ψ 1 , ψ 2 ) using the Hermite polynomials and = 10 −3 . The black dots correspond to values of T 32 (N(·, ·) Figure 4 : 4Plots of standardized skewness T 32 (F ) (a) and standardized kurtosis T 42 (F ) (b) versus scale T 2 (F ) for various beta distributions F ∼ B(ψ 1 , ψ 2 ) using the Laguerre polynomials and = 10 −3 . The black dots correspond to values of T 32 (N(·, ·)) and T 42(N(·, ·)). The colored dots illustrate values of T 32 (Be(T 1 (F ))) and T 42 (Be(T 1 (F ))) for each of the five fixed values of T 1 (F ). The deviations of the colored curves from the expected Bernoulli limits of T 32 and T 42 are larger compared to the Hermite polynomials ofFigure 3, due to the asymmetry of the Laguerre polynomials. Figure 5 : 5The conditional empirical quantile functionQ(p\|x) is plotted for each combination of age and sex, for year 2010 (solid lines), as well as the corresponding approximationQ appr (p\|x) based on m = 4 terms (dashed lines), using the Legendre polynomial series. Figure 7 : 7T42(F Y \|x ) Estimates of the four conditional L-functionals of location, scale, standardized skewness and standardized kurtosis for the bird migration data, for each combination x of year and the binary covariates age and sex. These conditional L-functionals are estimated from a linear quantile regression model, cf. 4 : 4The Mahalanobis distance M I (x 1 , x 2 ) between different pairs of subpopulations for the year 2000. These distances are computed for different collections I of conditional L-moments. In accordance with Figure 7 the subpopulations mainly differ in location, although some of them have quite large values of M I for scale as well. It can be seen that skewness differs the least and kurtosis the second least between subpopulations. 5 : 5The Mahalanobis distance M I (x 1 , x 2 ), for each subpopulation, between the years 1982 and 2019. The distances are computed for different collections I of conditional L-functionals. As for Table 4 the main differences are in location. However, standardized skewness and standardized kurtosis both have values of M I slightly below 4. If we regard 4 standard deviations as significant, these values are on the borderline of demonstrating significant differences in skewness and kurtosis between 1982 and 2019. Notice also that the Mahalanobis distances of I = {2, 32, 42} are larger than 4 for all four subpopulations, indicating that there are other changes other than those in location in the arrival distribution between 1982 and 2019. Table 1 : 1Values of how large a fraction (∆ 4 ) of the quantile functions Q that is explained by the first 4 terms of a polynomial series expansion, for selected distributions F , according to Table 2 : 2Variables used in the phenological data set. We fit the model with Julian day as response, and the other variables as covariates. Of primary interest is to study how the arrival distribution varies with year, while controlling for the effects of sex and age. number set I v. Adult female v. Adult male v. Juvenile male v. Adult maleSubpopulation comparison Order Juvenile female Juvenile male Juvenile female Adult female {1} 12.41 10.46 19.27 32.44 {2} 7.14 6.21 2.97 4.36 {32} 0.03 0.03 0.33 0.53 {42} 1.71 1.77 1.13 1.44 Table Subpopulation I SubpopulationJuvenile female Juvenile male Adult female Adult male{1} 23.66 19.46 29.81 27.11 {2} 2.41 2.05 2.80 2.46 {32} 3.20 3.01 4.13 3.57 {42} 3.27 2.86 3.73 3.17 Table Asymptotic properties of ideal linear estimators. C A Bennett, University of MichiganPh.D. dissertationBennett, C.A. (1952). Asymptotic properties of ideal linear estimators. Ph.D. dissertation, University of Michigan. Analysis of survival data by the proportional odds model. S Bennett, Statistics in Medicine. 2Bennett, S. (1983). Analysis of survival data by the proportional odds model. Statistics in Medicine 2, 273-277. On some robust estimators of location. P J Bickel, Annals of Mathematical Statistics. 36Bickel, P.J. (1965). On some robust estimators of location. 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[ "BOUNDARY REGULARITY FOR MINIMAL GRAPHS AND MEAN CURVATURE FLOWS WITH HIGHER CODIMENSION", "BOUNDARY REGULARITY FOR MINIMAL GRAPHS AND MEAN CURVATURE FLOWS WITH HIGHER CODIMENSION" ]	[ "Q I Ding ", "J Jost ", "Y L Xin " ]	[]	[]	In this paper, we derive global bounds for the Hölder norm of the gradient of minimal graphs, and as well as solutions of graphic mean curvature flows with arbitrary codimension. In particular, the minimal graphs obtained in [2, 15] satisfy a global C 1,γ -estimate for any γ ∈ (0, 1).	null	[ "https://arxiv.org/pdf/1706.01412v3.pdf" ]	119,634,097	1706.01412	61f701646bfa4b42d885f6a2130ce44b6180db46	BOUNDARY REGULARITY FOR MINIMAL GRAPHS AND MEAN CURVATURE FLOWS WITH HIGHER CODIMENSION 5 Jun 2017 Q I Ding J Jost Y L Xin BOUNDARY REGULARITY FOR MINIMAL GRAPHS AND MEAN CURVATURE FLOWS WITH HIGHER CODIMENSION 5 Jun 2017 In this paper, we derive global bounds for the Hölder norm of the gradient of minimal graphs, and as well as solutions of graphic mean curvature flows with arbitrary codimension. In particular, the minimal graphs obtained in [2, 15] satisfy a global C 1,γ -estimate for any γ ∈ (0, 1). Introduction Minimal graphs u = (u 1 , · · · , u m ) in R n+m over some domain Ω ⊂ R n satisfy a system of m quasilinear elliptic equations where m is the codimension. More precisely, we have (1.1) g i j ∂ i j u α = 0 in Ω, where (g i j ) is the inverse matrix of g i j = δ i j + α ∂ i u α ∂ j u α . One of the classical problems in the field is the Dirichlet problem, that is, to find solutions with (1.2) u α = ψ α on ∂Ω for some given ψ. As it turns out, in order to obtain the existence and regularity of solutions, some conditions on the geometry of the boundary of Ω and on the boundary data are needed. For m = 1, the problem is quite well understood, in particular thanks to the classical paper [5] of Jenkins and Serrin. For higher codimension, that is, for m > 1, the situation is more difficult and less well studied. A counterexample due to Lawson and Osserman [9] tells us that the situation is fundamentally different from the case m = 1. Important progress was made by M.-T. Wang [15] with an existence result for boundary values ψ that are close to 0 in some suitable norm. A crucial C 1,γ -estimate in that context was only provided later by Thorpe [12]. Thorpe shows (Lemma 5.2 in [12] which is formulated for maximal spacelike graphs in Minkowski space, but also works for minimal graphs in Euclidean space) that for C 3 -boundary data on a bounded smooth domain, a solution with small C 1 -norm satisfies a C 1,γ -estimate. Here, see Theorem 2.3 and Corollary 2.4, we can weaken the condition on the derivatives of u and need only the C 2 -norm of the boundary data. The proof relies on a blow-up argument that would lead to a contradiction with Allard's regularity theorem for varifolds if we had a sequence of solutions with unbounded Hölder norms for their derivatives. The same technique can also be applied to the interior curvature estimates of the mean curvature flow with Huisken's monotonicity formula, then we obtain a corresponding global C 1,γ -estimate in Theorem 3.3 with parabolic method in [10]. Equipped with these estimates, we can then also derive existence theorems for the Dirichlet problem for minimal graphs as in [2], see Theorem 4.2, and for the mean curvature flow, see Theorem 4.1, on mean convex domains for boundary data that are sufficiently small or do not deviate too much from codimension 1 data. 2. A priori C 1,γ -boundary estimate for minimal graphs Let R n be the standard n-dimensional Euclidean space, and R n + be a half space defined by {(x 1 , · · · , x n ) ∈ R n \| x n > 0}. Let B r (y) denote the ball in R n with radius r > 0 and centered at y ∈ R n . For any domain Ω ⊂ R n and any vector-valued function f = ( f 1 , · · · , f m ) ∈ C 1 (Ω, R m ), we define sup Ω 2 d f = sup x∈Ω 2 d f (x) = sup x∈Ω,1≤i< j≤n µ i (x)µ j (x), where {µ k (x)} n k=1 are the singular values of d f (x). Lemma 2.1. Let l α be an affine linear function in R n−1 for α = 1, · · · , m. Assume that u = (u 1 , · · · , u m ) ∈ C 1,γ (R n + , R m ) is a smooth solution of the minimal surface system (2.1)        g i j ∂ i j u α = 0 in R n + u α = l α on ∂R n + for α = 1, · · · , m, where (g i j ) is the inverse matrix of g i j = δ i j + α ∂ i u α ∂ j u α . If sup R n + 2 du < 1 and \|Du\| is uniformly bounded in R n + , then u is affine linear. Proof. By the standard Schauder theory for elliptic equations, u is smooth in R n + . For any considered point p ∈ ∂R n + , there are a constant σ > 0 and a domain V σ ⊂ R n+m containing p such that graph u ∩ V σ can be represented as a graph over R n + ∩ B σ (0) with the graphic functionû so that u = 0 on ∂R n + ∩ B σ (0). In particular,û ∈ C ∞ (R n + ∩ B σ (0), R m ). Letĝ i j = δ i j + α ∂ iû α ∂ jû α , and (ĝ −1 i j ) be the inverse matrix of (ĝ i j ). Let 0 ′ be the origin of R n−1 . Then (2.2) lim s→0ĝ i j (0 ′ ,s) = δ i j + δ in δ jn α (∂ x nû α ) 2 x=0 , and (2.3) lim s→0ĝ i j (0 ′ ,s) = δ i j − δ in δ jn α (∂ x nû α ) 2 1 + α (∂ x nû α ) 2 x=0 . Note that ∂ klû α = 0 for any 1 ≤ k ≤ n and 1 ≤ l ≤ n − 1 on ∂R n + ∩ B σ (0). By the minimal surface system (2.1), we have (2.4) ∂ 2 x n x nû α 1 + α (∂ x nû α ) 2 = 0 on ∂R n + ∩ B σ (0) for each α = 1, · · · , m. Therefore, ∂ 2 x n x nû α = 0, and then D 2ûα = 0 at the origin. Taking the derivative of the minimal surface system (2.1), it implies that D kûα = 0 at the origin for any k ≥ 3. By the definition ofû, we have D k u α = 0 on ∂R n + for any k ≥ 2. We extend u to R n − by setting u α (x ′ , −x n ) = −u α (x ′ , x n ) + 2l α (x ′ ) for x = (x ′ , x n ) ∈ R n + . Then u ∈ C ∞ (R n , R m ) by D k u α = 0 on ∂R n + for any k ≥ 2. In particular, u is a smooth solution of the minimal surface system in R n . Denote M = graph u {(x, u(x)) ∈ R n+m \| x ∈ R n }. Let M * be a tangent cone of M at infinity, which is a minimal cone. So there is a Lipschitz homogeneous function u * with sup R n + 2 du * < 1 and uniformly bounded gradient such that M * = graph u * = {(x, u * (x))\| x ∈ R n }. Moreover, u α * (x ′ , −x n ) = −u α * (x ′ , x n ) + 2l α (x ′ ) and u α * (x ′ , 0) = l α (x ′ ) . By Theorem 4.1 in [15], u * is smooth in R n \ R n−1 . Note that Du * (x ′ , −x n ) = Du * (x ′ , x n ), then the tangent cone of M * at each point in R n−1 is Euclidean. In other words, every point of M * is regular, which implies that u * is smooth in R n by Allard's regularity theorem (see [11] for instance). From Theorem A of [14], M * is an affine plane. Let B r (x) denote the ball in R n+m with radius r > 0 and centered at y ∈ R n . By the monotonicity of r −n Vol(M * ∩B r (0)) on r > 0, M is an affine plane. We complete the proof. Let Ω be a bounded domain in R n with C 2 -boundary, and let κ Ω be the maximal principal curvature of ∂Ω. Let us recall the local W 2,p -estimates for elliptic differential equations (see Theorem 9.4.1 and Theorem 11.3.2 in [8] for instance). Lemma 2.2. Let L = a i j D x i x j + b i D x i + c. Assume L is uniformly elliptic with λ · I n ≤ (a i j ) ≤ Λ · I n for some constants Λ > λ > 0, and a i j ∈ C 0 (Ω), b i , c ∈ L ∞ (Ω), f ∈ L p (Ω) for some 1 < p < ∞. Then there is a unique solution w ∈ W 2,p (Ω) to Lw = f a.e. in Ω. Moreover, there is a constant c 0 > 0 depending only on n, p, Λ/λ, R, κ Ω and the modulus of continuity of a i j such that for any x ∈ ∂Ω (2.5) \|\|w\|\| W 2,p (Ω∩B R (x)) ≤ c 0 \|\|w\|\| L p (Ω∩B 2R (x)) + \|\| f \|\| L p (Ω∩B 2R (x)) . Now we derive a priori C 1,γ -boundary estimates for minimal graphs with arbitrary codimension. Theorem 2.3. Let Ω be a bounded domain in R n with C 2 -boundary. Let u = (u 1 , · · · , u m ) ∈ C 1,γ (Ω, R m ) be a smooth solution of the minimal surface system (2.6)        g i j ∂ i j u α = 0 in Ω u α = ψ α on ∂Ω for α = 1, · · · , m, with g i j = δ i j + α ∂ i u α ∂ j u α . If sup Ω 2 du < 1 − ǫ for some ǫ ∈ (0, 1), then for any γ ∈ (0, 1), \|u\| C 1,γ (Ω) is bounded by a constant depending only on n, ǫ, \|u\| C 1 (Ω) , \|ψ\| C 2 (Ω) and κ Ω . Proof. Let us prove it by contradiction. Assume there are a sequence of domains Ω k with lim sup k κ Ω k < ∞ and a sequence of solutions u k ∈ C 1,γ (Ω k , R m ) to (2.6) with boundary data ψ k satisfying lim sup k \|ψ k \| C 2 (Ω k ) < ∞ so that sup Ω k \|Du k \| ≤ c, sup Ω k 2 du k < 1 − ǫ for some c > 0, ǫ ∈ (0, 1), and \|u k \| C 1,γ (Ω k ) → ∞ as k → ∞. Thus, the he Hölder norm of Du k on Ω: [Du k ] γ,Ω k = λ γ k for some sequence of numbers λ k converging to ∞. There are points z k ∈ Ω k such that [Du k ] γ,Ω k (z k ) ≥ 1 − k −1 λ γ k . Set u k (x) = λ k u k x λ k + z k − u k (z k ) , ψ k (x) = λ k ψ k x λ k + z k − ψ k (z k ) , and Ω k = λ k (Ω k − z k ). For any ǫ > 0, there are points y k ∈ Ω k such that \|Du k (y k ) − Du k (z k )\| ≥ 1 − (1 + ǫ)k −1 γ λ γ k \|y k − z k \| γ . So we have (2.7) \|D u k (λ k (y k − z k )) − D u k (0)\| = \|Du k (y k ) − Du k (z k )\| ≥ 1 − (1 + ǫ)k −1 γ λ γ k \|y k − z k \| γ = 1 − (1 + ǫ)k −1 λ k (y k − z k ) γ . For any ξ k , η k ∈ Ω k , (2.8) \|D u k (ξ k ) − D u k (η k )\| = Du k ξ k λ k + z k − Du k η k λ k + z k ≤λ γ k ξ k λ k − η k λ k γ = \|ξ k − η k \| γ . Hence we have [D u k ] γ, Ω k (0) ≥ 1 − k −1 , and [D u k ] γ, Ω k (x) ≤ 1 for each x ∈ Ω k . In particular, u k satisfies the minimal surface system with u k = ψ k on ∂ Ω k . It is clear that Ω k converges to a domain Ω ∞ which is R n or R n θ,τ {x ∈ R n \| x, θ < τ} for some (θ, τ) ∈ S n−1 × R ⊂ R n × R. Here, R n θ,τ is a half space perpendicular to the θ direction. De- note M k = graph u k = {(x, u k (x))\| x ∈ Ω k }. By the compactness of varifolds, there is a subsequence M i k of M k converging to a stationary varifold M ∞ in the sense of Radon measures, which can be represented as a graph over Ω ∞ with the Lipschitz graphic function u ∞ such that sup Ω ∞ \|Du ∞ \| ≤ c, [Du ∞ ] γ,Ω ∞ ≤ 1, sup Ω ∞ 2 du ∞ ≤ 1 − ǫ and u ∞ is linear on ∂Ω ∞ . From Theorem 4.1 of M.T. Wang [15], u ∞ is smooth in Ω ∞ . If Ω ∞ = R n θ,τ for some (θ, τ) ∈ S n−1 × R. , then u ∞ is a linear vector-valued function according to Lemma 2.1, and u i k converges to u ∞ in C 1 -norm. By the proof of Lemma 2.1, u ∞ is also a linear vector-valued function provided Ω ∞ = R n . Let us deduce the contradiction for the case of Ω ∞ = R n θ,τ first. For any R ≥ 4 max{c, τ}, sup Ω k \|D u k \| ≤ sup Ω k \|Du k \| ≤ c implies [D u k ] γ, Ω k ∩B R (0) (0) = [D u k ] γ, Ω k (0) ≥ 1 − k −1 . Since [D u k ] γ, Ω k ≤ 1, \| ψ k \| C 2 Ω k are uniformly bounded and the maximal principal curvature κ Ω k → 0, by Lemma 2.2 and the uniqueness theorem (see Theorem 8.1 in [4] for instance), u k ∈ W 2,p ( Ω k ) for p = 2n 1−γ , and \| u k \| W 2,p ( Ω k ∩B 2R (0)) is bounded independent of k from (2.5). Then the Sobolev imbedding theorem implies that there is a constant 0 < ǫ γ,R < 1 such that [ u k ] 1+γ 2 , Ω k ∩B R (0) ≤ 1/ǫ γ,R . Choosing ǫ γ,R sufficiently small if necessary, then there is a point ξ k ∈ Ω k ∩ B R (0) \ B ǫ γ,R (0) so that (2.9) \|D u k (ξ k ) − D u k (0)\| ≥ 1 − k −1 \|ξ k \| γ . However, (2.9) contradicts that u i k converges to a linear function in the C 1 -norm. Hence Ω ∞ R n θ,τ . For the case of Ω ∞ = R n , we can also get the contradiction from the above argument. This suffices to complete the proof. By Theorem 2.3, the solution u in Theorem 5.3 of [2] is C 1,γ for each γ ∈ (0, 1). For any vector-valued function f ∈ C 2 (Ω, R m ), set v f = det δ i j + α ∂ i f α ∂ j f α . Corollary 2.4. Let Ω be a bounded domain in R n with C 2 -boundary, and ψ ∈ C 2 (Ω, R m ). Let u = (u 1 , · · · , u m ) ∈ C 1,γ (Ω, R m ) be a smooth solution of the minimal surface system in Ω with u = ψ on ∂Ω. If sup Ω v u < 3, then for any γ ∈ (0, 1), \|u\| C 1,γ (Ω) is bounded by a constant depending only on n, ǫ, \|u\| C 1 (Ω) , \|ψ\| C 2 (Ω) and κ Ω . Proof. With the Bernstein theorem in higher codimension (see [6] [7]), it follows that any smooth solution u to the minimal surface system in R n + with linear boundary data and sup R n + v u < 3 must be linear. Following the proof of Theorem 2.3 step by step, we complete the proof. 3. A priori C 1,γ -boundary estimate for mean curvature flow For a point x = (x, t) ∈ R n × R = R n+1 , we set \|x\| = max{\|x\|, \|t\| 1/2 } and the cylinder Q R (x) = y = (y, s) ∈ R n+1 \| \|x − y\| < R, s < t . For a domain V ⊂ R n+1 , we define the parabolic boundary PV to be the set of all points x ∈ ∂V such that for any ǫ > 0, the cylinder Q ǫ (x) contains points not in V. Let Ω be a bounded domain in R n , and V Ω,T = Ω × (0, T ). Then PV Ω,T = Ω × {0} ∪ ∂Ω × [0, T ). For any set V ′ ⊂ R n+1 , γ 1 ∈ (0, 1], and a (vector-valued) function f defined on V ′ , we set [ f ] γ 1 ;V ′ (x) = sup y∈V ′ \{x} \| f (y) − f (x)\| \|y − x\| γ 1 on V ′ , and [ f ] γ 1 ;Ω = sup x∈V ′ [ f ] γ 1 ;Ω (x). For γ 2 ∈ (0, 2] and x = (x, t) ∈ V ′ , put f γ 2 ;V ′ (x) = sup (x,s)∈V ′ \{x} \| f (x, s) − f (x)\| \|s − t\| γ 2 /2 on V ′ , and f γ 2 ;Ω = sup x∈V ′ f γ 2 ;Ω (x). Now for any a > 0, we write a = k + γ with a nonnegative integer k and γ ∈ (0, 1). Let D denote the spatial and ∂ t the time derivative. Set (3.1) \| f \| a;V ′ = i+2 j≤k \|D i ∂ j t f \| V ′ + i+2 j=k [D i ∂ j t f ] γ;V ′ + i+2 j=k−1 D i ∂ j t f γ+1;V ′ . We say f ∈ H a (V ′ ) if \| f \| a;V ′ < ∞. Let B R denote the ball in R n+m centered at the origin with radius R > 0. Let us define a parabolic operator for f = ( f 1 , · · · , f m ) ∈ H 2 (V Ω,T ) by (3.2) L f α = ∂ f α dt − g i j D i j f α for α = 1, · · · , m, where (g i j ) is the inverse matrix of g i j = δ i j + α D i f α D j f α . We say L f = 0 if L f α = 0 for each α. By Proposition 2.2 in [15], L f = 0 implies that graph f (·,t) moves by mean curvature flow. Theorem 3.1. For R > 0, let f = ( f 1 , · · · , f m ) satisfy L f = 0 in Q R with f (0) = 0, where 0 is the origin of R n × R. If sup 2 d f < 1 − ǫ for some ǫ ∈ (0, 1), then there is a constant c = c(n, m, ǫ, \|D f \| 0 ) depending only on n, m, ǫ, \|D f \| 0 such that at the origin 0 (3.3) \|D 2 f \| ≤ cR −1 . Proof. By scaling, we only need to prove this Theorem with R = 1. Put Q = Q 1 and d Q (x) = inf y∈PQ \|x − y\|. Let us prove it by contradiction. Let f i be a sequence of smooth solutions of the mean curvature flow in Q 1 with f i (0) = 0 ∈ R m , sup 2 d f i ≤ 1 − ǫ and lim sup i \|D f i \| 0 < ∞ such that (3.4) lim i→∞       sup x∈Q d Q (x)\|D 2 f i (x)\|       = ∞. Denote R i = sup x∈Q d Q (x)\|D 2 f i (x)\|. There are points x i = (x i , t i ) ∈ Q such that R i = d Q (x i )\|D 2 f i (x i )\|. Set (3.5) f i (x, t) = 1 d Q (x i ) f i d Q (x i )x + x i , d 2 Q (x i )t + t i , then f i still satisfies L f i = 0 with R i = \|D 2 f i (0)\|. Moreover, sup 2 d f i ≤ 1−ǫ, lim sup i \|D f i \| 0 < ∞ and (3.6) R i = sup Q d Q (x i ) (x i ) d Q (x) d Q (x i ) \|D 2 f i \| d −1 Q (x i )(x−x i ),d −2 Q (x i )(t−t i ) = sup y∈Q d Q (y)\|D 2 f i (y)\|. Put M i t = graph f i (·,t) . Since M i t is a Lipschitz graph with uniform Lipschitz constant, then (3.7) M i t ∩B 1 e \|X\| 2 4t is uniformly bounded independent of i, t ∈ [−1, 0). For any sequence t j ∈ (0, 1] with t j → 0, there are sequences l i, j → ∞ as i → ∞ such that {l i, j } i is a subsequence of {l i, j−1 } i for each j ≥ 2, and the limit exists (and is not equal ∞) as j → ∞. By Huisken's monotonicity formula (see formula (1.2) in [1] for example), (3.10) −t k −t j (−t) n 2       M l i, j t ∩B 1/2 H M l i, j t − X 2t 2 e \|X\| 2 4t       dt ≤ t n 2 j M l i, j −t j ∩B 1/2 e − \|X\| 2 4t j − t n 2 k M l i, j −t k ∩B 1/2 e − \|X\| 2 4t k + c n −t k −t j (−t) n 2 M l i, j t ∩B 1 e \|X\| 2 4t , where c n is a constant depending only on n. Note that M i t is a Lipschitz graph with uniform Lipschitz constant. Then we infer (3.11) lim j→∞ 0 −t j (−t) n 2       lim i→∞ M l i, j t ∩B 1/2 H M l i, j t − X 2t 2 e \|X\| 2 4t       dt = 0. There is a sequence l j with l j ∈ {l i, j } i , such that we have and lim j→∞ R l j t j = ∞. Set (3.13) f i (x, t) = R l i f l i        x R l i , t R 2 l i        , and Σ i t = graph f i (·,t) . Then Σ i t is a sequence of mean curvature flow in B R l i (0)×R m with t ∈ [−R 2 l i , 0], sup 2 d f i ≤ 1 − ǫ and lim sup i \|D f i \| 0 < ∞ such that (3.14) R l i = sup x∈Q R l i d Q R l i (x)\|D 2 f i (x)\|. In particular, \|D 2 f i (x)\| ≤ 1 2 on Q R l i /2 . Hence from (3.12) we have (3.15) lim j→∞ 0 −R 2 l j t j (−t) n 2          Σ l j t ∩B R l j /2 R l j H Σ l j t − R l j X 2t 2 e − \|X\| 2 4t          dt = 0. Since f i satisfies L f i = 0, then \|∂ t f i (x)\| ≤ 1 2 n by \|D 2 f i (x)\| ≤ 1 2 on Q R l i /2 . By the Arzela-Ascoli Theorem, we can assume thatf i converges to f ∞ on any bounded domain K ⊂ Q R i /2 . Furthermore, sup 2 d f ∞ ≤ 1 − ǫ, \|D f ∞ \| 0 < ∞ and D f ∞ are (global) Lipschitz. Denote Σ ∞ t = graph f ∞ (·,t) . By the Fatou Lemma, we conclude (3.16) 0 −1 (−t) n 2       Σ ∞ t ∩B R H Σ ∞ − X 2t 2 e − \|X\| 2 4t       dt = 0 for any R > 0. Hence Σ ∞ t are self-shrinkers for each t < 0. Therefore, they are smooth by Allard's regularity theorem. From [3], Σ ∞ t is an n-plane for each t. Hence Σ i t converges to Σ ∞ t locally smoothly (see [16] for instance), but this contradicts \|D 2 f i (0)\| = 1. This suffices to complete the proof. Lemma 3.2. For R ∈ (0, 1), let f = ( f 1 , · · · , f m ) satisfy L f = 0 in Q 1 with f (0) = 0. If sup 2 d f < 1 − ǫ for some ǫ ∈ (0, 1), then there is a constant c = c(n, m, ǫ, \|D f \| 0 ) depending only on n, m, ǫ, \|D f \| 0 such that for any ξ ∈ R n × R m and ι ∈ R m (3.17) sup Q R/2 \|D f − ξ\| ≤ c       R −1 sup x∈Q R \| f (x) − ξ · x − ι\| + R       . Proof. By considering sup x∈Q d 2 Q (x)\|D 3 f i (x)\| instead of sup x∈Q d Q (x)\|D 2 f i (x)\| in (3.4), it is not hard to get that sup Q 3R/4 \|D 3 f \| ≤ cR −2 , where c = c(n, m, ǫ, \|D f \| 0 ) is a general constant depending only on n, m, ǫ, \|D f \| 0 . By L f = 0, it follows that sup Q 3R/4 \|D∂ t f \| ≤ c. Set g(x) = f (x) − ξ · x − ι for each ξ ∈ R n × R m and ι ∈ R m , then Dg = D f − ξ. With an interpolation inequality (see the proof of Lemma 4.1 of [10] for instance), for any x ∈ Q R/2 we have (3.18) \|Dg(x)\| ≤ c        R −1 sup y∈Q R/2 (x) \|g\| + R        , which suffices to complete the proof. Denote B r = B r (0) ⊂ R n for short, and B + r = B r ∩ R n + . Theorem 3.3. For any bounded domain Ω with C 2 -boundary and ψ = (ψ 1 , · · · , ψ m ) ∈ H 2 V Ω,T , let f = ( f 1 , · · · , f m ) ∈ H 2 V Ω,T be a smooth solution of L f = 0 in V Ω,T with f = ψ in PV Ω,T and Lψ = 0 on ∂Ω × {0}. If sup V Ω,T 2 d f < 1 − ǫ for some ǫ ∈ (0, 1), then there are constants γ ′ ∈ (0, 1 2 ] and C depending only on n, m, ǫ, \|D f \| V Ω,T , \|ψ\| 2;V Ω,T and κ Ω such that \| f \| 1+γ ′ ;V Ω,T ≤ C. Proof. We shall first derive Hölder estimates for D f on ∂Ω × (0, T ) by following the idea of the proof of Theorem 12.5 in [10]. From Appendix I, letf be a solution of (5.6) in B + 1 × (−1, 0) witĥ f = 0 in (∂B + 1 ∩ {x n = 0}) × (−1, 0]. Denote Q + r = Q r ∩ {x n > 0} for each r ∈ (0, 1]. For any x = (x, t) ∈ Q + r , put x ′ = (x, 0), ζ = Df α (x ′ ) and ζ n = D x nf α (x ′ ) . Then ζ, y = ζ n y n for any y = (y 1 , · · · , y n ) ∈ R n . From Lemma 7.32 in [10], there are a constant γ ′ ∈ (0, 1 2 ] and a general constant C depending only on n, m, ǫ, \|D f \| V Ω,T , \|ψ\| 2;V Ω,T and κ Ω such that (3.19) sup y∈Q r ζ n −f α (y) y n ≤ Cr γ ′ , which implies (3.20) sup y∈Q xn (x) f α (y) − ζ, y ≤ C x n r γ ′ . Let F be the mapping in Appendix I, and F(y) = (F(y), t y ) for each y = (y, t y ). From Lemma 3.2, it follows that (3.21) D f α F −1 (x) − Dψ α F −1 (x) − (DF) T F −1 (x) ζ ≤ C δx n sup y∈Q δxn (F −1 (x)) f α (y) − Dψ α F −1 (x) , y − ζ, DF F −1 (x) y − ζ, x . Here, δ is a positive constant ≤ 1 to be defined later. The bound of \|ψ\| 2;V Ω,T implies (3.22) ψ α (y) − Dψ α F −1 (x) , y ≤ Cδ 2 x 2 n for y ∈ Q δx n (F −1 (x)). With the definition of F, (3.23) ζ, F(y) − ζ, DF F −1 (x) y − ζ, x ≤ Cδ 2 x 2 n for y ∈ Q δx n (F −1 (x)). Combining the definition off in Appendix I and (3.22)(3.23), we conclude that (3.24) (DF) T F −1 (x) Df α x − (DF) T F −1 (x) ζ ≤ C δx n sup z∈F(Q δxn (F −1 (x))) f α (z) − ζ, z + Cδx n . We choose a suitable δ > 0 depending on κ Ω and get (3.25) \|Df α (x) − ζ\| ≤ C x n sup y∈Q xn (x) f α (y) − ζ, y + C x n . Combining the two inequalities (3.20) and (3.25) and r ∈ (0, 1] yields (3.26) Df α (x) − Df α (x ′ ) ≤ Cr γ ′ . From Lemma 7.32 in [10] again, (3.27) Df α (0) − Df α (x ′ ) ≤ Cr γ ′ . With (3.26) and (3.27), it follows that (3.28) Df α (x) − Df α (0) ≤ Cr γ ′ . Hence we complete the estimate of the C 1,γ ′ -norm of f on ∂Ω × (0, T ). For the Hölder estimates of D f on Ω × {0} and ∂Ω × {0}, see Theorem 12.7 and Theorem 12.9 in [10]. Combining the interior estimates of f (Theorem 3.1 and Lemma 3.2), we complete the proof. Existence theorems Let Ω be a bounded domain in R n with ∂Ω ∈ C 2 , and ϕ = (ϕ 1 , · · · , ϕ m ) be a vector-valued function in C ∞ (Ω, R n ). For any δ > 0, we choose a smooth function ξ δ (t) on R with compact support in (−δ, δ) such that ξ δ (0) = 0, ξ ′ δ (0) = 1 and −δ ≤ ξ ′ δ ≤ 1. Set a ϕ i j = δ i j + α ϕ α i ϕ α j and (a i j ϕ ) be the inverse matrix of (a ϕ i j ). Put ϕ α δ (x, t) = ϕ α (x) + ξ δ (t) i, j a i j ϕ ∂ i j ϕ α for each α = 1, · · · , m and (x, t) ∈ Ω × R. Denote V Ω = Ω × [0, ∞) . Then for any sufficiently small ǫ > 0 there is such a function ξ δ depending on ǫ, \|ϕ\| 4,Ω so that Lϕ δ = 0 on ∂Ω × {0}, (4.1) Lϕ α δ = ∂ ∂t ϕ α δ − i, j a i j ϕ δ ∂ i j ϕ α δ ≤ (1 + ǫ)n\|D 2 ϕ α \| in V Ω and \|ϕ δ \| 2;V Ω ≤ 2n\|ϕ\| 2,Ω , \|ϕ δ \| a;V Ω ≤ (1 + ǫ)\|ϕ\| a,Ω for any 1 ≤ a ≤ 3 2 . Let Ω be a mean convex domain with smooth boundary and (1 + 2ǫ)ϕ be a smooth function satisfying the inequality (1.3) in Theorem 1.1 of [2]. Let us consider the flow (4.2)          ∂ f α dt =g i j D i j f α in V Ω,T f α =ϕ α δ on PV Ω,T for α = 1, · · · , m, where (g i j ) is the inverse matrix of g i j = δ i j + α f α i f α j . Theorem 4.1. For sufficiently small δ > 0, there is a vector-valued function f = ( f 1 , · · · , f m ) ∈ H 2 (V Ω ) satisfying L f = 0 in V Ω with f = ψ in PV Ω . Moreover, there are constants C Ω > 0 and ǫ Ω ∈ (0, 1) depending only on n, m, κ Ω , \|ϕ\| 2,Ω as in Theorem 1.1 of [2] such that sup V Ω \|D f \| ≤ C Ω and sup V Ω 2 d f < 1 − ǫ. Proof. Let T be the maximal time of existence of the C 2 -solution to L f = 0 in V Ω,T such that sup V Ω,T \|D f \| ≤ C Ω and sup V Ω,T 2 d f < 1 − ǫ, where C Ω > 0 and ǫ Ω ∈ (0, 1) are constants depending only on n, m, κ Ω , \|ϕ\| 2,Ω as in Theorem 1.1 of [2]. It is clear that T > 0 by the shorttime existence of the flow (see Theorem 8.2 in [10]) and Schauder theory for linear parabolic equations. From Theorem 3.3, there are constants γ ′ , C ′ Ω depending only on n, m, κ Ω , \|ϕ\| 2,Ω such that \| f \| 1+γ ′ ;V Ω,T ≤ C ′ Ω . Hence for any t ∈ (0, min{δ, T }] sup Ω (\|D f (x, t)\| − \|Dϕ(x)\|) ≤ \| f \| 1+γ ′ ;V Ω,T t γ ′ /2 ≤ δ γ ′ /2 C ′ Ω . Combining (4.1) and \|ϕ δ \| 2;V Ω ≤ 2n\|ϕ\| 2,Ω , we choose a sufficiently small δ > 0 and fix it independently of T , and then get (4.3) ∂ ∂t ϕ α δ − i, j g i j ∂ i j ϕ α δ ≤ (1 + 2ǫ)n\|D 2 ϕ α \| in V Ω,min{δ,T } , where (g i j ) is the inverse matrix of g i j = δ i j + α f α i f α j . For t ∈ (min{δ, T }, T ), we have (4.4) ∂ ∂t ϕ α δ − i, j g i j ∂ i j ϕ α δ = i, j g i j ∂ i j ϕ α ≤ n\|D 2 ϕ α \|. Moreover, \|ϕ δ \| 1;V Ω ≤ (1 + ǫ)\|ϕ\| 1,Ω on V Ω,T . By the assumption on ϕ and the proof of Theorem 1.1 of [2], we conclude that T = ∞, and thus complete the proof. In the above Theorem, the solution f satisfies (4.5) \| f \| 1+γ ′ ;V Ω ≤ C ′ Ω . Theorem 4.2. For any mean convex bounded C 2 domain Ω, let ψ ∈ C 2 (Ω, R m ) be the vectorvalued function given in Theorem 1.1 of [2]. Then for any γ ∈ (0, 1) there is a solution u = (u 1 , · · · , u m ) ∈ C ∞ (Ω, R m ) ∩ C 1,γ (Ω, R m ) of the minimal surface system (4.6)        g i j u α i j = 0 in Ω u α = ψ α on ∂Ω for α = 1, · · · , m, with g i j = δ i j + α u α i u α j . Proof. There is a sequence of smooth mean convex domains Ω k converging to Ω such that ∂Ω k converges to ∂Ω in C 2 -norm. For example, these domains can be constructed by the level-set flow. Let ψ k = (ψ 1 k , · · · , ψ m k ) be a sequence of smooth vector-valued functions on Ω k converging to ψ in the C 2 -norm such that (1 + 2ǫ k )ψ k satisfies the inequality (1.3) in Theorem 1.1 of [2] for the domain Ω i instead of Ω. By Theorem 4.1, there is a smooth function f k = ( f 1 k , · · · , f m k ) solving (4.2) in V Ω i such that sup V Ω i \|D f k \| ≤ C Ω , \| f k \| 1+γ ′ ;V Ω i ≤ C ′ Ω , and sup V Ω i 2 d f k < 1 − ǫ. Here, C Ω , C ′ Ω and ǫ Ω ∈ (0, 1) are constants depending only on n, m, κ Ω , \|ϕ\| 2,Ω . Note that f k (·, t) converges to the solution u k of the minimal surface system (4.6) in Ω k . Then sup Ω i \|Du k \| ≤ C Ω , \|u k \| 1+γ ′ ,Ω i ≤ C ′ Ω , and sup Ω i 2 du k ≤ 1 − ǫ. By the compactness theorem of stationary varifolds or the Arzela-Ascoli Theorem, there is a subsequence of u k converging to u * , which is a solution of the minimal surface system (4.6) in Ω with sup Ω \|Du * \| ≤ C Ω , \|u * \| 1+γ ′ ,Ω ≤ C ′ Ω , and sup Ω 2 du * ≤ 1 − ǫ. By Lemma 2.2 and the uniqueness theorem (see Theorem 8.1 in [4] for instance), u * ∈ W 2,p (Ω) for p = n 1−γ and each γ ∈ (0, 1). Then the Sobolev imbedding theorem implies u * = (u 1 * , · · · , u m * ) ∈ C 1,γ (Ω, R m ). As a corollary, it is not hard to verify that the solution u in Theorem 1.1 in [15] is C 1,γ for each γ ∈ (0, 1). Corollary 4.3. Let Ω be a convex bounded C 2 -domain in R n , and ψ ∈ C 2 (Ω, R m ) and let β 0 be the constant in Theorem 1.2 in [2]. Then for any γ ∈ (0, 1) there is a solution u = (u 1 , · · · , u m ) ∈ C ∞ (Ω, R m ) ∩C 1,γ (Ω, R m ) of the minimal surface system (4.6) with boundary ψ such that sup Ω v u < β 0 . Appendix I For studying the boundary regularity of parabolic systems, we usually only need to consider a similar system on a portion of a half space by a coordinate transformation. Let B r be a ball with radius r and centered at the origin in R n . Let Ω be a domain in R n with C 2 -boundary. We assume that there is a coordinate change F : B 1 → F(B 1 ) ⊂ R n such that F, F −1 are C 2 -maps with F(B 1 ∩ ∂Ω) ⊂ {y\|y m = 0} and F(B 1 ∩ Ω) ⊂ {y\|y m > 0} and such that the matrix DFDF T has eigenvalues between two constants Λ −1 F and Λ F with 2 ≥ Λ F ≥ Λ −1 F ≥ 1 2 > 0. Hence Λ F converges to 1 as DF converges to the identity matrix. Moreover, we assume that (5.1) sup B 1 \|D 2 F\| ≤ Λ F − 1. For a C 2 vector-valued function f in V Ω,T = Ω × (0, T ), we define a new functionf byf (F(x), t) = f (y, t). Then D f = DF · Df . Put (5.2) A i j (y, Df (y, t)) = δ i j + ∂ y kf α (y, t)∂ x i F k (F −1 (y)) · ∂ y lf α (y, t)∂ x j F l (F −1 (y)). Now we assume that f satisfies the flow (5.3) ∂ f ∂t − g i j ∂ 2 x i x j f = 0 in V Ω,T with f = ψ on PV Ω,T , where (g i j ) is the inverse matrix of g i j = δ i j + α f α i f α j . Then (5.4) 0 = ∂ tf − ∂ x i F k A i j ∂ x j F l ∂ 2 y k y lf − A i j ∂ 2 x i x j F k ∂ y kf . Setψ by ψ =ψ • F, andf =f −ψ so thatf = 0 on F(∂Ω ∩ B 1 ) × [0, T ). Put (5.5) G kl (y, Df ) =A i j (y, Df + Dψ)∂ x i F k ∂ x j F l , Θ(y, Df ) =A i j (y, Df + Dψ)∂ 2 x i x j F k ∂ y kf + ∂ y kψ + G kl (y, Df )∂ 2 y k y lψ − ∂ tψ . Thenf satisfies the parabolic system (5.6) ∂ tf = G kl (y, Df (y))∂ 2 y k y lf + Θ(y, Df (y)). Hence there is a positive constant λ f depending only on n, m, Λ F , \|D f \| 0 and \|Dψ\| 0 such that (5.7) λ −1 f I n ≤ (G kl ) ≤ Λ F I n and \|Θ\| ≤ c n Λ F \|ψ\| 2;V Ω,T in V Ω,T . Here, c n is a constant depending only on n. any j. Up to the choice of the subsequence of t j , l i, j , we assume that the    dt = 0. Generic Mean Curvature Flow I; Generic Singularities. H Tobias, William P Colding, I I Minicozzi, Ann. of Math. 1752Tobias H. Colding and William P. Minicozzi II, Generic Mean Curvature Flow I; Generic Singularities, Ann. of Math., 175 (2) (2012), 755-833. Existence and non-existence of minimal graphs. J Qi Ding, Y L Jost, Xin, arXiv:1701.01674Qi Ding, J. Jost and Y.L. Xin, Existence and non-existence of minimal graphs, arXiv:1701.01674. On the self-shrinking systems in arbitrary codimensional spaces. Qi Ding, Zhizhang Wang, arXiv:1012.0429v2Qi Ding, Zhizhang Wang, On the self-shrinking systems in arbitrary codimensional spaces, arXiv:1012.0429v2, 2010. D Gilbarg, N Trudinger, Elliptic Partial Differential Equations of Second Order. Berlin-New YorkSpringer-VerlagD. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, (1983). The Dirichlet problem for the minimal sur-face equation in higher dimension. H Jenkins, J Serrin, J. Reine Angew. Math. 229H. Jenkins and J. Serrin, The Dirichlet problem for the minimal sur-face equation in higher dimension, J. Reine Angew. Math. 229 (1968), 170-187. Bernstein type theorems for higher codimension. J Jost, Y L Xin, Calc.Var. 9J. Jost, Y. L. Xin, Bernstein type theorems for higher codimension, Calc.Var. 9 (1999), 277-296. The Gauss image of entire graphs of higher codimension and Bernstein type theorems. J Jost, Y L Xin, Ling Yang, Calc. Var. Partial Differential Equations. 473-4J. Jost, Y. L. Xin and Ling Yang, The Gauss image of entire graphs of higher codimension and Bernstein type theorems, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 711-737. N , V , Krylov , Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Providence, Rhode IslandAmerican Mathematical Society96N,V, Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, Volume 96, American Mathematical Society, Providence, Rhode Island. Non-existence, non-uniqueness and irreg-ularity of solutions to the minimal surface system. H Lawson, R Osserman, Acta Math. 139H. Lawson and R. Osserman, Non-existence, non-uniqueness and irreg-ularity of solutions to the minimal surface system, Acta Math. 139 (1977), 1-17 Second order parabolic differential equations. G M Lieberman, World ScientificRiver Edge, N.J.G. M. Lieberman, Second order parabolic differential equations, World Scientific, River Edge, N.J., 1996. Leon Simon, Lectures on Geometric Measure Theory, Proceedings of the center for mathematical analysis Australian national university. 3Leon Simon, Lectures on Geometric Measure Theory, Proceedings of the center for mathematical analysis Aus- tralian national university, Vol. 3, 1983. The maximal graph Dirichlet problem in semi-Euclidean spaces. Benjamin Stuart, Thorpe , Comm. Anal. Geom. 202Benjamin Stuart Thorpe, The maximal graph Dirichlet problem in semi-Euclidean spaces, Comm. Anal. Geom. 20 (2012), no. 2, 255-270. Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Mu-Tao Wang, Invent. Math. 1483Mu-Tao Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (2002), no. 3, 525-543. On graphic Bernstein type results in higher codimension. M.-T Wang, Trans. Amer. Math. Soc. 3551M.-T. Wang, On graphic Bernstein type results in higher codimension, Trans. Amer. Math. Soc. 355 (2003), no. 1, 265-271. The Dirichlet Problem for the minimal surface system in arbitrary dimensions and codimensions. Mu-Tao Wang, Comm. Pure Appl. Math. 572Mu-Tao Wang, The Dirichlet Problem for the minimal surface system in arbitrary dimensions and codimensions, Comm. Pure Appl. Math. 57 (2004), no. 2, 267-281. A Local Regularity Theorem for Mean Curvature Flow. B White, Ann. of Math. 161B. White, A Local Regularity Theorem for Mean Curvature Flow, Ann. of Math. 161 (2005), 1487-1519. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany E-mail address: [email protected] Institute of Mathematics. ShanghaiFudan UniversityChina E-mail address: [email protected] Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany E-mail address: [email protected] Institute of Mathematics, Fudan University, Shanghai 200433, China E-mail address: [email protected]	[]
[ "Ergodicity of principal algebraic group actions", "Ergodicity of principal algebraic group actions" ]	[ "Hanfeng Li ", "Jesse Peterson ", "Klaus Schmidt " ]	[]	[]	An algebraic action of a discrete group Γ is a homomorphism from Γ to the group of continuous automorphisms of a compact abelian group X. By duality, such an action of Γ is determined by a module M = X over the integer group ring ZΓ of Γ. The simplest examples of such modules are of the form M = ZΓ/ZΓf with f ∈ ZΓ; the corresponding algebraic action is the principal algebraic Γ-action α f defined by f .In this note we prove the following extensions of results by Hayes [2]	10.1090/conm/631/12604	[ "https://arxiv.org/pdf/1312.3098v1.pdf" ]	5,984,999	1312.3098	20b248071664f1232c336673761c3743723dae3e	Ergodicity of principal algebraic group actions 11 Dec 2013 Hanfeng Li Jesse Peterson Klaus Schmidt Ergodicity of principal algebraic group actions 11 Dec 2013Contemporary Mathematics Dedicated to Shrikrishna Gopalrao Dani on the occasion of his 65th birthday An algebraic action of a discrete group Γ is a homomorphism from Γ to the group of continuous automorphisms of a compact abelian group X. By duality, such an action of Γ is determined by a module M = X over the integer group ring ZΓ of Γ. The simplest examples of such modules are of the form M = ZΓ/ZΓf with f ∈ ZΓ; the corresponding algebraic action is the principal algebraic Γ-action α f defined by f .In this note we prove the following extensions of results by Hayes [2] Principal Algebraic Group Actions Let Γ be a countably infinite discrete group with integral group ring ZΓ. Every g ∈ ZΓ is written as a formal sum g = γ g γ ·γ, where g γ ∈ Z for every γ ∈ Γ and γ∈Γ \|g γ \| < ∞. The set supp(g) = {γ ∈ Γ : g γ = 0} is called the H.L. was partially supported by the NSF grants DMS-1001625 and DMS-126623, and he would like to thank the Erwin Schrödinger Institute, Vienna, for hospitality and support while some of this work was done, J.P. was partially supported by the NSF grant DMS-1201565 and the Alfred P. Sloan Foundation, Both J.P. and K.S. would like to thank the University of Buffalo for hospitality and support while some of this work was done. c 0000 (copyright holder) (λ γ x) γ ′ = x γ −1 γ ′ , (ρ γ x) γ ′ = x γ ′ γ ,(1.1) for every γ ∈ Γ and x = (x γ ′ ) γ ′ ∈Γ ∈ T Γ . The Γ-actions λ and ρ extend to actions of ZΓ on T Γ given by λ f = γ∈Γ f γ λ γ , ρ f = γ∈Γ f γ ρ γ (1.2) for every f = γ∈Γ f γ · γ ∈ ZΓ. The pairing f, x = e 2πi γ∈Γ fγxγ , f = γ∈Γ f γ · γ ∈ ZΓ, x = (x γ ) ∈ T Γ , identifies ZΓ with the dual group T Γ of T Γ . We claim that, under this identification, X f := ker ρ f = x ∈ T Γ : ρ f x = γ∈Γ f γ ρ γ x = 0 = (ZΓf ) ⊥ ⊂ ZΓ = T Γ . (1.3) Indeed, h, ρ f x = h, γ ′ ∈Γ f γ ′ ρ γ ′ x = e 2πi γ∈Γ hγ γ ′ ∈Γ f γ ′ x γγ ′ = e 2πi γ∈Γ γ ′ ∈Γ h γγ ′−1 f γ ′ xγ = e 2πi γ∈Γ (hf )γ xγ = hf, x for every h ∈ ZΓ and x ∈ T Γ , so that x ∈ ker ρ f if and only if x ∈ (ZΓf ) ⊥ . Since the Γ-actions λ and ρ on T Γ commute, the group X f = ker ρ f ⊂ T Γ is invariant under λ, and we denote by α f the restriction of λ to X f . In view of this we adopt the following terminology. Definition 1.1. (X f , α f ) is the principal algebraic Γ-action defined by f ∈ ZΓ. In [2] the author calls a countably infinite discrete group Γ principally ergodic if every principal algebraic Γ-action α f , f ∈ ZΓ, is ergodic w.r.t. Haar measure on X f and proves that the following classes of groups are principally ergodic: torsion-free nilpotent groups which are not virtually cyclic, 1 free groups on more than one generator, and groups which are not finitely generated. In order to state our extensions of these results we denote by ℓ ∞ (Γ, R) ⊂ R Γ the space of bounded real-valued maps v = (v γ ) on Γ, where v γ is the value of v at γ, and we write v ∞ = sup γ∈Γ \|v γ \| for the supremum norm on ℓ ∞ (Γ, R). For 1 ≤ p < ∞ we set ℓ p (Γ, R) = {v = (v γ ) ∈ ℓ ∞ (Γ, R) : v p = γ∈Γ \|v γ \| p 1/p < ∞}. By ℓ p (Γ, Z) = ℓ p (Γ, R) ∩ Z Γ we denote the additive subgroup of integer-valued elements of ℓ p (Γ, R); for 1 ≤ p < ∞, ℓ p (Γ, Z) = ℓ 1 (Γ, Z) is identified with ZΓ by viewing each g = γ g γ · γ ∈ ZΓ as the element (g γ ) γ∈Γ ∈ ℓ 1 (Γ, Z). The group Γ acts on ℓ p (Γ, R) isometrically by left and right translations: for every v ∈ ℓ p (Γ, R) and γ ∈ Γ we denote byλ γ v andρ γ v the elements of ℓ p (Γ, R) satisfying (λ γ v) γ ′ = v γ −1 γ ′ and (ρ γ v) γ ′ = v γ ′ γ , respectively, for every γ ′ ∈ Γ. Note thatλ γγ ′ =λ γλγ ′ andρ γγ ′ =ρ γργ ′ for every γ, γ ′ ∈ Γ. The Γ-actionsλ andρ extend to actions of ℓ 1 (Γ, R) on ℓ p (Γ, R) which will again be denoted byλ andρ: for h = (h γ ) ∈ ℓ 1 (Γ, R) and v ∈ ℓ p (Γ, R) we setλ h v = γ∈Γ h γλ γ v,ρ h v = γ∈Γ h γρ γ v. (1.4) These definitions correspond to the usual convolutions λ h v = h · v,ρ h v = v · h * ,(1.5) where h → h * is the involution on ℓ 1 (Γ, C) defined as for ZΓ: h * γ = h γ −1 , γ ∈ Γ, for every h = (h γ ) ∈ ℓ 1 (Γ, C). For p = 2, the bounded linear operators λ h ,ρ h : ℓ 2 (Γ, R) −→ ℓ 2 (Γ, R) in (1.4) can be viewed as elements of the right (resp. left) equivariant group von Neumann algebra of Γ. (1) Γ contains a finitely generated amenable subgroup which is not virtually cyclic, or more generally, a finitely generated subgroup with a single end, (2) Γ is not finitely generated, (3) Γ contains an infinite property T subgroup, or more generally, a nonamenable subgroup Γ 0 with vanishing first ℓ 2 -Betti number β 1 (Γ 0 ) = 0. If f ∈ ZΓ is not a right zero-divisor, then the principal Γ-action α f on X f is ergodic (with respect to the normalized Haar measure of X f ). Theorem 1.3. Let Γ be a countably infinite discrete group which is not virtually cyclic. If f ∈ ZΓ satisfies that kerρ f * = {v ∈ ℓ 2 (Γ, R) :ρ f * (v) = v · f = 0} = {0}, (1.6) then the principal Γ-action α f on X f is ergodic. In view of the hypotheses on f in the Theorems 1.2 and 1.3 it is useful to recall the following result. Proposition 1.4. Let Γ be a countably infinite discrete amenable group. For every f ∈ ZΓ the following conditions are equivalent. (1) f is a right zero-divisor in ZΓ, (1) ⇔ (2): Taking * we see that (1) holds if and only if f * is a left zerodivisor in ZΓ. Applying (3) ⇔ (4) to f * , we see that the latter condition is equivalent to (2). Remark 1.5. Linnell's analytic zero-divisor conjecture is the conjectural statement that for any torsion-free discrete group Γ and any nonzero f ∈ CΓ, (2) {v ∈ ℓ 2 (Γ, R) : f * · v = 0} = {0}, (3) f is a left zero-divisor in ZΓ, (4) {v ∈ ℓ 2 (Γ, R) : f · v = 0} = {0}, (5) kerρ f = {v ∈ ℓ 2 (Γ, R) :ρ f (v) = 0} = {0}. Proof. (4) ⇔ (5): This follows from (f ·v) * = v * ·f * for all v ∈ ℓ 2 (Γ, R).kerρ f * = {0} [6, Conjecture 1]. Linnell has shown that this conjecture holds for Γ if G 1 is a normal subgroup of Γ, G 2 is a normal subgroup of G 1 , Γ is torsion-free, G 2 is free, G 1 /G 2 is elementary amenable, and Γ/G 1 is right orderable [7, Proposition 1.4]. If a countably infinite, torsion-free, and not virtually cyclic group Γ satisfies Linnell's analytic zero-divisor conjecture, then the principal Γ-action α f on X f is ergodic for every f ∈ ZΓ by Theorem 1.3. As a corollary to the Theorems 1.2 -1.3 and Remark 1.5 we obtain the following results by Hayes. (1) Γ is an infinite, torsion-free, nilpotent group not isomorphic to the integers, (2) Γ is the free group with k ≥ 2 generators. Then the principal Γ-action (X f , α f ) is ergodic for every f ∈ ZΓ. Proof. If f = 0, then α f is the left shift-action by Γ on X f = T Γ , which is obviously ergodic. Suppose therefore that f = 0. Since Γ is either free or torsion-free nilpotent, kerρ f * = {0} by Remark 1.5, so that α f is ergodic by either Theorem 1.2 or 1.3. Whereas the proofs of these results in [2] use structure theory of Γ, the proofs in this paper employ cohomological methods. Cohomological results Let Γ be a countably infinite discrete group and M a left ZΓ-module. A map c : Γ −→ M is a 1-cocycle (or, for our purposes here, simply a cocycle) if c(γγ ′ ) = c(γ) + γc(γ ′ ) (2.1) for all γ, γ ′ ∈ Γ. A cocycle c : Γ −→ M is a coboundary (or trivial) if there exists a b ∈ M such that c(γ) = b − γb (2.2) for every γ ∈ Γ. A finitely generated group G has two ends if and only if it is infinite and virtually cyclic, i.e., if and only if it contains a finite-index subgroup G ′ ∼ = Z. Stallings' theorem ( [13]) implies that a finitely generated group G has a single end whenever it is amenable and not virtually cyclic (see [8] for a short proof). Proposition 2.1. Let Γ be a countably infinite discrete group and ∆ ⊂ Γ a finitely generated subgroup with a single end. Then every cocycle c : ∆ −→ ZΓ is a coboundary. Proof. By [3,Theorem 4.6] if ∆ has a single end, then every 1-cocycle ∆ −→ Z∆ is a coboundary. 2 It follows that for each γ ∈ Γ there is some b γ ∈ Z[∆γ] such that the restriction of c(δ) on ∆γ is equal to b γ − δb γ for all δ ∈ ∆. For each δ ∈ ∆, there is a finite set W δ of right cosets of ∆ in Γ such that the support of c(δ) is contained in ∆γ∈W δ ∆γ. If F is a finite symmetric set of generators of ∆, then for any ∆γ ∈ δ ′ ∈F W δ ′ , one has (1 − δ) · b γ = 0 for every δ ∈ F and hence for every δ ∈ ∆. Therefore c(δ) is equal to 0 on ∆γ for all ∆γ ∈ δ ′ ∈F W δ ′ and δ ∈ ∆. Set b = ∆γ∈ δ ′ ∈F W δ ′ b γ ∈ ZΓ. Then c(δ) = (1 − δ) · b for all δ ∈ ∆. Next we prove an analogous result for nonamenable groups with vanishing first ℓ 2 -Betti number, e.g., infinite property T groups [1, Corollary 6]. Proposition 2.2. Let Γ be a countably infinite discrete group and ∆ ⊂ Γ a nonamenable subgroup with β for every γ ∈ Γ. : γ → U γ of Γ. A map c : Γ −→ H is a 1-cocycle for U if c(γγ ′ ) = c(γ) + U γ c(γ ′ ) (2.3) for all γ, γ ′ ∈ Γ, The following lemma is well-known (cf. [11,Proposition 1.6]). For convenience of the reader, we give a proof here. Proof. Since U does not weakly contain the trivial representation of Γ, we can find a finite subset F ⊂ Γ and some ε > 0 such that δ∈F v − U δ v ≥ ε v for all v ∈ H. Let c be an approximate coboundary of Γ taking values in H. Let (b n ) n≥1 be a sequence in H such that the coboundaries c n (γ) = b n − U γ b n , γ ∈ Γ, approximate c in the sense of (2.5). Then Consider the affine isometric action V of Γ on H defined by V γ v = U γ v + c(γ) for all γ ∈ Γ and v ∈ H. Set Y = {c(γ ′ ) : γ ′ ∈ Γ}, and let γ ∈ Γ. Since V γ (Y ) = Y , we obtain that V γ (center(Y )) = center(Y ) and hence that U γ (center(Y )) + c(γ) = center(Y ). Thus c(γ) = center(Y ) − U γ (center(Y )) for all γ ∈ Γ, so that c is a coboundary. Proof of Proposition 2.2. By [1] in the finitely generated case, and [9, Corollary 2.4] in general, if ∆ is nonamenable and β (2) 1 (∆) = 0, then every 1-cocycle ∆ −→ ℓ 2 (∆, R) for the left regular representation is a coboundary. It follows that for each γ ∈ Γ there is some b γ ∈ ℓ 2 (∆γ, R) such that the restriction of c(δ) on ∆γ is equal to b γ − δb γ for all δ ∈ ∆. Since c(δ) has finite support for each δ ∈ ∆, we conclude that the cocycle c : ∆ −→ ℓ 2 (Γ, R) is an approximate coboundary. Because ∆ is nonamenable, its left regular representation on ℓ 2 (∆, R) does not contain the trivial representation weakly. Since the restriction of the left regular representation of Γ on ℓ 2 (Γ, R) to ∆ is a direct sum of copies of the left regular representation of ∆, it does not contain the trivial representation of ∆ weakly either. By Lemma 2.3 there exists v ∈ ℓ 2 (Γ, R) satisfying c(δ) = v −λ δ v = (1 − δ)v (2.6) for every δ ∈ ∆. Since ∆ is nonamenable, it is infinite. It follows that that v ∈ ℓ 2 (Γ, Z) = ZΓ. If a subgroup ∆ ⊂ Γ has more than one end then there exist nontrivial cocycles c : ∆ −→ Z∆ (cf. [12,5.2. Satz IV] or [14,Lemma 3.5]), which immediately implies the existence of nontrivial cocycles c : ∆ −→ ZΓ. For example, if ∆ is the free group on k ≥ 2 generators, it has nontrivial cocycles. However, Proposition 2.4 below guarantees triviality of cocycles which become trivial under right multiplication by an element f ∈ ZΓ satisfying (1.6) (cf. Remark 1.5). Proposition 2.4. Let Γ be a countably infinite discrete group, ∆ ⊂ Γ a nonamenable subgroup, and let f ∈ ZΓ satisfy that kerρ f * = {0}. If c : ∆ −→ ZΓ is a cocycle such that cf is a coboundary, then c is a coboundary. Lemma 2.5. Let Γ be a countably infinite discrete group, ∆ ⊂ Γ a nonamenable subgroup, and let f ∈ ZΓ. We writeλ ∆ for the unitary representation of ∆ obtained by restricting the left regular representationλ of Γ on (1.6)), then c is a coboundary. ℓ 2 (Γ, C) to ∆. If c : ∆ −→ ℓ 2 (Γ, C) is a cocycle forλ ∆ such that c · f =ρ f * c is a coboundary and c(∆) is contained in the orthogonal complement V of kerρ f * in ℓ 2 (Γ, C) (cf. Proof. By assumption there exists a b ∈ ℓ 2 (Γ, C) such that (1 − δ) · b = c(δ) · f for every δ ∈ ∆. Letρ f * = U H be the polar decomposition [4, Theorem 6.1.2] ofρ f * , where U is a partial isometry on ℓ 2 (Γ, C), H = ρ f f * 1/2 = (ρ fρf * ) 1/2 , and both U and H lie in the left-equivariant group von Neumann algebra N Γ. Note that ker H = kerρ f * . We write H = ρ f * 0 λ dE λ for the spectral decomposition of the positive self-adjoint operator H and consider, for each 0 < ε < ρ f * , the projection operator P ε = P − E ε , where P is the orthogonal projection ℓ 2 (Γ, C) −→ V . Then one has P ε → P in the strong operator topology as ε ց 0. Put Q ε = U P ε U * for every ε with 0 < ε < ρ f * . Then P ε (c(δ)) · f =ρ f * P ε (c(δ)) = U HP ε (c(δ)) = U P ε H(c(δ)) = Q ε U H(c(δ)) = Q ερ f * (c(δ)) = Q ε (c(δ) · f ) = Q ε ((1 − δ) · b) = (1 − δ) · Q ε (b) for every δ ∈ ∆. Since ρ f * v ≥ ε v for every v ∈ range(P ε ), there exists V ε ∈ N Γ vanishing on the orthogonal complement of range(ρ f * P ε ) and satisfying that V ερ f * v = v for every v ∈ range(P ε ). Therefore P ε (c(δ)) = V ερ f * P ε (c(δ)) = V ε Q ε ((1 − δ) · b)) = (1 − δ) · V ε Q ε (b). The 1-cocycle δ → P ε c(δ) = (1 − δ) · V ε Q ε (b) forλ ∆ is thus a coboundary. Since P ε (c(δ)) → c(δ) in ℓ 2 (Γ, C) as ε ց 0 for every δ ∈ ∆, we conclude that the 1-cocycle c : ∆ −→ ℓ 2 (Γ, C) forλ ∆ is an approximate coboundary. Since ∆ is nonamenable, the left regular representation of ∆ on ℓ 2 (∆, C) does not weakly contain the trivial representation of ∆. Thus, the representationλ ∆ of ∆ on ℓ 2 (Γ, C), as a direct sum of copies of the left regular representation of ∆, does not weakly contain the trivial representation of ∆. From Lemma 2.3 we conclude that there is some b ∈ ℓ 2 (Γ, C) satisfying c(δ) = (1 − δ)b for every δ ∈ ∆. Proof of Proposition 2.4. Suppose that f ∈ ZΓ satisfies (1.6), and that c : ∆ −→ ZΓ is a 1-cocycle such that cf is a coboundary. Then cf is also a coboundary when c is viewed as an ℓ 2 (Γ, C)-valued cocycle for the unitary representationλ ∆ on ℓ 2 (Γ, C). Lemma 2.5 shows that there exists a b ∈ ℓ 2 (Γ, C) such that c(δ) = (1 − δ) · b for every δ ∈ ∆. In order to prove that b ∈ ZΓ we set, for every ε > 0, F ε (b) = {γ ∈ Γ : \|b γ \| ≥ ε}. Then F ε is finite, and so is the set {δ ∈ ∆ : \|(δ · b) γ \| = \|b δ −1 γ \| ≥ ε} = {δ ∈ ∆ : δ −1 γ ∈ F ε } = γF −1 ε ∩ ∆ for every γ ∈ Γ. Since ∆ is nonamenable, it is infinite, and by varying ε we see that lim δ→∞ (δ · b) γ = 0 for every γ ∈ Γ. Since c(δ) γ = b γ − (δ · b) γ ∈ Z we conclude, by letting δ → ∞, that b γ ∈ Z for every γ ∈ Γ. This completes the proof of the proposition. Ergodicity of principal actions We recall the following result from [10, Lemma 1.2 and Theorem 1.6]. Theorem 3.1. If α is an algebraic action of a countably infinite discrete group Γ on a compact abelian group X with dual groupX, then α is ergodic if and only if the orbit {α γ a : γ ∈ Γ} is infinite for every nontrivial a ∈X. Corollary 3.2. Let Γ be a countably infinite discrete group, f ∈ ZΓ, and let α f be the principal algebraic Γ-action on the group X f with Haar measure µ f (cf. Definition 1.1). For a ∈ ZΓ/ZΓf = X f let S(a) = {γ ∈ Γ : γ · a = a} be its stabilizer. Then α f is ergodic with respect to µ f if and only if S(a) has infinite index in Γ for every nonzero a ∈ ZΓ/ZΓf . Proof of Theorem 1.2. Suppose that f ∈ ZΓ is not a right zerodivisor, but that α f is nonergodic. By Corollary 3.2 there exists an h ∈ ZΓ such that h / ∈ ZΓf and the Γ-orbit D = {γh + ZΓf : γ ∈ Γ} of a = h + ZΓf in ZΓ/ZΓf is finite. We denote by ∆ = {δ ∈ Γ : δh − h ∈ ZΓf } (3.1) the stabilizer of a, which has finite index in Γ by hypothesis, and consider the cocycle c : ∆ −→ ZΓ given by h − δh = c(δ)f (3.2) for every δ ∈ ∆ (here we are using that f is not a right zero-divisor). If ∆ 0 ⊂ ∆ is an infinite subgroup on which c is a coboundary then c(δ) = b−δb for some b ∈ ZΓ and every δ ∈ ∆ 0 . Hence c(δ)f = (1 − δ)bf = (1 − δ)h for every δ ∈ ∆ 0 . Since ∆ 0 is infinite, this implies that h = bf ∈ ZΓf , contrary to our choice of h. In other words, if c is a coboundary when restricted to any infinite subgroup, we run into a contradiction with our assumption that α f is nonergodic. Proof of (1). If Γ 0 ⊂ Γ is a finitely generated subgroup with a single end, then the same is true for its finite-index subgroup ∆ ∩ Γ 0 where ∆ is from (3.1). Proposition 2.1 shows that c is a coboundary on ∆ ∩ Γ 0 . As was explained at the beginning of the proof of this theorem this contradicts the non-ergodicity of α f . Proof of (2). This is [2,Theorem 2.4.1]. For convenience of the reader we include the proof. Let Γ 0 ⊂ Γ be the subgroup generated by supp(h) ∪ supp(f ). Since Γ is not finitely generated there exists an increasing sequence of subgroups Γ n ⊂ Γ, n ≥ 1, such that Γ n+1 is generated over Γ n by a single element γ n+1 ∈ Γ n+1 Γ n . Put D = {γh + ZΓf : γ ∈ Γ} ⊂ ZΓ/ZΓf and D n = {γh + ZΓf : γ ∈ Γ n } ⊂ ZΓ/ZΓf, n ≥ 0. Then \|D 0 \| ≤ \|D 1 \| ≤ · · · ≤ \|D n \| ≤ · · · ≤ \|D\| < ∞. Hence there exists an N ≥ 0 with γ N +1 h + ZΓf = γ ′ h + ZΓf for some γ ′ ∈ Γ N . Then (γ N +1 − γ ′ )h = gf for some g ∈ ZΓ. We write g = g 1 + g 2 with supp(g 1 ) ⊂ Γ N and supp(g 2 ) ∩ Γ N = ∅. Then γ N +1 h − g 2 f = g 1 f + γ ′ h. (3.3) All the terms on the right hand side of (3.3) are supported in Γ N , whereas the supports of the terms on the left hand side of (3.3) are disjoint from Γ N . Hence both sides of (3.3) have to vanish, which means that γ N +1 h = g 2 f and h ∈ γ −1 N +1 g 2 f ∈ ZΓf , contrary to our choice of h. As explained above, this contradiction proves the ergodicity of α f . Proof of (3). If Γ 0 ⊂ Γ is a nonamenable subgroup with β (2) 1 (Γ 0 ) = 0, then the same is true for its finite-index subgroup ∆∩Γ 0 . By Proposition 2.2, the cocycle c : ∆ ∩ Γ 0 −→ ZΓ is a coboundary, which leads to a contradiction as in (1). Proof of Theorem 1.3. If Γ is amenable, use Theorem 1.2 (1) or (2). If Γ is nonamenable, combine the argument at the beginning of the proof of Theorem 1.2 with Proposition 2.4. 1 A discrete group Γ is virtually cyclic if it has a cyclic finite-index subgroup. Virtually cyclic groups can obviously not be principally ergodic: if Γ = Z, and if ZΓ is identified with the ring of Laurent polynomials Z[u ±1 ] in the obvious manner, then the principal algebraic Z-action α f defined by f = 1 − u is trivial -and hence nonergodic -on X f = T. Theorem 1 . 2 . 12Let Γ be a countably infinite discrete group which satisfies one of the following conditions: ( 2 ) 2⇔ (3) ⇒ (4): This is part of [5, Proposition 4.16]. Suppose that Γ satisfies either of the following conditions. . Then every cocycle c : ∆ −→ ZΓ is a coboundary. For the proof of Proposition 2.2 we have to discuss cocycles of Γ which take values in a Hilbert space H carrying a unitary action U and such a cocycle is a coboundary if and only if there exists a b ∈ H with c(γ) = b − U γ b (2.4) for every γ ∈ Γ. The cocycle c is an approximate coboundary if there exists a sequence (c n ) n≥1 of coboundaries c n : Γ −→ H such that lim n→∞ c n (γ) − c(γ) = 0 (2.5) Lemma 2. 3 . 3Let U be a unitary representation of Γ on H which does not contain the trivial representation weakly. Then every approximate coboundary c : Γ −→ H for U is a coboundary. δ∈F c(δ) = lim n→∞ δ∈F c n (δ) ≥ ε lim sup n→∞ b n , γ ∈ Γ. For a bounded subset Y of H and v ∈ H, set d(v, Y ) = sup y∈Y v − y .Since H is a Hilbert space, the function v → d(v, Y ) on H takes a minimal value at exactly one point, namely the Chebyshev center of Y , which we denote by center(Y ). The authors are grateful to Andreas Thom for alerting us to this reference. Group cohomology, harmonic functions and the first L 2 -Betti number. M E B Bekka, A Valette, Potential Anal. 64M.E.B. Bekka and A. Valette, Group cohomology, harmonic functions and the first L 2 -Betti number, Potential Anal. 6 (1997), no. 4, 313-326. Ergodicity of nilpotent group actions, Gauss's lemma and mixing in the Heisenberg group. B R Hayes, SeattleUniversity of WashingtonSenior ThesisB.R. Hayes, Ergodicity of nilpotent group actions, Gauss's lemma and mixing in the Heisenberg group, Senior Thesis, University of Washington, Seattle, 2009. Ends of groups and the associated first cohomology groups. C H Houghton, J. London Math. Soc. 2C.H. Houghton, Ends of groups and the associated first cohomology groups, J. London Math. Soc. (2) 6 (1972), 81-92. R V Kadison, J R Ringrose, Advanced Theory, Graduate Studies in Mathematics. Providence, RIAmerican Mathematical SocietyIIFundamentals of the Theory of Operator AlgebrasR.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II. Advanced Theory, Graduate Studies in Mathematics, 16. American Mathemat- ical Society, Providence, RI, 1997. Entropy, determinants, and L 2 -torsion. H Li, A Thom, 10.1090/S0894-0347-2013-00778-XJ. Amer. Math. Soc. to appear in printH. Li and A. Thom, Entropy, determinants, and L 2 -torsion, J. Amer. Math. Soc. DOI:10.1090/S0894-0347-2013-00778-X (to appear in print). Zero divisors and L 2 (G). P A Linnell, C. R. Acad. Sci. Paris Sér. I Math. 3151P.A. Linnell, Zero divisors and L 2 (G), C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 1, 49-53. Division rings and group von Neumann algebras. P A Linnell, Forum Math. 56P.A. Linnell, Division rings and group von Neumann algebras, Forum Math. 5 (1993), no. 6, 561-576. Non-properness of amenable actions on graphs with infinitely many ends. S Moon, A Valette, Ischia Group Theory. Hackensack, NJS. Moon and A. Valette, Non-properness of amenable actions on graphs with infinitely many ends, In: Ischia Group Theory 2006, pp: 227-233, World Sci. Publ., Hackensack, NJ, 2007. Group cocycles and the ring of affiliated operators. J Peterson, A Thom, Invent. Math. 1853J. Peterson and A. Thom, Group cocycles and the ring of affiliated operators, Invent. Math. 185 (2011), no. 3, 561-592. Dynamical Systems of Algebraic Origin. K Schmidt, Birkhäuser, BaselK. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, Basel, 1995. Rigidity of commensurators and irreducible lattices. Y Shalom, Invent. Math. 1411Y. Shalom. Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1-54. Die erste Cohomologiegruppe vonÜberlagerungen und Homotopie-Eigenschaften dreidimensionaler Mannigfaltigkeiten. E Specker, Comment. Math. Helv. 23E. Specker, Die erste Cohomologiegruppe vonÜberlagerungen und Homotopie- Eigenschaften dreidimensionaler Mannigfaltigkeiten, Comment. Math. Helv. 23 (1949), 303-333. On torsion-free groups with infinitely many ends. J R Stallings, Ann. of Math. 88J.R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88 (1968), 312-334. Groups of cohomological dimension one. R G Swan, J. Algebra. 12R.G. Swan, Groups of cohomological dimension one, J. Algebra 12 (1969), 585-610. Chongqing 401331. Hanfeng Li, China and Department of Mathematics, SUNY at Buffalo. Department of Mathematics, Chongqing UniversityNY 14260-2900, U.S.A. E-mail address: [email protected] Li: Department of Mathematics, Chongqing University, Chong- qing 401331, China and Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, U.S.A. E-mail address: [email protected]	[]
[ ") Ontario Institute for Cancer Research", ") Ontario Institute for Cancer Research" ]	[ "Surya Saha ", "; Scott Cain ; Ethalinda ", "K S Cannon ; Nathan Dunn " ]	[]	[ "Agricultural Research Service, Corn, Insects, and Crop Genetics Research Unit" ]	The GFF3 format is a common, flexible tab-delimited format representing the structure and function of genes or other mapped features (https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md). However, with increasing re-use of annotation data, this flexibility has become an obstacle for standardized downstream processing. Common software packages that export annotations in GFF3 format model the same data and metadata in different notations, which puts the burden on end-users to interpret the data model.The AgBioData consortium is a group of genomics, genetics and breeding databases and partners working towards shared practices and standards. Providing concrete guidelines for generating GFF3, and creating a standard representation of the most common biological data types would provide a major increase in efficiency for AgBioData databases and the genomics research community that use the GFF3 format in their daily operations.The AgBioData GFF3 working group has developed recommendations to solve common problems in the GFF3 format. We suggest improvements for each of the GFF3 fields, as well as the special cases of modeling functional annotations, and standard protein-coding genes. We welcome further discussion of these recommendations. We request the genomics and bioinformatics community to utilize the github repository (https://github.com/NAL-i5K/AgBioData_GFF3_recommendation) to provide feedback via issues or pull requests.	null	[ "https://arxiv.org/pdf/2202.07782v1.pdf" ]	246,867,000	2202.07782	201cac52b7cb31a7bb1cd53b7c7826ab87d222a3	) Ontario Institute for Cancer Research Andrew FarmerCopyright Andrew Farmer Surya Saha ; Scott Cain ; Ethalinda K S Cannon ; Nathan Dunn ) Ontario Institute for Cancer Research Agricultural Research Service, Corn, Insects, and Crop Genetics Research Unit Rex Nelson; Boyce Thompson Institute, Ithaca, NY; Toronto, Canada; Truveta, Seattle, WAAndrew Farmer1Authors Iowa State University, Ames, IA, (7) European Molecular Biology Laboratory, European Bioinformatics Institute, Wellcome Genome Campus, Hinxton, UK, (8) Lawrence Berkeley National Lab, Berkeley, California, USA, (9) USDA, Agricultural Research Service, National Agricultural Library) The GFF3 format is a common, flexible tab-delimited format representing the structure and function of genes or other mapped features (https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md). However, with increasing re-use of annotation data, this flexibility has become an obstacle for standardized downstream processing. Common software packages that export annotations in GFF3 format model the same data and metadata in different notations, which puts the burden on end-users to interpret the data model.The AgBioData consortium is a group of genomics, genetics and breeding databases and partners working towards shared practices and standards. Providing concrete guidelines for generating GFF3, and creating a standard representation of the most common biological data types would provide a major increase in efficiency for AgBioData databases and the genomics research community that use the GFF3 format in their daily operations.The AgBioData GFF3 working group has developed recommendations to solve common problems in the GFF3 format. We suggest improvements for each of the GFF3 fields, as well as the special cases of modeling functional annotations, and standard protein-coding genes. We welcome further discussion of these recommendations. We request the genomics and bioinformatics community to utilize the github repository (https://github.com/NAL-i5K/AgBioData_GFF3_recommendation) to provide feedback via issues or pull requests. Introduction The GFF3 format is a commonly used tab-delimited format representing the structure and function of genes or other mapped features (https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md). The format's flexibility allows scientists to easily manipulate GFF3 files, and it helps accurately represent the complex biological information being captured. However, with increasing re-use of annotation data, in particular from different sources (software output from custom datasets, and/or reference datasets provided by databases), this flexibility has become an obstacle for downstream processing. Common software packages that export annotations in GFF3 format model the same data and metadata in different notations, which puts the burden on end-users to understand possibly undocumented assumptions about the data model, then to convert the data for downstream applications. For example, the CDS phase field is commonly misinterpreted by both dataset generators and consumers, which can lead to vastly different and erroneous amino acid sequences derived from the same GFF3 file. The AgBioData consortium (https://www.agbiodata.org) is a group of genomics, genetics and breeding databases and partners working towards shared practices and standards 1 . Almost every AgBioData database uses the GFF3 format in some capacity, either for content ingest (into the database or associated tools, such as JBrowse 2 ), analysis, distribution, or all of the above. AgBioData members report that much of their data wrangling time is spent reformatting and correcting GFF3 files that model the same data types in different ways. Providing concrete guidelines for generating GFF3, and creating a standard representation of the most common biological data types in GFF3 that would be compatible with the most commonly used tools, would provide a major increase in efficiency for all AgBioData databases. The AgBioData GFF3 working group has developed new recommendations to solve common problems in the GFF3 format. We have referred to and in some cases adopted guidelines developed by the Alliance of Genome Resources (https://docs.google.com/document/d/1yjQ7lozyETeoGkPfSMTAT8IN3ZIAuy5YkbsBdjGeLww/edit), and NCBI (https://www.ncbi.nlm.nih.gov/datasets/docs/v1/reference-docs/file-formats/aboutncbi-gff3/; https://www.ncbi.nlm.nih.gov/sites/genbank/genomes_gff/). Below, we suggest improvements for each of the GFF3 fields, as well as the special cases of modeling functional annotations, and standard protein-coding genes. We welcome debate and discussion of these recommendations from the larger community -these recommendations will only be helpful if they are refined and then adopted by many. Our goal is to clarify the GFF3 specification and limit ambiguity for AgBioData and other databases and resources. source (column 2) -Change level: no change -Summary: There are no major changes from the previous SO specification. We recommend that the source field is used to define the source of the sequence feature concisely. Source is used to extend the feature ontology by adding a qualifier to the type field. -Proposed changes to specification: none -Rationale: The values used for this field vary widely as it's a free text field which can lead to parsing and interpretation issues for downstream software and data loading. -Best Practices: -We recommend that programs generating and consuming GFF3 follow the constraints outlined below and account for the fact that the feature can be a result of multiple tools in a pipeline. -Optionally, a pragma can specify the source. We recommend following the VCF specification for Info/ID: https://samtools.github.io/hts-specs/VCFv4.3.pdf. We discourage verbose use of this pragma. -Validation: -It is not necessary to specify a source. If there is no source, put a "." (a period) in this field. -Note that only spaces are allowed to represent whitespace. In general, follow the formatting requirements of the specification. From the GFF3 specification: "Literal use of tab, newline, carriage return, the percent (%) sign, and control characters must be encoded using RFC 3986 Percent-Encoding; no other characters may be encoded." -If there are multiple sources, use a literal comma to separate them (NOT %2C). Source names should not include literal commas. -Specify the tool, method or pipeline used to generate this annotation or the database it was acquired from. Mention the version number if available. -The feature should be a well defined output of the tool or database specified. If there is any ambiguity or post-processing, it should be clearly explained in an optional pragma stanza. -The pragma will not be validated. -Example in Supplementary data type (column 3) -Change level: no change -Summary: We endorse the Alliance recommendations for the 'type' field when modeling hierarchical gene features. This aligns with the SO specification that expects this to be "either a term from the Sequence Ontology or an SO accession number". -Proposed changes to specification: none -Rationale: Software interpreting the type column can run into difficulties with complex cases. Software is easier to develop and maintain if we can make some simplifying assumptions about how genes are typically modeled. Using simple terms will additionally improve human readability and interpretation. -Best practice: Top-level feature types can include 'gene' and 'pseudogene'. Optionally include a so_term_name attribute in column 9 to specify the child (type) of gene -e.g. protein_coding_gene, ncRNA_gene, miRNA_gene and snoRNA_gene (http://purl.obolibrary.org/obo/SO_0000704). Transcript features should include the appropriate SO term in column 3 (e.g. mRNA, snoRNA, etc). -Validation: -Must be a valid SO term or SO accession number -All child rows should use a type within the hierarchy of the parent -A list of the SO terms and the hierarchy in OBO format is fetched from http://song.cvs.sourceforge.net/viewvc/checkout/song/ontology/sofa.obo by default -Example in Supplementary data start, end (column 4,5) -Change level: no change -Summary: No changes from the SO specification. -Proposed changes to specification: none -Rationale: Consistent use of start and end coordinates are essential. -Best practice: We recommend that programs generating and consuming GFF3 be aware of the existence of circular chromosomes, which will require alternate interpretation of the end coordinates. -Validation: -start and end are 1-based coordinates. -start must always be less than or equal to end. -A feature with no length, for example, an insertion site, is indicated by start = end. The insertion site is to the right of the position. There is no recommendation for representing an insertion at the beginning, that is, before the first base as 0 is an invalid coordinate. -A feature that is one base in length, e.g., a SNP, is also indicated by start = end. Distinguishing between one-base and zero-length features will have to rely on other fields, such as the type field. -In the absence of the 'Is_circular=true' attribute in column 9, end indicates the terminal coordinate of the feature. -If 'Is_circular=true' appears in column 9, start gives the beginning coordinate of the feature and end is start + (feature length -1). This means the value for end may be larger than the chromosome size -Example in Supplementary data 5. score (column 6) a. Change level: moderate b. Summary: There is no clear guidance on how to interpret the score column. Therefore, define how the score was calculated in a pragma. c. Proposed changes to specification: Define score calculation via pragma. d. Rationale: There is currently no standard for providing metadata or context for the score column, rendering the score essentially meaningless. e. Best practice: Optionally, define how the score was calculated in a pragma. May have multiple parts, such as score name, program, version, range, and whether quality increases or decreases or is constant with increasing values. Use EDAM ontology (https://edamontology.org) where possible. The score itself must be a floating point number. This recommendation considers the score column only when representing gene models. f. Validation: i. A period indicates no score. ii. If any record has a value in the score column: -Example: NA phase (column 8) -Change level: recommendation only -Summary: Programs generating and consuming gff3 should pay close attention to the phase field and validate it, as phase is often incorrect. -Proposed changes to specification: none -Rationale: We have identified three main problems with the phase field. 1) Phase is often ignored or misinterpreted, both by programs generating gff3 and programs that consume it. When recorded manually, phase is often incorrect. This is problematic for programs that calculate the CDS and protein sequence using the combination of CDS coordinates and phase. Other methods that are frequently used to calculate the CDS and protein sequence (e.g. longest ORF, identifying start and stop codons) make critical assumptions that can also generate incorrect sequence, in particular for fragmented genomes where gene models may not have start and/or stop codons. 2) Even if the phase is correct, a translation table is required to correctly calculate the protein sequence, and there may be multiple translation tables needed for a given gff3, for example when both nuclear and organellar sequence is represented. 3) Even when the phase and translation tables are correct, the correct sequence may not be inferred due to post-translational modifications (e.g. selenocysteines) or problematic reference genome assemblies. NCBI represents these edge cases via the 'transl_except' attribute. -Best practices: -We recommend that programs generating and consuming gff3 pay close attention to this value and validate it; however, validation may still fail in complex cases. Phase may be an example where the GFF3 format has reached its limit. In cases where the correct sequence may not be inferred due to post-translational modifications ( 8. attributes (column 9): ID -Change level: recommendation only -Summary. The ID attribute's role is to specify relationships between parent and child features within the GFF3. However, it is often -but not always -also used to specify a globally unique, persistent identifier. This second interpretation causes many problems with downstream software and validators. We recommend NOT using the ID attribute to specify the globally unique, persistent identifier, but instead using a separate attribute, such as Dbxref or gene_id. -Proposed changes to specification: Recommend additional attributes (reserved or nonreserved) to specify the globally unique, persistent identifier -Rationale. The GFF3 specification requires an ID attribute to define parent-child (part of) relationships for hierarchically modeled features (see specification for a detailed definition). In practice, the ID attribute is often also used to stand in for a globally unique, persistent identifier (in particular for genes, transcripts and proteins; see this article for definitions of and best practices on persistent identifiers: https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.2001414). This second interpretation becomes problematic when users or software assume it is true in all cases, as they expect additional meaning beyond a generic, unique string. While a dualuse ID attribute may seem convenient, it is not always clear which level of a gene feature may have the 'true' persistent identifier -the gene, transcript, protein, even exon? In addition, it is not always possible to accurately model parent-child relationships if it is also a globally unique, persistent identifier. -Best practices. Use the ID attribute as originally intended by the GFF3 specification, and do not assume that it contains a globally unique persistent identifier. For these, use an additional attribute. The attribute may depend on the use case -for example, if the persistent identifier is maintained by another database, Dbxref may be used (however, it may be confusing if multiple Dbxref identifiers are specified). The Alliance for Genome Resources uses the unreserved attributes gene_id, transcript_id, and protein_id, where the values of these are curies (https://en.wikipedia.org/wiki/CURIE). NCBI's RefSeq uses the unreserved attributes gene, transcript_id, and protein_id. We recommend using either of these conventions, with an additional 'feature_id' attribute for features that are neither genes, transcripts, or proteins. While this recommendation may require software and databases to adjust, this is simpler than forging a way for the ID attribute to meet all downstream tools' and users' needs. Note the still unresolved yet related problem of the corresponding FASTA definition line -there are no guidelines in terms of correspondence between the FASTA defline and identifying information in the GFF3 file. -Various taxonomic communities have nomenclature standards that aim to provide consistent naming across genes. Please identify and follow these standards for your community. -Name should not be mistaken for a unique identifier, or dbxref -Note that different, non-reserved attributes are sometimes used instead of Name. For example, NCBI uses product for the protein product name, gene_desc for the full gene name, and symbol as the gene abbreviation. -Summary: There are no major changes from the previous SO specification. The primary function of Alias appears to be for human consumption, display, indexing, and tracking synonyms or prior names. If there are additional functions that you know the value should be used for, we recommend that you choose a different attribute that is more specific, and could be consumed programmatically. The validator won't check Alias values. -Rationale: The Alias field is used for alternate names or identifiers, and there are no real constraints on what these may be. Some important cases are as follows. If a gene is merged, or if the type of a genomic feature is changed, the name of the original feature may need to change. Conversely, if a gene is split, we should retain a reference to its original name. If two papers cite the same gene using a different name, we should be able to search for either one even if only one is an official one. -Proposed changes to specification: None -Best practices: Use Alias for human consumption of alternate or historical names and identifiers (e.g., gene merge), but do not assume that this field will be consumed programmatically. Alias should not be a replacement for Dbxref, and valid CURIE in Dbxref should be housed within Dbxref and not in Alias. Commas, tabs, and pipes should be avoided in alias names. It is possible to have multiple Alias values. It is recommended that these be separated via a comma. -Validation: Any set of symbols would be appropriate and does not need to be unique. Alias values can not include a semicolon. -Example in Supplementary data 11. attributes (col 9): Dbxref -Change level: Recommendation only -Summary: 1) Use the Dbxref field to cross-reference the same entity at a database -the field is sometimes mis-interpreted to reference related information, but not the same entity. 2) The Dbxref should result in a resolvable URL. -Proposed changes to specification: Recommend use of the field for global entity references and crosslinks between databases. can be instantly forwarded to https://www.rcsb.org/structure/2gc4. The syntax of Compact Identifiers includes three parts: a provider code, a namespace prefix, and an accession. The provider code and namespace prefix are manually curated and stable, and can be easily looked up at its web site. Note that the databases represented by these four resources may overlap. -Best practices: -The dbxref must refer to the same entity (not related information) in an external database. The format of a dbxref record may take the form "dbxref=database + identifier", where "database" is an abbreviated database name registered on a known dbxref list (above), affixed without space with a specific path leading to the database agent that accepts the "identifier" for information retrieval. The "database:identifier" construct is called unique resource identifier (URI -Validation: There may be a challenge when writing the validators, given that there are multiple independently maintained lists, and there may be lags in information updates. The validator could check whether a HTTP response status code 200 is returned based on the url built from the dbxref registry in the directive. Semicolons (";") are not permitted in the URL as it's used as the delimiter between column 9 "name=value" pairs.. -Example: See section "18. Progamas" for details. 12. attributes (col 9): Derives_from -Change level: minor -Summary: The most common use for the Derives_from attribute is to describe the relationship between CDS and polypeptide features. However, 1) not all software recognizes this relationship, and 2) we do not recommend modeling polypeptide features in GFF3 (see recommendations for 'Modeling hierarchical relationships of a proteincoding gene'). Avoid modeling polypeptide features in general to prevent downstream interpretation problems of Derives_from. -Proposed changes to specification: None. -Rationale: The Derives_from attribute (http://purl.obolibrary.org/obo/RO_0001000) is used in situations where the relationship between features is temporal, and therefore the part_of relationship implied by the 'Parent' attribute is not appropriate (e.g. polypeptides are derived from CDS features, or in the case of polycistronic genes). In practice most programs that consume or create GFF3 do not check whether implied part_of relationships are actually valid per the Sequence Ontology (a notable exception is Genometools (http://genometools.org/cgi-bin/gff3validator.cgi)). -Best practices: To avoid breaking software that consumes GFF3, we recommend not specifying a polypeptide feature if you're modeling a typical protein-coding gene based on genomic coordinates (see also recommendations for 'Modeling hierarchical relationships of a protein-coding gene' below). -Proposed changes to specification: None -Rationale: Sometimes the Note attribute can contain irrelevant information. -Best practices: Use primarily for notes relevant for public consumption (e.g., important errata for downstream users). Use as a last resort and at your own risk. In some cases a custom tag may be more appropriate. A common use-case would be for curation notes relevant to external users. -Summary: Target is an attribute intended to encode a parseable relationship between a region on the sequence given in column 1 and a region on another sequence, possibly (but not necessarily) another sequence referenced in column 1 of another record in the same gff file. Gap is an associated attribute encoding the alignment (i.e. gapping structure) needed to put the elements of the two sequences into homologous correspondence. -Proposed changes to specification: None -Rationale: The GFF3 specification describes the purpose of these two attributes as being for the representation of sequence alignments. Since the time that specification was written, numerous alternative formats for storing sequence alignments have been developed in response to the proliferation of sequence data brought about by the advent of next generation sequencing technologies. Therefore, although there may be cases in which the use of such attributes to represent alignments in the context of GFF files is convenient for a specific purpose (e.g. a gene prediction program seeking to represent the evidence behind its models), in general we advocate the use of alternative file formats such as BAM or PAF for representing alignments between sequences. -There are additional contexts in which the representation of homology between regions on sequences is desirable without the fine grained structure of base pair correspondence. For example, syntenic relationships (SO:0005858) between regions of chromosomes that are derived from collinear blocks of homologous genes may be computed without having the full details of the alignment of the genomic regions. In this case, since the relationship is likely going to be an inter-genomic comparison (e.g. between species), it is important to be able to represent the information as a single record bearing the information about the paired genomic regions. Target can be used in such cases to represent the relationship, following the space-delimited structure suggested in the GFF3 specification "target_id start end [strand]". We note that although the GFF3 specification indicates that spaces in the target_id must be hex-encoded, we would recommend that the target_id represent a sequence identifier such as would be found in Column 1 of the same (or another) GFF3 file containing annotations of the target sequence. -Best practices: We advocate the use of alternative file formats such as BAM or PAF for representing alignments between sequences. -Validation: The components of the encoded attribute must be single-space delimited and must consist of not less than 3 and not more than 4 fields. Field 1 must conform to the same syntax and semantics as specified for Column 1 (seqid). Fields 2 and 3 must conform to the syntax and semantics for Columns 4 and 5 (start and stop) respectively. Field 4 is optional and conforms to the syntax and semantics of Column 7 (strand), though a missing value will be indicated by absence of the field rather than using "." -Example in Supplementary data 16. Attributes (col 9): complex metadata (e.g. functional annotations) -Change level: Major change -Summary: In some instances we may need to model complex sets of metadata within the GFF3. -Proposed changes to specification: Provide a format for including richer metadata. -Rationale: In general, modeling functional annotations in GFF3 should be avoided if other mechanisms (GPAD, GPI, all spec formats and versions ) are available. However, in some instances other formats are not sufficient or readily available for tooling. In those instances, the ability to include information such as functional annotations that track annotation provenance, gene products, or properly annotated GO evidence may be necessary. -Best practices: -We discourage the use of GO terms, or any functional annotation that requires an evidence code, without supplying the evidence code. -GO term or functional annotations should never be incorporated into gff3 within the Dbxref or Ontology_term fields. Another file format should be used, e.g. GAF or GPAD, if possible, otherwise modeling using <complex metadata> is recommended. -In the context where metadata such as functional annotations must be included in GFF3 column 9, the general format we would suggest and has been adopted in Apollo (https://github.com/gmod/apollo) and Artemis (https://github.com/sanger-pathogens/Artemis) for GO (go_annotations), Gene Product (gene_product), and Provenance (provenance) annotations is <type>=<type annotation>;. Each type is only included once, but can include multiple type annotations. Note that <type annotation> is URL encoded. -<type_annotations> are URL encoded and of the format: rank=<rankA>;<key1>=<value1A>;<key2>=<value2A>,rank=<rankB>; <key1>=<value1B>;<key2>=<value2B> . With multiple annotations, provide a rank to indicate which would go first, though this may not always be relevant. Multiple annotations are comma-delimited. Multiple key/value pairs are semicolon delimited. -GO terms should be updated annually -you might not want to include this as 'static' information. -Validation: -<type> should be lower-case and be of the form <type1>=<type1 annotations>;<type2>=<type2 annotations>; etc. -<type annotation> entries are URL encoded -A <type> can have multiple <type annotations>, which are separated by a comma (url-encoded %3B). -Each <type annotation> has multiple key-value pairs, separated by a semi-colon (url-encoded as %2C). -<rank> is not necessary, but it is preferred if more than one annotation for a type exists. -<type>, <rank>, <key> should all be lower-case. -Example in Supplementary data 17. Modeling hierarchical relationships of a protein-coding gene -Change level: Recommendation only -Summary: The primary purpose of GFF3 is to model gene structure. -Rationale: We wish to provide a standard way to render this type of information as there are many valid ways to render the same protein-coding gene. -Proposed changes to specification: None -Best practices: -Strongly encourage only one parent per feature. However, parsers and validators should still support multiple parents per feature, in particular for elegance and backwards compatibility and to support more "non-standard" protein structures especially within non-eukaryotic organisms. -Edge cases that can't follow this standard recommendation exist, e.g. ribosome slippage, trans-splicing, features split across scaffolds due to assembly problems -further recommendations need to be developed. Conclusions and Future Work These recommendations are targeted at ameliorating challenges with protein-coding gene models represented on a common coordinate system. To complement the recommendations described by this paper, an open source software validator needs to be developed by the community that can be extended as new issues are reported by users. Future work on GFF3 specifications and recommendations should include modeling miRNAs and QTL data and addressing features located in a pan genome coordinate space. The GFF3 specification would also benefit from pragmas for describing provenance of and/or workflow used to generate the data. Given the often tight coupling of FASTA and GFF3 data, there would be a benefit in outlining best practices for FASTA definition lines to complement the GFF3 specification. We request the genomics and bioinformatics community to utilise the AgBioData GFF3 recommendation github repository (https://github.com/NAL-i5K/AgBioData_GFF3_recommendation) to actively contribute to the development of the GFF3 recommendations. This update to the original specification creates a framework for the community to collaborate and update the venerable GFF3 file format to handle feature types and data integration challenges yet unseen. Modeling hierarchical relationships of a protein-coding gene ##gff-version 3 ##sequence-region chr9 1 138394717 chr9 . gene 99206226 99222116 . -. [email protected];ID=98674a30-c148-4d05-8b17-53e3d09645ac;date_last_modified=2020-04-09;Name=ALG2-002a;date_creation=2019-11-05 chr9 . mRNA 99206226 99222116 . -. [email protected];Parent=98674a30-c148-4d05-8b17-53e3d09645ac;ID=7dd2784f-6357-4852-9c9b-21d8483255d8;orig_id=transcript:ENST00000476832;date_last_modified=2020-03-02;Name=ALG2-002a-00001;date_creation=2019-10-25 chr9 . CDS 99217934 99218680 . -0 Parent=7dd2784f-6357-4852-9c9b-21d8483255d8;ID=b0fc5d5b-65e6-41a0-a65e-7afadfe1d8d1;Name=b0fc5d5b-65e6-41a0-a65e-7afadfe1d8d1 chr9 . exon 99221130 99222116 . -. Parent=7dd2784f-6357-4852-9c9b-21d8483255d8;ID=7aa73c7e-bb44-4e86-aeb3-921340d60e45;Name=7aa73c7e-bb44-4e86-aeb3-921340d60e45 chr9 . exon 99206226 99218846 . -. Parent=7dd2784f-6357-4852-9c9b-21d8483255d8;ID=9baa5521-2f79-4493-9435-0120aa4f96d0;Name=9baa5521-2f79-4493-9435-0120aa4f96d0 chr9 . mRNA 99217367 99218836 . -. [email protected];Parent=98674a30-c148-4d05-8b17-53e3d09645ac;ID=6eebcdae-fb8b-42af-a178-f70a0e8ff5da;orig_id=transcript:ENST00000476832;date_last_modified=2020-04-09;Name=ALG2-002a-00003;date_creation=2020-04-09 chr9 . CDS 99217934 99218836 . -0 Parent=6eebcdae-fb8b-42af-a178-f70a0e8ff5da;ID=6eebcdae-fb8b-42af-a178-f70a0e8ff5da-CDS;Name=6eebcdae-fb8b-42af-a178-f70a0e8ff5da-CDS chr9 . exon 99217367 99218836 . -. Parent=6eebcdae-fb8b-42af-a178-f70a0e8ff5da;ID=cbf89b7d-32de-40a0-be12-1332afe2a527;Name=cbf89b7d-32de-40a0-be12-1332afe2a527 chr9 . mRNA 99217372 99218836 . -. [email protected];Parent=98674a30-c148-4d05-8b17-53e3d09645ac;ID=3da41f24-d14b-4f01-8e27-c15f86912acb;orig_id=transcript:ENST00000319033;date_last_modified=2020-04-09;Name=ALG2-002a-00004;date_creation=2020-04-09 chr9 . exon 99217372 99218836 . -. Parent=3da41f24-d14b-4f01-8e27-c15f86912acb;ID=b5adb98d-82c9-4b5b-a320-1c044bced6e0;Name=b5adb98d-82c9-4b5b-a320-1c044bced6e0 chr9 . CDS 99217934 99218836 . -0 Parent=3da41f24-d14b-4f01-8e27-c15f86912acb;ID=3da41f24-d14b-4f01-8e27-c15f86912acb-CDS;Name=3da41f24-d14b-4f01-8e27-c15f86912acb-CDS chr9 . mRNA 99217367 99221956 . -. [email protected];Parent=98674a30-c148-4d05-8b17-53e3d09645ac;ID=7cf3b8f6-7241-4145-9822-00bbb8fd4a87;orig_id=transcript:ENST00000476832;date_last_modified=2020-04-08;Name=ALG2-002a-00002;date_creation=2020-04-08 chr9 . CDS 99221547 99221954 . -0 Parent=7cf3b8f6-7241-4145-9822-00bbb8fd4a87;ID=010f5cfb-3f84-4e74-9ad0-6d54c1463502;Name=010f5cfb-3f84-4e74-9ad0-6d54c1463502 chr9 . CDS 99217934 99218836 . -0 Parent=7cf3b8f6-7241-4145-9822-00bbb8fd4a87;ID=010f5cfb-3f84-4e74-9ad0-6d54c1463502;Name=010f5cfb-3f84-4e74-9ad0-6d54c1463502 chr9 . exon 99221547 99221956 . -. Parent=7cf3b8f6-7241-4145-9822-00bbb8fd4a87;ID=58e7b74f-f60a-438f-a869-e8a0fc13bd3a;Name=58e7b74f-f60a-438f-a869-e8a0fc13bd3a chr9 . exon 99217367 99218836 . -. Parent=7cf3b8f6-7241-4145-9822-00bbb8fd4a87;ID=d39ce049-51aa-48b4-a45b-ce8422b7693b;Name=d39ce049-51aa-48b4-a45b-ce8422b7693b ### - Change level: No change. -Summary: No change from the original specification. -Rationale: NA -Best practices: Follow the original specification. -Validation: Values should include '+', '-', '.'. '?' can be used when the strand is relevant but unknown. - Validation. ID attribute: only validate whether IDs for each feature are unique within the scope of the GFF file, with the exception of discontinuous features. Persistent identifiers specified via the Dbxref attribute can be validated according to Dbxref rules. -Example in Supplementary data 9. attributes (col 9): Name -Change level: recommendation only -Summary: A designation for the given feature used for display. -Proposed changes to specification: None -Rationale: Naming standards exist and should be followed when possible. -Best practices: - Validation: refer to different community standards. No automated validation currently possible. -Example in Supplementary data 10. attributes (col 9): Alias -Change level: Recommendation only - Validation: This needs further analysis and discussion. -Example in Supplementary data 13. attributes (col 9): Note -Change level: No change -Summary: No changes relative to the original definition. -- Validation: May not include a semicolon. May be repeated. -Example in Supplementary data 14. attributes (col 9): Ontology_term -Change level: Recommendation only. -Summary: Avoid. -Proposed changes to specification: None -Rationale: See section about functional annotation and metadata below. -Best practices: Avoid. In general, do not use gff3 for functional annotation if possible. Use GAF, GPAD or other instead. If you do have to use it, use a CURIE backed ontology term (e.g. GO:0000077). In general, supplying an ontology term without underlying evidence, reference, or other context will be under-defined functional annotation and will be incomplete for downstream users and of little utility. -Validation: Should be a CURIE that resolves into a published ontology term. The validator should issue a warning that Ontology_term should be avoided. -Example in Supplementary data Change level: Recommendation only. Table 1 : 1Summary of recommendationsWe recommend that developers and databases follow the Sequence Ontology GFF3 specifications (https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md) with emphases and additions below. Each field contains information in the following categories:•Change level: The level of change relative to the SO specification. Values are 'No change', 'Recommendation only','minor', 'moderate', 'major' • Summary: A summary of the GFF3 working group's findings. • Proposed changes to specification: A list of the proposed changes to the SO specification. • Rationale: The rationale behind these changes. • Best Practices: Recommended best practices for this field. • Validation: How software would validate whether the field is used correctly. • Example: An example implementation of the field. Examples are listed in the Appendix. 1. Seqid (column 1) -Change Level. Recommendation only -Summary. Optionally, provide an Alias table to specify alternate identifiers/aliases for the seqid. -Proposed Changes to Specifications. Institute a new pragma for alias table link. -Rationale. Sequences often have aliases (multiple identifiers, human-readable names that are not globally unique), and users prefer human-readable display names when viewing sequences in browsers. -Recommendation. Optionally provide a machine-and human-readable 'alias' table to specify identifiers and their aliases, which is provided by GenBank, and requires INSDC If the GenBank Alias table is not available, then a separate alias table can be provided. A pragma line, ##alias-table [columns] indicates where the table begins, and the definitions of the columns provided. All identifiers in column 1 of the GFF3 file must be uniquely present in column 1 of the alias table. The columns in the alias table should be tab delimited. There will be no validation for an alternate alias table. -Example in Supplementary dataColumn Change level Attributes Change level Seqid (column 1) Recommendation ID Recommendation Source (column 2) No change Name Recommendation Type (column 3) No change Alias Recommendation Start, end (column 4, 5) No change Dbxref Recommendation Score (column 6) Moderate Derives_from Recommendation Strand (column 7) No change Note No change Phase (column 8) Recommendation Ontology_term Recommendation Modeling protein- coding genes Recommendation Target, Gap Recommendation Functional annotations Major change Specific recommendations //www.ncbi.nlm.nih.gov/genbank/genomes_gff/) -Optionally provide a phase pragma, if any sequences in the GFF3 file do not use the standard genetic code (id = 1, see https://www.ncbi.nlm.nih.gov/IEB/ToolBox/C_DOC/lxr/source/data/gc.prt#0107 ). The pragma should provide the translation table id, and the reference sequences in the GFF3 that will use that translation table id, e.g. ##phase <RefSeq ID> <translation table ID>; -Validation: For a description of what phase means in the context of a single CDS line, see the 'Column 8: "phase' section of the current GFF3 specification. Optionally provide the translation table id, and the reference sequences in the GFF3 that will use that translation table id, in a phase pragma. The validator will use the translation tables in https://www.ncbi.nlm.nih.gov/IEB/ToolBox/C_DOC/lxr/source/data/gc.prt. If no phase pragma is given, or if not all reference sequences are specified in the phase pragma, the validator will use the Standard genetic code (id 1). Protein and CDS fasta are optional but highly recommended. Validator will generate CDS/protein sequences based on the phase specified in the gff3 file. -If no CDS/protein fasta is available: -Check for internal stops in the protein sequence -validation fails if stops are present -If CDS and protein fasta are available: -Compare given sequence to sequence generated from gff3 -validation fails if sequences are not identical -Example in Supplementary data for the following use cases -Pragma specifying translation table: specify exemptions to standard code only -Example of incorrect vs. correct phasee.g. selenocysteines) or problematic reference genome assemblies, use the 'transl_except' convention developed by NCBI on the CDS feature (transl_except=(pos:<base_range>%2Caa:<amino_acid>); https: -Rationale: The database cross-reference (dbxref) links a particular feature in the GFF record to a specific external database record by identity, source, association, or ontology links. To date, there are four established lists with hundreds of registered databases that offer external links by dbxref. The GO consortium list is maintained on GitHub with defined schema and format validation tools.It currently lists 266 databases (http://amigo.geneontology.org/xrefs). The UniProt Knowledgebase cross-reference list contains 183 databases (https://www.uniprot.org/docs/dbxref) flagged to represent 18 categories of databases (https://www.uniprot.org/database/) for better utility. The NCBI- GenBank db_xrefs list was developed in 1997 (https://doi.org/10.1038/ng0497-339), however it has only 129 databases listed to date (https://www.ncbi.nlm.nih.gov/genbank/collab/db_xref/). In addition, identifiers.org (https://identifiers.org/) provides a free service for looking up and referencing a data ID to one of the 714 pre-curated life science database locations using Compact Identifiers syntax. For example, the uniform resource identifier (URI) http://identifiers.org/pdb/2gc4 ). The URI must be specific to a record. Both dbxref record and the resolved URL should be a continuous string compliant to RFC-3986 [https://datatracker.ietf.org/doc/rfc3986/]. The composed URL must be unique and exist. Multiple dbxref items may be allowed for one GFF record. We recommend using identifiers.org for identifier resolution. -Pragma: format: dbxref=URI; where URI is concatenated (without space) with the follow components: -Protocol://domain name/ + [namespace] + identifier -Name space is the path to the identifier handler; -Identifier: unique accession of an entity -Sort order. This pertains to having child features come after parent features in the gff; most loaders don't care about order of the features by coordinates. Child features should be listed after parent features. -Child coordinates that are not contained within parent coordinates often indicate an error and should trigger a warning in a gff3 validator. -The gff3 format assumes that parent and child features have a part_of relationship type. -There can be a ### directive between gene models. -Do not list multiple values in column 1 (for features split across scaffolds) -Polypeptide features are not required or recommended -Introns can be annotated, but are not necessary and are implied. -Type should be specified and validated as part of the sequence ontology cv terms (see also notes on column 3, type, above): http://www.sequenceontology.org/miso -Validation: -Entries that are non-parent entries should have a valid parent entry via the ID. -IDs should be internally resolvable. -Example in Supplementary data 18. Pragmas (also called directives) provide information about the entire dataset represented in the GFF3 document. Pragma lines begin with ##. Here, we suggest modifying the definition of two pragmas in the GFF3 specification. Example: ##dbxref=ncbiprotein:CAA71118.1; (which resolves to https://identifiers.org/ncbiprotein:CAA71118.1) -Addendum: This requires that the database providing the xref register at and obtain a namespace from Identifiers.org. -Ontology URIs -In the current GFF3 specification, ontology URIs, for example ##featureontology URI, can be specified via cv URLs (e.g. http://song.cvs.sourceforge.net/checkout/song/ontology/sofa.obo?revisi on=1.6). These URLs should be avoided. Instead, we recommend using the official OBO version IRI PURLs, for example http://purl.obolibrary.org/obo/so.obo. -Example: ## feature-ontology http://purl.obolibrary.org/obo/so.obo -Species -The current specification recommends using NCBI URLs to specify the species that annotations are derived from in the ##species pragma. We recommend using an OBO CURIE, instead. -Example: ##species NCBITaxon:9606Pragmas -Dbxref -This pragma is optional. -Format: ##dbxref=<URI> - AcknowledgementsWe thank Margaret Woodhouse for the inspiration for this working group, Vamsi Kodali and Terence Murphy for input on various aspects of the GFF3 specifications, and many other contributors from AgBioData and the larger research community who have provided feedback on earlier versions.This research was supported in part by the US. Department of Agriculture, Agricultural Research Service. Mention of trade names or commercial products in this publication is solely for the purpose of providing specific information and does not imply recommendation or endorsement by the U.S. Department of Agriculture.Attributes (col 9): AliasIn the case of pax6a (http://zfin.org/ZDB-GENE-990415-200) there are multiple "aliased names" associated with it under previous names: pax-a Pax6.1 pax6 pax[zf-a] (1) paxzfa zfpax-6a cb280 (1) etID309716.25 (1) fc20e07 wu:fc20e07(1)-. [email protected];provenance=rank%3D1%3Bfield%3DDESCRIPTION%3Bdb_xr ef%3D:%3Bevidence%3DECO:0000501%3Bnote%3D["Description of provenance of an individual field within an annotation. "]%3Bbased_on%3D[]%3Blast_updated%3D2021-01-11 16:39:32.454%3Bdate_created%3D2021-01-11 16:39:32.454;Parent=c9637c84-1c18-4320-8c58-7277ef768fd9;go_annotations=rank%3D1%3Baspect%3DMF%3Bterm%3DGO:0004381%3Bdb_x ref%3DPMID:171711%3Bevidence%3DECO:0000315%3Bgene_product_relationship%3DRO:00 02327%3Bnegate%3Dfalse%3Bnote%3D["This is a made up example."]%3Bbased_on%3D["UniProt:`123141"]%3Blast_updated%3D2021-01-11 Department of Agriculture prohibits discrimination in all its programs and activities on the basis of race, color, national origin, age, disability, and where applicable, sex, marital status, familial status, parental status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or part of an individual's income is derived from any public assistance program. (Not all prohibited bases apply to all programs.) Persons with disabilities who require alternative means for communication of program information (Braille, large print, audiotape, etc.) should contact USDA's TARGET Center at (202) 720-2600 (voice and TDD). U S The, 1400To file a complaint of discrimination. write to USDA, Director, Office of Civil RightsThe U.S. Department of Agriculture prohibits discrimination in all its programs and activities on the basis of race, color, national origin, age, disability, and where applicable, sex, marital status, familial status, parental status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or part of an individual's income is derived from any public assistance program. (Not all prohibited bases apply to all programs.) Persons with disabilities who require alternative means for communication of program information (Braille, large print, audiotape, etc.) should contact USDA's TARGET Center at (202) 720-2600 (voice and TDD). To file a complaint of discrimination, write to USDA, Director, Office of Civil Rights, 1400 20250-9410, or call (800) 795-3272 (voice) or (202) 720-6382 (TDD). S W Independence Avenue, D C Washington, USDA is an equal opportunity provider and employerIndependence Avenue, S.W., Washington, D.C. 20250-9410, or call (800) 795-3272 (voice) or (202) 720-6382 (TDD). USDA is an equal opportunity provider and employer. AgBioData consortium recommendations for sustainable genomics and genetics databases for agriculture. L Harper, Harper, L. et al. AgBioData consortium recommendations for sustainable genomics and genetics databases for agriculture. Database 2018, (2018). JBrowse: a dynamic web platform for genome visualization and analysis. R Buels, 16:45:45.133%3Bdate_created%3D2021-01-11 16:45:450000077%3Bdb_xref%3DAsp GD_REF:ASPL0000000005%3Bevidence%3DECO:0000501%3Bgene_product_relationship%3DR O:0002331%3Bnegate%3Dfalse%3Bnote%3D. 1766%2C"This is an example GO functional annotation created withinBuels, R. et al. JBrowse: a dynamic web platform for genome visualization and analysis. Genome Biol. 17, 66 (2016). 16:45:45.133%3Bdate_created%3D2021-01-11 16:45:45.133%2Crank%3D2%3Baspect%3DBP%3Bterm%3DGO:0000077%3Bdb_xref%3DAsp GD_REF:ASPL0000000005%3Bevidence%3DECO:0000501%3Bgene_product_relationship%3DR O:0002331%3Bnegate%3Dfalse%3Bnote%3D["This was pulled from AMIGO: http://amigo.geneontology.org/amigo/gene_product/AspGD:ASPL0000108267"%2C"This is an example GO functional annotation created within	[ "https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md).", "https://github.com/NAL-i5K/AgBioData_GFF3_recommendation)", "https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md).", "https://github.com/gmod/apollo)", "https://github.com/sanger-pathogens/Artemis)", "https://github.com/NAL-i5K/AgBioData_GFF3_recommendation)", "https://github.com/The-Sequence-Ontology/Specifications/blob/master/gff3.md)" ]
[ "Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation", "Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation" ]	[ "Le Xu \nDepartment of Electrical and Electronic Engineering\nThe University of Hong Kong\n\n", "Lei Cheng [email protected] \nCollege of Information Science and Electronic\nZhejiang University\n388 Yuhangtao RoadHangzhouChina\n", "Ngai Wong \nDepartment of Electrical and Electronic Engineering\nThe University of Hong Kong\n\n", "Yik-Chung Wu \nDepartment of Electrical and Electronic Engineering\nThe University of Hong Kong\n\n" ]	[ "Department of Electrical and Electronic Engineering\nThe University of Hong Kong\n", "College of Information Science and Electronic\nZhejiang University\n388 Yuhangtao RoadHangzhouChina", "Department of Electrical and Electronic Engineering\nThe University of Hong Kong\n", "Department of Electrical and Electronic Engineering\nThe University of Hong Kong\n" ]	[]	Tensor train (TT) decomposition, a powerful tool for analyzing multidimensional data, exhibits superior performance in many machine learning tasks. However, existing methods for TT decomposition either suffer from noise overfitting, or require extensive fine-tuning of the balance between model complexity and representation accuracy. In this paper, a fully Bayesian treatment of TT decomposition is employed to avoid noise overfitting, by endowing it with the ability of automatic rank determination. In particular, theoretical evidence is established for adopting a Gaussian-product-Gamma prior to induce sparsity on the slices of the TT cores, so that the model complexity is automatically determined even under incomplete and noisy observed data. Furthermore, based on the proposed probabilistic model, an efficient learning algorithm is derived under the variational inference framework. Simulation results on synthetic data show the success of the proposed model and algorithm in recovering the ground-truth TT structure from incomplete noisy data. Further experiments on real-world data demonstrate the proposed algorithm performs better in image completion and image classification, compared to other existing TT decomposition algorithms.	null	[ "https://arxiv.org/pdf/2010.06564v2.pdf" ]	248,377,296	2010.06564	5d34aea1db12915455e129c9c33a9ed8db9f368d	Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation 25 Apr 2022 Le Xu Department of Electrical and Electronic Engineering The University of Hong Kong Lei Cheng [email protected] College of Information Science and Electronic Zhejiang University 388 Yuhangtao RoadHangzhouChina Ngai Wong Department of Electrical and Electronic Engineering The University of Hong Kong Yik-Chung Wu Department of Electrical and Electronic Engineering The University of Hong Kong Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation 25 Apr 2022Preprint submitted to Elsevier April 26, 2022Bayesian InferenceTensor CompletionTensor Train * Corresponding Author Tensor train (TT) decomposition, a powerful tool for analyzing multidimensional data, exhibits superior performance in many machine learning tasks. However, existing methods for TT decomposition either suffer from noise overfitting, or require extensive fine-tuning of the balance between model complexity and representation accuracy. In this paper, a fully Bayesian treatment of TT decomposition is employed to avoid noise overfitting, by endowing it with the ability of automatic rank determination. In particular, theoretical evidence is established for adopting a Gaussian-product-Gamma prior to induce sparsity on the slices of the TT cores, so that the model complexity is automatically determined even under incomplete and noisy observed data. Furthermore, based on the proposed probabilistic model, an efficient learning algorithm is derived under the variational inference framework. Simulation results on synthetic data show the success of the proposed model and algorithm in recovering the ground-truth TT structure from incomplete noisy data. Further experiments on real-world data demonstrate the proposed algorithm performs better in image completion and image classification, compared to other existing TT decomposition algorithms. Introduction Driven by the increasing demand for multidimensional big data analysis, tensor decomposition has come up as an emerging technique that shows superior performance in a variety of data analytic tasks, including image completion [1,2,3,4], classification [5,6,7], and neural network compression [8,9,10]. Among the many tensor decomposition formats, the tensor train (TT) decomposition has made remarkable success in the above-mentioned applications owning to its particular algebraic format. In addition to multidimensional data mining, TT decomposition has witnessed a unique advantage in solving originally formidable large scale optimization problems by reformulating them using the TT format with suitable TT ranks, thus significantly saving the computational resources [11,12,13,14,15]. A number of algorithms have been put forward to solve the TT decomposition problems from the perspective of multi-linear algebra and optimization. For example, if the tensor is fully observed, TT-SVD [16] finds the TT cores based on truncated SVDs to the unfolding data matrices of the observed tensor. Although simple to implement, it cannot handle missing data, and is prone to overfitting of noises when the required reconstruction accuracy is set too high. On the other hand, when a tensor is not fully observed, alternating optimization methods, which update one TT-core in each iteration while keeping others fixed, are often adopted [17,18,19,20]. However, these methods require the knowledge of TT structure, or TT ranks explicitly, which unfortunately is unknown in practice. To bypass the challenge of presetting TT ranks, recent works [2,21] proposed a low-rank pursuing scheme, which targets at minimizing the weighted TT ranks, or incorporates the optimization of the TT ranks in a regularization term. While the tensor structure is not a prerequisite anymore, these schemes require fine-tuning the weightings in the objective function or regularization parameters, which may take different values under different applications, thus being very time-consuming in finding the right set of parameters. Therefore, it becomes a crucial problem to determine the TT model complexity, i.e., TT ranks, to avoid overfitting. To overcome this challenge, Bayesian method, which has been shown to effectively avoid noise overfitting without parameter fine-tuning [22,3,23,24], would be helpful. However, existing Bayesian models cannot be straightforwardly extended to the TT format because of TT's unique algebraic structure in the coupling of adjacent TT cores through the ranks. Fortunately, we might draw some inspirations from Bayesian Tucker Decomposition (TD) [24], where the ranks of different factor matrices are coupled in the central core tensor. In particular, extending the modeling in Bayesian TD, we adopt a Gaussianproduct-Gamma prior for the TT model, where the variance of each element of the TT cores is determined by two hyper Gamma distributed parameters. With the model for probabilistic TT decomposition established, variational inference is adopted to approximate the posterior distribution of the unknown parameters. The contribution of this paper includes: • A Gaussian-product-Gamma prior is adopted for tensor train representation. Different from previous works using similar models [24,25], it is proved for the first time that such model could theoretically lead to sparsity in TT slices, thus establishing the legitimacy of such modeling. • The proposed variational inference algorithm can automatically select the TT ranks, thus determining a suitable model complexity, which helps to achieve high recovery accuracy even in the presence of noise. • Extensive experimental results on synthetic data and real-world datasets have demonstrated the superior performance of the proposed algorithm in multiple applications, namely RGB and hyper-spectral image completion with random and structured missing data, and image classification, over the recent TT decompostion algorithms. A conference version of this work is in [26]. This paper significantly extends the results in [26] with the following technical and practical contributions: i) Thorough theoretical analyses and proofs are provided to explain the proposed probabilistic model. ii) Insights on the updating equations are provided, which inspires a new and straightforward strategy of rank selection with a convergence guarantee. iii) The complexity of the proposed algorithm is analysed, together with the derivation of an efficient update equation that substantially reduces the computation when the data are fully observed. iv) More comprehensive experimental assessments are presented, including validation of the accuracy of rank estimation on synthetic data, and image completion under different noise/missing patterns on different datasets. The rest of the paper is organized as follows. Section 2 gives a brief review on the TT decomposition and Bayesian methods. Section 3 analyzes the Gaussian-product-Gamma distribution and shows how it is adopted in the TT model. The inference algorithm is presented in Section 4. Numerical results are reported in Section 5, and the conclusions are drawn in Section 6. Notation: Boldface lowercase and uppercase letters are used to denote vectors and matrices, respectively. Boldface capital calligraphic letters are used for tensors. I n denotes the identity matrix with size n × n. Notation diag(a) is a diagonal matrix, with a on its diagonal. An element of a matrix or tensor is specified by the subscript, e.g., Y i,j,k denotes the (i, j, k)-th element of tensor Y, while Y :,:,k includes all elements in the first and second modes with k fixed at the third mode. The operator ⊗ denotes the Kronecker product, and • denotes the entry-wise product of two tensors with the same size. E[.] represents the expectation of the variables. N (µ, Σ) denotes the Gaussian distribution with mean µ and variance Σ, and Gamma(α, β) denotes the Gamma distribution with shape α and rate β. Preliminaries Tensor Train Decomposition A D-th order tensor X ∈ R J 1 ×...×J D can be represented in the tensor train format as X j 1 j 2 ...j D = R 2 r 2 =1 . . . R D r D =1 G (1) 1,r 2 ,j 1 . . . G (D) r D ,1,j D = G(1) :,:,j 1 G :,:,j 2 , . . . G (D) :,:,j D G (1) , G (2) , . . . , G (D) j 1 j 2 ...j D .(1) From the first line of (1), it is stated that the (j 1 , j 2 , . . . , j D )-th element of the D-th order tensor can be expressed as a multiple summation of the product of elements from a number of 3-dimensional tensors {G (d) ∈ R R d ×R d+1 ×J d } D d=1 , known as TT cores, with their size parameters {R d } D+1 d=1 termed as TT ranks (with R 1 and R D+1 fixed as 1). Furthermore, from the second line of (1), it can be seen that X j 1 j 2 ...j D can also be interpreted as consecutive matrix products among frontal matrix slices in the TT cores. For the TT ranks {R d } D+1 d=1 , they control the size of TT cores, and thus are deemed as the hyper-parameters that controls the model complexity. An example of the TT decomposition of a 3rd-order tensor is illustrated in Fig. 1. In particular, X 2,3,1 = G (1) :,:,2 G(2) :,:,3 G :,:,1 is highlighted in the figure, where the associated matrix slice index in the d-th TT core is the same as the d-th index of X 2,3,1 . Consequently, the goal of TT decomposition/completion is to learn 3dimensional tensor cores {G (d) } D d=1 from the D-dimensional tensor data Y, with the following optimization problem [16,17,18,19,20]: min G (1) ,G (2) ,...,G (D) O • Y− G (1) , G (2) , . . . , G (D) 2 F ,(2) where O is the observation tensor with its element being 1 if the corresponding data element is observed and 0 otherwise. Bayesian Model in Matrix and Tensor Factorization The Gaussian-Gamma distribution is commonly used in Bayesian modeling for matrix and tensor factorization. Taking matrix decomposition U = BA T as an example, in which U ∈ R M ×N , B ∈ R M ×L and A ∈ R N ×L . A common prior modeling is to assume each column of the factor matrix A and B follows a zero-mean Gaussian distribution with its precision following a Gamma distribution [22]. Mathematically, p(A) = L =1 N (A :, \|0, λ −1 I N ) p(B) = L =1 N (B :, \|0, λ −1 I M ), p(λ) = L =1 Gamma(λ \|α , β ),(3) in which λ serves as the precision of factors A :, and B :, . An important property of (3) is that the marginal distribution of A :, and B :, is a student's t distribution [27]. With α and β tending to zero, the marginal distribution will highly peak at zero and have a heavy tail, which indicates the prior belief that the many elements in A and B are zero. Moreover, during the inference procedure, it is found that very large λ leads A :, and B :, close to zero, which induces group sparsity in columns of A and B and therefore facilitates low-rank matrix decomposition. Similar modeling can be found in tensor canonical polyadic decomposition (CPD) [3,23], which is a high order extension of matrix decomposition. Applications using the above sparsity promoting prior modeling have shown that it can reduce noise overfitting and determine the model complexity at the same time. However, this model is not suitable for the TT decomposition, since matrix factorization or tensor CPD is only controlled by a single rank, while TT decomposition consists of multiple ranks {R 2 , . . . , R D }, with each rank constraining the dimensions of two adjacent TT cores. Take the 3rd-order tensor in Fig. 1 as an example, the intermediate TT core G (2) interacts with both G (1) and G (3) , with its size determined by R 2 as well as R 3 . To extend the idea in (3) to the TT decomposition, a strategy similar to Bayesian Tucker Decomposition [24] may be employed. The idea is that the precision of the coupled components is represented as multiplication of the Gamma variables. However, currently there is no theoretical guarantee for such modeling. Therefore, in the next section, we will first investigate such a model, which we term Gaussian-product-Gamma model, in the uni-variate case, then apply it on TT decomposition and see how TT decomposition benefits from it. Probabilistic Model for tensor train representation Theoretical Analysis for Gaussian-Product-Gamma Model As discussed in Section 2.2, to extend the traditional Gaussian-Gamma prior for TT decomposition, the Gaussian-product-Gamma prior will be adopted, with its univariate form given by p(x\|λ 1 , λ 2 ) = N (x\|0, (λ 1 λ 2 ) −1 ), p(λ 1 ) = Gamma(λ 1 \|α 1 , β 1 ), p(λ 2 ) = Gamma(λ 2 \|α 2 , β 2 ).(4) Although empirically shown to be effective in denoising and rank selection [24,25], the effect of the product of precision parameters on the Gaussian-Gamma prior pair has never been theoretically analyzed. As the prior (4) will be assigned to each element of the TT cores in the next subsection, we start to gain more insights into this prior by establishing the following propositions. Proposition 1. The Gamma distribution is conditionally conjugate to the Gaussian distribution in (4). That is, p(λ 1 \|x, λ 2 , α 1 , β 1 , α 2 , β 2 ) = Gamma(λ 1 \|α 1 + 1 2 , λ 2 2 x 2 + β 1 ),(5) and a similar result holds for λ 2 . While (5) is not the full conjugate property, it still ensures a closed-form update in the inference procedure. Proof: Firstly, multiplying the equations in (4), the joint distribution of x, λ 1 and λ 2 is obtained as P (x, λ 1 , λ 2 \|α 1 , β 1 , α 2 , β 2 ) = β α 1 1 β α 2 2 √ 2πΓ(α 1 )Γ(α 2 ) λ α 1 − 1 2 1 λ α 2 − 1 2 2 × exp − ( λ 1 λ 2 2 x 2 + β 1 λ 1 + β 2 λ 2 ) .(6) Then λ 1 can be integrated out by observing that the joint distribution (6) with respect to λ 1 follows the form of a Gamma distribution, with shape and rate parameters being α 1 + 1 2 and λ 2 x 2 2 + β 1 respectively, and we obtain P (x, λ 2 \|α 1 , β 1 , α 2 , β 2 ) = β α 1 1 β α 2 2 λ α 2 − 1 2 2 Γ(α 1 + 1 2 ) √ 2πΓ(α 1 )Γ(α 2 ) ( λ 2 2 x 2 + β 1 ) −(α 1 + 1 2 ) × exp − β 2 λ 2 .(7) Finally, (5) is obtained by dividing (6) by (7). Proposition 2. The prior distribution of x in model (4) is p(x\|α 1 , β 1 , α 2 , β 2 ) = β α 1 1 β α 2 2 Γ(α 1 + 1 2 ) √ 2πΓ(α 1 )Γ(α 2 ) exp − β 2 λ 2 λ α 2 − 1 2 2 ( ( √ λ 2 x) 2 2 + β 1 ) −(α 1 + 1 2 ) dλ 2 .(8) Proof: Proposition 2 is obtained by further integrating out λ 2 in (7). Proposition 3. With the hyperparameter set {α 1 , β 1 } or {α 2 , β 2 } tends to 0, the prior distribution of x would be proportional to 1 \|x\| , which highly peaks at zero, and has a heavy tail. Proof: The result is obtained by observing that the term ( √ λ 2 x) 2 + β 1 ) −(α 1 + 1 2 ) in (8) tends to 1 √ λ 2 \|x\| as α 1 and β 1 both tend to 0. A similar result holds when α 2 and β 2 tend to zero, as another valid form of (8) can be obtained by exchanging the positions of {λ 1 , α 1 , β 1 } and {λ 2 , α 2 , β 2 }, which corresponds to the integration of (6) firstly with respect to λ 2 , and λ 1 . Proposition 3 can be understood from two perspectives. On one hand, the prior distribution highly peaks at zero, revealing the initial belief that x is most likely to be zero. On the other hand, it has heavy tails, which allows x to take a very large value. Moreover, if such modeling is applied to a set of variables, then most of the variables will become zero, while the remaining few take large values, thus inducing sparsity. An illustration of Property 3 is shown in Fig. 2, which plots the prior distribution of x when α 1 = α 2 = 10 −6 , β 1 = β 2 = 10 −6 . Likelihood and Priors for the Tensor Train Model The observed tensor can be expressed as A = O • (Y + W),(9)where Y = G (1) , G (2) , . . . , G (D) is the ground-truth tensor which is assumed to be in the TT format, and W is a noise tensor, with each element modeled as independent and identically distributed (i.i.d.) Gaussian variables with mean 0 and precision τ (i.e., the inverse of the variance). Correspondingly, the logarithm of the likelihood function of the observed tensor is ln p(A\|O, {G (d) } D d=1 , τ ) = \|Ω\| 2 ln τ − τ 2 O • A− G (1) , G (2) , . . . , G (D) 2 F + const,(10) where Ω denotes the set of indices of the observed entries, and \|Ω\| denotes the cardinality of Ω, which equals the number of observed entries. It is not difficult to see that the result of maximizing (10) will be the same as that of solving problem (2), regardless what the noise variance τ −1 is. However, as discussed in the last section, minimizing the square error works well only when the tensor train ranks are suitably chosen. Thus, instead of maximizing the log-likelihood, we build a hierarchical probabilistic model by treating {G (d) } D d=1 as variables, which enhances the expressive power of the model, and allows automatic rank determination. As has been discussed in the last section, the Gaussian-product-Gamma prior (4) can be adopted for the TT cores to induce sparsity: p(G (d) \|λ (d) , λ (d+1) ) = L d k=1 L d+1 =1 N G (d) k, ,: \|0, (λ (d) k λ (d+1) ) −1 I J d , ∀d ∈ {1, 2, . . . , D},(11)p(λ (d) \|α (d) , β (d) ) = L d k=1 Gamma(λ (d) k \|α (d) k , β (d) k ), ∀d{2, . . . , D},(12) where (1) and λ (D+1) are scalars and set as 1 so that the expression in (11) is applicable for the first and last TT cores. Furthermore, λ (d) = [λ (d) 1 , . . . , λ (d) L d ] for d = 2, . . . , D, λα (d) = [α (d) 1 , . . . , α (d) L d ] and β (d) = [β (d) 1 , . . . , β (d) L d ] for d = 2, . . . , D are hyperparameters of the Gamma distributions, with probabilistic density function Gamma(x\|α, β) = β α x α−1 e −βx /Γ(α). In (11) (11) is depicted in Fig. 3. For the noise precision τ in (10), we model it as a Gamma distribution with hyperparameters α τ and β τ : p(τ \|α τ , β τ ) = Gamma(τ \|α τ , β τ ).(13) Since we have no information about the distribution of {λ (d) } and τ , we (12) and (13) non-informative [28]. The hierarchical probabilistic model for the TT decomposition is shown in Fig. 4. Notice that the Gaussian-product-Gamma prior for the TT cores is more complicated than that in (4) since the λ (d+1) k in (11) and (12) controls the k-th lateral slice of G (d) and the k-th horizontal slice of G (d+1) , while λ 1 and λ 2 in (4) only controls a univariate x. Despite the difference, the sparsity promoting property still exists as shown in the next subsection. set α (d) = β (d) = 10 −6 × 1 L d ×1 and α τ = β τ = 10 −6 to make Sparsity Analysis of the TT Model To see how the Gaussian-product-Gamma prior modeling benefits the TT decomposition, the marginal distribution of a horizontal or lateral slice of the TT-core can be investigated, and the following proposition holds. λ (2) λ (3) . . . model (11) and (12) become λ (D) α (2) , β (2) α (3) , β (3) . . . α (D) , β (D) G (1) G (2) . . . G (D−1) G (D) W τ ατ , βτ A . . .p(G (d) k,:,: ) ∝ L d+1 =1 ( J d j d G (d) k, ,j d 2 ) − J d 2 ,(14)p(G (d+1) :,m,: ) ∝ L d+1 =1 ( J d j d G (d+1) ,m,j d 2 ) − J d 2 ,(15) respectively. Proof: Take the row slice G (d) k,:,: as an example. The joint distribution of all the TT cores and hyperparameters is firstly obtained by multiplying (11) and (12). Integrating out irrelevant TT cores and {λ (n) \|n = d or d + 1}, the following distribution can be obtained p(G (d) k,:,: , λ (d) k , λ (d+1) ) ∝ λ (d) k J d 2 +α (d) k −1 exp − β (d) k λ (d) k × L d+1 =1 λ (d+1) J d 2 +α (d+1) −1 × exp − ( λ (d) k J d j d G (d) k, ,j d 2 2 + β (d+1) )λ (d+1) ,(16) which is organized to reveal that each λ (d+1) follows a Gamma distribution with a new set of hyperparameters. Further integrating out all λ (d+1) , the k is obtained as p(G (d) k,:,: , λ (d) k ) ∝ λ (d) k J d 2 +α (d) k −1 exp − β (d) k λ (d) k × L d+1 =1 ( λ (d) k J d j d G (d) k, ,j d 2 2 + β (d+1) ) −( J d 2 +α (d+1) ) .(17) It is observed from (17) that when α (d+1) and β (d+1) tend to zero, we obtain (14). , thus reducing the model complexity. A demonstration of (14) with L d+1 = 2 is shown in Fig. 5a, and as a comparison, the marginal distribution under standard Gaussian prior is shown in Fig. 5b. Inference Algorithm The goal of Bayesian inference is to find the posterior distribution of the unknown variables Θ : = {G (d) } D d=1 , {λ (d) } D d=2 , τ , and subsequently the marginal distribution of each of the variables. However, since the marginal distribution p(A) = p(A, Θ)dΘ is difficult to compute due to the complex model, the posterior p(Θ\|A) = p(A, Θ)/p(A) is also intractable. To bypass this problem, we turn to variational inference (VI), which uses a variational distribution q(Θ) to approximate the true posterior p(Θ\|A), by minimizing their Kullback-Leibler (KL) divergence: min q(Θ) KL q(Θ) \|\| p(Θ\|A) = q(Θ) ln q(Θ) p(Θ\|A) dΘ.(18)Mean-field approximation q(Θ) = S s=1 q(Θ s ) with Θ s ⊂ Θ, ∪ S s=1 Θ s = Θ, and Θ s ∩ Θ t = ∅ for s = t, is commonly adopted to simplify (18). Under this approximation, the optimal variational distribution of each variable set Θ s can be obtained as [29, pp. 737]: ln q * (Θ s ) = E Θ\Θs [ln p(A, Θ)] + const,(19) where E Θ\Θs means expectation with respect to Θ except Θ s . It is obvious that when computing for q * (Θ s ), we need to know q(Θ t ) where t = s. Therefore, the variational distribution needs to be updated alternatingly for different s. Since (19) is a convex problem with respect to each q(Θ s ) [30, pp. 466], the convergence of the iterative updating procedure is guaranteed. For the proposed TT model, we impose the mean-field approximation as q(Θ) = q(τ ) D+1 d=1 q(λ (d) ) D d=1 L d k=1 L d+1 =1 q(G (d) k, ,: ). Following (19), the closed-from update for each variable set can be derived, with the key steps sketched below and the whole algorithm summarized in Algorithm 1. Update G (d) from d = 1 to D Each fiber of the TT core follows a Gaussian distribution q(G (d) k, ,: ) = J d j d =1 N (E[G (d) k, ,j d ], υ G (d) k, ,j d ) . The variance and mean of each element from the filber can be derived as υ G (d) k, ,j d = E[τ ] j 1 ,j 2 ,...,j D \j d O j 1 j 2 ...j D E[b (<d) (k−1)L d +k ]E[b (>d) ( −1)L d+1 + ] + E[λ (d) k ]E[λ (d+1) ] −1 ,(20)E[G (d) k, ,j d ] = υ G (d) k, ,j d E[τ ] j 1 ,j 2 ,...,j D \j d O j 1 j 2 ...j D A j 1 j 2 ...j D E[t (<d) k ]E[t (>d) ] − L d k =1 k =k L d+1 =1 = E[b (<d) (k−1)L d +k ]E[G (d) k , ,j d ]E[b (>d) ( −1)L d+1 + ] ,(21) in which we used the following notations to make the expression more concise, E[t (<d) ] = d−1 n=1 E[G (n) :,:,jn ],(22)E[b (<d) ] = d−1 n=1 E[G (n) :,:,jn ⊗ G (n) :,:,jn ],(23) and E[t (>d) ] and E[b (>d) ] in a similar fashion. Update λ (d) from d = 2 to D The variational distribution of λ (d) k is a Gamma distribution q(λ (d) k ) = Gamma(α (d) k ,β (d) k ), with E[λ (d) k ] =α (d) k /β (d) k and α (d) k = J d L d+1 2 + J d−1 L d−1 2 + α (d) k ,(24)β (d) k = 1 2 J d j d =1 L d+1 =1 (E[G (d) k, ,j d 2 ]E[λ (d+1) ]) + 1 2 J d−1 j d−1 =1 L d−1 =1 (E[G (d−1) ,k,j d−1 2 ]E[λ (d−1) ]) + β (d) k .(25) Update τ The variational distribution of τ is a Gamma distribution q(τ ) = Gamma(α τ ,β τ ), } D d=2 , {β (d) } D d=2 , α τ , β τ ; while Not Converged do Update the TT cores via (20) and (21); (24) and (25); Update τ via (26) and (27); Rank selection; end Update {λ (d) } D d=2 viawith E[τ ] =α τ /β τ , andα τ = \|Ω\| 2 + α τ ,(26)β τ = 1 2 O • A 2 F − 2 J 1 j 1 =1 . . . J D j D =1 O j 1 ...j D A j 1 ...j D D d=1 E[G (d) :,:,j d ] + J 1 j 1 =1 . . . J D j D =1 O j 1 ...j D D d=1 E[G (d) :,:,j d ⊗ G Further discussion Initialization. Since we set the hyperparameters of the prior Gamma dis- tributions {α (d) } D d=2 , {β (d) } D d=2 , α τ , β τ to be 10 −6 , the initialization of other variables can be set accordingly. In particular, the initialization of E[λ = E[λ (d) k ]E[λ (d+1) ]I J d = I J d . For the mean of the TT-cores, we complete the observed tensor with random values drawn from N (0, 1), decompose it using TT-SVD, and apply the results as the initialization. For the initial ranks, we adopt the maximal possible ranks r max ∈ R (D+1)×1 , with its d-th element (r max ) d being the rank of the unfolding matrix A [d] ∈ R d d =1 J d × D d =d+1 J d [16], whose [j 1 + d d =2 (j d − 1) d −1 i=1 J d −i , j d+1 + D d =d+2 (j d − 1) d −d−1 i=1 J d −i ] -th element is the [j 1 , . . . , j D ]-th element in A. However, since the middle element of r max commonly grows exponentially with respect to the order of the tensor, which might be too big for high order tensor, we further set an upper bound for the initial TT rank, which is 15 times the tensor dimension. Consequently, the initial TT rank is chosen as L d = min((r max ) d , 15J d ). Rank Selection. As can be seen from (20), the variance of the TT cores is affected by {λ (d) } D d=2 . If λ (d) k is large enough, the variance of all elements in both slices G will tend to zero. Moreover, (21) shows that a small variance will further force the mean of a slice to be zero, and in such a way {λ (d) } contributes to rank selection. From another perspective, it can be seen from (25) that the update of λ becomes very large, thus acts like an indicator that decides whether to prune the slices. In practice, the rank selection can be implemented by discarding the slices whose λ (d) k is much larger than others, e.g., 100 times larger than the smallest one. Convergence Analysis As discussed in the begining of this section where the VI algorithm is introduced, the convergence of the proposed algorithm is guaranteed. Moreover, the rank pruning can be done after each iteration while not affecting the convergence of the algorithm, since every time when a slice is eliminated, it is equivalent to restarting the VI algorithm with a smaller model size and with the current variational distribution serving as a new initialization. O ( D d=1 J d + D 2 )\|Ω\|(L 2 + L 4 ) . Though the complexity could be large if the initial TT ranks are large, it should be noticed that pruning is operated in each iteration of the algorithm, thus the computational burden at later iterations would be smaller, especially for tensors with low true TT-ranks. Efficiency improvement. According to the complexity analysis above, in Algorithm 1, (20) and (27) constitutes a great part of the complexity, due to the sum of products of the elements from O and the Kroneckers of the TT cores. Especially, when the number of observed elements grows, the complexity grows as well, which leads to a huge computational burden when there is no missing data. Below, we reveal that this problem can be solved by exchanging the execution order of the summation and the product when a tensor is fully observed. Firstly, the most computationally part from (20) is taken out as follows, c(j d ) = j 1 ,j 2 ,...,j D \j d O j 1 j 2 ...j D E[b (<d) (k−1)L d +k ] × E[b (>d) ( −1)L d+1 + ],(28) which takes time complexity O(\|Ω\|DL 4 ). Substitute (23) into the above equation, the following equation is obtained, c(j d ) = j 1 E[G(1) :,:,j 1 ⊗ G :,:,j 1 ] j 2 E[G(2) :,:,j 2 ⊗ G :,:,j 2 ] . . . in which we further change the sequence of the product of the kroneckers and the summation. Then it can be shown that when a tensor is fully observed (i.e., O j 1 j 2 ...j D = 1 for any indice), we have c(j d ) = j 1 E[G(1) :,:,j 1 ⊗ G :,:, j 1 ] × j 2 E[G(2) :,:,j 2 ⊗ G :,:, j 2 ] × . . . × j D E[G (D) :,:,j D ⊗ G (D) :,:,j D ] ,(29) as j 1 , j 2 , . . . , j D are not coupled anymore. The overall complexity to calculate (29) is O(DL 4 ), which is a great efficiency improvement compared to calculating (28) directly. Furthermore, when a large percentage of a tensor is observed, e.g., 90 percent, (28) can be calculated by subtracting extra factors from (29), which takes time complexity O(( D d=1 J d − \|Ω\|)DL 4 )) and is also more efficient. Numerical Experiments Validation on Synthetic Data We first test the capability of the proposed algorithm to estimate the TT-ranks. A synthetic tensor Y = G (1) , G (2) , G (3) with size [20,20,20] is considered, where each element of G (d) ∈ R R d ×R d+1 ×20 is drawn from a normal distribution N (0, 1). Since the synthetic tensors are of order 3 and R 1 = R 4 = 1, we only need to determine the second and third TT-ranks. The original tensor is contaminated by additive Gaussian noise tensor W, with W j 1 j 2 j 3 ∼ N (0, σ 2 ). The observed tensor is A = O • (Y + W), where O is an indicator tensor with its elements drawn from the binomial distribution with certain missing rate. The performance of the proposed probabilistic TT model is tested under different signal-to-noise ratios (SNR) defined as 20 log( A F / W F ), missing ratios and true TT-ranks. Each testing condition is simulated for 100 times. The results are presented in bar figures with the average estimated TT-ranks represented by the heights of the bars, and a pair of lines showing the one standard deviation. Furthermore, the estimation accuracy is also shown on the top of each figure. Fig. 6a shows the TT ranks estimation results under different missing rates. The true TT ranks are set as [1,5,5,1], and the SNR is set as 20dB. It can be seen that the estimation accuracy is higher than 90% in all cases, and achieves 100% when missing rate is equal or less than 20%. When the accuracy is not 100%, the standard deviation becomes larger as the missing rate becomes higher. The reason is that under higher missing rate, when the rank estimation goes wrong, it would give a rank that differs a lot from 5, like 20, while under lower missing rate, the wrongly estimated rank is commonly around the true rank 5. Fig. 6b shows the rank accuracy under different SNRs, with the true TT ranks [1, 5, 5, 1] and missing rate 0%. It is observed that when the SNR is equal or larger than 5dB, the rank accuracy is 100%. However, when the SNR becomes 0dB, the proposed method fails to give accurate estmation of the TT ranks, which is understandable as the strong noise masks the underlying TT structure. Fig. 6c shows the estimated rank accuracy under different true TT ranks, with the missing rate set as 20% and the SNR set as 20dB. When the true TT ranks are [ accuracy. When the TT ranks are set as [1,15,15,1] and [1,20,20,1], the estimation accuracy becomes lower, and generally the estimated ranks are smaller than the true ranks. The same phenomenon was also observed in Bayesian matrix decomposition [31] and CPD [32], and the reason might be the strong sparsity promoting property of the Gaussian-Gamma model, and the adopted Gaussian-product-Gamma model inherits such property. Next, we examine the tensor recovery ability using synthetic data, where Table 1 shows the relative standard error (RSE) of the recovered tensorÂ: A −Â F / A F . In particular, the effects of the missing rate (with SNR = 20dB, TT ranks = [1,5,5,1]), SNR (with 0% missing rate, TT ranks = [1,5,5,1]), and the true TT ranks (with 20% missing rate, SNR = 20dB) are shown in Table 1a, 1b and 1c respectively. The results of the proposed method is compared with those of sparse tensor-train optimization (STTO) [4], simple low-rank tensor completion via TT (SiLRTC-TT) [2], tensor completion by parallel matrix factorization via TT (TMAC-TT) [2] and TT-SVD [16]. In addition, the tensor recovery performance of a non-TT-format tensor completion method -fully Bayesian CP factorization (FBCP) [3] is also compared. The tuning parameters of the above algorithms have been finely tuned to achieve the best performance. From Table 1, it can be seen that the proposed method outperforms other methods in all cases. In particular, TT-SVD is not applicable when there are missing entries, as shown in Table 1a, while STTO, SiLRTC-TT, and TMAC-TT perform poorly when there is noise, as shown in Table 1b. For STTO, when there is little noise and the true ranks match its rank parameters, its performance is better than other optimization-based TT methods (i.e., SiLRTC-TT and TMAC-TT), but is still not as good as the performance of the proposed method, as Table 1a shows. On the other hand, when the true ranks are different from its rank parameters, STTO performs much worse than other methods, as shown in Table 1c. Finally, for FBCP, its performance is poor when the missing rate is high or the TT ranks are high, which is due to the mismatch between the assumed CPD format in FBCP and the more complicated TT structure in the synthetic data. Image Completion In this subsection, the results of image completion experiments on RGB images and hyperspectral images (HSI) are presented. The performance of the proposed algorithm is compared with those of SiLRTC-TT [2], TMAC-TT [2], STTO [4], TTC-TV [33], FBCP [3], and FaLRTC [1], with their parameters fine-tuned to achieve the best performance. Before performing image completion, we adopt the tensor augmentation on the images, which is firstly introduced in [34], and further proved to be effective in image completions in [2]. The core concept of tensor augmentation is to fold a matrix into a high-order tensor. In particular, given a matrix A ∈ R M ×N , its dimension M and N can be written as M = f i=1 M i and N = f i=1 N i for some integers {M i } and {N i }. Then, we construct a tensor with elements A (m 1 n 1 ),(m 2 n 2 ),...,(m f n f ) = A m 1 m 2 ...m f ,n 1 n 2 ...n f , and finally get an f -th order tensor A ∈ R M 1 N 1 ×M 2 N 2 ×...×M f N f . Tensor augmentation is equivalent to have the original matrix cut into multiple small matrix blocks, and then treat each matrix block as a fiber of the reordered tensor. An example of tensor augmentation is illustrated in the upper part of Fig. 7. In our experiments, we further improve the tensor augmentation by padding the images and using overlapping windows, which is shown in the bottom part of Fig. 7. In particular, the boundaries of the image is first replicated. Then, instead of folding the basic M 1 × N 1 matrix block, overlapping (M 1 + 1) × (N 1 + 1) matrix blocks (with vertical stride M 1 and horizontal stride N 1 ) is folded as columns of the new tensor. The reason for doing so is that the neighbourhood information of the separated matrix blocks will be retained after reordering the entries. Random missing observations In this experiment, 13 RGB images with size 256 × 256 × 3 are tested, with 80 percent randomly missing. We add boundary padding to the original image of size 2 8 × 2 8 × 3, and then reshape it to a 9-dimensional tensor with size 16 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 3. Table 2 and Table 3 show the peak signal-to-noise ratio (PSNR) defined as 20 ln max(A) √ 3M N /\|\|A −Â\|\| F and the structural similarity index measure (SSIM) [35] of the recovered images under no noise and under Gaussian noise with variance 0.1, respectively. It can be seen that the proposed algorithm achieves the overall best performance in terms of both PSNR and SSIM, no matter there is noise or not. When there is no noise, in most cases the proposed algorithm recovers images with more than 1dB higher PSNR than other algorithms. Furthermore, for the 'airplane' and 'jellybeans' figures, the improvement is over 3dB and 5dB in PSNR respectively compared to the second best, and the better PSNR is also visually evident in the recovered images shown in Fig. 8. When there is Gaussian noise with variance 0.1, the performance of the proposed algorithm achieves about 2dB higher PSNR and more than 0.1 higher SSIM than the second best, and recovers recognizable figures as shown in Fig. 8. One thing worth to notice is that SiLRTC-TT, TMAC-TT and FaLRTC all setÂ = A as a constraint in their optimization schemes. Thus, they inherently cannot handle noisy data, and this is clearly corroborated in the poorly recovered images in Fig. 8. Table 4 shows the MPSNR and MSSIM of the recovered images, which are the mean PSNR and SSIM for all bands, respectively. It can be seen that the proposed method outperforms the competitors, especially when compared with the non-Bayesian methods like TMAC-TT or FaLRTC, since they cannot handle noise. For FBCP, though it has the ability to overcome noise and achieves similar MPSNR and MSSIM with the proposed method, it tends to generate over-smoothed images. As can be seen from Fig.9, the rift of the yellow feather can be barely seen from the recovered image by FBCP, while it is obvious for the other recovered images. Stripe missing observations Image Classification In this subsection, we present the performance of the proposed algorithm on image classification. In particular, after TT decomposition, the TT cores of the decomposed data are fed into the support tensor train machine (STTM) [7], which extends the support vector machine (SVM) to tensors and adopts TT-structured weighting parameters for classification. The original STTM uses TT-SVD to decompose data for training and testing, while in our experiments the proposed algorithm is adopted. Additionally, the performance of SVM is also tested for comparison. The cifar-10 dataset (https://www.cs.toronto.edu/~kriz/cifar.html) [36] is used, from which different sample sizes are used for training (specified in Fig. 10) and 1000 images are used for testing. Three scenarios are considered: clean data for training and testing, noisy data for training and testing, training on clean data and testing on noisy data. Gaussian noise with zero mean and variance 0.01 is added to generate the noisy data. Since both STTM and SVM are binary classifiers, but there are totally 10 classes in the dataset, we perform classification for every two classes, and calculate the average error rate. The results are presented in Fig. 10. From Fig. 10a it can be seen that for the clean data, STTM combined with TT-SVD and the proposed algorithm achieve similar performance, under all training sample sizes, and both outperform SVM. However, as illustrated in Fig. 10b and 10c, when there is noise, no matter in both training and testing data, or in only the testing data, STTM combined with TT-SVD performs much worse than the proposed algorithm. This is because that the proposed algorithm can recover the underlying TT structure of the data even under noise perturbation, while TT-SVD cannot. Conclusions In this paper, a probabilistic TT model, with the capability of automatic rank determination was proposed to avoid noise overfitting. The legitimacy of the proposed model was verified by establishing the sparsity promoting property of the adopted Gaussian-product-Gamma prior. Learning algorithm was derived under variational inference framework. Simulation results on synthetic data demonstrated the ability of the proposed algorithm to accurately recover the underlying TT structure from incomplete noisy data. Further-more, experiments on image completion and image classification showed the proposed algorithm leads to higher recovery or classification accuracy than other state-of-the-art TT decomposition algorithms. Appendix A. Derivation for the VI algorithm In this section, we present the derivation of the variational inference (VI) algorithm for the proposed TT model. Especially, the detailed expression of the expectation terms are provided. Following the probabilistic model proposed in Section 3.2, the logarithm of the joint distribution of the observed tensor and the unknown variables is ln (p(A, Θ)) = \|Ω\| 2 ln τ − τ 2 O • A− G (1) , G (2) , . . . , G (D) 2 F + 1 2 D d=1 L d k L d+1 J d ln(λ (d) k λ (d+1) ) − λ (d) k λ (d+1) J d j d =1 G (d) k, ,j d 2 + D d=2 L d k=1 α (d) k − 1 ln λ (d) k − β (d) k λ (d) k + (α τ − 1) ln τ − β τ τ + const. (A.1) where A is the observed tensor, O is the indicator tensor, and Θ defined as {G (d) } D d=1 , {λ (d) } D d=2 , τ . To learn the posterior distribution of the unknown variables, the mean field variational inference (MFVI) is adopted, which assumes that the non-overlapping subsets {Θ s \|Θ s ⊂ Θ, ∪ S s=1 Θ s = Θ, Θ s ∩ Θ t = ∅ for s = t} are mutually independent. As discussed in Section 4, the optimal variational distribution of Θ s is ln q * (Θ s ) = E Θ\Θs [ln p(A, Θ)] + const. (A.2) By alternatively updating Θ s by (A.2), an approximation of the posterior is obtained when it converges. For the proposed TT model, we impose the mean-field approximation as q(Θ) = q(τ ) D+1 d=1 q(λ (d) ) D d=1 L d k=1 L d+1 =1 q(G (d) k, ,: ), and the optimal variational distribution at each iteration is derived as below. Update q(G − τ 2 O • A− G (1) , G (2) , . . . , G (D) 2 F − 1 2 λ (d) k λ (d+1) J D j d =1 G (d) k, ,j d 2 + const. (A.3) For the Frobenius norm, it can be expanded as where the subscripts of t (<d) , t (>d) , b (<d) , and b (>d) are decided by finding the coupled factors with G (d) k, ,j d . Since different cores are assumed to be independent in the mean-field approximation, it is obtained that Update q(λ (d) ) for d ∈ {2, . . . , D} O • A− G (1) , G (2) , . . . , G (D) 2 F = J 1 j 1 =1 . . . J D j D =1 O j 1 ...j D − 2A j 1 ...j D G(1)E[t (<d) ] = E[G The update for q(λ (d) ) is calculated by substituting (A.1) into (A.2) with Θ s set as λ (d) : ln q λ (d) = const + L d k=1 E Θ\λ (d) ln λ (d) k J d L d+1 2 + J d−1 L d−1 2 + α (d) k − 1 − 1 2 J d j d =1 L d+1 =1 G (d) k, ,j d 2 λ (d+1) + 1 2 J d−1 j d−1 =1 L d−1 =1 G (d−1) ,k,j d−1 2 λ (d−1) + β (d) k λ (d) k . (A.11) From (A.11) it is observed that q(λ (d) ) = L d k=1 Gamma(λ 13) in which E[G 2) with Θ s set as τ , the variational distribution for τ is as follows, (d) k \|α (d) k ,β (d) k ) with parametersα (d) k = J d L d−1 2 + J d−1 L d−1 2 + α (d) k , (A.12) β (d) k = 1 2 J d j d =1 L d+1 =1 (E[G (d) k, ,j d 2 ]E[λ (d+1) ]) + 1 2 J d−1 j d−1 =1 L d−1 =1 (E[G (d−1) ,k,j d−1 2 ]E[λ (d−1) ]) + β (d) k , (A.ln q (τ ) = E Θ\τ − 1 2 O • A 2 F − 2 J 1 j 1 =1 . . . which clearly shows that τ obeys a Gamma distribution, with parameterŝ α τ = \|Ω\| 2 + α τ , (A.15) β τ = 1 2 O • A 2 F − 2 J 1 j 1 =1 . . . J D j D =1 O j 1 ...j D A j 1 ...j D × D d=1 E[G Figure 1 : 1An example of the tensor train decomposition. Figure 2 : 2Marginal distribution (8) with α 1 = α 2 = 10 −6 , β 1 = β 2 = 10 −6 . Figure 3 : 3Coupling of {λ (d) } in the TT model. and (12), {L d } D d=1 are the assumed TT ranks, which are chosen as large numbers such that automatic selection of important tensor core slices G ,: is possible during model inference. A simple demonstration of the coupling in Proposition 4 . 4With α (d+1) and β (d+1) tend to zero for all , the marginal distribution of the horizontal slice G (d) k,:,: and the lateral slice G (d+1) :,m,: under Figure 4 : 4Probabilistic model for the TT decomposition. Figure 5 : 5Demonstration of the marginal distribution of the slices of TT cores under different prior modeling. Proposition 4 indicates that with very small hyperparameters {α (d+1) } and {β (d+1) } for from 1 to L d+1 , the probabilistic density of the prior of the TT core slices G (d) k,:,: and G (d+1):,m,: will mostly concentrate around zero, which leads the posterior of the TT cores to flavor sparse solutions. Moreover, as can be seen in next section, sparsity of slices G :,j d ] + β τ . α τ /β τ = 1, respectively. Consequently, the covariance of the TT-cores is initialized as υ G (d) k, ,: 4. 2 . 2Complexity analysis and efficiency improvement with fully observed tensors Complexity Analysis. The complexity of the proposed algorithm comes from the update of{G (d) } D d=1 , {λ (d) } D d=1and τ . For simplicity, we suppose that all TT-ranks are initially set as L. For the updating of each TT core G (d) , it takes O DL 2 to get t (<d) and t (>d) , O DL 4 to get b (<d) and b (>d) , then O \|Ω\|DL 2 to get the mean and O \|Ω\|DL 4 to get the variance of a TT core in(21) and(20) respectively. Furthermore, each λ (d) requires O (J d + J d−1 )L 2 to compute, and τ requires O \|Ω\|D(L 4 + L 2 ) to compute. It can be seen that the complexity is dominated by computing {G (d) } D d=1 , and the overall complexity is :,j D ] . . . , Figure 6 : 6Performance of rank estimation under different conditions. Figure 7 : 7Illustrations on tensor augmentation. Figure 8 : 8Examples of the recovered images under different noise settings. From left to right: The original image, the observed image, the images recovered by: the proposed algorithm, SiLRTC-TT, TMAC-TT, STTO, TTC-TV, FBCP, FaLRTC. The first three lines: 'airplane', 'jellybeans', and 'splash' with 80% missing and no noise. The last two lines: 'sailboat' and 'tree' with 80% missing and noise variance 0.1. Figure 9 : 9Recovered 'feathers' HSIs using different tensor completion methods, with bands[25,15,5] set as the R, G, B layers. From left to right: The original image, the mask, the observed image, the images recovered by: the proposed algorithm, TMAC-TT, STTO, FBCP, FaLRTC. using clean data and testing on noisy data. Figure 10 : 10Testing Accuracy with respect to training size. Acknowledgement Funding: This work was supported by the National Natural Science Foundation of China [grant number 62001309]; the Guangdong Basic and Applied Basic Research Foundation [grant number 2019A1515111140]; and the General Research Fund (GRF) from the Hong Kong Research Grant Council [project number 17206020 and 17207018]. ,: ) for d ∈ {1, . . . , D}, k ∈ {1, . . . , L d }, ∈ {1, . . . , L d+1 } Substituting (A.1) into (A.2) with Θ s set as G in (A.3), the expectation is taken over all variables except G :,j D ) 2 , since the slices of different TT cores are multiplied and then squared. Fortunately, notice that (G :,j D ). Using the mixed-product property of theKronecker product (A 1 . . . A D ) ⊗ (B 1 . . . B D ) = (A 1 ⊗ B 1 ) . . . (A D ⊗ B D ),...,j D \j d O j 1 ...j D − 2A j 1 ...j D G distribution.Moreover, it can be seen that the elements indexed by different j d are independent of each other, and thus can be updated separately. Therefore, ,j 2 ,...,j D \j dO j 1 j 2 ...j D E[b (<d) results hold for t (>d) and b(>d) . For the expectation of G :,je ])] is the Kronecker-form covariance. Again, because q(G (e) block matrices with dimension L d × L d+1 and the (k, )-th block matrix contains only one nonzero element υ G (e) k, ,je at the (k, )-th position. be obtained by picking up the (kL d + k, L d+1 + )the property of the Gamma distribution [37, pp. 70]. Update q(τ ) Substitute (A.1) into (A. O j 1 ...j D A j 1 .. :,j d ] is as (A.10). 1,5,5,1] and[1,10,10,1], the proposed method gives 100%100% 100% 100% 100% 100% 93% 99% 92% 97% 95% 0 0.2 0.4 0.6 0.8 Missing rate 0 2 4 6 8 10 Estimated TT-rank 2nd rank 3rd rank (a) with respect to missing rate (SNR = 20dB, true TT ranks = [1, 5, 5, 1]). 0% 0% 100% 100% 100% 100% 100% 100% 0 5 10 15 SNR(dB) 0 5 10 15 20 25 Estimated TT-rank 2nd rank 3rd rank (b) with respect to SNR (0% missing rate, true TT ranks = [1, 5, 5, 1]). 100% 100% 100% 100% 9% 87% 0% 40% 5 10 15 20 True TT ranks 0 5 10 15 20 25 Estimated TT-ranks 2nd rank 3rd rank (c) with respect to the true TT-ranks (20% missing rate, SNR = 20dB). Table 1 : 1RSE = A −Â F / A F of tensor recovery from incomplete and noisy observed tensors. a) with respect to the missing rate (SNR = 20dB, true TT ranks =[1,5,5,1]). with respect to TT ranks (20% missing rate, SNR = 20dB).(missing rate 0% 20% 40% 60% 80% proposed method 8.22e-4 1.10e-3 1.50e-3 2.60e-3 4.12e-2 STTO 1.00e-2 8.30e-3 6.90e-3 6.00e-3 6.89e-2 SiLRTC-TT 1.00e-2 1.00e-2 1.58e-2 7.38e-2 5.27e-1 TMAC-TT 1.00e-2 1.63e-2 2.44e-2 1.26e-2 5.45e-1 FBCP 2.20e-3 2.60e-3 4.00e-3 7.60e-3 3.26e-1 TT-SVD 9.80e-3 Not applicable (b) with respect to SNR (0% missing rate, true TT ranks = [1, 5, 5, 1]). SNR\dB 0 5 10 15 proposed method 7.90e-2 2.52e-2 8.10e-3 2.60e-3 STTO 1.01 3.17e-1 1.00e-1 3.12e-2 SiLRTC-TT 1.00 3.16e-1 1.00e-1 3.16e-2 TMAC-TT 1.00 3.17e-1 1.00e-1 3.16e-2 FBCP 1.77e-1 5.95e-2 2.06e-2 7.00e-3 TT-SVD 1.01 3.21e-1 9.90e-2 3.11e-2 (c) True TT ranks 5 10 15 20 proposed method 1.01e-3 3.90e-3 1.38e-2 7.40e-2 STTO 8.30e-3 9.83e-2 1.57e-1 1.63e-1 SiLRTC-TT 1.00e-2 1.90e-2 5.33e-2 9.17e-2 TMAC-TT 1.63e-2 4.62e-2 6.96e-2 8.32e-2 FBCP 2.60e-3 1.07e-2 1.11e-1 1.97e-1 TT-SVD Not applicable Table 2 : 2Performance of image completion without noise. Table 3 : 3Performance of image completion under Gaussian noise with variance 0.1.Proposed Algorithm SiLRTC-TT TMAC-TT STTO TTC-TV FBCP FaLRTC PSNR SSIM PSNR SSIM PSNR SSIM PSNRSSIM PSNRSSIM PSNRSSIM PSNRSSIM airplane 18.51 0.452 14.13 0.181 10.49 0.078 15.11 0.207 15.00 0.224 16.57 0.218 13.50 0.163 baboon 18.56 0.276 14.76 0.193 10.52 0.091 15.35 0.209 15.23 0.218 16.67 0.180 14.00 0.170 barbara 18.98 0.409 14.85 0.183 10.60 0.073 15.45 0.204 15.42 0.235 16.60 0.239 13.92 0.162 couple 19.00 0.341 16.76 0.153 12.80 0.058 16.11 0.137 16.23 0.174 18.14 0.239 16.10 0.139 facade 18.76 0.295 14.91 0.203 11.56 0.112 15.33 0.222 15.58 0.318 19.470.471 15.21 0.323 goldhill 19.66 0.361 15.32 0.178 11.27 0.075 15.59 0.190 15.57 0.222 17.76 0.254 14.55 0.165 house 19.97 0.502 14.92 0.155 10.20 0.058 15.83 0.176 15.57 0.195 17.41 0.257 14.09 0.136 jellybeens 20.31 0.675 14.23 0.150 10.07 0.055 14.88 0.166 14.83 0.176 18.11 0.334 13.68 0.133 lena 19.72 0.464 15.07 0.176 11.05 0.067 15.58 0.181 15.64 0.208 16.99 0.233 14.08 0.152 peppers 17.76 0.432 14.04 0.182 9.63 0.061 14.51 0.205 14.85 0.228 15.09 0.215 12.94 0.149 sailboat 17.38 0.358 14.04 0.200 9.14 0.078 14.91 0.232 15.04 0.266 15.95 0.255 13.68 0.202 splash 19.11 0.449 15.14 0.172 10.45 0.057 16.11 0.204 15.96 0.209 18.04 0.307 14.92 0.172 tree 17.41 0.361 13.99 0.211 9.19 0.075 14.88 0.242 15.04 0.274 15.36 0.208 13.23 0.182 Table 4 : 4Performance of HSI completion with structured missing and noise variance 0.01.Proposed Algorithm TMAC-TT STTO FBCP FaLRTC MPSNRMSSIM MPSNRMSSIM MPSNRMSSIM MPSNRMSSIM MPSNRMSSIM balloon 30.93 0.833 21.06 0.149 20.95 0.144 30.82 0.801 21.04 0.147 beads 25.22 0.664 21.18 0.330 20.42 0.301 24.39 0.614 21.01 0.319 feathers 28.15 0.692 21.33 0.205 21.15 0.197 28.33 0.673 21.29 0.202 is written as (8 × 4 × 4 × 4) × (8 × 4 × 4 × 4) × (5 × 6). After boundary padding on both spatial and spectral dimensions, each HSI is folded with size 100 × 16 × 16 × 16 × 7 × 6. (k−1)L d +k ] k, ,j d ] = υ G (d) k, ,j d E[τ ] j 1 ,j 2 ,...,j D \j d O j 1 j 2 ...j D A j 1 j 2 ...j D E[t× E[b (>d) ( −1)L d+1 + ] + E[λ (d) k ]E[λ (d+1) ] −1 , (A.6) E[G (d) (<d) k ]E[t (>d) ] − L d k =1 k =k L d+1 =1 = E[b (<d) (k−1)L d +k ]E[G (d) k , ,j d ]E[b (>d) ( −1)L d+1 + ] , (A.7) HSIs from the CAVE data set (http://www.cs.columbia.edu/CAVE/ database/multispectral/) are tested in this experiment. The original HSIs are with size 512 × 512 × 31, and the first 30 bands are selected for testing. For each band of the images, stripe missing pattern is adopted, and the Gaussian noise with variance 0.01 is added. The visual effects of the stripe missing pattern and the observed images are shown inFig.9. The image size Tensor completion for estimating missing values in visual data. J Liu, P Musialski, P Wonka, J Ye, IEEE transactions. 351J. Liu, P. 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Zeng, Fast low-rank bayesian matrix completion with hierarchical gaussian prior models, IEEE Transactions on Signal Processing 66 (11) (2018) 2804-2817. Learning nonnegative factors from tensor data: Probabilistic modeling and inference algorithm. L Cheng, X Tong, S Wang, Y.-C Wu, H V Poor, IEEE Transactions on Signal Processing. 68L. Cheng, X. Tong, S. Wang, Y.-C. Wu, H. V. Poor, Learning non- negative factors from tensor data: Probabilistic modeling and inference algorithm, IEEE Transactions on Signal Processing 68 (2020) 1792-1806. Fast and accurate tensor completion with total variation regularized tensor trains. C.-Y Ko, K Batselier, L Daniel, W Yu, N Wong, IEEE Transactions on Image Processing. C.-Y. Ko, K. Batselier, L. Daniel, W. Yu, N. Wong, Fast and accurate tensor completion with total variation regularized tensor trains, IEEE Transactions on Image Processing (2020). . J I Latorre, quant-ph/0510031Image compression and entanglement. arXiv preprintJ. I. Latorre, Image compression and entanglement, arXiv preprint quant-ph/0510031 (2005). Image quality assessment: from error visibility to structural similarity. Z Wang, A C Bovik, H R Sheikh, E P Simoncelli, IEEE transactions on image processing. 134Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE transac- tions on image processing 13 (4) (2004) 600-612. Learning multiple layers of features from tiny images. A Krizhevsky, G Hinton, A. Krizhevsky, G. Hinton, et al., Learning multiple layers of features from tiny images (2009). Hand-book on statistical distributions for experimentalists. C Walck, 10University of StockholmC. Walck, et al., Hand-book on statistical distributions for experimen- talists, University of Stockholm 10 (2007). He is currently pursuing the Ph.D. degree at the University of Hong Kong. His research interests include tensor decomposition, Bayesian inference, and their applications in machine learning and wireless communication. Nanjing, ChinaLe Xu received the B.Eng. degree from Southeast UniversityLe Xu received the B.Eng. degree from Southeast University, Nanjing, China, in 2017. He is currently pursuing the Ph.D. degree at the University of Hong Kong. His research interests include tensor decomposition, Bayesian inference, and their applications in machine learning and wireless communi- cation. He received the B.Eng. degree from Zhejiang University in 2013, and the Ph.D. degree from the University of Hong Kong. His research interests are in Bayesian machine learning for tensor data analytics, and interpretable machine learning for information systems. Hangzhou, ChinaZJU Young Professor) in the College of Information Science and Electronic Engineering at Zhejiang UniversityHe was a research scientist in Shenzhen Research Institute of Big DataLei Cheng is an Assistant Professor (ZJU Young Professor) in the College of Information Science and Electronic Engineering at Zhejiang University, Hangzhou, China. He received the B.Eng. degree from Zhejiang University in 2013, and the Ph.D. degree from the University of Hong Kong in 2018. He was a research scientist in Shenzhen Research Institute of Big Data from 2018 to 2021. His research interests are in Bayesian machine learning for tensor data analytics, and interpretable machine learning for information systems. He is currently an Associate Professor with the Department of Electrical and Electronic Engineering at HKU. His research interests include electronic design automation (EDA), model order reduction, tensor algebra, linear and nonlinear modeling & simulation. Ngai Wong, SM, IEEE) received his B.Eng and Ph.D. in EEE from The University of Hong Kong (HKU), and he was a visiting scholar with Purdue University. West Lafayetteand compact neural network designNgai Wong (SM, IEEE) received his B.Eng and Ph.D. in EEE from The University of Hong Kong (HKU), and he was a visiting scholar with Purdue University, West Lafayette, IN, in 2003. He is currently an Associate Profes- sor with the Department of Electrical and Electronic Engineering at HKU. His research interests include electronic design automation (EDA), model or- der reduction, tensor algebra, linear and nonlinear modeling & simulation, and compact neural network design. a member of technical staff. Since 2006, he has been with HKU, where he is currently an Associate Professor. He was a Visiting Scholar with Princeton University in 2017. His research interests include general areas of signal processing, machine learning, and communication systems. He was an Editor for the IEEE COMMUNICATIONS LETTERS and the IEEE TRANSAC-TIONS ON COMMUNICATIONS. Yik-Chung Wu, ; , Hong Kong, Senior Member, IEEE) received the B.Eng. (EEE) and the M.Phil. degrees from The University of Hong Kong (HKU). College Station, TX, USA; Princeton, NJ, USA, ashe was with the Thomson Corporate Research. He is currently an Editor for the Journal of Communications and NetworksYik-Chung Wu (Senior Member, IEEE) received the B.Eng. (EEE) and the M.Phil. degrees from The University of Hong Kong (HKU), Hong Kong, in 1998 and 2001, respectively, and the Ph.D. degree from Texas A&M Uni- versity, College Station, TX, USA, in 2005. From 2005 to 2006, he was with the Thomson Corporate Research, Princeton, NJ, USA, as a member of technical staff. Since 2006, he has been with HKU, where he is cur- rently an Associate Professor. He was a Visiting Scholar with Princeton University in 2017. His research interests include general areas of signal pro- cessing, machine learning, and communication systems. He was an Editor for the IEEE COMMUNICATIONS LETTERS and the IEEE TRANSAC- TIONS ON COMMUNICATIONS. He is currently an Editor for the Journal of Communications and Networks.	[]
[ "Note on the quasi-proper direct image with value in a Banach analytic set", "Note on the quasi-proper direct image with value in a Banach analytic set" ]	[ "Daniel Barlet " ]	[]	[]	We give a rather simple proof of the generalization of Kuhlmann's quasi-proper direct image theorem to the case of a map with values in a Banach analytic set. The proof uses a generalization of the Remmert-Stein's theorem to this context. AMS Classification. 32 H 02 -32 K 05 -32 C 25.	null	[ "https://arxiv.org/pdf/1609.04933v1.pdf" ]	119,311,946	1609.04933	5b3d85406137e4bca27c9509031d72106187b6c7	Note on the quasi-proper direct image with value in a Banach analytic set 16 Sep 2016 Daniel Barlet Note on the quasi-proper direct image with value in a Banach analytic set 16 Sep 201614/09/16Quasi-proper map Quasi-proper direct image Banach analytic set We give a rather simple proof of the generalization of Kuhlmann's quasi-proper direct image theorem to the case of a map with values in a Banach analytic set. The proof uses a generalization of the Remmert-Stein's theorem to this context. AMS Classification. 32 H 02 -32 K 05 -32 C 25. Introduction The aim of the present Note is to give a rather simple proof for the generalization of Kuhlmann's quasi-proper direct image theorem to the case of a map with values in Banach analytic set. N. Kuhlmann (and D. Mathieu in the Banach case, see [K.64], [K.66] and [M.00]) proved this direct image result for a "semi-proper" holomorphic map; this is a weaker hypothesis than quasi-proper, so their result is better than the one presented here. But the quasi-proper case fits well with the situation we are mainly interested in the study of f-analytic family of cycles (see [B.08], [B.13] and [B.15] 1 ). Our simpler argument via the generalization of the Remmert-Stein's theorem does not work for a semi-proper map (see the remark following proposition 2.0.3). In the appendix we give an easy proof of the Remmert's direct image theorem in the proper finite case (with values in an open set of a Banach space) which is used in our proof to make this Note self-contained modulo the theorem III 7.4.1. in [B-M 1]. A simple proof of Kuhlmann's quasi-proper direct image theorem In order to show the strategy of proof for the generalization of Kuhlmann's theorem with values in an open set of a Banach space, we shall begin by a simple proof of the finite dimensional case using the "usual" Remmert-Stein's theorem. Proof. The first point is to prove that f (M) is closed in N. Let y n = f (x n ) be a sequence in f (M) and suppose that y n converges to a point y ∈ N. Let V be a neighbourhood of y in N and K a compact subset in M such that for any z ∈ V each irreducible component of f −1 (z) meets K. Then we may assume that for n large enough y n lies in V and so that x n is chosen in K. So, up to pass to a sub-sequence we may assume that the sequence (x n ) converges to x ∈ K. Then the continuity of f implies hat y = f (x). So f (M) is closed. Consider now the integer p := max{dim x f −1 (y), y ∈ N} and define the set Z := {y ∈ N / dim(f −1 (y)) = p}. We want to show that Z is a closed analytic subset in N. To show that Z is closed consider a sequence (y n ) in Z converging to a point y in N. Let V be a neighbourhood of y in N and K a compact subset in M such that for any z ∈ V each irreducible component of f −1 (z) meets K. Then for each n large enough in order that y n lies in V , we may find a point x n ∈ K such that f (x n ) = y n and with dim xn f −1 (y n ) = p. So, up to pass to a subsequence we may assume that the sequence x n converges to x ∈ K. We have f (x) = y ; consider now a p−scale E := (U, B, j) around x adapted to f −1 (f (x)). Then the map g := f × (p U • j) : j −1 (U × B) → W × U is proper and finite if W ⊂ V is an open neighbourhood of x which is small enough. For each x n ∈ W the image of g contains {x n } × U. So it contains also {x} × U and the fibre f −1 (y) has dimension p in W . So y is in Z and Z is closed. To prove the analyticity of Z we argue as above and we remark that for any holomorphic function h : U × B → C m with zero set j(W ) ∩ (U × B) we have h(y, t) ≡ 0 for t ∈ U when y is in Z. This will give (infinitely many) holomorphic equations for Z in W using a Taylor expansion in t of the holomorphic function h. This proves the analyticity of Z. Remark that Z has dimension ≤ m − p where m := dim M. Now we shall prove the theorem by induction on the integer p ≥ 0. Note that, for p = 0 the result is already proved. So let assume that the result is proved for p = q − 1 and we shall prove it for p = q. If the closed analytic set Remarks. 1. The semi-properness of f would be enough to get the fact that f (M) is closed. 2. The restriction of a quasi-proper map to a closed saturated analytic subset X in M 2 is again quasi-proper. This is also true for a semi-proper map. 3. When f is quasi-proper to have quasi-properness for f restricted to closed analytic subset X ⊂ M it is enough for X to be an union of some irreducible components of f −1 (y) for y ∈ f (X). The case with values in a Banach analytic set Definition 2.0.2 Let f : M → S be a holomorphic map from a reduced complex space M to a Banach analytic set S. We say that f is quasi-proper when for each point s ∈ S there exists a neighbourhood V of s in S and a compact set K in M such that, for each σ ∈ V , each irreducible component of f −1 (σ) meets K. Note that f quasi-proper implies semi-proper 3 as the definition above implies that f (K) ∩ V = f (M) ∩ V . In particular this condition implies that f (M) is closed in S. So our result is local on S and it is enough to prove the Kulmann's theorem with value in an open set of a Banach space. Proposition 2.0.3 Let f : M → S be a holomorphic map from a reduced and irreducible complex space M to a Banach analytic set S. We assume that f is quasi-proper. Let p := max{dim f −1 (s), s ∈ S}, and define Z := {s ∈ S / dim f −1 (s) = p}. Then Z is a closed analytic subset in S which is finite dimensional (so Z is a reduced complex space embedded in S; see the theorem III 7.4.1 de [B-M 1]). remark. This result is not true in general for a semi-proper holomorphic map: Let π : M → N be an infinite connected cover of a complex manifold N of dimension n ≥ 2. Let y ∈ N and let V be a relatively compact open set containing y such that π admit a continuous section σ :V →W where W is a relatively compact open set in M. Let (y ν ) be a sequence of points in V \ {y} with limit y and let (x ν ) be a discrete sequence in M \ W such that π(x ν ) = y ν for each ν ∈ N. Let τ :M → M be the blow-up of M at each point x ν and putπ := τ • π. Thenπ is semi-proper and the subset Z of M where the fibre ofπ has dimension n − 1 is exactely the subset {y ν , ν ∈ N} which is not closed in N. First step of the proof: Z is closed in S. Let (s ν ) ν≥0 be a sequence in Z which converges to a point s in S. Let V be an open neighbourhood of s and K a compact subset of M such that for any σ ∈ V each irreducible component of f −1 (σ) meets K. For ν ≫ 1 we have s ν ∈ V , and if Γ ν is a p−dimensional irreducible component of f −1 (s ν ) , the intersection Γ ν ∩ K is not empty, and we may choose some x ν ∈ Γ ν ∩ K. Up to pass to a sub-sequence, we may also assume that the sequence (x ν ) converges to a point x ∈ K. Of course we shall have x ∈ f −1 (s). Choose now a p−scale E := (U, B, j) such that x is in j −1 (U × B) and such that j −1 (Ū × ∂B) ∩ f −1 (s) = ∅. This is possible because we know that dim f −1 (s) ≤ p. Then, up to shrink V around s, we may assume that for any σ ∈ V we have f −1 (σ) ∩ j −1 (Ū × ∂B) = ∅. This means that for ν ≫ 1 the scale E is adapted to Γ ν and with deg E (Γ ν ) = k ν ≥ 1. Define on the open set j −1 (U × B) ∩ f −1 (V ) ⊂ M the holomorphc map g := (p U • j) × f : j −1 (U × B) ∩ f −1 (V ) → U × V where p U : U × B → U is the projection and where V is the open set in S defined by the condition s ∈ f (j −1 (Ū × ∂B)). Second step: g is a closed map with finite fibres. The finiteness of fibres is obvious because for σ ∈ V the intersection f −1 (σ) ∩ j −1 (Ū × ∂B) is empty, so j(g −1 (t, σ)) ⊂ {t} × B is a compact analytic subset in a polydisc, so a finite set. To show the closeness of g, choose a closed set F in f −1 (V ) ∩ j −1 (U × B) and a sequence (t ν , σ ν ) in g(F ) ∩ (U × V ) converging to a point (t, σ) ∈ U × V , and let (x ν ) be a sequence in F such that g(x ν ) = (t ν , σ ν ). As j −1 (Ū ×B) is compact, we may assume, up to pass to a sub-sequence, that the sequence (x ν ) converges to some x ∈ j −1 (Ū ×B) ⊂ M. But the limit of g(x ν ) = (t ν , σ ν ) is in U × V by assumption and this implies that x is in j −1 (U ×B). As it cannot be in j −1 (U × ∂B) because f (x) is in V , x lies in f −1 (V ) ∩ j −1 (U × B) and so x is in F . This proves the closeness of g. Now the Remmert's theorem in the proper finite case, but with values in a Banach analytic set 4 , applies and shows that the image of g is a reduced complex space. In this case it is clear that the cardinal of the fibres is locally bounded, so k ν is bounded up to shrink U and V . So, up to pass to a subsequence, we may assume that k ν is constant equal to k ≥ 1. Then it is easy to see, again up to pass to a sub-sequence and to shrink U, that the sequence j(Γ ν ) converges to a multiform graph. This implies that f −1 (s) has dimension p, so s is in Z. We shall denote Ξ ⊂ (U × V ) the image of g. Third step. We shall prove now that, assuming that we choose V small enough around the given point s in Z, the set Z ′ := {σ ∈ V / {σ} × U ⊂ Ξ} is a closed analytic subset in V . Remark that Z ′ is contained in Z but it may be smaller that Z because the fibre of a point s ∈ V \ Z ′ may be of dimension p via a component of f −1 (s) which does not meet j −1 (U × B). Up to shrink V and U we may assume that V is a closed analytic subset of an open set U in some Banach space, and that we have a holomorphic map Φ : U → Hol(Ū, F ) where F is a Banach space, such that the associated holomorphic map Φ : U × U → F satisfiesΦ −1 (0) ∩ (V × U) = Ξ. Then it is clear that we have Z ′ = Φ −1 (0). Now remark that Z ′ ×U is a closed analytic subset of Ξ which is finite dimensional. So this implies that Z ′ is also finite dimensional and we have dim Z ′ ≤ dim M − p. If we cover the compact set f −1 (s) ∩ K by finitely many p−scales as above, we obtain that Z is locally a finite union of such Z ′ as above and then Z is a finite dimensional analytic set of dimension ≤ dim M − p near the point s; and, as we already know that Z is closed, it is a closed finite dimensional analytic set of dimension ≤ dim M − p in S. Theorem 2.0.4 Let f : M → N a quasi-proper holomorphic map between an irreducible complex space M and a Banach analytic set S. Then f(M) is a closed analytic subset in S which is locally finite dimensional . Proof. We shall prove the theorem by induction on the integer p defined in the previous proposition. Note that for a given M, we have always p ≤ dim M < +∞ as we assume M irreducible. The case p = 0 is clear because in this case we have Z = f (M). Assume the theorem proved when p ≤ q − 1 for some q ≥ 1 and we shall prove it for p = q. From the previous steps we know that Z is a closed analytic subset in S and that it has finite dimension ≤ dim M − p. Let us consider now the map f : M \ f −1 (Z) → S \ Z induced by f . It is clearly quasi-proper and for this map we havep ≤ q − 1. So, by the induction hypothesis, the image off is a closed analytic subset in S \ Z which is irreducible of finite dimension dim Im(f ) ≥ dim M −p ≥ dim M − q + 1 ≥ dim Z + 1. Now we want to apply the Remmert-Stein theorem to conclude. This is clear in the case where S is a finite dimensional complex space because the dimensions satisfy the desired inequality, but we want now to treat the case where S is a Banach analytic set. As the problem is local, we may replace S by a open set in a Banach space, and we apply the generalization of Remmert-Stein theorem obtained in the next section to conclude. Here is a first step of the proof of this theorem. Lemma 3.0.6 Let X ⊂ U be a closed analytic subset in an open set U in a Banach space E. Assume that X has finite pure dimension ≤ n and that X is countable at infinity. Let a ∈ U \ X and x ∈ X. Assume also that a n−dimensional linear subspace L containing a is given. Then there exists a linear closed codimension n subspace P in E containing a and x, such P is transversal to L at a and that the set X ∩ P is discrete and countable. Proof. We shall make an induction on n ≥ 0. The case n = 0 is trivial. So let assume that the case n − 1 is proved and consider X of pure dimension d ≤ n. Choose for each irreducible component X ν of X a point x ν ∈ X ν which is not equal to x 5 . Now choose an hyperplane H containing a and x and such that x ν is not in H for each ν, and that H is transversal to L at a. Such a hyperplane exists thanks to Baire's theorem. Then X ∩ H is purely (d − 1)-dimensional and contains x. The induction hypothesis gives us a co-dimension n − 1 subspace Π containing a and x, transversal at a to L ∩ H, such that Π ∩ (X ∩ H) is discrete and countable. Now the co-dimension of P := Π ∩ H is exactly equal to n and is transversal to L at a. This completes the proof. Proof of the theorem 3.0.5. We follow the proof of the proposition II 4.8.3 case (i) given in [B-M 1] but with an ambiant infinite dimensional complex Banach space. The theorem 3.0.5 follows from this result as in the finite dimensional case. First, the following point is not completely obvious : the local compactness of X ∪A. Of course A and X are locally compact by assumption. So it is enough to show that this union is locally compact near a point a ∈ A. Remark first that we have ∂X ⊂ ∂U ∪ A. and X ∪A is closed in U. So if we consider ε > 0 small enough, we haveB(a, 2ε) ⊂ U andB(a, ε) ∩ A is a compact set. Now we want to show thatB(a, ε) ∩ (X ∪ A) is compact. So it is enough to show that any sequence (x ν ) inB(a, ε) ∩ X admits a converging sub-sequence to a point inB(a, ε). Let δ ν := d(x ν , A ∩B(a, ε)). 5 As n ≥ 1 an irreducible component is not equal to {x}. If δ ν goes to 0 for some sub-sequence, then there is a Cauchy sub-sequence converging to some a ′ ∈B(a, ε) ∩ (X ∪ A). If δ ν ≥ α > 0 for all large enough ν, then the sequence is contained in the subset B (a, ε) \ {x / d(x, A ∩B(a, ε)) ≥ α)} ∩ X. This subset of X is compact, because it is closed and a sequence in it cannot approach neither ∂U (asB(a, 2ε) ⊂ U, any point z ∈B(a, ε) satisfies d(z, ∂U) ≥ ε) nor A, and X is locally compact (so any discrete sequence has to approach the boundary of X). Assume that X has dimension d + k with k ≥ 1. For a ∈ A a fixed smooth point, let V be a d + k sub-manifold through a containing A (we use here the theorem III 7.4.1 in [B-M 1]) and choose a co-dimension d + k plane P transversal to V at a meeting some point in X ∩ B(a, ε) and such that P ∩ X is discrete. Up to shrink V and taking a small ball B 0 with center a in P , we may assume that an open neighbourhood of a in B(a, ε) is isomorphic to the product V × B 0 . And from our construction we have X ∩ ({a} × B 0 ) which is discrete. Claim. We can choose two small balls B ′ ⊂ B in P with center a contained in B 0 ⊂ B(a, ε) ∩ P such that (B \ B ′ ) ∩ X = ∅ and then we may find an open neighbourhood U of a in V in order that the projection on U of the set X ∩ (U × B ′ ) = (X ∪ A) ∩ (U × B) is proper (this means closed with compact fibres) and that its restriction to X ∩ ((U \ A) × B) has finite fibres. proof. The choice of two arbitrary small balls B ′ ⊂ B with the first condition (B \ B ′ ) ∩ X = ∅ is easy because we know that X ∩ ({a} × B 0 ) is discrete : the distances to a of the points in X ∩ ({a} × B 0 ) are in a discrete subset in ]0, ε[ and we can choose r ′ < r with r arbitrary small such that ]r ′ , r[ avoids these values. But the subset (X ∪ A) ∩B(a, ε) is compact and then, as A ∩ (B \ B ′ ) is empty, the subset X ∩ ({a} × B) is compact. Then there exists an open neighbourhood U of a in V such that the projection ofX ∩ (U × B) on U is proper : Let K be a compact neighbourhood inX ∩ (V × B) of the compact setX ∩ ({a} × B) (remember that we proved thatX = X ∪ A is locally compact). Then for K small enough the distance of a point in K to the closed set {a} × (B \ B ′ ) is bounded below by a positive number, and so there exists an open neighbourhood U of a in V such that U × (B \ B ′ ) does not meet K. This is enough to prove the claim thanks to the following simple remark, asX ∩ ((U \ A) × B) = X ∩ ((U \ A) × B) because A ≃ A × {a} ⊂ U × B ′ . Remark. Any compact analytic subset in X ∩ B(a, ε) is finite: assume that Z is a connected component of such a compact analytic subset. As compactness implies that there are at most finitely many Z, it is enough to prove that Z has at most one point. Assume that z 1 = z 2 are two points in Z. Choose a continuous linear form on E such that l(z 1 ) = l(z 2 ). As the map l \|Z : Z → C is holomorphic, it has to be constant (maximum principle and connectness). Conclusion : Z has at most one point. The end of the proof of the theorem 3.0.5 is now analogous to the end of the proof in the finite dimensional setting (we are in the "easy" case where the dimension of X is strictly bigger than d the dimension of A ; see the remark before the lemma 4.8.7 in [B-M 1] ch.III). Remarks. Theorem 1 .0. 1 11Let f : M → N a quasi-proper holomorphic map between irreducible complex spaces. Then f (M) is a closed analytic subset of N. Z introduced above is equal to N, the result is clear because f (M) = N. If not N \Z is an open dense set in N because N is irreducible and also M \f −1 (Z) is also an open dense set in M. Moreover the mapf : M \f −1 (Z) → N \Z induced by f is again quasiproper. The induction hypothesis gives that f (M \ f −1 (Z)) is a closed analytic subset in N \ Z and it has dimension bigger or equal to m − (p − 1) = m − p + 1. As Z has dimension ≤ m − p we may apply Remmert-Stein theorem to obtain that the closure of f (M \ f −1 (Z)) in N is again an analytic set. This conclude the proof as we have f (M) = f (M \ f −1 (Z)) ∪ Z, because we know that f (M) is closed. Let U an open set in a Banach space and let A ⊂ U be a closed analytic susbset of dimension d. Let X ⊂ U \ A a closed irreducible analytic subset in U \ Z of finite dimension ≥ d + 1. ThenX is a closed irreducible analytic subset in U of finite dimension equal to dim X.3 The Remmert-Stein' theorem in a Banach space Theorem 3.0.5 1 . 1Along the same line, it is not difficult to prove the analog of the Remmert-Stein's theorem with equality of dimensions, assuming there exists an open set in U meeting each irreducible component of A and in which X ∪ A is an analytic subset. 2. In our proof we use in a crucial way the metric in the ambient Banach space. If the extension of this proof to an open set in a Frechet space may be easy, the case of a non metrizable e.l.c.s.s. (for instance a dual of Frechet space ) does not seem clear. Note that the proper case is proved with values in an open set of any e.l.c.s.s. in [B-M 1] ch.III. Appendix: the direct image theorem in the proper finite case Notation. An e.l.c.s.s. is a locally convex topological vector space which is separated and sequentially complete. For the definition we shall use of a holomorphic map on a reduced complex space M with values in a e.l.c.s.s. see [B-M 1] ch. III definition 7.1.1. Note that the general proper case of this result is given in loc. cit. th. III 7.4.3 but the proof of the proper finite case is much more simple. To have a simple and self-contained proof (modulo the theorem III 7.4.1. in [B-M 1]) of the quasi-proper direct image theorem discussed in this Note, we shall give a easy proof of the direct image theorem in the proper finite case with values in an e.l.c.s.s.(this result has been used in the Banach case in our proof). This allows to use the proof of the previous section also in the proper case, thanks to the following lemma. Lemma 4.0.7 Let f : M → S a holomorphic map from a reduced complex space to an open set of an e.l.c.s.s. Assume that f is closed with compact fibres (so f is proper). Then f is quasi-proper.4 Nevertheless we prove the semi-proper case of the direct image theorem with values in C f n (M ), the space of finite type n−dimensional cycles in the complex space M in [B.15] th. 2.3.2. . This property of X is not enough, in general, in the case of a semi-proper map f . This is precisely what happen for the subset Z introduced below or in the proposition 2.0.3.2 A subset X of M is saturated for f if x ∈ X implies f −1 (x) ⊂ X. Recall that a continuous map f : M → S between Hausdorff topological spaces is semi-proper if for any s ∈ S there exists a neighbourhood V of s in S and a compact set K in M such that for any σ ∈ V the fibre f −1 (σ) meets K. This is a topological notion contrary to the quasi-properness which asks that M is a complex space and that the fibres of f are (closed) analytic subsets in M (but no complex structure in needed on S). So we have to use the proper case with finite fibres of the Remmert' theorem with values in an open set of a complex Banach space to prove this proposition. To make this Note self-contained (modulo the theorem III 7.4.1. in [B-M 1]) we give a simple proof of this case in the appendix (section 4). See also [B-M 1] chapter III for the general case. −1 (σ ν,K ) which does not meet K. Now, for a fixed K we may choose the points σ ν,K to be distinct. Let W be a relatively compact open set in M containing the compact fibre f −1 (s) and let K :=W . Then for each ν we can find a point x ν which is in Γ ν,K . So the point x ν is in the closed set M \ W . Consider the subset F := {x ν } in M \ W . It is a closed set in M \ W (and also in M) because if x ∈F \ F , the point x is limit of an ultra-filter of points in F , and so f (x) is equal to s, because the intersection of the V ν for any ultra-filter is reduced to {s}. K Of, If f is not quasi-proper at s ∈ S there exists. But as x ∈ M \ W and f −1 (s) ⊂ W , this is a contradiction. So the map f is quasi-proper at any point s ∈ S. If f is not quasi-proper at s ∈ S there exists, for each open neighbourhood V ν of s and for each compact K in M a point σ ν,K ∈ V ν and an irreducible component Γ ν,K of f −1 (σ ν,K ) which does not meet K. Now, for a fixed K we may choose the points σ ν,K to be distinct. Let W be a relatively compact open set in M containing the compact fibre f −1 (s) and let K :=W . Then for each ν we can find a point x ν which is in Γ ν,K . So the point x ν is in the closed set M \ W . Consider the subset F := {x ν } in M \ W . It is a closed set in M \ W (and also in M) because if x ∈F \ F , the point x is limit of an ultra-filter of points in F , and so f (x) is equal to s, because the intersection of the V ν for any ultra-filter is reduced to {s}. But as x ∈ M \ W and f −1 (s) ⊂ W , this is a contradiction. So the map f is quasi-proper at any point s ∈ S. Let f : M → S be a holomorphic map of a complex reduced space M to an open set S in a e.l.c.s.s. Assume that f is closed with finite fibres (so proper and finite). Then f (M) is a closed analytic subset which is locally of finite dimension in SProposition 4.0.8 Let f : M → S be a holomorphic map of a complex reduced space M to an open set S in a e.l.c.s.s. Assume that f is closed with finite fibres (so proper and finite). Then f (M) is a closed analytic subset which is locally of finite dimension in S. As f (M) is the locally finite union of the sets f (M i ), i ∈ I, where M i , i ∈ I is the set of irreducible components of M because f is quasi-proper and finite thanks to the previous lemma. we may assume that M is irreducible. Let m := dim M. As f (M) is a closed set in S, it is enough to show that f (M) is an finite dimensional analytic subset near a point s = f (x) in f (M). proof. As f (M) is the locally finite union of the sets f (M i ), i ∈ I, where M i , i ∈ I is the set of irreducible components of M because f is quasi-proper and finite thanks to the previous lemma, we may assume that M is irreducible. Let m := dim M. As f (M) is a closed set in S, it is enough to show that f (M) is an finite dimensional analytic subset near a point s = f (x) in f (M). c.s.s. G such that the point s is isolated in H ∩ f (M) which is maximal for this property. Such a H exists because it is easy to construct a sequence (H q ) of closed co-dimension q affine subspaces in G containing s such that the germ of f −1 (H q ) at f −1 (s) gives a strictly decreasing sequence of analytic germs as long as f −1 (H q ) is strictly bigger than the finite set f −1 (s). Then at the last step q 0 we have the equality of germs f −1 (H q 0 ) = f −1 (s) in M. If H q 0 is not maximal with this property. Let G be the ambiant vector space of S. Choose a finite co-dimensional closed affine subspace H containing s in the e.l.. replace H q 0 by a maximal closed affine subspace containing s and containing H q 0 with this propertyLet G be the ambiant vector space of S. Choose a finite co-dimensional closed affine subspace H containing s in the e.l.c.s.s. G such that the point s is isolated in H ∩ f (M) which is maximal for this property. Such a H exists because it is easy to construct a sequence (H q ) of closed co-dimension q affine subspaces in G containing s such that the germ of f −1 (H q ) at f −1 (s) gives a strictly decreasing sequence of analytic germs as long as f −1 (H q ) is strictly bigger than the finite set f −1 (s). Then at the last step q 0 we have the equality of germs f −1 (H q 0 ) = f −1 (s) in M. If H q 0 is not maximal with this property, replace H q 0 by a maximal closed affine subspace containing s and containing H q 0 with this property. (s)) → (C n , p(s)) is finite by construction. So its image is an analytic germ. If it is not equal to (C n , p(s)) we can find a line ∆ through p(s) in C n such that g −1 (p(s)) = f −1 (s). Then the affine space p −1 (∆) contradicts the maximality of H. So the germ g is surjective and we can find an open polydisc U with center p(s) in C n such that g : M ′ := g −1 (U) → U is a k−sheeted branch covering. So we have a holomorphic map ϕ : U → Sym k (M ′ ) classifying the fibres of g. Composed with the holomorphic map induced by Sym k (p H •f ) we obtain that f (M ′ ) is a multiform graph of U contained in U × H (see. G Write, = C N ⊕ H, G → H be the projections. Then the germ of analytic map g : (M, f −. 1As we have f (M ′ ) = p −. M in a neighbourhood of s, we conclude that f (M) is a finite dimensional analytic subset in G in a neighbourhood of sWrite G = C n ⊕ H 0 where H 0 is the closed co-dimension vector subspace in G directing H and let p : G → C n and p H : G → H be the projections. Then the germ of analytic map g : (M, f −1 (s)) → (C n , p(s)) is finite by construction. So its image is an analytic germ. If it is not equal to (C n , p(s)) we can find a line ∆ through p(s) in C n such that g −1 (p(s)) = f −1 (s). Then the affine space p −1 (∆) contradicts the maximality of H. So the germ g is surjective and we can find an open polydisc U with center p(s) in C n such that g : M ′ := g −1 (U) → U is a k−sheeted branch covering. So we have a holomorphic map ϕ : U → Sym k (M ′ ) classifying the fibres of g. Composed with the holomorphic map induced by Sym k (p H •f ) we obtain that f (M ′ ) is a multiform graph of U contained in U × H (see [B-M 1] chapter III section 7.2). As we have f (M ′ ) = p −1 (U) ∩ M in a neighbourhood of s, we conclude that f (M) is a finite dimensional analytic subset in G in a neighbourhood of s. Reparamétrisation universelle de familles f-analytiques de cycles. Comment. Helv. Bibliography • [B.08] Barlet, D83Bibliography • [B.08] Barlet, D. Reparamétrisation universelle de familles f-analytiques de cycles ... Comment. Helv. 83 (2008), pp. 869-888. Barlet Daniel, Quasi-proper meromorphic equivalence relations. Math. Z. 273B.13• [B.13] Barlet Daniel, Quasi-proper meromorphic equivalence relations, Math. Z. (2013), vol. 273, pp. 461-484. Strongly quasi-proper maps and the f-flattening theorem. D Barlet, arXiv:1504.0157941 pages• [B.15] Barlet, D. Strongly quasi-proper maps and the f-flattening theorem, math-arXiv:1504.01579 (41 pages) Cycles analytiques complexes. I. Théorèmes de préparation des cycles. D Barlet, J Et Magnússon, Cours Spécialisés, 22. Société Mathématique de FranceParis• [B-M 1] Barlet, D. et Magnússon, J. Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés, 22. Société Mathématique de France, Paris (2014). Über holomorphe Abbildungen komplexer Raüme Arch. der Math. 15• [K.64] Kuhlmann, N.Über holomorphe Abbildungen komplexer Raüme Arch. der Math. 15, (1964) pp. 81-90. Bemerkungenüber holomorphe Abbildungen komplexer Raüme Wiss. N Kuhlmann, Abh. Arbeitsgemeinschaft Nordrhein-Westfalen. 33Festschr. Gedäachtnisfeier K. Weierstrass• [K.66] Kuhlmann, N. Bemerkungenüber holomorphe Abbildungen komplexer Raüme Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedäachtnisfeier K. Weierstrass, (1966) pp.495-522. Universal reparametrisation of a family of cycles : a new approach to meromorphic equivalence relations. D Mathieu, Ann. Inst. Fourier (Grenoble). 504• [M.00] Mathieu, D. Universal reparametrisation of a family of cycles : a new approach to meromorphic equivalence relations Ann. Inst. Fourier (Grenoble) vol. 50 fasc. 4, (2000) pp. 1155-1189.	[]
[ "SPHERICAL AND HYPERBOLIC LENGTHS OF IMAGES OF ARCS", "SPHERICAL AND HYPERBOLIC LENGTHS OF IMAGES OF ARCS" ]	[ "T K Carne " ]	[]	[]	Let f : D → C be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length ρ is then o(ρ 1/2 ). In this note we consider the corresponding results when f maps into the unit disc with the hyperbolic metric or the Riemann sphere with the spherical metric. Similar but not identical results hold.	null	[ "https://arxiv.org/pdf/0711.0170v1.pdf" ]	16,077,695	0711.0170	2c820c3824feeb8ee5e5fb3b8d0abd5f3a4be003	SPHERICAL AND HYPERBOLIC LENGTHS OF IMAGES OF ARCS 1 Nov 2007 T K Carne SPHERICAL AND HYPERBOLIC LENGTHS OF IMAGES OF ARCS 1 Nov 2007arXiv:0711.0170v1 [math.CV] Let f : D → C be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length ρ is then o(ρ 1/2 ). In this note we consider the corresponding results when f maps into the unit disc with the hyperbolic metric or the Riemann sphere with the spherical metric. Similar but not identical results hold. Introduction Let f : D → C be an analytic function on the unit disc D which is in the Dirichlet class, so the Euclidean area of the image f (D), counting multiplicity, is finite. Keogh [K] showed that a radial arc [0, z] in the disc is mapped to an arc with Euclidean length E that satisfies E = o(ρ(0, z) 1/2 ) where ρ is the hyperbolic length in D. The paper [BC] explored the result in some detail. In this paper we will consider the corresponding results when the image domain is either the disc or the extended complex plane. For these we will use the natural Riemannian metrics: the hyperbolic metric on the unit disc and the spherical metric on the extended complex plane. Similar results hold in these cases, essentially because we can localise the result to a small disc with an Euclidean image. However, there are interesting differences and the arguments make the importance of the hyperbolic metric still more apparent. We will consider domains A with a Riemannian metric ds = λ A (z)\|dz\|. Here λ A is a strictly positive function on A giving the density of the metric. If f : A → B is an analytic map, then the derivative has norm \|\|f ′ (z o )\|\| A→B = \|f ′ (z o )\| λ B (f (z o )) λ A (z o ) . This is the factor by which f changes infinitesimal lengths at the point z o for the metrics on A and B. The area is changed by the square of this factor. On the complex plane C we will use the Euclidean metric \|dz\| with density 1. On the unit disc D we will use the hyperbolic metric with density λ H (z) = 2 1 − \|z\| 2 . This has constant curvature −1. On the extended complex plane we will use the spherical metric with density λ S (z) = 2 1 + \|z\| 2 . This has constant curvature +1 and is isometric, under stereographic projection, with the unit sphere in Euclidean R 3 . This is the Riemann sphere and we will denote it by P. The subscripts H, E, S will be used to specify the hyperbolic, Euclidean and spherical metrics respectively. So d H , d E , d S are the distances on these three spaces and A H , A E , A S are the area measures. Let f : D → B be an analytic map into some domain B with density λ. The area of the image, counting multiplicity is 1 0 2π 0 \|f ′ (re iθ )\| 2 λ(f (re iθ )) 2 dθ r dr = D \|\|f ′ (z)\|\| 2 E→E dA E (z) . We will abbreviate this to A B (f (D)) when there is no chance of confusion. It is more natural to write this as an area integral over the unit disc using the hyperbolic metric: A B (f (D)) = D \|\|f ′ (z)\|\| 2 H→B dA H (z) . Let f : D → C be an analytic map with A E (f (D)) < ∞, so f is in the Dirichlet class. In [BC] it was shown that \|\|f ′ (z)\|\| H→E A E (f (D)) 4π 1/2 . (1.1) Choose a direction from the origin and consider a radial arc of hyperbolic length ρ from 0 in this direction. The image of this arc will have Euclidean length L E (ρ). By integrating (1.1) along the radial arc we can show that L E (ρ) = o(ρ 1/2 ) (1.2) as ρ ր ∞, that is as the radial arc extends to the boundary. The purpose of this note is to consider the corresponding results for maps f : D → D or f : D → P. For the hyperbolic image, both (1.1) and (1.2) hold. In the spherical case, (1.1) fails when the area A S (f (D)) is large. However, it does hold when this area is small and this is sufficient to establish (1.2). In all cases, the arguments are very similar in spirit to those in [BC] and so they are not laboured. That paper considers many analogues and extensions of the results and these, similarly, can be established for hyperbolic and spherical images. The norm of the derivative We wish to establish bounds on the derivative \|\|f ′ (z)\|\| H→B in terms of the area A B (f (D)). In the Euclidean case this is very simple and is already done in Keogh's paper [K]. We include a proof to compare with later results for the hyperbolic and spherical images. Proposition 2.1 Let f : D → C be an analytic map with the Euclidean area of the image A E (f (D)), counting multiplicity, finite. Then \|\|f ′ (z o )\|\| H→E A E (f (D)) 4π 1/2 . Proof: Composing f with a hyperbolic isometry will not alter the area of the image. Hence we can assume that z o = 0. Let f (z) = a n z n be the power series for f . Then \|\|f ′ (0)\|\| H→E = \|f ′ (0)\|/2 = \|a 1 \|/2. The usual integration of the corresponding Fourier series gives A E (f (D)) = π ∞ n=1 n\|a n \| 2 . So we certainly have \|\|f ′ (0)\|\| 2 H→E = \|a 1 \| 2 4 A E (f (D)) 4π as required. The inclusion map f : D ֒→ C shows that the constant 1/4π in this result is the best possible. Let r o = tanh 1 2 ρ o (so the disc of hyperbolic radius ρ o centred on 0 has Euclidean radius r o ). Then the Möbius transformation T : z → r o z + z o 1 + z o r o z maps the unit disc conformally onto ∆. So we can apply the Proposition to g = f • T to obtain r o \|\|f ′ (z o )\|\| H→E A E (f (∆)) 4π 1/2 . This gives us a way of localising the result and hence of applying it when the image domain has a different metric. Proposition 2.2 Let f : D → D be an analytic map with the hyperbolic area of the image A H (f (D)), counting multiplicity, finite. Then \|\|f ′ (z o )\|\| H→H A H (f (D)) 4π 1/2 . Proof: By composing with hyperbolic isometries, we may assume that z o = 0 and f (z o ) = 0. Then \|\|f ′ (z o )\|\| H→H = \|f ′ (0)\|. The hyperbolic density in D is always at least 2, so A E (f (∆)) 1 4 A H (f (∆)). Therefore, Proposition 2.1 shows that \|\|f ′ (0)\|\| H→H = 2\|\|f ′ (0)\|\| H→E 2 A E (f (D)) 4π 1/2 A H (f (D)) 4π 1/2 as required. For the map f : z → εz we have \|\|f ′ (0)\|\| H→H = ε while A H (f (D)) = 4πε 2 /(1 − ε 2 ). So we see that the constant 1/4π in the proposition is the best possible. The corresponding result for the spherical metric fails. For consider a univalent map k : D → P with k(0) = 0 and whose image has spherical area 4π = A S (P). For example, the Koebe function k : z → z/(1 − z) 2 . This has A S (k(D)) = 4π and \|\|k ′ (0)\|\| H→S = \|k ′ (0)\| = 0 . For any λ = 0, the map f (z) = λk(z) also has A S (k(D)) = 4π but \|\|f ′ (0)\|\| H→S = \|λ\|. So we can not have any inequality of the form \|\|f ′ (0)\|\| 2 H→S cA S (f (D)) . Nonetheless, the result does hold provided that the spherical area of the image is sufficiently small. Proposition 2.3 Let f : D → P be an analytic map with the spherical area of the image A S (f (D)), counting multiplicity, less than 2π. Then \|\|f ′ (z o )\|\| H→S cA E (f (D)) 1/2 for some constant c. Proof: We may assume that z o = 0 and f (z o ) = 0. So \|\|f ′ (0)\|\| H→S = \|f ′ (0)\|. Choose δ > 0 so that the spherical area of the ball B S (c, δ) = {z ∈ P : d S (c, z) < δ} is less than 1 2 π. For this δ, find a maximal set of points c 0 , c 1 , c 2 , . . . , c K with c 0 = 0, the other points c 1 , c 2 , c K in P \ f (D), and all of the distances d S (c i , c j ) δ for i = j. Since K is maximal, there can be no point of P \ f (D) lying outside the balls B S (c j , δ). Hence these K + 1 balls cover all of P \ f (D) and so (K + 1)A S (B S (0, δ)) A S (P \ f (D)) 2π . This implies that K + 1 2π A S (B(0, δ)) > 4 . So we can find 3 points w 0 , w 1 , w ∞ in P \ f (D) which satisfy d S (0, w i ) > δ , d S (w i , w j ) > δ (2.1) for i = j. We can now compare the spherical metric on P with the hyperbolic metric on P\{w 0 , w 1 , w ∞ }. The three punctured sphere P \ {w 0 , w 1 , w ∞ } has a hyperbolic metric and the conditions (2.1) allow us to estimate its properties uniformly. From now on, consider P \ {w 0 , w 1 , w ∞ } with this metric. This hyperbolic metric can be written as λ(z)ds S (z) where ds S denotes the infinitesimal spherical metric and λ is the density relative to this spherical metric. The function λ is bounded away from 0 and tends to +∞ at the three punctures. The conditions (2.1) imply that there are constant K, K ′ , K ′′ > 0, depending only on δ, with the following properties: (a) λ(0) K; (b) the spherical balls B S (w i , K ′ ) are disjoint and at least hyperbolic distance 1 from 0 in P \ {w 0 , w 1 , w ∞ }; (c) The hyperbolic density λ(z) K ′′ for all points z within a hyperbolic distance 1 from 0 in P \ {w 0 , w 1 , w ∞ }. (Compare this with the estimates in [A], 1-9. There is a Möbius transformation T : P → P that maps our three punctured sphere P\{w 0 , w 1 , w ∞ } onto the standard three punctured sphere P\{0, 1, ∞}. The conditions (2.1) show that this transformation only distorts the metrics by a controlled amount.) Now f : D → P \ {w 0 , w 1 , w ∞ } is an analytic map between two hyperbolic domains so the Schwarz -Pick lemma implies that f is a contraction for the hyperbolic metrics. This implies that f maps the hyperbolic disc ∆ = B H (0, 1) with hyperbolic radius 1 into the region {w ∈ P \ {w 0 , w 1 , w ∞ } : d H (0, w) < 1} . Condition (a) above shows that \|\|f ′ (0)\|\| H→S 2 K \|\|f ′ (0)\|\| H→H . Condition (c) shows that the hyperbolic area A H (f (∆)) is at most K ′′2 times the spherical area A S (f (∆)). Finally, we can apply Proposition 2.2 to the map f \| ∆ : ∆ → P \ {w 0 , w 1 , w ∞ } (or, more properly, to its lift to the universal cover of P \ {w 0 , w 1 , w ∞ }). This gives r o \|\|f ′ (0)\|\| H→H A H (f (∆)) 4π 1/2 K ′′2 A S (f (∆)) 4π 1/2 for r o = tanh 1 2 . Putting all of these together gives \|\|f ′ (0)\|\| H→S 2K ′′ r o K A S (f (D)) 4π 1/2 as required. The argument used in the proof shows that, for any C < A S (P) = 4π, there is a constant K(C) with \|\|f ′ (z o )\|\| H→S K(C)A S (f (D)) 1/2 provided that A S (f (D)) C. The lengths of image arcs We will need to introduce some notation that will apply throughout the remainder of the paper. We wish to apply the propositions of §2 to the function f restricted to hyperbolic discs B H (γ(t), δ) for some hyperbolic radius δ. This disc lies inside the ball B H (0, t + δ) and outside the disc B H (0, t − δ). So the area A B (f (∆)) A(t + δ) − A(t − δ). Initially, let us consider the Euclidean case f : D → C. For this Proposition 2.1 gives us \|\|f ′ (γ(t))\|\| H→E 1 tanh 1 2 δ A(t + δ) − A(t − δ) 4π 1/2 . So integrating and applying the Cauchy -Schwarz inequality gives L E (ρ o ) = ρ0 0 \|\|f ′ (γ(t))\|\| H→E dt ρ0 0 1 tanh 1 2 δ A(t + δ) − A(t − δ) 4π 1/2 dt ρ0 0 1 4π tanh 2 1 2 δ (A(t + δ) − A(t − δ)) dt 1/2 ρ0 0 1 dt 1/2 ρ0 0 1 4π tanh 2 1 2 δ (A(t + δ) − A(t − δ)) dt 1/2 ρ 1/2 o Suppose that the area A E (f (D)) is finite. Then A(t) increases to its limiting value A E (f (D)). So the integral ρ0 0 1 tanh 2 1 2 δ A(t + δ) − A(t − δ) 4π dt is bounded independently of ρ o . Hence L E (ρ o ) = O(ρ 1/2 o ) . More carefully, we can apply the above result to the arc from some value u o up to ρ o . The integrand A(t + δ) − A(t − δ) is then no more than A E (f (D)) − A(u o − δ). This tends to 0 as u o ր ∞, so we see that L E (ρ) = o(ρ 1/2 o ) as ρ 0 → ∞. Thus we have reproved Keogh's Theorem as in [BC]. L E (ρ o ) = o(ρ 1/2 o ) as ρ o → ∞. Note that the proof given above only required the application of Proposition 2.1 to functions where the area of the image is small, for we knew that A(t) ր A E (f (D)) as t ր ∞. Hence Propositions 2.2 and 2.3 give the corresponding results for maps into the hyperbolic plane and the Riemann sphere. In [BC] examples were constructed of functions f : D → C showing that the power 1 2 in Theorem 2.1 is the best possible. Since these examples had f bounded and the hyperbolic, Euclidean and spherical metrics are Lipschitz equivalent on any compact region inside unit disc, these examples also show that the power is best possible in Theorems 3.2 and 3.3. It is easy to adapt the argument used above to more general situations. Suppose, for example, that the area A(t) grows to infinity but we have control on the rate of growth. Then we can use the same ideas to obtain a bound L(ρ o ) = O(ρ α o ) for suitable exponents α. In more detail, fix α > 1. A different splitting of the integrands in the appeal to the Cauchy -Schwarz inequality above gives us L E (ρ o ) ρ0 0 1 tanh 1 2 δ A(t + δ) − A(t − δ) 4π 1/2 dt ρ0 0 1 4πt α−1 tanh 2 1 2 δ (A(t + δ) − A(t − δ)) dt 1/2 ρ0 0 t α−1 dt 1/2 ρ0 0 1 4πt α−1 tanh 2 1 2 δ (A(t + δ) − A(t − δ)) dt 1/2 ρ α o α 1/2 For the first integrand, we can ignore the behaviour near 0 and write ρ0 1 t α−1 tanh 2 1 2 δ (A(t + δ) − A(t − δ)) dt = = ρ0−δ 1 (t − δ) α−1 tanh 2 1 2 δ A(t) dt − ρ0+δ 1 (t + δ) α−1 tanh 2 1 2 δ A(t) dt . So we want the integral ∞ 1 (t − δ) α−1 − 1 (t + δ) α−1 A(t) tanh 2 1 2 δ dt to converge. Then it will follows that L E (ρ o ) = o(ρ α/2 o ) as ρ 0 ր ∞. The mean value theorem immediately gives 1 (t − δ) α−1 − 1 (t + δ) α−1 2δ(α − 1)(t − δ) −α . So we see that we require the area A(t) to grow sufficiently slowly that ∞ δ tanh 1 2 δ A(t) (t − δ) α dt converges. Using Proposition 2.2 in place of Proposition 2.1 gives us the same results for functions into the unit disc. In order to use Proposition 2.3 to obtain corresponding results for meromorphic functions into the Riemann sphere we need to ensure that A(t + δ) − A(t − δ) is sufficiently small for 2.3 to apply. In order to achieve this the radius δ usually needs to decrease as t increases. The natural class of functions to consider here is those of finite order, so the Nevanlinna characteristic T (r) satisfies lim sup r→1 T (r) log 1 1−r < ∞ . (See [T].) This implies that ∞ T (r)(1 − r) k−1 dr < ∞ for some k. The derivative T ′ (r) is S(r)/r where S(r) = A S (f ({z : \|z\| < r})) 4π = A S (t) 4π for t = log(1 + r)/(1 − r), which is the hyperbolic radius corresponding to the Euclidean radius r. So, integrating by parts gives ∞ A S (t) exp(−(k + 1)t) dt < ∞ . In order to obtain estimates for these functions we would need to take the radius δ tending exponentially to 0 as t increased to ∞. The details do not seem inspiring. Examples As in [BC], it is useful to consider examples that limit what can happen to the lengths of the images of arcs. As there we can construct many examples defined on a region S = {x + iy ∈ C : \|y\| < h(x)} for some slowly increasing function h. This will be a hyperbolic simply-connected domain, so it is conformally equivalent to the unit disc. Let q : D → S be the conformal map which fixes the origin. The hyperbolic metric is Lipschitz equivalent to the pseudo-hyperbolic metric which has density 1/d E (z, ∂S). The positive real axis is then a hyperbolic geodesic in S and the length of the segment from the origin to t o is approximately ρ o = to 0 1 h(t) dt . The hyperbolic disc B(0, ρ o ) is then certainly contained in the part of S to the left of {x+iy ∈ S : y < t o }. In [BC] the function q itself was considered. For us, we need to follow q by a mapping that is an isometry from the positive real axis with the Euclidean metric to the hyperbolic plane or the Riemann sphere with their metrics. For this we follow q by an exponential map. This gives us corresponding examples for maps into the hyperbolic plane or the Riemann sphere. It may also be worth considering analogous examples for maps into the Riemann sphere where we constrain the Nevanlinna characteristic rather than the spherical area of the image. If the analytic map f : D → P has A(f (D)) finite, then we certainly have T (r) = T ( 1 2 ) + r 1 2 A(f (D)) 4πs ds = T ( 1 2 ) + A(f (D)) 4π log 2r . So the Nevanlinna characteristic is bounded. However, we do not have L S (ρ o ) = o(ρ 1/2 o ) for every function f with bounded characteristic. Consider the universal cover of an annulus {z ∈ C : R −1 < \|z\| < R}, say q : R 2 + → {z ∈ C : R −1 < \|z\| < R} defined on the upper half-plane R 2 + . We can arrange for the hyperbolic geodesic {iy : y > 0} to be mapped to the unit circle. If the segment from i to Ki is mapped to one complete circuit of the circle, then so are the segments from K n i to K n+1 i for every integer n. Consequently we see that the hyperbolic geodesic from i to i∞ has an image with L S (ρ o ) ∼ ρ o and not a half power. An almost identical argument gives the same conclusion for the Blaschke product B(z) = −1 n=−∞ (−1) 2 n i − z 2 n i + z ∞ n=0 2 n i − z 2 n i + z with zeros evenly spaced hyperbolically along the imaginary axis. The image of the positive imaginary axis is now the curve that traces out repeatedly the line segment between the two critical values of B. This shows that we can not hope for a better inequality than L S (ρ o ) = O(ρ o ) for functions with bounded Nevanlinna characteristic. Even this is untrue, as the following example shows. Let (y n ) be a strictly increasing sequence of strictly positive real numbers with 1/y n convergent. Then there is a Blaschke product B(z) = iy n − z iy n + z with zeros at the points iy n . This product is symmetric about the imaginary axis with B(−z) = B(z) . So, between any two successive zeros iy n and iy n+1 , there is a single critical point. We will be interested in the case where the y n converge slowly to ∞, so the hyperbolic distances d H (iy n , iy n+1 ) = log y n+1 /y n decrease to 0 as n → ∞. The Blaschke product is a contraction for the hyperbolic metric because of the Schwarz -Pick lemma, so the images under f of the critical points must converge to 0. The image of the positive imaginary axis then traces out line segments on the real axis between successive critical values and converges to 0. Now consider the geodesic γ(t) = −1 + ie t in the upper half-plane. As t → ∞, this becomes closer, in the hyperbolic metric, to the positive imaginary axis. Hence we see that image B(γ) traces out a path that never takes the value 0 and winds (negatively) about 0. Since d H (γ(t), ie t ) < 1/e t , we see that, for large n, the path B(γ[iy n , iy n+1 ]) completes approximately half a circuit about 0. Define f : R 2 + → P by f (z) = B(z + 1) B(z − 1) . This is an analytic function with bounded Nevanlinna characteristic because of Fatou's theorem, which says that the ratio of two bounded analytic functions has bounded characteristic (see [N] or Proposition 4.1 later). For a point iy on the imaginary axis we have f (iy) = B(1 + iy) B(−1 + iy) = B(1 + iy) B(1 + iy) . So the path f (γ[iy n , iy n+1 ]) lies on the unit circle and completes approximately one circuit about 0 for each sufficiently large n. If we set ρ o = log y n to be the hyperbolic distance from i to iy n , then we have L S (ρ o ) = L S (log y n ) ∼ 2πn . For example, take y n = n 2 . Then L S (ρ o ) ∼ 2π exp 1 2 ρ o . So we certainly do not have L S (ρ o ) = O(ρ 0 ). The issue here is that the Nevanlinna characteristic depends crucially on the choice of the origin. When we try to apply the arguments of Theorem 3.3 we take each point of the radial arc as the origin and so we require a bound on the Nevanlinna characteristic independent of the position of the origin. We will see that such a stronger, uniform condition is enough to give L P (ρ o ) = O(ρ o ). It will be useful in this context to have a precise form of Fatou's Theorem: Proposition 4.1 Fatou For a meromorphic function f : D → P the following conditions are equivalent. (a) f has bounded characteristic with T (1) K; (b) There are two analytic functions f 0 , f ∞ : D → C with f = f 0 /f ∞ , \|f 0 (z)\| 2 + \|f ∞ (z)\| 2 1 and \|f 0 (0)\| 2 + \|f ∞ (0)\| 2 e −2K for all z ∈ D. Proof: First we will consider this result when the function f extends analytically across the boundary D with no zeros or poles on {z : \|z\| = 1}. We can then obtain the general result by applying this to the functions restricted to discs of radius r < 1. We will need to use the chordal distance k(w, w ′ ) on the Riemann sphere. This is the length of the chord in R 3 joining the points w, w ′ ∈ P, so k(w, w ′ ) = 2\|w − w ′ \| 1 + \|w\| 2 1 + \|w ′ \| 2 and k(w, w ′ ) = 2 sin 1 2 d S (w, w ′ ) . Write f (z) = B 0 (z) B ∞ (z) exp h(z) where B 0 , B ∞ are finite Blaschke products on the zeros and poles of f and h is analytic on the closed unit disc. Let u 0 , u ∞ be continuous functions on the closed disc, harmonic on the interior and with boundary values u 0 (ζ) = log k(f (ζ), 0) , u ∞ (ζ) = log k(f (ζ), ∞) for \|ζ\| = 1. (4.1) This means that the pair of functions f 0 , f ∞ are Corona data. Proof: Suppose that f : D → P has bounded characteristic. Then we can write f = f 0 /f ∞ where f 0 = 1 2 B 0 exp(u 0 + iũ 0 ) ; f ∞ = 1 2 B ∞ exp(u ∞ + iũ ∞ ) and u 0 , u ∞ are the harmonic functions with boundary values (4.1). Theorem 4.1 shows that exp −2T (f zo ; 1) = \|f 0 (z 0 )\| 2 + \|f ∞ (z o )\| 2 . Hence, \|f 0 (z o )\| 2 + \|f ∞ (z o )\| 2 δ 2 if and only if T (f zo ; 1) − log δ. We can now prove that the inequality L S (ρ o ) = O(ρ o ) does hold for functions f with uniformly bounded Nevanlinna characteristic. . Proof: The corollary shows that we can write f as f 0 /f ∞ where f 0 , f ∞ are both bounded analytic functions. Moreover the function F : D → C 2 ; z → (f 0 (z), f ∞ (z)) will satisfy δ \|\|F (z)\|\| 1 for the Euclidean norm \|\| \|\| on C 2 . The Schwarz -Pick lemma, applied to f 0 and f ∞ , shows that \|\|F ′ (z)\|\|(1 − \|z\| 2 ) 2 . Also, a simple calculation gives \|\|f ′ (z)\|\| H→S = 2\|f ′ (z)\| 1 + \|f (z)\| 2 1 − \|z\| 2 2 = \|f ′ 0 (z)f ∞ (z) − f 0 (z)f ′ ∞ (z)\| \|f 0 (z)\| 2 + \|f ∞ (z)\| 2 (1 − \|z\| 2 ) \|\|F ′ (z)\|\| \|\|F (z)\|\| \|\|F (z)\|\| 2 (1 − \|z\| 2 ) \|\|F ′ (z)\|\| \|\|F (z)\|\| (1 − \|z\| 2 ) . So we have \|\|f ′ (z)\|\| H→S 2 \|\|F (z)\|\| 2 δ . Integrating this along a hyperbolic geodesic give the result. Let f : D → B be an analytic map into one of the domains B = D, C, P. Let γ : [0, ∞) → D be a radial hyperbolic geodesic with unit speed and argument θ, so γ(t) = re iθ where r = tanh 1 2 t. Consider the arc γ[0, ρ o ] with hyperbolic length ρ o . This has an image of length L B (ρ o ) = ρo 0 \|\|f ′ (γ(t)\|\| H→B dt in the metric for B.For each ρ < ∞, the image of the hyperbolic ball B H (0, ρ) = {z ∈ D : d H (0, z) < ρ} has finite area, which we denote by A(ρ) = A B (f (∆)). Then A(ρ) is an increasing function. If the area of the entire image A B (f (D)) is finite, then A(ρ) is bounded above and converges to A B (f (D)) as ρ ր ∞. Let f : D → C be an analytic map with A E (f (D)) finite. For any argument θ, the Euclidean length L E (ρ o ) of the image under f of the radial arc [0, r o e iθ ] of hyperbolic length ρ 0 (so r o = tanh 1 2 ρ o ) satisfies Let f : D → D be an analytic map with A H (f (D)) finite. For any argument θ, the hyperbolic length L H (ρ o ) of the image under f of the radial arc [0, r o e iθ ] of hyperbolic length ρ 0 satisfiesL H (ρ o ) = o(ρ 1/2 o ) as ρ o → ∞.Theorem 3.3 Let f : D → P be an analytic map with A S (f (D)) finite. For any argument θ, the spherical length L S (ρ o ) of the image under f of the radial arc [0, r o e iθ ] of hyperbolic length ρ 0 satisfies L S (ρ o ) = o(ρ 1/2 o ) as ρ o → ∞. Let f : D → P be an analytic function with uniformly bounded Nevanlinna characteristic. Then L S (ρ o ) = O(ρ o ) We can also apply this result to analytic maps f : ∆ → C defined on any hyperbolic domain ∆. In particular, let ∆ be the ball of hyperbolic radius ρ o about z o :Then u 0 (ζ) − u ∞ (ζ) = log k(f (ζ), 0) k(f (ζ), ∞) = log \|f (ζ)\| = Re h(ζ) and so u 0 (z) − u ∞ (z) = Re h(z) for each z ∈ D. We can choose harmonic conjugatesũ o andũ ∞ withThis certainly gives f = f 0 /f ∞ . For \|ζ\| = 1 we haveThe points 0, ∞ and f (z) are the vertices of a right-angle triangle in P, so Pythagoras' theorem showsThe First Nevanlinna Theorem shows thatand, since u 0 is harmonic, this isThis shows that (a) implies (b). It is a little simpler to reverse this argument to prove that (b) implies (a).Finally, for any analytic function f : D → P we can apply the above result tofor those r < 1 with no zeros or poles of f on {z : \|z\| = r}. By taking locally uniform limits as r ր 1 we obtain the proposition in general.We will say that an analytic function f : D → P has uniformly bounded characteristic if there is a constant C with each of the functionshaving characteristic T (f zo ; r) at most C for every r < 1. This is saying that the Nevanlinna characteristic is bounded independently of the origin z o we choose in D. The smallest value for C is clearly invariant under composing f with hyperbolic isometries of D. Fatou's theorem immediately gives us:Corollary 4.2A meromorphic function f : D → P has uniformly bounded characteristic if and only if f = f 0 /f ∞ for two bounded analytic functions f 0 , f ∞ with δ < (\|f 0 \| 2 + \|f ∞ \| 2 ) 1/2 1 for some δ > 0. L V Ahlfors, Conformal invariants, topics in geometric function theory. McGraw-HillL.V. Ahlfors, Conformal invariants, topics in geometric function theory, McGraw-Hill, 1973. Euclidean and hyperbolic lengths of images of arcs. A F Beardon, T K Carne, Proc. London Math. Soc. to appearA.F. Beardon and T.K. Carne, Euclidean and hyperbolic lengths of images of arcs, Proc. London Math. Soc., (2007) [to appear]. A property of bounded schlicht functions. F R Keogh, J. London Math. Soc. 29F.R. Keogh, A property of bounded schlicht functions, J. London Math. Soc., 29 (1954), 379-382. Eindeutige analytische Funktionen. R Nevanlinna, Springer VerlagBerlinR. Nevanlinna, Eindeutige analytische Funktionen, Springer Verlag, Berlin , 1936. Potential theory in modern function theory. M Tsuji, Department of Pure Mathematics and Mathematical Statistics. Centre for Mathematical Sciences. CB3 0WB UK. [email protected]. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge. CB3 0WB UK. [email protected]	[]
[ "Doublon-Holon Binding Effects on Mott Transitions in Two-Dimensional Bose Hubbard Model", "Doublon-Holon Binding Effects on Mott Transitions in Two-Dimensional Bose Hubbard Model" ]	[ "Hisatoshi Yokoyama \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n", "Masao Ogata \nDepartment of Physics\nUniversity of Tokyo\nBunkyo-ku113-0033TokyoJapan\n" ]	[ "Department of Physics\nTohoku University\n980-8578SendaiJapan", "Department of Physics\nUniversity of Tokyo\nBunkyo-ku113-0033TokyoJapan" ]	[]	A mechanism of Mott transitions in a Bose Hubbard model on a square lattice is studied, using a variational Monte Carlo method. Besides an onsite correlation factor, we introduce a four-body doublon-holon factor into the trial state, which considerably improves the variational energy and can appropriately describe a superfluidinsulator transition. Its essense consists in binding (and unbinding) of a doublon to a holon in a finite short range, identical with the cases of fermions. The features of this transition are qualitatively different from those of Brinkman-Rice-type transitions.	10.1016/j.jpcs.2008.06.087	[ "https://arxiv.org/pdf/0708.2765v1.pdf" ]	16,089,245	0708.2765	f13a4ffdcdd88b7d627432394d60afd36acff644	Doublon-Holon Binding Effects on Mott Transitions in Two-Dimensional Bose Hubbard Model 21 Aug 2007 Hisatoshi Yokoyama Department of Physics Tohoku University 980-8578SendaiJapan Masao Ogata Department of Physics University of Tokyo Bunkyo-ku113-0033TokyoJapan Doublon-Holon Binding Effects on Mott Transitions in Two-Dimensional Bose Hubbard Model 21 Aug 2007Bose Hubbard modelMott transitiondoublon-holon binding factorvariational Monte Carlo methodsquare A mechanism of Mott transitions in a Bose Hubbard model on a square lattice is studied, using a variational Monte Carlo method. Besides an onsite correlation factor, we introduce a four-body doublon-holon factor into the trial state, which considerably improves the variational energy and can appropriately describe a superfluidinsulator transition. Its essense consists in binding (and unbinding) of a doublon to a holon in a finite short range, identical with the cases of fermions. The features of this transition are qualitatively different from those of Brinkman-Rice-type transitions. Introduction: After early theoretical studies of Mott or superfluid-insulator transitions in interacting Bose systems [1], an experimental example has been realized using an ultracold dilute gas of bosonic atoms in an optical lattice [2]. The essence of this system is considered to be captured [3] by a Bose Hubbard model. This basic model has been studied with various methods; for square lattices, properties of T c , superfluid density, etc. were studied, applying a quantum Monte Carlo method to small systems (mainly 6 × 6 square lattice) [4], and a ground-state phase diagram in a plane of chemical potential and interaction strength was obtained, using a strong-coupling expansion [5]. These studies estimated the critical interaction strength of Mott transitions at U c /t = 16.4 ± 0.8 and 16.69, respectively, for the particle density of n = 1 (n = N e /N with N e : particle number, N : site number) at T = 0. Thus, the existence of a Mott transition has been embodied, but the mechanism of the transition is still not clear. Variational Monte Carlo approaches are very useful to understand the mechanism of the Mott transition, because one can directly and exactly treat wave functions. For the Bose Hubbard model, wave functions with only onsite correlation factors, which corresponds to the celebrated Gutzwiller wave function (GWF, Ψ G ) for fermions [6], were studied first [7]. In contrast to for fermions, GWF for bosons is solved analytically without additional mean-field-type approximations [8] for arbitrary dimensions, and yield a Brinkman-Rice-type (BR) transition [9] at U = U BR . In BR transitions, all the lattice sites are occupied with exactly one particle and the hopping completely ceases in the insulating regime, namely, Ψ G → N j=1 b † j \|0 and E = 0 for U > U BR . This result is caused by an oversimplified setup of the wave function, in which the effect of density fluctuation should be included. In this work, we introduce a doublonholon binding correlation factor into the trial function, following previous studies for fermions. Thereby, we can describe a superfluid-insulator transition more appropriately. 2. Formulation: We consider a spinless Bose-Hubbard model on a square lattice, H = −t ij (b † i b j + b † j b i ) + U 2 j n j (n j − 1),(1) where b † j is a creation operator of a boson at site j, n j = b † j b j and t, U > 0. Here, ij denotes a nearest-neighbor-site pair; the definition of t is a half of what was given in some literatures [4,7]. In this work, we restrict to n ∼ 1, because we would like to consider the most simple case of Mott transitions. The cases of other commensurate densities (n ≥ 2) must be essentially identical. We study this model eq. (1) through a variation theory. As a trial wave function, we use a Jastrow type of Ψ Q = P Q P G Φ (QWF), following previous studies for fermions [10,11]. Here, Φ is the ground state of noninteracting (completely coherent) bosons, namely Φ = 1, P G is an onsite projector corresponding to the famous Gutzwiller factor for fermions [6]: P G (g) = g D , with D = i n i (n i − 1)/2. As we repeatedly showed [11,12], intersite correlation factors are indispensable for appropriate descriptions of interacting systems. In particular near half filling, a four-body doublonholon correlation factor P Q is crucial to explain the mechanism of Mott transitions [13,14,15]. For S = 1/2 fermions, PQ is explicitly written as, PQ(µ) = (1 − µ)Q(2) with 0 ≤ µ ≤ 1 and Q = i τ [d i (1 − e i+τ ) + e i (1 − d i+τ )] ,(3)where d i (= n i↑ n i↓ ) and e i [= (1 − n i↑ )(1 − n i↓ )] are the doublon and holon operators respectively, i runs over all the sites, and τ the four nearestneighbor sites of the site i. When the variational parameter µ vanishes, Ψ Q is reduced to the GWF, Ψ G = P G Φ, in which doublons and holons can move independently. On the other hand, in the limit of µ → 1, a doublon becomes bound to a holon in nearest-neighbor sites. Consequently, plus (holons) and minus (doublons) charge carriers completely cancels, indicating a metal-insulator transition. In this work, we extend eq. (2) to bosons, and include the correlation between diagonal (secondnearest)-neighbor sites: P Q (µ, µ ′ ) = (1 − µ) Q (1 − µ ′ ) Q ′ ,(4) where the primes ( ′ ) denote the cases of diagonal neighbors. In eq. (4), Q has the same form asQ in eq. (3), but the doublon operator d i is replaced by a multiplon operator m i , which yields 1 (0) if the site i is multiply occupied (otherwise). Near Mott transitions, P Q of eq. (4) is substantially a doublon-holon factor, because the probability density of multiplon with more than two bosons is almost zero for such large values of U/t. To estimate expectation values accurately, we use a variational Monte Carlo method [16,17,18] of fixed particle numbers. First, we optimize the variational parameters, g, µ and µ ′ , simultaneously for each set of U/t, n and L, and then calculate physical quantities with the optimized parameters. Fig. 1. Comparison of total energy per site between GWF and QWF as a function of U/t for several system sizes. The particle density is commensurate (n = 1.0). For GWF, a Brinkman-Rice-type transition occurs at U = U BR . The dash-dotted line is a curve proportional to −t/U . ¡ ¢ ¡ £ ¡ ¤ ¡ ¥ ¥ ¡ ¦ § ¡ ¨ © ! " " # $ " % & ' ( ) 0 1 2 3 4 1 5 5 6 7 3 8 9 @ A B C D E F G H I P Q P R Through the optimization process, we average substantially several million samples, which reduces statistical errors in the total energy typically to the order of 10 −4 t. To check system-size dependence, particularly near phase transitions, we employ square lattices of N = L × L sites up to N = 1024 (L = 32) with the periodic-periodic boundary conditions. 3. Results: We start with comparison of the total energies E/t between GWF and QWF, which are shown in Fig. 1 for n = 1. As mentioned, in GWF [7], a BR transition occurs at U BR /t = 12 + 8 √ 2 = 23.31...; for U > U BR , each site is occupied by exactly one particle and hopping or density fluctuation completely ceases. Consequently, E/t vanishes in the insulating regime. On the side of Bose fluid (U < U BR ), E/t vanishes as ∝ (1 − U/U BR ) 2 , meaning this transition is a continuous type. The difference of the two functions is small for small U/t ( < ∼ 15), but it becomes conspicuous as U/t approaches Mott critical values (U/t > ∼ 20). Thus, QWF is a considerably improved function in the point of the variation principle. As we will discuss in detail shortly, QWF also exhibits a superfluid-insulator transition, but its behavior is qualitatively different from that of GWF. In contrast to E of GWF, E of QWF in the insulating regime (U > U c ) does not vanish but is proportional to −t 2 /U as seen in Fig. 1, which behavior is expected from strong-correlation theories, namely, the density fluctuation does not cease but is re- stricted to a finite short range. This is the essence of the mechanism of Mott transition owing to the doublon-holon binding. In Fig. 2, we show the magnification of E/t of QWF for U ∼ U c . For large L (> 20), we find, near U = U c , double-minimum structure of E/t in the space of variational parameters, meaning this transition is first order, although such structure cannot be confirmed for L ≤ 14 by our VMC calculations. The first-order features are more easily found in the variational parameters and in some quantities. In Fig. 3 S T S U V W V S V V V X V Y ` W a Y ` W a X b ` W a¡ ¢ ¡ £ ¤ ¥ ¤ ¦ § ¥ § ¨ © § £ ª £ « ¬ ® ¯ ° ± ® ² ² ³ ´ ° F ig. 5. The occupation rate of the k = (0, 0) level for the two wave functions is depicted as a function of U/t. Data for several system sizes are simultaneously plotted. holon are tightly bound in nearest-neighbor sites. In Fig. 4, we show the average of D, which is, near U c , virtually identical with the doublon density, namely, an order parameter of Mott transitions. The discontinuities of this quantity at U c corroborate a first-order Mott transition. Finally, let us consider a couple of properties of this Mott transition. In Fig. 5, we show the occupation rate, ρ(k) = b † k b k /N , of the lowest-energy level, k = 0 = (0, 0) versus U/t. Although ρ(0) does not directly indicate the superfluid density, it must be a good index of superfluidity. For the noninteracting case (U/t = 0), all the particles fall in the k = 0 level. As the interaction becomes strong, ρ(0) decreases at first gradually and drops discontinuously to the order of 1/N at U c . Thus, the superfluidity vanishes at the transition. In Fig. 5, we plot the chemical potential, ζ = ∂E/∂n, estimated 6. The Behavior of chemical potential, ζ/t, as a function of particle density near n = 1 is shown for two values of U/t, 16 in the superfluid regime and 24 in the insulating regime. Data of three system sizes are simultaneously plotted for each U/t. The singular behavior in ζ at n = 1 for U > Uc stems from the singularly low E at n = 1 as a function of n. µ ¶ · ¸ ¸ ¶ ¸ µ ¸ µ ¹ µ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Á Ä Ã AE Ã Â Ç È Ç É Ê Ë Fig. from finite differences, as a function of n. In the superfluid regime slightly below U c (U/t = 16, gray symbols), ζ is a smooth function of n even at n = 1, indicating the state is gapless in density excitation. On the other hand in the insulating regime slightly above U c (U/t = 24, black symbols), ζ has a large discontinuity at n = 1; a density excitation gap opens for U > U c at n = 1. The gap behavior is also confirmed by the density correlation function N (q) for small \|q\| (not shown). Discussions: In this proceedings, we have found that a wave function with doublon-holon correlation factor, Ψ Q , qualitatively improve the description of a Mott transition also in a Bose system. Thus, it is probable that the mechanism of Mott transitions for bosons is basically identical to that for fermions. It is urgent to compare theoretical results with experiments particularly of optical lattices. We have left many issues to be discussed, which will be published elsewhere soon. When main calculations here were finished, we became aware that a similar wave function had been studied recently [19]. Fig. 3 . 3Behavior of doublon-holon binding parameter near Uc. Data for five system sizes are plotted; for L ≥ 20, discontinuities are observed at U = Uc. Solid and open symbols have the same meaning as in Fig. 2. Fig. 4 . 4, we plot the optimized doublon-holon binding parameter µ near U c . For L ≥ 20, there are clear discontinuities at U = U c ; the large values of µ for U > U c indicate that a doublon Behavior of D , which is substantially the doublon density and an order parameter of Mott transitions. Data are shown only for the optimized states. . M P A For Instance, Fisher, Phys. Rev. B. 40546For instance, M. P. A. Fisher et al. , Phys. Rev. B 40, 546 (1989). . M Greiner, Nature. 41539M. Greiner et al. , Nature 415, 39 (2002). . D Jaksch, Phys. Rev. B. 813108D. Jaksch et al. , Phys. Rev. B 81, 3108 (1998). . W Krauth, N Trivedi, Europhys. Lett. 14627W. Krauth, N. Trivedi, Europhys. Lett. 14, 627 (1991). . N Elstner, H Monien, Phys. Rev. B. 5912184N. Elstner and H. Monien, Phys. Rev. B 59, 12184 (1999). . M Gutzwiller, Phys. Rev. Lett. 10159M. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963). . W Krauth, M Caffarel, J. -P Bouchaud, Phys. Rev. B. 453137W. Krauth, M. Caffarel, J. -P. 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[ "About the globular homology of higher dimensional automata", "About the globular homology of higher dimensional automata" ]	[ "Philippe Gaucher Résumé " ]	[]	[]	On introduit un nouveau nerf simplicial d'automate parallèle dont l'homologie simpliciale décalée de un fournit une nouvelle définition de l'homologie globulaire. Avec cette nouvelle définition, les inconvénients de la construction de [9] disparaissent. De plus les importants morphismes qui associentà tout globe les zones correspondantes de branchements et de confluences de chemins d'exécution deviennent ici des morphismes d'ensembles simpliciaux.One of the contributions of[11]is the introduction of two homology theories as a starting point for studying branchings and mergings in higher dimensional automata (HDA) from an homological point of view. However these homology theories had an important drawback : roughly speaking, they were not invariant by subdivisions of the observation. Later in [9], using a model of concurrency by strict globular ω-categories borrowed from [19], two new homology theories are introduced : the negative and positive corner homology theories H − and H + , also called the branching and the merging homologies. It is proved in [8] that they overcome the drawback of Goubault's homology theories.Another idea of [9] is the construction of a diagram of abelian groups like inFigure 1, where H gl * is a new homology theory called the globular homology. Geometrically, the non-trivial cycles of the globular homology must correspond to the oriented empty globes of C, and the non-trivial cycles of the branching (resp. the merging) homology theory must correspond to the branching (resp. merging) areas of execution paths. And the morphisms h − and h + must associate to any globe its corresponding branching area and merging area of execution paths. Many potential applications in computer science of these morphisms are put forward in[9].Globular homology was therefore created in order to fulfill two conditions :• Globular homology must take place in a diagram of abelian groups like inFigure 1. And the geometric meaning of h − and h + must be exactly as above described.• Globular homology must be an invariant of HDA with respect to reasonable deformations of HDA, that is of the corresponding ω-category.What is a reasonable deformation of HDA was not yet very clear in[9]. This question is discussed with much more details in[10].The old globular homology (i.e. the construction exposed in [9]) satisfied the first condition, and the second one was supposed to be satisfied by definition (cf. Definition 8.2 of two homotopic ω-categories in [9]), even if some problems were already mentioned, particularly the non-vanishing of the "old" globular homology of I 3 , and more generally of I n for any n 1 in strictly positive dimension. This latter problem is disturbing because the n-cube I n (i.e. the corresponding automaton which consists of n 1-transitions carried out at the same time) can be deformed by crushing all the p-faces with p > 1 into an ω-category which has only 0-morphisms and 1-morphisms and because the globular homology is supposed to be an invariant by such deformations. The philosophy exposed in [10] tells us similar things : using S-deformations and T-deformations, the n-cube and the oriented line must be the same up to homotopy, and therefore must have the same globular homology.	null	[ "https://arxiv.org/pdf/math/0002216v5.pdf" ]	6,365,873	math/0002216	9b3b75291394f3c7ceb06e07b6e6256f9f670c4e	About the globular homology of higher dimensional automata 23 May 2001 Philippe Gaucher Résumé About the globular homology of higher dimensional automata 23 May 2001 On introduit un nouveau nerf simplicial d'automate parallèle dont l'homologie simpliciale décalée de un fournit une nouvelle définition de l'homologie globulaire. Avec cette nouvelle définition, les inconvénients de la construction de [9] disparaissent. De plus les importants morphismes qui associentà tout globe les zones correspondantes de branchements et de confluences de chemins d'exécution deviennent ici des morphismes d'ensembles simpliciaux.One of the contributions of[11]is the introduction of two homology theories as a starting point for studying branchings and mergings in higher dimensional automata (HDA) from an homological point of view. However these homology theories had an important drawback : roughly speaking, they were not invariant by subdivisions of the observation. Later in [9], using a model of concurrency by strict globular ω-categories borrowed from [19], two new homology theories are introduced : the negative and positive corner homology theories H − and H + , also called the branching and the merging homologies. It is proved in [8] that they overcome the drawback of Goubault's homology theories.Another idea of [9] is the construction of a diagram of abelian groups like inFigure 1, where H gl * is a new homology theory called the globular homology. Geometrically, the non-trivial cycles of the globular homology must correspond to the oriented empty globes of C, and the non-trivial cycles of the branching (resp. the merging) homology theory must correspond to the branching (resp. merging) areas of execution paths. And the morphisms h − and h + must associate to any globe its corresponding branching area and merging area of execution paths. Many potential applications in computer science of these morphisms are put forward in[9].Globular homology was therefore created in order to fulfill two conditions :• Globular homology must take place in a diagram of abelian groups like inFigure 1. And the geometric meaning of h − and h + must be exactly as above described.• Globular homology must be an invariant of HDA with respect to reasonable deformations of HDA, that is of the corresponding ω-category.What is a reasonable deformation of HDA was not yet very clear in[9]. This question is discussed with much more details in[10].The old globular homology (i.e. the construction exposed in [9]) satisfied the first condition, and the second one was supposed to be satisfied by definition (cf. Definition 8.2 of two homotopic ω-categories in [9]), even if some problems were already mentioned, particularly the non-vanishing of the "old" globular homology of I 3 , and more generally of I n for any n 1 in strictly positive dimension. This latter problem is disturbing because the n-cube I n (i.e. the corresponding automaton which consists of n 1-transitions carried out at the same time) can be deformed by crushing all the p-faces with p > 1 into an ω-category which has only 0-morphisms and 1-morphisms and because the globular homology is supposed to be an invariant by such deformations. The philosophy exposed in [10] tells us similar things : using S-deformations and T-deformations, the n-cube and the oriented line must be the same up to homotopy, and therefore must have the same globular homology. H gl (C) I I I I I I I I I H − (C) H + (C) Figure 1: Associating to any globe its two corners The non-vanishing of the second globular homology group of I 3 (see Figure 2(c)) is due for instance to the 2-dimensional globular cycle (R(−00) * 0 R(0 + +)) * 1 (R(−0−) * 0 R(0 + 0)) − (R(−00) * 0 R(0 + +)) − (R(−0−) * 0 R(0 + 0)) It is the reason why it was suggested in [9] to add the relation A * 1 B = A + B at least to the 2-dimensional stage of the old globular complex. But there is then no reason not to add the same relation in the rest of the definition of the old globular complex. For example, if we take the quotient of the old globular complex by the relation A * 1 B = A + B for any pair (A, B) of 2-morphisms, then the ω-category defined as the free ω-category generated by the globular set generated by two 3-morphisms The formal globular homology of Definition 9.3 is exactly equal to the quotient of the old globular complex by these missing relations. It is conjectured (see conjecture 9.5) that this homology theory will coincide for free ω-categories generated by semi-cubical sets with the homology theory of Definition 5.2, this latter being the simplicial homology of the globular simplicial nerve N gl shifted by one. We claim that Definition 5.1 (and its simplicial homology shifted by one) cancels the drawback of the old globular homology at least for the following reasons : • It is noticed in [9] that both corner homologies come from the simplicial homology of two augmented simplicial nerves N − and N + ; there exists one and only natural transformation h − (resp. h + ) from N gl to N − (resp. N + ) preserving the interior labeling (Theorem 6.1). α A B B B / / β (a) Composition of two 2-morphims (0) (2) (02) > > (12) / / (1) \ d @ @ @ @ @ @ @ @ @ @ @ @ @ @ • In homology, h − and h + induce two natural linear maps from H gl * to resp. H − * and H + * which do exactly what we want. • The globular homology (formal or not) of I n vanishes in strictly positive dimension for any n 0. The globular homology of ∆ n (the n-simplex) and of 2 n (the free ω-category generated by one n-dimensional morphism) as well. • Using Theorem 9.7 explaining the exact mathematical link between the old construction and the new one, one sees that one does not lose the possible applications in computer science pointed out in [9]. • The new globular homology, as well as the new globular cut are invariant by Sdeformations, that is intuitively by contraction and dilatation of homotopies between execution paths. We will see however that it is not invariant by T-deformations, that is by subdivision of the time, as the old definition and this problem will be a little bit discussed. This paper is two-fold. The first part introduces the new material. The second part justifies the new definition of the globular homology. After Section 2 which recalls some conventions and some elementary facts about strict globular ω-categories (non-contracting or not) and about simplicial sets, the setting of simplicial cuts of non-contracting ω-categories and that of regular cuts are introduced. The first notion allows to enclose the new globular nerve of this paper and both corner nerves in one unique formalism. The notion of regular cuts gives an axiomatic framework for the generalization of the notion of negative and positive folding operators of [8]. Section 4 is an illustration of the previous new notions on the case of corner nerves. In the same section, some non-trivial facts about negative folding operators are recalled. Section 5 provides the definition of the globular nerve of a non-contracting ω-category. The organization of the rest of the paper follows the preceding explanations. First in Section 6, the morphisms h − and h + are constructed. Section 7 proves that the globular cut is regular. In particular, we get the globular folding operators. Section 8 proves the vanishing of the globular homology of the n-cube, the n-simplex and the free ω-category generated by one n-morphism. At last Section 9 makes explicit the exact relation between the new globular homology and the old one. Section 10 speculates about deformations of ω-categories considered as a model of HDA and the construction of the bisimplicial set of [10] is detailed. Conventions and notations HDA means higher dimensional automaton. In this paper, this is another term for semi-cubical set, or the corresponding free ω-category generated by it. Various homology theories (see the diagram of Theorem 9.7) will appear in this paper. It is helpful for the reader to keep in mind that the total homology of a semi-cubical set is used nowhere in this work. Non-contracting ω-category Let C be an ω-category. We want to define an ω-category PC (P for path) obtained from C by removing the 0-morphisms, by considering the 1-morphisms of C as the 0-morphisms of PC, the 2-morphisms of C as the 1-morphisms of PC etc. with an obvious definition of the source and target maps and of the composition laws (this new ω-category is denoted by C [1] in [10]). The map P : C → PC does not induce a functor from ωCat to itself because ω-functors can contract 1-morphisms and because with our conventions, a 1-source or a 1-target can be 0-dimensional. Hence the following definition Proposition and definition 2.6. For a globular ω-category C, the following assertions are equivalent : (i) PC is an ω-category ; in other terms, * i , s i and t i for any i 1 are internal to PC and we can set * PC i = * C i+1 , * PC i = * C i+1 and * PC i = * C i+1 for any i 0. (ii) The maps s 1 and t 1 are non-contracting, that is if x is of strictly positive dimension, then s 1 x and t 1 x are 1-dimensional (a priori, one can only say that s 1 x and t 1 x are of dimension lower or equal than 1) If Condition (ii) is satisfied, then one says that s 1 and t 1 are non-contracting and that C is non-contracting. Proof. Suppose s 1 and t 1 non-contracting. Let x and y be two morphisms of strictly positive dimension and p 1. Then s 1 s p x = s 1 x therefore s p x cannot be 0-dimensional. If x * p y then s 1 (x * p y) = s 1 x if p = 1 and if p > 1 for two different reasons. Therefore x * p y cannot be 0-dimensional as soon as p 1. Definition 2.7. Let f be an ω-functor from C to D. The morphism f is non-contracting if for any 1-dimensional x ∈ C, the morphism f (x) is a 1-dimensional morphism of D (a priori, f (x) could be either 0-dimensional or 1-dimensional). Definition 2.8. The category of non-contracting ω-categories with the non-contracting ωfunctors is denoted by ωCat 1 . Notice that in [9], the word "non-1-contracting" is used instead of simply "non-contracting". Since [10], the philosophy behind the idea of deforming the ω-categories viewed as models of HDA is better understood. In particular, the idea of not contracting the morphisms is relevant only for 1-dimensional morphisms. So the "1" in "non-1-contracting" is not anymore necessary. Definition 2.9. Let C be a non-contracting ω-category. Then the ω-category PC above defined is called the path ω-category of C. The map C → PC induces a functor from ωCat 1 to ωCat. Here is a fundamental example of non-contracting ω-category. Consider a semi-cubical set K and consider the free ω-category Π(K) := n∈ K n .I n generated by it where • I n is the free ω-category generated by the faces of the n-cube, whose construction is recalled in Section 4. • the integral sign denotes the coend construction and K n .I n means the sum of "cardinal of K n " copies of I n (cf. [15] for instance). Cut of globular higher dimensional categories Before introducing the globular nerve of an ω-category, let us introduce the formalism of regular simplicial cuts of ω-categories. The notion of simplicial cuts enables us to put together in the same framework both corner nerves constructed in [9,8] and the new globular nerve of Section 5. The notion of regular cuts enables to generalize the notion of negative (resp. positive) folding operators associated to the branching (resp. merging) nerve (cf. [8]). It is also an attempt to finding a way of characterizing these three nerves. There are no much more things known about this problem. Definition 3.1. [5] An augmented simplicial set is a simplicial set ((X n ) n 0 , (∂ i : X n+1 −→ X n ) 0 i n+1 , (ǫ i : X n −→ X n+1 ) 0 i n ) together with an additional set X −1 and an additional map ∂ −1 from X 0 to X −1 such that ∂ −1 ∂ 0 = ∂ −1 ∂ 1 . A morphism of augmented simplicial set is a map of N-graded sets which commutes with all face and degeneracy maps. We denote by Sets ∆ op + the category of augmented simplicial sets. The "chain complex" functor of an augmented simplicial set X is defined by C n (X) = ZX n for n −1 endowed with the simplicial differential map (denoted by ∂) in positive dimension and the map ∂ −1 from C 0 (X) to C −1 (X). The "simplicial homology" functor H * from the category of augmented simplicial sets Sets ∆ op + to the category of abelian groups Ab is defined as the usual one for * 1 and by setting H 0 (X) = Ker(∂ −1 )/Im(∂ 0 − ∂ 1 ) and H −1 (X) = ZX −1 /Im(∂ −1 ) whenever X is an augmented simplicial set. Definition 3.2. A (simplicial) cut is a functor F : ωCat 1 → Sets ∆ op + together with a family ev = (ev n ) n 0 of natural transformations ev n : F n −→ tr n P where F n is the set of n-simplexes of F. A morphism of cuts from (F, ev) to (G, ev) is a natural transformation of functors φ from F to G which makes the following diagram commutative for any n 0 : F n φn evn / / tr n P G n evn < < y y y y y y y y The terminology of "cuts" is borrowed from [21]. It will be explained later : cf. the explanations around Figure 3 and also Section 10. There is no ambiguity to denote all ev n by the same notation ev in the sequel. The map ev of N-graded sets is called the evaluation map and a cut (F, ev) will be always denoted by F. If F is a functor from ωCat 1 to Sets ∆ op + , let C F n+1 (C) := C n (F(C)) and let H F n+1 be the corresponding homology theory for n −1. Let M F n : ωCat 1 −→ Ab be the functor defined as follows : the group M F n (C) is the subgroup generated by the elements x ∈ F n−1 (C) such that ev(x) ∈ tr n−2 PC for n 2 and with the convention M F 0 (C) = M F 1 (C) = 0 and the definition of M F n is obvious on non-contracting ω-functors. The elements of M F * (C) are called thin. Let CR F n : ωCat 1 −→ Comp(Ab) be the functor defined by CR F n := C F n /(M F n + ∂M F n+1 ) and endowed with the differential map ∂. This chain complex is called the reduced complex associated to the cut F and the corresponding homology is denoted by HR F * and is called the reduced homology associated to F. A morphism of cuts from F to G yields natural morphisms from H F * to H G * and from HR F * to HR G * . There is also a canonical natural transformation R F from H F * to HR F * , functorial with respect to F, that is making the following diagram commutative : 2. F 0 := tr 0 P. H F * R F / / HR F * H G * R G / / HR G 3. ev • ǫ i = ev. 4. for any natural transformation of functors µ from F n−1 to F n with n 1, and for any natural map from tr n−1 P to F n−1 such that ev • = Id tr n−1 P , there exists one and only one natural transformation µ. from tr n P to F n such that the following diagram commutes tr n P Id tr n P & & µ. / / F n evn / / tr n P tr n−1 P Id tr n−1 P 8 8 in O O / / F n−1 ev n−1 / / µ O O tr n−1 P in O O where i n is the canonical inclusion functor from tr n−1 P to tr n P. 6. Let Φ F n := F n • ev be a natural transformation from F n−1 to itself ; then Φ F n induces the identity natural transformation on CR F n . 7. if x, y and z are three elements of F n (C), and if ev(x) p ev(y) = ev(z) for some 1 p n, then x + y = z in CR F n+1 (C) and in a functorial way. let If F is a regular cut, then the natural transformation Φ F n is called the n-dimensional folding operator of the cut F. By convention, one sets F 0 = Id F −1 and Φ F 0 = Id F −1 . There is no ambiguity to set Φ F (x) := Φ F n+1 (x) for x ∈ F n (C) for some ω-category C. So Φ F defines a natural transformation, and even a morphism of cuts, from F to itself. However beware of the fact that there is really an ambiguity in the notation F : so this latter will not be used. Condition 3 tells us that the ǫ i operations are really degeneracy maps. Condition 4 ensures the existence and the uniqueness of the folding operator associated to the cut. Condition 5 tells us several things. A priori, a natural transformation like ev∂ i F n from tr n−1 P to tr n−2 P is necessarily of the form d α p for some p n − 1 and for some α ∈ {−, +}. Indeed consider the free ω-category 2 n (A) generated by some n-morphism A. Then ev∂ i ev∂ i F n = ev∂ i F n d β n−1 for some β ∈ {−, +} = ev∂ i F n i n−1 d β n−1 = ev∂ i ǫ n−2 F n−1 d β n−1 by construction of F n = ev F n−1 d β n−1 = d β n−1 by construction of F n Therefore ev∂ n−2 F n , ev∂ n−1 F n ⊂ {s n−1 , t n−1 } always holds. Condition 5 states more precisely that these latter sets are actually equal. In other terms, the operator F n concentrates the "weight" on the faces ∂ n−2 Condition 6 explains the link between the thin elements of the cut and the folding operators. Intuitively, the folding operators move the labeling of the elements of the cuts in a canonical position without changing the total sum on the source and target sides. What is exactly this canonical position is precisely described by Proposition 3.5. Conditions 5 and 7 ensure that by moving the labeling of an element, we stay in the same equivalence class modulo thin elements. Now here are some trivial remarks about regular cuts : • Let f be a natural set map from tr 0 PC = C 1 to itself. Let 2 1 be the ω-category generated by one 1-morphism A. Then necessarily f (A) = A and therefore f = Id. So the above axioms imply that ev 0 = Id. • The map Φ F n induces the identity natural transformation on HR F n . • For any n 1, there exists non-thin elements x in F n−1 (C) as soon as C n = ∅. Indeed, if u ∈ C n , ev F n (u) = u, therefore F n (u) is a non-thin element of F n−1 (C). We end this section by some general facts about regular cuts. Proposition 3.4. Let f be a morphism of cuts from F to G. Suppose that F and G are regular. Then Φ G • f = f • Φ F as natural transformation from F to G. In other terms, the following diagram is commutative : F f / / Φ F G Φ G F f / / G Proof. Let n 0 and let P (n) be the property : "for any ω-category C and any x ∈ tr n PC, then f F n+1 (x) = G n+1 x." One has Φ F 1 := Id F 0 , Φ G 1 := Id G 0 and necessarily f 0 = Id by definition of a morphism of cuts. Therefore P (0) holds. Now suppose P (n) proved for some n 0. One has evf F n+2 = ev F n+2 = Id tr n+1 P since f is a morphism of cuts and f F n+2 i n+1 = f (ǫ n . F n+1 )i n+1 = f ǫ n F n+1 by definition of ǫ n . F n+1 = ǫ n f F n+1 since f morphism of simplicial sets = ǫ n G n+1 by induction hypothesis Therefore the natural transformation f F n+2 from tr n+1 P to G n+1 can be identified with ǫ n . G n+1 which is precisely G n+2 . Therefore P (n + 1) is proved. At last, if x ∈ F n (C), then Φ G f (x) = G n+1 evf (x) by definition of folding operators = G n+1 ev(x) since f preserves the evaluation map = f F n+1 ev(x) since P (n) holds = f Φ F (x) by definition of folding operators Proposition 3.5. If u is a (n + 1)-morphism of C with n 1, then F n+1 u is an homotopy within the simplicial set F(C) between F n s n u and F n t n u. Proof. The natural map ev ∂ i F n+1 for 0 i n from tr n P to tr n−1 P is of the form d α i m i for m i n with m i n − 1 for 0 i n − 2 and ev∂ n−1 F n+1 , ev∂ n F n+1 = {s n , t n }. Therefore for 0 i n − 2, ∂ i F n+1 = ∂ i F n+1 s n = ∂ i F n+1 t n by rule 5b of Definition 3.3. And by construction of F n+1 , one obtains ∂ i F n+1 = ǫ n−2 ∂ i F n s n = ǫ n−2 ∂ i F n t n . Corollary 3.6. If x ∈ CR F n+1 (C), then ∂x = ∂ F n+1 x = F n s n x − F n t n x in CR F n (C). In other terms, the differential map from CR F n+1 (C) to CR F n (C) with n 1 is induced by the map s n − t n . The cuts of branching and merging nerves We see now that the corner nerves N η defined in [9] are two examples of regular cuts with the correspondence η n := N η n , Φ η n := Φ N η n , H η n := H N η n , HR η n := HR N η n and ev(x) = x(0 dim(x) ). Let us first recall the construction of the free ω-category I n generated by the faces of the n-cube. The faces of the n-cube are labeled by the words of length n in the alphabet {−, 0, +}, one word corresponding to the barycenter of one face. We take the convention that 00 . . . 0 (n times) =: 0 n corresponds to its interior and that − n (resp. + n ) corresponds to its initial state − − · · · − (n times) (resp. to its final state + + · · · + (n times)). If x is a face of the n-cube, let R(x) be the set of faces of x. If X is a set of faces, then let R(X) = x∈X R(x). Notice that R(X ∪ Y ) = R(X) ∪ R(Y ) and that R({x}) = R(x). Then I n is the free ω-category generated by the R(x) with the rules 1. For x p-dimensional with p 1, s p−1 (R(x)) = R(s x ) and t p−1 (R(x)) = R(t x ) where s x and t x are the sets of faces defined below. 2. If X and Y are two elements of I n such that t p (X) = s p (Y ) for some p, then X ∪ Y belongs to I n and X ∪ Y = X * p Y . The set s x is the set of subfaces of the faces obtained by replacing the i-th zero of x by (−) i , and the set t x is the set of subfaces of the faces obtained by replacing the i-th zero of x by (−) i+1 . For example, s 0+00 = {-+00, 0++0, 0+0-} and t 0+00 = {++00, 0+-0, 0+0+}. Figure 2(c) represents the free ω-category generated by the 3-cube. The branching and merging nerves are dual from each other. We set ωCat(I n+1 , C) η := {x ∈ ωCat(I n+1 , C), d η 0 (u) = η n+1 and dim(u) = 1 =⇒ dim(x(u)) = 1} where η ∈ {−, +} and where η n+1 is the initial state (resp. final state) of I n+1 if η = − (resp. η = +). For all (i, n) such that 0 i n, the face maps ∂ i from ωCat(I n+1 , C) η to ωCat(I n , C) η are the arrows ∂ η i+1 defined by ∂ η i+1 (x)(k 1 . . . k n+1 ) = x(k 1 . . . [η] i+1 . . . k n+1 ) and the degeneracy maps ǫ i from ωCat(I n , C) η to ωCat(I n+1 , C) η are the arrows Γ η i+1 defined by setting Γ − i (x)(k 1 . . . k n ) := x(k 1 . . . max(k i , k i+1 ) . . . k n ) Γ + i (x)(k 1 . . . k n ) := x(k 1 . . . min(k i , k i+1 ) . . . k n ) with the order − < 0 < +. Proposition and definition 4.1. [9] Let C be an ω-category. The N-graded set N η (C) together with the convention N η −1 (C) = C 0 , endowed with the maps ∂ i and ǫ i above defined with moreover ∂ −1 = s 0 (resp. ∂ −1 = t 0 ) if η = − (resp. η = +) and with ev(x) = x(0 n ) for x ∈ ωCat(I n , C) is a simplicial cut. It is called the η-corner simplicial nerve N η of C. Set H η n+1 (C) := H n (N η (C)) for n −1. These homology theories are called branching and merging homology respectively and are exactly the same homology theories as that defined in [9] and studied in [8]. And we have Theorem 4.2. [8] The simplicial cut N η is regular. The associated folding operator N η n coincides with the operator η n defined in [8]. And therefore the associated homology theory HR N η n coincide with the reduced corner homology HR η n defined in [8]. It is useful for the sequel to remind some important properties of the folding operators associated to corner nerves. Theorem 4.3. [8] Let C be an ω-category. Let x be an element of N − n (C) . Then the following two conditions are equivalent : 1. the equality x = Φ − n (x) holds 2. for 1 i n, one has ev∂ + i x = ∂ + i x(0 n ) is 0-dimensional and for 1 i n − 2, one has ∂ − i x ∈ Im(Γ − n−2 . . . Γ − i ). Another operator coming from [8] which matters for this paper is the operator θ − i . Sketch of proof. Consider the θ − 1 , . . . , θ − n−1 of [8]. One has Definition 4.4. Let x ∈ N − n (C) for some C such that for any 1 j n + 1, ∂ + j x is 0-dimensional.∂ + j θ − i = θ − i−1 ∂ + j if j < i θ − i ∂ + j if j > i + 2 ∂ + i θ − i = v ψ − i ∂ + i ∂ + i+1 θ − i = ǫ i+1 ∂ + i+1 ∂ − i + i ǫ i+1 ∂ + i+1 ∂ + i+1 ∂ + i+2 θ − i = v ψ + i ∂ + i+2 where, for the last formula, v ψ ± i are other operators which is not important to explicitly define here : the only important thing is that ∂ + i θ − i remains 0-dimensional if the argument is 0-dimensional. Hence property 1. As for property 2, it is enough to check it for i = 1. And in this case, y is a thin 4-cube satisfying ∂ + 1 y = v ψ − 2 Γ − 1 ∂ + 1 x ∂ + 2 y = Γ − 2 ∂ + 2 x ∂ + 3 y = ǫ 3 (Γ − 1 ∂ + 2 ∂ − 1 x + 1 ǫ 2 ∂ + 2 ∂ + 2 x) ∂ + 4 y = v ψ + 2 Γ − 2 ∂ + 3 x Once again, we refer to [8] for the precise definition of the operators involved in the above formulas. The only thing that matters here is the dimension of ∂ + i y. By [8], we know that Φ − = Θ • Ψ when Θ is a composite of θ − i and such that for x negative, Ψx = x. Hence property 3. The graded set (ωCat(I n , C)) n 0 endowed with the operations ∂ ± i above defined and by the maps ǫ i (x)(k 1 . . . k n+1 ) = x(k 1 . . . k i . . . k n+1 ) for x ∈ ωCat(I n , C) and 1 i n + 1 is a cubical set and is usually known as the cubical singular nerve of C [4]. The use of the same notation ǫ i for the degeneracy maps of the cubical singular nerve and the degeneracy maps of the three simplicial nerves appearing in this paper is very confusing. Fortunately, we will not need the degeneracy maps of the cubical singular nerve in this work except for Theorem 4.5 right above. The globular cut The most direct way of constructing a cut of ω-categories consists of using the composite of both functors P : C → PC and N where N is the simplicial nerve functor defined by Street 1 . Let us start this section by recalling the construction of the free ω-category ∆ n generated by the faces of the n-simplex. The faces of the n-simplex are labeled by the strictly increasing sequences of elements of {0, 1, . . . , n}. The length of a sequence is equal to the dimension of the corresponding face plus one. If x is a face of the n-simplex, its subfaces are all increasing sequences of {0, 1, . . . , n} included in x. If x is a face of the n-simplex, let R(x) be the set of faces of x. If X is a set of faces, then let R(X) = x∈X R(x). Notice that R(X ∪ Y ) = R(X) ∪ R(Y ) and that R({x}) = R(x) . Then ∆ n is the free ω-category generated by the R(x) with the rules 1. For x p-dimensional with p 1, s p−1 (R(x)) = R(s x ) 1 Of course, the functor N can be viewed as a functor from ωCat1 to Sets ∆ op + , but a "good" cut should not be extendable to a functor from ωCat to Sets ∆ op + . and t p−1 (R(x)) = R(t x ) where s x and t x are the sets of faces defined below. 2. If X and Y are two elements of ∆ n such that t p (X) = s p (Y ) for some p, then X ∪ Y belongs to ∆ n and X ∪ Y = X * p Y . where s x (resp. t x ) is the set of subfaces of x obtained by removing one element in odd position (resp. in even position). Sometimes we will write (for instance) (0 < 4 < 5 < 8 < 9) instead of simply (04589). Figure 2(b) gives the example of the 2-simplex. Let x ∈ ωCat(∆ n , C). Then consider the labeling of the faces of respectively ∆ n+1 and ∆ n−1 defined by : • ǫ i (x)(σ 0 < · · · < σ r ) = x(σ 0 < · · · < σ k−1 < σ k − 1 < · · · < σ r − 1) if σ k−1 < i and σ k > i. • x(σ 0 < · · · < σ k−1 < i < σ k+1 − 1 < · · · < σ r − 1) if σ k−1 < i, σ k = i and σ k+1 > i + 1. • x(σ 0 < · · · < σ k−1 < i < σ k+2 − 1 < · · · < σ r − 1) if σ k−1 < i, σ k = i and σ k+1 = i + 1. and ∂ i (x)(σ 0 < · · · < σ s ) = x(σ 0 < · · · < σ k−1 < σ k + 1 < · · · < σ s + 1) where σ k , . . . , σ s i and σ k−1 < i. It can be checked that ǫ i (x) (resp. ∂ i (x)) are ω-functors from ∆ n+1 (resp. ∆ n−1 ) to C [23]. By construction, the map [n] → ∆ n induces then a functor from the wellknown category ∆ whose associated presheaves are the simplicial sets to ωCat. Therefore N (C) = (ωCat(∆ * , C), ∂ i , ǫ i ) is a simplicial set which is called the simplicial nerve of C. Definition 5.1. The globular cut N gl (or the globular nerve) is the functor from ωCat 1 to Geometrically, the elements of N gl n (C) are full (n+1)-globes. Figure 3 depicts a 2-simplex in the globular nerve. The simplexes seen by the globular cut are intuitively transverse to the execution paths, as well as those of corner nerves. Hence the terminology of cuts. Sets ∆ op + defined by N gl n (C) = ωCat(∆ n , PC) for n 0 and with N gl −1 (C) = C 0 × C 0 , and endowed with the augmentation map ∂ −1 from N gl 0 (C) = C 1 to N gl −1 (C) = C 0 × C 0 defined by ∂ −1 x = (s 0 x, t 0 x). The evaluation map ev is defined by ev(x) = x((0 . . . n)) for x ∈ ωCat(∆ n , Here is now the new definition of the globular homology of a globular ω-category C : for n −1 and this homology theory is called the globular homology of C. Associating to any globe its corners The purpose of the rest of the paper is to justify that Definition 5.2 is the right definition. This is not a mathematical statement of course ! We follow the order of the remarks at the very end of Section 1 which explain what kind of conditions the globular homology must fulfill. So we have first to construct h − and h + and we must verify that geometrically, in homology, h − and h + do what we expect to find. In fact, we refer to [10] for intuitive explanations of h − and h + . We only recall here Figure 4 as an illustration and care only about the construction of h − . Theorem 6.1. Let α ∈ {−, +}. There exists one and only one morphism of cuts h α from N gl to N α . Moreover, for any non-contracting ω-category C, both morphisms h α from N gl (C) to N α (C) are injective. The rest of the section is devoted to the proof of Theorem 6.1. The following sequence of propositions establishes the existence of h − . The term cub n denotes the set of faces of the n-cube, as described in Section 4. We briefly recall how filling shells in the cubical singular nerve. This technical tool already appears in [4] for ω-groupoids and in [1] for ω-categories. A particular case can be found in [9]. If x ± i is a n-shell, then it induces a labeling x on the set of faces of dimension at most n of the (n + 1)-cube in the following manner : let k 1 . . . k n+1 be a face of dimension at most n ; then there exists i such that k i = 0 ; then let x(k 1 . . . k n+1 ) := x i (k 1 . . . k i . . . k n+1 ). The axiom satisfied by an n-shell ensures the coherence of the definition. w u C > > A / / v B O O XA A A A A A \ d A A A A A A (a) A 2-globular simplex X t 0 u / / t; C t 0 u / / t 0 u / / A ; C t 0 u ? ? t 0 u K S Proposition and definition 6.3. Let x ± i be an (n − 1)-shell with n 1. • The labeling of the faces of dimension at most (n − 1) of I n defined by x ± i always induces an ω-functor and only one from I n \{R(0 n )} to C. Denote it by x. • The n-shell (x ± i ) is said fillable if there exists a morphism u of C such that s n−1 u = x (s n−1 R(0 n )) and t n−1 u = x (t n−1 R(0 n )). In this case, there exists a unique ωfunctor x from I n to C such that ∂ ± i x = x ± i for 1 i n and x(0 n ) = u. Proof. Using the freeness of I n , the construction in the proof of [9] Proposition 5.1 yields the ω-functor x from I n \{R(0 n )} to C. The hypotheses stated in [9] were too strong indeed. If moreover the shell is fillable in the above sense, one concludes still as in the proof of [9] Proposition 5.1. Now we can construct h − . Theorem 6.4. Let x be an n-simplex of the globular simplicial nerve of C. Then the map h − n (x) from cub n+1 to C defined by 1. + ∈ {k 1 . . . k n+1 } implies h − n (x)(k 1 . . . k n+1 ) = t 0 x((0)) (notice that (0) is the final state of ∆ n ) 2. {k 1 , . . . , k n+1 } ⊂ {−, 0} and {k 1 , . . . , k n+1 } ∩ {0} = {k σ 0 +1 , . . . , k σr +1 } with σ 0 < · · · < σ r implies h − n (x)(k 1 . . . k n+1 ) = x((σ 0 . . . σ r )) 3. h − n (x)(− n+1 ) = s 0 x((n)) (notice that (n) is the initial state of ∆ n ) yields an ω-functor from I n+1 to C. Moreover, h − induces a morphism of simplicial sets from the globular nerve of C to its negative corner nerve. And the map from N gl −1 (C) to N − −1 (C) defined by (x, y) → x extends the previous morphism to the corresponding augmented simplicial nerves. Moreover for n 0, h − n is a one-to-one map and the image of h − n contains exactly all cubes x of the negative corner nerve such that as soon as ∂ + i x exists, then it is 0-dimensional. There is no ambiguity to set h − (x) = h − n (x) if x is an n-simplex of the globular cut. In the sequel, in order to make easier the reading of the calculations, we suppose that an expression like (σ 0 < σ j k < σ j+1 < ... < σ r ) is the same thing as (σ 0 < σ j < σ j+1 < ... < σ r ) in ∆ * but with an additional information given within the calculation itself : here that σ j k < σ j+1 holds. Proof. One proves by induction on n the following property P (n) : " For any n-simplex x of the globular simplicial nerve of any ω-category C, the map h − (x) from cub n+1 to C induces an ω-functor and moreover an element of ωCat(I n+1 , C) − ." Let x be a 0-simplex of the globular nerve of C. Then x is an ω-functor from ∆ 0 to PC, and therefore it can be identified with the 1-morphism x((0)) of C. Therefore h − (x)(0) = x((0)) by rule 2 h − (x)(+) = t 0 x((0)) by rule 1 h − (x)(−) = s 0 x((0)) by rule 3 Therefore P (0) is proved. Now suppose that P (n) is proved for n 0. Let x be a (n + 1)-simplex of the globular simplicial nerve of some ω-category C. If + ∈ {k 1 , . . . , k n+1 }, then with σ 0 < · · · < σ r . For a given i such that 1 i n + 2, set ∂ − i (h − (x))(k 1 . . . k n+1 ) = h − (x)(k 1 . . . k i−1 − k i . .w 1 . . . w n+2 = k 1 . . . k i−1 − k i . . . k n+1 as word. Then let {w 1 , . . . , w n+2 } ∩ {0} = {w τ 0 +1 , . . . , w τr +1 } with τ 0 < · · · < τ r . The relation between the sequence of σ j and the sequence of τ j is as follows : σ j + 1 i − 1 =⇒ σ j = τ j σ j + 1 i =⇒ σ j + 1 = τ j And we have ∂ − i (h − (x))(k 1 . . . k n+1 ) = h − (x)(k 1 . . . k i−1 − k i . . . k n+1 ) by definition of ∂ − i = x((τ 0 . . . τ r )) by rule 2 = x((σ 0 < · · · < σ j 0 i − 2 < i − 1 < σ j 0 +1 + 1 < · · · < σ r + 1)) = (∂ i−1 x)((σ 0 . . . σ r )) by definition of ∂ i−1 = h − (∂ i−1 x)(k 1 . . . k n+1 ) by rule 2 Therefore ∂ − i (h − (x)) = h − (∂ i−1 x). And by rule 1, ∂ + i (h − (x)) is the constant ω-functor from cub n+1 to C which sends any face of I n+1 on t 0 x((0)). Therefore (∂ ± i (h − (x))) 1 i n+1 is a (n + 1)-shell in the cubical nerve of C which is fillable. By Proposition 6.3, the labeling h − (x) of cub n+2 induces an ω-functor from I n+2 to C and P (n + 1) is proved. By construction, the equality ∂ − i (h − (x)) = h − (∂ i−1 x) holds for any n-simplex x of the globular nerve and for 1 i n + 1. It remains to check that for such a simplex x, with σ 0 < · · · < σ r . For a given i such that 1 i n + 1, Γ − i (h − (x)) = h − (ǫ i−1 x) for i 1 n + 1. Consider a face k 1 . . . k n+2 of the (n + 2)-cube. If + ∈ {k 1 , . . . , k n+2 }, then Γ − i (h − (x))(k 1 . . . k n+2 ) = h − (x)(k 1 . . . max(k i , k i+1 ) . . . k n+2 ) by definition of Γ − i = t 0 x((0)) by rule 1 = h − (ǫ i−1 x)(k 1 . . .{k 1 , . . . , max(k i , k i+1 ), . . . , k n+2 } ⊂ {−, 0} and set w 1 . . . w n+1 = k 1 . . . max(k i , k i+1 ) . . . k n+2 as word. Then let {w 1 , . . . , w n+1 } ∩ {0} = {w τ 0 +1 , . . . , w τs+1 } with τ 0 < · · · < τ s . One has to calculate Γ − i (h − (x))(k 1 . . . k n+2 ) = h − (x)(k 1 . . . max(k i , k i+1 ) . . . k n+2 ) by definition of Γ − i = x((τ 0 . . . τ s )) by definition of h − for some 1 i n + 2. The situation can be decomposed in three mutually exclusive cases : 1. k i = k i+1 = 0. In this case, there exists a unique j 0 such that σ j 0 + 1 = i, s = r − 1 and σ j + 1 i − 1 =⇒ σ j = τ j (in this case, j < j 0 ) τ j 0 + 1 = i = σ j 0 + 1 σ j + 1 i + 2 =⇒ σ j − 1 = τ j−1 (in this case, j > j 0 + 1) Then σ j 0 +2 i + 1 and x((τ 0 . . . τ s )) = x((σ 0 < · · · < σ j 0 = i − 1 < σ j 0 +2 − 1 < · · · < σ s+1 − 1)) = (ǫ i−1 x)(σ 0 . . . σ j 0 σ j 0 +1 σ j 0 +2 . . . σ s+1 ) by definition of ǫ i and since σ j 0 +1 = i = (h − (ǫ i−1 x))(k 1 . . . k n+2 ) by definition of h − 2. k i = k i+1 = −. In this case, s = r and σ j + 1 i − 1 =⇒ σ j = τ j σ j + 1 i + 2 =⇒ σ j − 1 = τ j Then for some k, x((τ 0 . . . τ s )) = x((σ 0 < · · · < σ k < i − 1 < σ k+1 − 1 < · · · < σ r − 1)) = (ǫ i−1 x)((σ 0 . . . σ k σ k+1 . . . σ r )) by definition of ǫ i = (h − (ǫ i−1 x))(k 1 . . . k n+2 ) by definition of h − 3. k i = k i+1 . Now s = r and since {k i , k i+1 } ⊂ {−, 0} , then there exists a unique j 0 such that σ j 0 + 1 ∈ {i, i + 1} and we have σ j + 1 i − 1 =⇒ σ j = τ j (in this case, j < j 0 ) τ j 0 + 1 = i σ j + 1 i + 2 =⇒ σ j − 1 = τ j (in this case, j > j 0 ) There are two subcases : σ j 0 + 1 = i and σ j 0 + 1 = i + 1. In the first situation, x((τ 0 . . . τ s )) = x((σ 0 < · · · < σ j 0 −1 < σ j 0 = i − 1 < σ j 0 +1 − 1 < · · · < σ r − 1)) = x((σ 0 < · · · < σ j 0 −1 < σ j 0 < σ j 0 +1 − 1 < · · · < σ r − 1)) = (ǫ i−1 x)((σ 0 < · · · < σ j 0 < σ j 0 +1 < · · · < σ r )) by definition of ǫ i = (h − (ǫ i−1 x))(k 1 . . . k n+2 ) by definition of h − In the second situation, x((τ 0 . . . τ s )) = x((σ 0 < · · · < σ j 0 −1 < σ j 0 − 1 = i − 1 < σ j 0 +1 − 1 < · · · < σ r − 1)) = x((σ 0 < · · · < σ j 0 −1 < σ j 0 − 1 < σ j 0 +1 − 1 < · · · < σ r − 1)) = (ǫ i−1 x)((σ 0 < · · · < σ j 0 < σ j 0 +1 < · · · < σ r )) by definition of ǫ i = (h − (ǫ i−1 x))(k 1 . . . k n+2 ) by definition of h − Notice that h − induces a natural transformation from CR gl * to CR − * which is not injective. Consider for example the ω-category consisting of two composable 1-morphisms u and v with t 0 u = s 0 v. The 0-simplexes u and u * 0 v of N gl 0 have indeed the same image by h − in CR − 1 . To see that, consider the thin square c from I 2 to C defined by c(−0) = u * 0 v, c(0+) = t 0 v, c(0−) = u, c(+0) = v and c(00) = u * 0 v. Now we arrive at : Theorem 6.5. There exists one and only one morphism of cuts from N gl to N − . The proof of this theorem uses Theorem 8.3 assertion 1 as shortcut. There is no vicious circle because the uniqueness of h − and h + is used nowhere in this paper. The only fact which is used is that Theorem 6.4 provides a natural transformation from N gl to N − which is injective on the underlying sets. Proof. Let h and h ′ be two morphisms of cuts from N gl to N − . One proves by induction on n that h n and h ′ n from N gl n to N − n coincide. For n = 0, N gl 0 = N − n = tr 0 P. The only natural transformation from tr 0 P to itself is Id tr 0 P , therefore h 0 = h ′ 0 . Suppose P (n) proved for some n 0. Then for any x ∈ N gl n+1 (C), and for any 0 i n + 1, ∂ − i+1 h n+1 (x) = h n (∂ i x) since h morphism of simplicial sets = h ′ n (∂ i x) by induction hypothesis = ∂ − i+1 h ′ n+1 (x) since h ′ morphism of simplicial sets Now with 1 j n + 2, (∂ + j h n+1 (x))(− n+1 ) = h n+1 (x)(− · · · − [+] j − · · · −) = h n+1 (x) (t 0 R(− · · · − [0] j − · · · −)) = t 0 (h n+1 (x)(R(− · · · − [0] j − · · · −))) since h n+1 (x) ω-functor = t 0 (∂ − 1 . . . ∂ − j . . . ∂ − n+2 h n+1 (x))(0) = t 0 h 0 (∂ 0 . . . ∂ j−1 . . . ∂ n+1 x)(0) since h morphism of simplicial sets = t 0 (∂ 0 . . . ∂ j−1 . . . ∂ n+1 x)((0)) So the 0-morphism ∂ + j h n+1 (x))(− n+1 ) is the value of the constant map t 0 •x of Theorem 8.3 (denoted by T (x) in Section 10). Let D be the unique ω-category such that PD = ∆ n+1 and with D 0 = {α, β}, s 0 (PD) = {α}, t 0 (PD) = {β} and α = β. And consider Id ∆ n+1 ∈ N gl n+1 (D). Suppose that + ∈ {k 1 , . . . , k n+2 } ⊂ {−, +} and suppose that at least two k i are equal to +. Then there exists a 1-morphism u of I n+2 such that s 0 u = ℓ 1 . . . ℓ n+2 with exactly one ℓ i equal to + and such that t 0 u = k 1 . . . k n+2 . Then s 0 (h n+1 (Id ∆ n+1 )(u)) = h n+1 (Id ∆ n+1 )(ℓ 1 . . . ℓ n+2 ) = β by the previous calculation. Since β is the unique morphism of D with 0-source β, then h n+1 (Id ∆ n+1 )(u) = β and therefore h n+1 (Id ∆ n+1 )(k 1 . . . k n+2 ) = β. Suppose now that + ∈ {k 1 , . . . , k n+2 } with perhaps some 0 in the set. Then s 0 (h n+1 (Id ∆ n+1 )(k 1 . . . k n+2 )) = β and therefore ev • h n+1 (Id ∆ n+1 )(k 1 . . . k n+2 ) = β = T (Id ∆ n+1 ) . The ω-functor x from ∆ n+1 to PC induces a non-contracting ω-functor x from D to C with x(α) = S(x) (S(x) being the value of the constant map s 0 • x by Theorem 8.3) and x(β) = T (x) which sends Id ∆ n+1 ∈ N gl n+1 (D) on x ∈ N gl n+1 (C). So by naturality, ev • h n+1 (x)(k 1 . . . k n+2 ) = T (x). Therefore for any 1 j n + 2, ∂ + j h n+1 (x) = ∂ + j h ′ n+1 (x). By hypothesis, ev(h n+1 (x)) = ev(x) = ev(h ′ n+1 (x) ). So h n+1 (x) and h ′ n+1 (x) induce the same labeling of the faces of I n+2 and P (n + 1) is proved. Without explanation, here is the construction of h + : Proposition 6.6. Let x be an n-simplex of the globular simplicial nerve of C. Then the map h + n (x) from cub n+1 to C defined by 1. − ∈ {k 1 . . . k n+1 } implies h + n (x)(k 1 . . . k n+1 ) = s 0 x((n)) (notice that (n) is the initial state of ∆ n ) 2. {k 1 , . . . , k n+1 } ⊂ {+, 0} and {k 1 , . . . , k n+1 } ∩ {0} = {k σ 0 +1 , . . . , k σr +1 } with σ 0 < · · · < σ r implies h + n (x)(k 1 . . . k n+1 ) = x((σ 0 . . . σ r )) 3. h + n (x)(+ n+1 ) = t 0 x((0)) (notice that (0) is the final state of ∆ n ) yields an ω-functor from I n+1 to C. Moreover, h + induces a morphism of simplicial sets from the globular nerve of C to its positive corner nerve. And the map from N gl −1 (C) to N + −1 (C) defined by (x, y) → y extends the previous morphism to the corresponding augmented simplicial nerves. Moreover for n 0, h + n is a one-to-one map and the image of h + n contains exactly all cubes x of the positive corner nerve such that as soon as ∂ − i x exists, then it is 0-dimensional. Question 6.7. Is it possible to find an appropriate setting where the globular cut would be an initial object ? Is it possible to characterize the diagram of cuts of Figure 1 ? As immediate corollary of the construction of h − and its injectivity, let us introduce the analogue of Proposition 6.3 in the globular nerve. Definition 6.8. In a simplicial set A, a n-shell is a family (x i ) i=0,...,n+1 of (n + 2) nsimplexes of A such that for any 0 i < j n + 1, ∂ i x j = ∂ j−1 x i . Proposition 6.9. Let C be a non-contracting ω-category. Consider a n-shell (x i ) i=0,...,n+1 of the globular simplicial nerve of C. Then 1. The labeling defined by (x i ) i=0,...,n+1 yields an ω-functor x (and necessarily exactly one) from ∆ n+1 \{(01 . . . n + 1)} to PC. 2. Let u be a morphism of C such that s n u = x (s n R((01 . . . n + 1))) and t n u = x (t n R((01 . . . n + 1))) Then there exists one and only one ω-functor still denoted by x from ∆ n+1 to PC such that for any 0 i n + 1, ∂ i x = x i and x((01 . . . n + 1)) = u. Regularity of the globular cut This section is devoted to the proof of the following theorem. Theorem 7.1. The globular cut is regular. The principle of this proof is to use the injectivity of the natural transformation h − from N gl to N − and to use the regularity of N − . The folding operator Φ gl n := Φ N gl n is called the n-dimensional globular folding operator and we set gl n := N gl n . It is clear that rule 1 and rule 2 of Definition 3.3 are satisfied. We have to check the rest of it. Theorem 7.2. For any natural transformation of functors µ from N gl n−1 to N gl n with n 1, and for any natural map from tr n−1 P to N gl n−1 such that ev • = Id tr n−1 P , there exists one and only one natural transformation denoted by µ. from tr n P to N gl n such that the following diagram commutes where i n is the canonical inclusion functor from tr n−1 P to tr n P. Proof. The natural transformation h − from tr n−1 P to N − n−1 can be lifted to a natural transformation (h − (µ)).(h − ) from tr n P to N − n since the cut N − is regular. Since h − (µ. ) = (h − (µ)).(h − ) and since h − is one-to-one in positive degree, there is at most one solution for this lifting problem. tr n P h − (µ).(h − ) & & N gl n h − / / N − n tr n−1 P h − 7 7 in O O / / N gl n−1 h − / / µ O O N − n−1 h − (µ) O O Let x ∈ C n+1 . For 0 i n, the natural transformation ev ∂ i h − (µ).(h − ) : tr n P → tr n−1 P is of the form d α i m i for some α i ∈ {−, +} and some m i n. Therefore ∂ i h − (µ).(h − ) = ∂ i h − (µ).(h − ) i n d α i m i by Definition 3.3 rule 5b = ∂ i h − (µ)h − d α i m i by hypothesis = ∂ i h − µ d α i m i = h − ∂ i µ d α i m i since h − morphism of simplicial sets So ∂ i (h − (µ).(h − )) (x) ∈ h − (N gl n−1 (C)) for any 0 i n and by Proposition 6.9, (h − (µ).(h − )) (x) ∈ h − (N gl n (C)). Let ′ (x) be the unique element of N gl n (C) such that h − ′ (x) := h − (µ).(h − ) (x) Then ′ is a solution. Corollary 7.3. The equalities h − Φ gl = Φ − h − and h + Φ gl = Φ + h + hold. Proof. It is a consequence of the naturality of h − and h + and of Proposition 3.4. Now here is a characterization of globular folding operators : Proposition 7.4. Let x be a n-simplex of the globular nerve of C. Then x = Φ gl (x) if and only if for 0 i n − 2, ∂ i x ∈ Im(ǫ n−2 . . . ǫ i ). Proof. The equality x = Φ gl (x) implies h − (x) = Φ − (h − (x)) , implies by Theorem 4.3 that for 1 i n − 1, h − (∂ i−1 x) = ∂ − i (h − (x)) = Γ − n−1 . . . Γ − i − i d (−) i h − (x)(0 n+1 ) = h − ǫ n−2 . . . ǫ i−1 gl i s i x((0 . . . n)) therefore ∂ i−1 x ∈ Im(ǫ n−2 . . . ǫ i−1 ). Conversely, if for 0 i n − 2, ∂ i x ∈ Im(ǫ n−2 . . . ǫ i ), then h − (x) = Φ − h − (x) = h − Φ gl (x) and therefore x = Φ gl (x). Theorem 7.5. The globular folding operator Φ gl induces the identity map on the globular reduced chain complex CR gl * . Proof. Consider the θ − i operators of Theorem 4.5. If x ∈ N gl n , then h − x is negative. So θ − i h − x is also negative by Theorem 4.5(1) and determines a unique element θ gl i x ∈ N gl n such that h − θ gl i x = θ − i h − x. It is clear that these operators θ gl i induces the identity map on the reduced globular complex by Theorem 4.5 (2). Since Φ − h − x is also negative, then by Theorem 4.5(3), Φ − h − x = θ − i 1 . . . θ − is h − x for some sequence i 1 , . . . , i s . Therefore by the injectivity of h − , Φ gl x = θ gl i 1 . . . θ gl is x Theorem 7.6. In the reduced globular complex, one has gl n (x * p y) = gl n (x) + gl n (y) for any morphisms x and y of C of dimension n and for 1 p n − 1. Sketch of proof. One has h − ( gl n (x * p y)) = − n (x * p y) = − n (x) + − n (y) + t 1 + ∂ − t 2 = h − ( gl n (x)) + h − ( gl n (y)) + t 1 + ∂ − t 2 with t 1 a thin (n + 1)-cube and t 2 a thin (n + 2)-cube. The proof made in [8] shows that t 1 and t 2 are in the image of h − . Indeed, the existence of t 1 and t 2 comes from the vanishing of some globular nerve. Therefore t 1 = h − (T 1 ) and t 2 = h − (T 2 ) where T 1 is a thin n-simplex and T 2 a thin (n + 1)-simplex. This completes the proof. In fact one can explicitly verify that if x and y are two n-morphisms of C, then gl n (x * n−1 y) − gl n (x) − gl n (y) is a boundary in the normalized globular complex. It suffices to consider the thin (n + 1)-cube B n n−1 (x, y) of [8] which turns to be in the image of h − because it is negative. Therefore with b(x, y) ∈ ωCat(∆ n , PC) defined by ∂ i b(x, y) = ǫ n−2 . . . ǫ i gl i+1 d (−) i+1 i+1 x for 0 i n − 3 (observe that d (−) i+1 i+1 x = d (−) i+1 i+1 y), ∂ n−2 b(x, y) = gl n y, ∂ n−1 b(x, y) = gl n (x * n−1 y), ∂ n b(x, y) = gl n x, one has ∂b(x, y) = ± gl n (x * n−1 y) − gl n (x) − gl n (y) + degenerate elements. Example of calculations of globular homology The main goal of this section is to prove the vanishing of the globular homology of the ncube in positive dimension for all n 0. However we also study the case of the ω-category 2 n generated by one n-morphism and pose some questions about the globular homology of the ω-category generated by a composable pasting scheme in the sense of [12]. 1. Let x be an ω-functor from ∆ n to PC for some n 0. Then the set maps (σ 0 . . . σ r ) → s 0 x((σ 0 . . . σ r )) and (σ 0 . . . σ r ) → t 0 x((σ 0 . . . σ r )) from the underlying set of faces of ∆ n to C 0 are constant. The unique value of s 0 • x is denoted by S(x) and the unique value of t 0 • x is denoted by T (x). 2. For any pair (α, β) of 0-morphisms of C, for any n 1, and for any 0 i n, then ∂ i N gl n (C[α, β]) ⊂ N gl n−1 (C[α, β]). 3. For any pair (α, β) of 0-morphisms of C, for any n 0, and for any 0 i n, then ǫ i N gl n (C[α, β]) ⊂ N gl n+1 (C[α, β]). 4. By setting, G α,β N gl n (C) := N gl n (C[α, β]) for n 0 and G α,β N gl −1 (C) := {(α, β), (β, α)}, one obtains a (C 0 × C 0 )-graduation on the globular nerve ; in particular, one has the direct sum of augmented simplicial sets N gl * (C) = (α,β)∈C 0 ×C 0 G α,β N gl * (C) and G α,β N gl * (C) = N gl * (C[α, β]). Proof. The only non-trivial part is the first assertion. Let P (n) be the property : "for any non-contracting ω-category C and any ω-functor x from ∆ n to PC, the set map (σ 0 . . . σ r ) → s 0 x((σ 0 . . . σ r )) from the set of faces of ∆ n to C 0 is constant." There is nothing to check for P (0). For P (1), if x is an ω-functor from ∆ 1 to PC, then s 1 x((01)) = x((1)) and t 1 x((01)) = x((0)) in C. Therefore s 0 x((0)) = s 0 t 1 x((01)) = s 0 x((01)). Therefore P (1) is true. Suppose P (n) proved for some n 1 and let us prove P (n + 1). For any 1 i n, the ω-functor x : ∆ n+1 → PC induces an ω-functor on the ω-category ∆ n+1 i generated by the face (0 . . . i . . . n + 1) and its subfaces. One has an isomorphism of ω-categories ∆ n ∼ = ∆ n+1 i . Therefore the restriction of s 0 • x to the faces of ∆ n+1 i is constant by induction hypothesis. Now it is clear that ∆ n+1 i ∩ ∆ n+1 i+1 ∼ = ∆ n−1 = ∅ since n 1. Therefore the set map s 0 • x restricted to ∆ n+1 i ∪ ∆ n+1 i+1 is constant. Therefore the restriction of the set map s 0 • x to the faces of dimension at most n of ∆ n+1 is constant. We know that s n R((01 . . . n + 1)) = Ψ(X 0 , X 1 , . . . , X s ) where X 0 , X 1 , . . . , X s are faces of ∆ n+1 of dimension at most n. So s 0 x((01 . . . n + 1)) = s 0 s n+1 x((01 . . . n + 1)) = s 0 x (s n R((01 . . . n + 1))) since x ω-functor = s 0 xΨ(X 0 , X 1 , . . . , X s ) where Ψ is a function using only the compositions of ∆ n+1 . Then xΨ(X 0 , X 1 , . . . , X s ) = Ψ ′ (x(X 0 ), x(X 2 ), . . . , x(X s )) where Ψ ′ is obtained from Ψ by replacing * i by * i+1 since x is an ω-functor from ∆ n+1 to PC. So s 0 x((01 . . . n + 1)) = Ψ ′ (s 0 x(X 0 ), s 0 x(X 2 ), . . . , s 0 x(X s )) = s 0 x(X 0 ) with the axioms of ω-categories. Therefore P (n + 1) is proved. Proof. By proceeding as in Theorem 8.6, we see that it suffices to prove that More generally, as in [8], one sees that if C is a non-contracting ω-category such that PC is the free ω-category generated by a composable pasting scheme in the sense of [12], then H gl p (C) = 0 for p 1. This is related to the problem of the existence of the derived pasting scheme of a given composable pasting scheme [14]. Conjecture 8.8. Let C be an ω-category which is the free ω-category generated by a composable pasting scheme (therefore C is non-contracting). Then for any p > 0, H gl p (C) = 0. Relation between the new globular homology and the old one First of all, recall the definition of both formal corner homology theories from [8]. Definition 9.1. Let C be a non-contracting ω-category. Set • CF − 0 (C) := ZC 0 • CF − 1 (C) := ZC 1 • CF − n (C) = ZC n /{x * 0 y = x, x * 1 y = x + y, . . . , x * n−1 y = x + y mod Ztr n−1 C} for n 2 with the differential map s n−1 − t n−1 from CF − n (C) to CF − n−1 (C) for n 2 and s 0 from CF − 1 (C) to CF − 0 (C) . This chain complex is called the formal negative corner complex. The associated homology is denoted by HF − (C) and is called the formal negative corner homology of C. The map CF − * (resp. HF − * ) induces a functor from ωCat 1 to Comp(Ab) (resp. Ab). and symmetrically Definition 9.2. Let C be a non-contracting ω-category. Set • CF + 0 (C) := ZC 0 • CF + 1 (C) := ZC 1 • CF + n (C) = ZC n /{x * 0 y = y, x * 1 y = x + y, . . . , x * n−1 y = x + y mod Ztr n−1 C} for n 2 with the differential map s n−1 − t n−1 from CF + n (C) to CF + n−1 (C) for n 2 and t 0 from CF + 1 (C) to CF + 0 (C). This chain complex is called the formal positive corner complex. The associated homology is denoted by HF + (C) and is called the formal positive corner homology of C. The map CF + * (resp. HF + * ) induces a functor from ωCat 1 to Comp(Ab) (resp. Ab). The maps ± n from C n to C ± n (C) induce a natural transformation from CF ± * to CR ± * and a natural transformation from HF ± * to HR ± * . Definition 9.3. Let C be a non-contracting ω-category. Set • CF gl 0 (C) := ZC 0 ⊗ ZC 0 ∼ = Z(C 0 × C 0 ) • CF gl 1 (C) := ZC 1 • CF gl n (C) = ZC n /{x * 1 y = x + y, . . . , x * n−1 y = x + y mod Ztr n−1 C} for n 2 with the differential map s n−1 − t n−1 from CF gl n (C) to CF gl n−1 (C) for n 2 and s 0 ⊗ t 0 from CF gl 1 (C) to CF gl 0 (C). This chain complex is called the formal globular complex. The associated homology is denoted by HF gl (C) and is called the formal globular homology of C. By Theorem 7.6 and Corollary 3.6, we see that the globular folding operators induce a natural morphism of chain complex from CF gl * to CR gl * , and therefore a natural transformation from HF gl * to HR gl * . Question 9.4. When is the natural morphism of chain complexes R gl from CF gl * (C) to CR gl * (C) a quasi-isomorphism ? Conjecture 9.5. (About the thin elements of the globular complex) Let C be a globular ω-category which is either the free globular ω-category generated by a semi-cubical set or the free globular ω-category generated by a globular set. Let x i be elements of C gl n (C) and let λ i be natural numbers, where i runs over some set I. Suppose that for any i, ev(x i ) is of dimension strictly lower than n (one calls it a thin element). Then i λ i x i is a boundary if and only if it is a cycle. The above conjecture is clear for C gl 2 because all thin elements are degenerate. In higher dimension, there is enough room to have thin elements which are composition of degenerate elements, but which are not degenerate themselves. The above conjecture is equivalent to claiming that the globular homology and the reduced one are equivalent for free globular ω-categories generated by either a semi-cubical set or a globular set. Now we are in position to give the exact statement relating the old globular homology of [9] and the new one. (C) = ZC n , ∂ old−gl (x) = (s 0 x, t 0 x) if x ∈ ZC 1 and for n 1, x ∈ ZC n+1 implies ∂ old−gl (x) = s n x − t n x. This complex is called the old globular complex of C and its corresponding homology the old globular homology. Instead of C old−gl 0 (C) = ZC 0 ⊕ ZC 0 , we set C old−gl 0 (C) = Z(C 0 ⊗ C 0 ) with the differential ∂ old−gl (x) = s 0 x ⊗ t 0 x for x ∈ C 1 . This makes H old−gl 1 slightly change. It does not matter because there is no influence on any potential applications. The difference appears in a situation like that of Figure 5. With C old−gl 0 (C) = ZC 0 ⊕ ZC 0 , u + x − w − v is a old globular cycle. With C old−gl 0 (C) = Z(C 0 ⊗ C 0 ), this fake 1-globular cycle is killed. → H gl * is the canonical map induced by x → gl n (x) from C n to N gl n−1 (C) • the map H old−gl * → HF gl * is the canonical map making all identifications like A * n B = A + B for any n 1 and any p-morphisms A and B with p n + 1 • the map HF gl * → HF ± * is the canonical map making the supplemental identification x = x * 0 y or y = x * 0 y depending on the sign ± • the map HF ± * → HR ± * is the canonical map induced by the folding operators ± of [8] (which is likely to be an isomorphism for any strict globular ω-category), and the map HF gl * → HR gl * is the canonical map induced by the folding operators gl (which is also likely to be an isomorphism for any strict globular ω-category) • the maps R gl,± are the canonical maps from the globular or corner homology to the corresponding reduced homology (which are conjecturally an isomorphism for any free ω-category generated by a semi-cubical set or a globular set). Proof. This is due to the fact that for n 1, the natural map (h ± n ) old is induced by the set map − n from C n to ωCat(I n , C) − ([9] Proposition 7.4). The difference between H old−gl Intuitively, the globular nerve of C contains all achronal cuts in the middle of all globes, whereas the negative and positive corner simplicial nerves contain all achronal cuts close to respectively the negative and the positive corners of the automaton. The expression "achronal" is borrowed from [6] and [7]. In these papers, HDA are modeled by local pospaces, and an achronal subspace Y of a local pospace is a topological subspace such that x y and x, y ∈ Y imply x = y. The remarkable point is that the set of all achronal cuts of a given type can be enclosed into a simplicial set. This could mean that the whole geometry of the free ω-category C generated by a semi-cubical set (i.e. a HDA) would be contained in the following diagram of augmented simplicial sets N − (C) N + (C) and in its temporal graph tr 1 C. This latter contains the information about the temporal structure of the HDA. A problem, already mentioned in [10], is the question of the invariance of the globular homology of an ω-category up to a choice of a cubification 2 of the corresponding HDA. There are two types of deformations : the spatial deformations or S-deformations and the temporal deformations or T-deformations. The globular cut is invariant by S-deformation, that is by deformations of p-morphisms with p 2. This is simply due to the fact that such a deformation corresponds in the globular cut to a deformation of any simplex containing it as label. Therefore such a deformation corresponds to a deformation up to homotopy, in the usual sense, of the globular cut. Unlike the corner homologies, the globular homology turns indeed to depend on the subdivision of time. The reason is contained in Figure 6. The obvious 1-functor from the left to the right such that u → u 1 * 0 u 2 should leave the globular homology invariant. This is not the case because the first globular homology is for the left member the free Z-module generated by v −w and u * 0 v −u * 0 w, and for the right member the free Z-module generated by v − w and u 2 * 0 v − u 2 * 0 w and u 1 * 0 u 2 * 0 v − u 1 * 0 u 2 * 0 w. However in Figure 6, one can subdivide as many times as one wants for example v, and the globular homology will not change. One way to overcome this problem is exposed in the last sections of [10], devoted to the description of a generic way to produce T-invariants starting from the globular nerve. Let us prove [10] Claim 5.1 which enables to introduce the bisimplicial set mentioned in that paper. Let C be a non-contracting ω-category. Using Theorem 8.3, recall that for some ωfunctor x from ∆ n to PC, one calls S(x) the unique element of the image of s 0 • x and T (x) the unique element of the image of t 0 • x. If (α, β) is a pair of N gl −1 (C), set S(α, β) = α and T (α, β) = β. α u / / β v w F F γ (a) C α 1 u 1 / / α 2 u 2 / / β v w F F γ (b) Subdivision of u in C Proposition 10.1. Let C be a non-contracting ω-category. Let x and y be two ω-functors from ∆ n to PC with n 0. Suppose that T (x) = S(y). Let x * y be the map from the faces of ∆ n to C defined by (x * y)((σ 0 . . . σ r )) := x((σ 0 . . . σ r )) * 0 y((σ 0 . . . σ r )). Then the following conditions are equivalent : 1. The image of x * y is a subset of PC. 2. The set map x * y yields an ω-functor from ∆ n to PC and ∂ i (x * y) = ∂ i (x) * ∂ i (y) for any 0 i n. On contrary, if for some (σ 0 . . . σ r ) ∈ ∆ n , (x * y)((σ 0 . . . σ r )) is 0-dimensional, then x * y is the constant map S(x) = T (y). Proof. We have to prove that Condition 1 implies Condition 2. Let us consider P (n) : "for any non-contracting ω-category C and any ω-functor x and y from ∆ n to PC such that T (x) = S(y) and such that the image of x * y is a subset of PC, then x * y yields an ω-functor from ∆ n to PC and ∂ i (x * y) = ∂ i (x) * ∂ i (y) for any 0 i n." Property P (0) is obvious. Suppose P (n − 1) proved for n 1. For any 0 i n, ∂ i (x) * ∂ i (y) is a set map from ∆ n−1 to PC satisfying the hypothesis of the proposition, so by induction hypothesis, ∂ i (x) * ∂ i (y) yields an ω-functor from ∆ n−1 to PC. Let z i := ∂ i (x) * ∂ i (y). For 0 j < i n, ∂ j (z i ) = (∂ j ∂ i (x)) * (∂ j ∂ i (y)) by induction hypothesis = (∂ i−1 ∂ j (x)) * (∂ i−1 ∂ j (y)) = ∂ i−1 (∂ j (x) * ∂ j (y)) by induction hypothesis = ∂ i−1 z j Therefore (z i ) 0 i n is an (n − 1)-shell. So it provides a unique ω-functor z : ∆ n \{(01 . . . n)} → PC by Proposition 6.9. It remains to check that z (s n−1 R((01 . . . n))) = s n ((x * y)((01 . . . n))) and z (t n−1 R((01 . . . n))) = t n ((x * y)((01 . . . n))) to complete the proof. Let us check the first equality. One has s n−1 R((01 . . . n)) = Ψ(X 1 , . . . , X s ) where Ψ uses only composition laws and where X 1 , . . . , X s are faces of ∆ n of dimension at most n − 1. Denote by Ψ ′ the same function as Ψ with * i replaced by * i+1 . Then z (s n−1 R((01 . . . n))) = zΨ(X 1 , . . . , X s ) = Ψ ′ (z(X 1 ), . . . , z(X s )) since z ω-functor = Ψ ′ (x(X 1 ) * 0 y(X 1 ), . . . , x(X s ) * 0 y(X s )) by definition of z = Ψ ′ (x(X 1 ), . . . , x(X s )) * 0 Ψ ′ (y(X 1 ), . . . , y(X s )) by interchange law = (xΨ(X 1 , . . . , X s )) * 0 (yΨ(X 1 , . . . , X s )) since x and y ω-functors = (xs n−1 R((01 . . . n))) * 0 (ys n−1 R((01 . . . n))) = (s n xR((01 . . . n))) * 0 (s n yR((01 . . . n))) since x and y ω-functors = s n (xR((01 . . . n)) * 0 yR((01 . . . n))) by interchange law = s n ((x * y)((01 . . . n))) Now let us suppose that (x * y)((σ 0 . . . σ r )) is 0-dimensional in C for some (σ 0 . . . σ r ). Then s 1 x((σ 0 . . . σ r )) * 0 s 1 y((σ 0 . . . σ r )) is 0-dimensional. Either s 0 (σ 0 . . . σ r ) = (n) (the initial state of ∆ n ) or there exists a 1-morphism U of ∆ n such that s 0 U = (n) and t 0 U = s 0 (σ 0 . . . σ r ). In the first case, x((n)) * 0 y((n)) is 0-dimensional. In the second case, x(t 0 U ) * 0 y(t 0 U ) = t 1 x(U ) * 0 t 1 y(U ) = t 1 (x(U ) * 0 y(U )) is 0-dimensional. Then x(U ) * 0 y(U ) is 0-dimensional as well as x((n)) * 0 y((n)) = s 1 (x(U ) * 0 y(U )) . For any face (τ 0 . . . τ r ) of ∆ n \{(n)}, there exists a 1-morphism V from ((n)) to s 0 (τ 0 . . . τ r ) or t 0 (τ 0 . . . τ r ) : let us say s 0 (τ 0 . . . τ r ). Since s 1 (x * y)(V ) = (x * y)((n)) is 0-dimensional, then (x * y)(V ) is 0-dimensional, as well as t 1 (x * y)(V ) = (x * y)(s 0 (τ 0 . . . τ r )) = s 1 (x * y)((τ 0 . . . τ r )). Therefore (x * y)((τ 0 . . . τ r )) is 0-dimensional. In the sequel, we set (α, β) * (β, γ) = (α, γ), S(α, β) = α and T (α, β) = β. If x is an ω-functor from ∆ n to PC, and if y is the constant map T (x) (resp. S(x)) from ∆ n to C 0 , then set x * y := x (resp. y * x := x). Theorem 10.2. Suppose that C is an object of ωCat 1 . Then for n 0, the operations S, T and * allow to define a small category N gl n (C) whose morphisms are the elements of N gl n (C) ∪ {constant maps ∆ n → C 0 } and whose objects are the 0-morphisms of C. If N gl −1 (C) is the small category whose morphisms are the elements of C 0 × C 0 and whose objects are the elements of C 0 with the operations S, T and * above defined, then one obtains (by defining the face maps ∂ i and degeneracy maps ǫ i in an obvious way on {constant maps ∆ n → C 0 }) an augmented simplicial object N gl * in the category of small categories. Proof. Equalities S(x) = ∂ i S(x), S(x) = ǫ i S(x), T (x) = ∂ i T (x), T (x) = ǫ i T (x) are consequences of Proposition 8.3. Equality ∂ i (x * y) = ∂ i x * ∂ i y is proved right above. The verification of ǫ i (x * y) = ǫ i x * ǫ i y is straightforward. A and B such that t 1 A = s 1 B gives rise to a 3-dimensional globular cycleA * 1 B − A − B because s 2 (A * 1 B − A − B) = s 2 A * 1 s 2 B − s 2 A − s 2 B = 0 and t 2 (A * 1 B − A − B) = t 2 A * 1 t 2 B − t 2 A − t 2 B =0. So putting the relation A * 1 B − A − B = 0 in the old globular complex for any pair of morphisms (A, B) of the same dimension sounds necessary. Similar considerations starting from the calculation of the (n − 1)-th globular homology group of I n entail the relations A * n B − A − B for any n 1 and for any pair (A, B) of p-morphisms with p n + 1 in the old globular chain complex. Figure 2 : 2Some ω-categories (a k-fold arrow symbolizes a k-morphism) Definition 3. 3 . 3A cut F is regular if and only if it satisfies the following properties : 1. For any ω-category C, the set F −1 (C) only depends on tr 0 C = C 0 : i.e. for any ω-categories C and D, C 0 = D 0 implies F −1 (C) = F −1 (D). F 1 : 1= Id F 0 and F n := ǫ n−2 . . . . ǫ 0 . F 1 a natural transformation from tr n−1 P to F n−1 for n 2 ; then the natural transformations ∂ i F n for 0 i n−1 from tr n−1 P to F n−2 satisfy the following properties (a) ev∂ n{s n−1 , t n−1 }. (b) if for some ω-category C and some u ∈ C n , ev∂ i Fn (u) = d α p u for some p n and for some α ∈ {−, +}, then ∂ i Fn (A) ∈ 2 n (A) and therefore ev∂ i Fn 2 2(A) = d α p (A) for some p and some α. By naturality, this implies that ev∂ i F n = d α p . If 0 i < n − thin. Now if n − 2 i n − 1, then Then x is called a negative element of the branching nerve. Theorem 4 . 5 . 45Let n 2. There exists natural transformations θ − 1 , . . . , θ − n−1 from N − n to itself satisfying the following properties : 1. If x is a negative element of N − n (C), then for any 1 i n − 1, θ − i x is a negative element as well. 2. If x is a negative element of N − n (C), then for any 1 i n − 1, there exists a thin negative element y i of N − n+1 (C) such that ∂ − y i − x is a linear combination of thin negative elements. 3. There exists a composite of θ − 1 , . . . , θ − n−1 which coincides with the negative folding operators on negative elements of N − n . For instance, s (04589) = {(4589), (0489), (0458)} and t (04589) = {(0589), (0459)}. PC). The homology theory H gl n := H N gl n is called the globular homology and HR gl n := HR N gl n the reduced globular homology. Figure 3 : 3Globular 2-simplex Definition 5.2. Let C be a non-contracting ω-category. We set H gl n+1 (C) := H n (N gl (C)) Figure 4 : 4Illustration of h − Definition 6.2. A n-shell in the cubical singular nerve is a family of 2(n + 1) elements x ± i of ωCat(I n , C) − such that ∂ α i x β j = ∂ β j−1 x α i for 1 i < j n + 1 and α, β ∈ {−, +}. . k n+1 ) by definition of ∂ − i for 1 i n + 2 = t 0 x((0)) by rule 1 = h − (∂ i−1 x)(k 1 . . . k n+1 ) again by rule 1 If + / ∈ {k 1 , . . . , k n+1 }, i.e. if {k 1 , . . . , k n+1 } ⊂ {−, 0}, set {k 1 , . . . , k n+1 } ∩ {0} = {k σ 0 +1 , . . . , k σr+1 } 1 , . . . , k n+2 }, i.e. if {k 1 , . . . , k n+2 } ⊂ {−, 0}, set {k 1 , . . . , k n+2 } ∩ {0} = {k σ 0 +1 , . . . , k σr+1 } Theorem 8. 1 . 1For any p > 0 and any n 0, H gl p (2 n ) = 0. Proof. For p = 1, it is obvious. For p > 1, one has H gl p (2 n ) ∼ = H p−1 (P2 n ) ∼ = H p−1 (2 n−1 ) = 0 where H * (D) means the simplicial homology of the simplicial nerve of the ω-category D. Definition 8.2. [9] Let C be an ω-category and let α and β be two 0-morphisms of C. Then the bilocalization of C with respect to α and β is the ω-subcategory of C obtained by keeping in dimension 0 only α and β and by keeping in positive dimension all morphisms x such that s 0 x = α and t 0 x = β. It is denoted by C[α, β]. Theorem 8.3. Let C be a non-contracting ω-category. s 0 x((01)) = s 0 s 1 x((01)) = s 0 x((1)) and Definition 8 . 4 . 84Let C be a non-contracting ω-category with exactly one initial state α and one final state β. Then the bilocalization C[α, β] is also non-contracting and one can set ΩC = P(C[α, β]). pair ((r), (s)) of 0-morphisms of ∆ n and for n 2. However, ∆ n [(r), (s)] is nonempty if and only if r > s with our conventions and in this case, ∆ n [(r), (s)] ∼ = ∆ r−s [(r − s), (0)]. Therefore H gl p (∆ n [(r), (s)]) ∼ = H p−1 (I r−s−1 ) by Theorem 8.5. Figure 5 : 5A false 1-globular cycle in the old globular homology Definition 9.6.[9] Let (C old−gl * (C), ∂ old−gl ) be the chain complex defined as follows : C old−gl 0 (C) = ZC 0 ⊕ ZC 0 and for n 1, C old−gl n 0 and 0H gl 0 is also not important. The group H old−gl 0 was indeed only introduced to define the morphisms h − and h + in dimension 0. But H old−gl 0 does not have any computer-scientific meaning and is not involved in any potential applications. 10 Globular homology and deformation of HDAThe following table summarizes how the globular nerve may be understood and compared with the two corner nerves of C. Figure 6 : 6Subdivision of time Then one has oneProposition 2.10. For any semi-cubical set K, Π(K) is a non-contracting ω-category. The functor Π : Sets semi op → ωCat from the category of semi-cubical sets to that of ω-Proof. The characterization of Proposition 2.6 gives the solution.categories yields a functor from Sets semi op to the category of non-contracting ω-categories ωCat 1 . Theorem 9.7. We have the following commutative diagram of natural transformations for * 0 where • the map H old−gl H gl h ± / / R gl H ± * R ± H old−gl * 7 7 n n n n n n n n n n n n n (h ± ) old ( ( / / ' ' P P P P P P P P P P P P P HR gl * h ± / / HR ± * HF gl * gl O O h ± / / HF ± * ± O O N gl (C) h − z z t t t t t t t t t h + Some authors[11] [21] use the term cubicalation : this means decomposing a HDA in cubes. Theorem 8.5.[18,2,16]Let n 1. Then Ω∆ n = I n−1 and ΩI n−1 = P n−1 where P n−1 is the free ω-category generated by the composable pasting scheme of the faces of the (n − 1)-dimensional permutohedron. Let α and β be two 0-morphisms of I n such that I n [α, β] contains other morphisms than α and β. Then in particular it contains some 1-morphisms from α to β which is a composite of 1-dimensional faces of I n . Suppose that α = k 1 . . . k n . Then β is obtained from α by replacing some k i equal to − by +. Let k σ 1 , . . . , k σr be these k i . Thenas ω-category. Therefore it suffices to prove that H gl p (I n [− n , + n ]) vanishes. The vanishing of H gl 1 (I n [− n , + n ]) is obvious. One hasfor p 2 by Theorem 8.5 and H p−1 (P n ) = 0 because the simplicial nerve of a composable pasting scheme is contractible[12].Theorem 8.7. For any n 0, and any p > 0, H gl p (∆ n ) = 0. one obtains a bisimplicial set which seems to be well-behaved with respect to subdivision of time. Indeed the first total homology groups associated to both ω-categories of Figure 6 are equal to Z. Further explanations will be given in future papers. To conclude, let us point out that in reasonable cases, i.e. when the p-morphisms (with p 2) of a non-contracting ω-category C are invertible with respect to the composition laws * i of C for i 1. By composing by the classifying space functor of small categories (cf. for example [20] for further details). then PC becomes a globular ω-groupoid in the sense of Brown-HigginsBy composing by the classifying space functor of small categories (cf. for example [20] for further details), one obtains a bisimplicial set which seems to be well-behaved with respect to subdivision of time. Indeed the first total homology groups associated to both ω-categories of Figure 6 are equal to Z. Further explanations will be given in future papers. To conclude, let us point out that in reasonable cases, i.e. when the p-morphisms (with p 2) of a non-contracting ω-category C are invertible with respect to the composition laws * i of C for i 1, then PC becomes a globular ω-groupoid in the sense of Brown-Higgins. And therefore in such a case, it is well-known that the globular nerve of C satisfies the Kan property (see [23] or a generalization in [24]). 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[ "LiDAR-Inertial 3D SLAM with Plane Constraint for Multi-story Building", "LiDAR-Inertial 3D SLAM with Plane Constraint for Multi-story Building" ]	[ "Jiashi Zhang ", "Chengyang Zhang ", "Jun Wu ", "Jianxiang Jin ", "Qiuguo Zhu " ]	[]	[]	The ubiquitous planes and structural consistency are the most apparent features of indoor multi-story Buildings compared with outdoor environments. In this paper, we propose a tightly coupled LiDAR-Inertial 3D SLAM framework with plane features for the multi-story building. The framework we proposed is mainly composed of three parts: tightly coupled LiDAR-Inertial odometry, extraction of representative planes of the structure, and factor graph optimization. By building a local map and inertial measurement unit (IMU) pre-integration, we get LiDAR scan-to-local-map matching and IMU measurements, respectively. Minimize the joint cost function to obtain the LiDAR-Inertial odometry information. Once a new keyframe is added to the graph, all the planes of this keyframe that can represent structural features are extracted to find the constraint between different poses and stories. A keyframe-based factor graph is conducted with the constraint of planes, and LiDAR-Inertial odometry for keyframe poses refinement. The experimental results show that our algorithm has outstanding performance in accuracy compared with the state-of-the-art algorithms.Index Terms-SLAM, range sensing, mapping, sensor fusion.• We propose the method of finding and constructing the	null	[ "https://arxiv.org/pdf/2202.08487v1.pdf" ]	246,904,557	2202.08487	64da7a7a030010f3d82ba18d27ae41ddd233f952	LiDAR-Inertial 3D SLAM with Plane Constraint for Multi-story Building Jiashi Zhang Chengyang Zhang Jun Wu Jianxiang Jin Qiuguo Zhu LiDAR-Inertial 3D SLAM with Plane Constraint for Multi-story Building 1Index Terms-SLAMrange sensingmappingsensor fusion The ubiquitous planes and structural consistency are the most apparent features of indoor multi-story Buildings compared with outdoor environments. In this paper, we propose a tightly coupled LiDAR-Inertial 3D SLAM framework with plane features for the multi-story building. The framework we proposed is mainly composed of three parts: tightly coupled LiDAR-Inertial odometry, extraction of representative planes of the structure, and factor graph optimization. By building a local map and inertial measurement unit (IMU) pre-integration, we get LiDAR scan-to-local-map matching and IMU measurements, respectively. Minimize the joint cost function to obtain the LiDAR-Inertial odometry information. Once a new keyframe is added to the graph, all the planes of this keyframe that can represent structural features are extracted to find the constraint between different poses and stories. A keyframe-based factor graph is conducted with the constraint of planes, and LiDAR-Inertial odometry for keyframe poses refinement. The experimental results show that our algorithm has outstanding performance in accuracy compared with the state-of-the-art algorithms.Index Terms-SLAM, range sensing, mapping, sensor fusion.• We propose the method of finding and constructing the I. INTRODUCTION With the development of quadruped robots' motion control and environmental perception capabilities, the scenarios they can explore are also expanding from 2D to 3D compared with wheeled mobile robot. Accurate state estimation and mapping are the basic premises for applying robots in the real world. As for indoor environments, especially multi-story buildings, the multi-planar feature can help the robot achieve low-drift simultaneous localization and mapping (SLAM) due to the structural consistency. A 3D LiDAR based on scanning mechanism has the advantages of textureless, invariant to the illumination, and broad horizontal of view (FOV) of 360°, which is generally used in indoor environments [1], [2]. Under normal circumstances, LiDAR-aided SLAM mainly uses extracting corner points and surf points method [3]- [6], Normal Distributions Transform (NDT) [7] scan matching, or floor extraction [8] methods to achieve SLAM for a single floor. Although many algorithms implement SLAM by extracting planes in indoor environments, most only use plane constraints in the odometry part and achieve accurate SLAM algorithms by finding the scan-toscan plane correspondence. However, when the robot explores from bottom to top in a multi-story building, the existing algorithms cannot achieve accurate state estimation on the robot's 6-DOF, due to long-distance and loop closure does not work. In the multi-story SLAM, due to the consistency of structure between different floors, some planes on different floors can represent the same building structure. Here we call these planes structural representative planes (SRP). Fig. 1 shows an example of SRP, where the same SRP is displayed in the same color on different stories. Finding the correspondence between SRP within the scope is the key to achieving low-drift SLAM in multi-story scenes. This paper presents a tightly-coupled LiDAR-Inertial 3D SLAM framework using planes to build global constraints. Our framework has three parts: tightly coupled LiDAR-Inertial odometry, extraction of representative planes of the structure, and factor graph optimization. The odometry is obtained by jointly optimizing the relative pose of the scan-to-localmap and the inertial measurement unit (IMU) pre-integration measurements. According to the odometry information, all the SRP will be extracted as candidates for the global plane constraint once a new keyframe is selected. Transform the global SRP to the keyframe coordinate system, and construct the global constraint relationship between keyframes according to the direction of planes' normals and the distance to the coordinate origin. Add odometry information and constraint information from planes to the factor graph, perform global optimization, and get the accurate pose of each keyframe. The main contributions of this paper can be summarized as follows: global constraints of SRP in the multi-story blocks to achieve accurate 6-DOF state estimation of the robot when the loop closure is not possible. • We propose a tightly coupled LiDAR-Inertial, keyframebased SLAM framework to get the dense 3D point cloud maps of multi-story blocks. • We validate the algorithm using the data collected from Velodyne VLP-16 and Xsens Mti-300 mounted on a real quadruped robot (Jueying Robot). Compared with other state-of-the-art algorithms, better results are obtained. II. RELATED WORK LiDAR Inertial odometry In recent years, 3D LiDAR and IMU have been widely used in SLAM, both indoors and outdoors. The fusion methods of LiDAR and IMU are mainly divided into two categories: loosely coupled and tightly coupled. In the field of loosely coupled, LOAM [3] is a classic loosely coupled framework. It uses the orientation calculated by the IMU de-skew the point cloud and as prior information in the optimization process. The same method is also applied to its variants LeGO-LOAM [4]. Zhen W et al. [9] integrate IMU measurements and LiDAR estimations from a Gaussian particle filter (GPF) and a pre-built map with error state Kalman filter (ESKF). The more popular loosely coupled method is the extended Kalman filters (EKF). [10]- [12] propose some generic EKF-based frameworks for robot state estimation, which can integrate the measurements of LiDAR and IMU, as well as global position system (GPS). LIO-Mapping [13] realized LiDAR-Inertial tightly coupled algorithm by optimizing the cost function that includes both LiDAR and inertial measurements. However, the optimization process is carried out in a sliding window, so the timeconsuming calculations make it impossible to maintain realtime performance. In their follow-up work, R-LINS [14], they use iterated-ESKF for the first time to achieve LiDAR-Inertial tightly coupled fusion and propose an iterated Kalman filter [15] to reduce wrong matchings in each iteration. A tightly coupled framework based on iterated Kalman filter is presented in [16], similar to R-LINS. An incremental k-d tree data structure is adopted to ensure cumulative updates and dynamic balance to ensure fast and robust LiDAR mapping. LIO-SAM [5] proposed by Shan T et al. optimizes the measurements of LiDAR and IMU by factor graph, and at the same time, estimates the bias of the IMU. SLAM related to plane features Whether in vision-based SLAM or LiDAR-based SLAM, plane-related features are widely used to improve state estimation accuracy. In LiDARbased SLAM, LOAM [3] proposed extracting feature points from planar surface patches and sharp edges based on curvature calculation and improved the iterative closest point (ICP) [17] method based on the extracted feature points demonstrating the superb LiDAR odometry effect. Koide K et al. [8] realize SLAM in a large-scale environment by detecting the ground, assuming that the indoor environment is a single flat floor. But this assumption is not applicable in all scenes and can only limit the height on the z-axis. Kaess [18] introduces an efficient parametrization of planes based on quaternion, suitable for least-squares estimation. Besides, he presents a relative plane formulation to speed up the convergence process. LIPS [19] extract the plane in the three-axis direction of the point cloud, not only the ground plane, and combine the plane and IMU measurements in a graph-based framework. At the same time, the closets point (CP) is used to represent the plane to solve the singularity. K Pathak et al. [20] present a new algorithm called minimally uncertain maximal consensus (MUMC) to determine the unknown plane correspondences in the front-end. π-LSAM, an indoor environment SLAM system using planes as landmarks, was proposed by Zhou L et al. [21]. They adopt plane adjustment (PA) as the back-end to optimize plane parameters and poses of keyframes, similar to bundle adjustment (BA) in visual SLAM. Their subsequent work [22] extended this by using first-order Taylor expansion to replace the Levenberg Marquardt (LM) [23] method. To achieve faster computational speed, they define the integrated cost matrix (ICM) for each plane and achieve outstanding SLAM effects in a single-layer indoor environment. All of the above frameworks use a single LiDAR or a loosely coupled method of LiDAR and IMU as the front-end. On the contrary, we use a tightly coupled LiDAR-Inertial method as the front-end, which can obtain a more accurate prior pose of the keyframe, making it more precise when looking for the corresponding between the planes. III. SYSTEM OVERVIEW Our algorithm consists of three parts, LiDAR-Inertial Odometry, SRP Constraint and Graph Optimization, as shown in Fig. 2. The LiDAR-Inertial Odometry performs pre-integration on high-frequency IMU data and corrects the motion distortion of the point cloud. Find the relative measurements of LiDAR by extracting feature points from the point cloud. Optimize the cost function that includes pre-integration and LiDAR relative measurements to obtain the odometry. The SRP Constraint extracts SRP from the point cloud of each keyframe and matches with the global SRP to form the edge between the vertices in the factor graph. The graph optimization performs optimization every time a new SRP constraint is formed and adjusts the poses of the keyframes. IV. LIDAR-INERTIAL ODOMETRY A. IMU Pre-integration The LiDAR and IMU reference frames at time t are noted L t and I t , respectively. The state X W It of IMU to be estimated in the world frame W and the extrinsic matrix T L I from IMU to LiDAR can be written as: X W It = p W It T v W It T q W It T b at T b gt T T T L I = p L I T q L I T T(1) where p W It , v W It , and q W It are the position, velocity, and orientation of IMU in the world frame W at time t. b at and b gt are the bias of accelerometer and gyroscope of IMU. Usingâ t andω t to represent the raw measurements of acceleration and angular velocity in frame L t , respectively, and they can be defined as: a t = R It W a W i − g W + b at + n â ω t = ω t + b wt + n w(2) where R It W is the rotation matrix from world frame W to frame I t and g W is the constant gravity vector in world frame W . n a and n w are the noises of IMU, modeled as Gaussian white noises. With the continuous-discrete IMU inputs, the position, velocity, and orientation of the IMU at time t + ∆t can be calculated according to Eq. (2). p t+∆t = p t + v t ∆t + 1 2 g W ∆t 2 + 1 2 R W It (â t − b at − n a ) ∆t 2 v t+∆t = v t + g W ∆t + R W It (â t − b at − n a ) ∆t q t+∆t = q t ⊗ 1 2 Ω (ω t − b wt − n w ) q t ∆t(3) where ⊗ is used for the multiplication of two quaternions, and Ω(ω) is: Ω(ω) = − ω × ω −ω T 0(4) where · × ∈ R 3×3 stands for the skew-symmetric matrix, let t i and t j be the starting time and ending time of a raw LiDAR scanS i , respectively, so the pre-integration measurements ∆p ij , ∆v ij , ∆q ij of IMU from time t i to t j can be computed as: ∆p ij = j−1 k=i ∆v ik ∆t + 1 2 ∆R ik (â k − b a k − n a ) ∆t 2 ∆v ij = j−1 k=i ∆R ik (â k − b a k − n a ) ∆t ∆q ij = j−1 k=i δq k = j−1 k=i 1 2 ∆t (ω k − b w k − n w ) 1(5) Readers can refer to [24] for detailed derivation from Eq. (2) to Eq. (5). B. Scan Deskewing and Feature Extraction Due to the relative movement between the laser and the robot, there will be motion distortion for the raw LiDAR outputS i , whereS i represents the point cloud starting from time t i to time t j . Every point x(t) ∈S i is transformed to the correct position by linear interpolation to T L ij according to its timestamp, where t ∈ [t i , t j ). T L ij can be obtained by IMU pre-integration and extrinsic matrix T L I , and the undistorted scan can be represented by S i . To improve the efficiency of calculation, only the feature points that can reflect the characteristics of the surrounding environment are selected to find the relative pose of the LiDAR. Here we use the method of extracting feature points located on sharp edges and planar surfaces proposed by LOAM. The extracted edge and planar feature points from S i are denoted as F Li e and F Li p , respectively. C. LiDAR Relative Measurements When the new feature points F Li e and F Li p are extracted, the measurements of LiDAR need to be found to jointly perform the optimization with IMU. 1) Building Local Map: Since the points of a single scan are not dense enough, to obtain more accurate LiDAR measurements, we use a sliding window to construct a local map. The sliding window contains n LiDAR frames from time t i−1 to time t i−n . Since we have extracted planar points and edge points separately, we transform F to fit a plane in the frame L i−1 and express in Hesse normal form: x T n − d = 0 (6) where n is the unit normal vector of plane, and d is the distance from plane to the origin of frame L i−1 . So for each plane point x Li p ∈ F Li p , the residual can be expressed as the point-plane distance: T Li−1 Li = R Li−1 Li p Li−1 Li 0 1 r P (T Li−1 Li ) = R Li−1 Li x p Li + p Li−1 Li T n − d(7) Similar to the calculation method of the plane point, the Hesse normal form can also describe the line in R 2 . For each edge point, the residual can be represented as the point-line distance: r E (T Li−1 Li ) = R Li−1 Li x e Li + p Li−1 Li T n − d(8) D. Front-End Optimization We build a cost function including IMU measurements and LiDAR measurements jointly. To get more accurate odometry for each frame of LiDAR, we optimize all the states in the sliding window iteratively. For a sliding window of size n at time t i , the states need to be optimized is X i = T i−n i , ..., T i−n i−(n−1) , and the final cost function is described as: min Xi 1 2 α∈{i−n,...,i−1} r I (z α α+1 , X i ) 2 C Iα I α+1 + x p L i ∈F p L i β∈{i−n,...,i−1} r P (X i ) 2 p C L i−n L β+1 + x e L i ∈F e L i γ∈{i−n,...,i−1} r E (X i ) 2 e C L i−n L γ+1(9) where X 2 C = X T CX and r I (X i ) is the residual of IMU measurements, which is defined in [13]. r P (X i ) and r E (X i ) are the residuals of planar points matching and edge points matching. C Iα Iα+1 , C Li−n L β+1 , C Li−n Lγ+1 represent the covariance matrix. This non-linear least squares problem can be solved using the Levenberg-Marquardt algorithm [23]. V. SRP CONSTRAINT AND GRAPH OPTIMIZATION In this part, we extract keyframes based on the LiDAR-Inertial odometry and extract all SRP from the LiDAR scan in the keyframe coordinate system, find the correspondence in the entire graph and construct constraints as demonstrated in Fig. 3. LiDAR frames LiDAR key frames graph vertices SRP LiDAR-Inertial Odomtry SRP Constraints Fig. 3. The structure of the factor graph. The system selects keyframes based on the odometry as the vertices of the factor graph. The edges between the vertices are formed by LiDAR-Inertial odometry (blue curve) and SRP constraints (red line). A. SRP Extraction For the calculation efficiency, we select keyframes as vertices of the factor graph according to the odometry of the front-end. Since we are using a LiDAR based on scanning mechanism, the change of the yaw angle does not affect the selection of keyframes. The new keyframe will be selected only when the distance between the new frame and the previous keyframe exceeds 1m or the pitch angle or roll angle exceeds 10°. We extract all SRP from the corrected LiDAR scan S i for each newly added keyframe K i . Here we define the plane as π(n, d) through the Hesse normal form described by Eq. (6). n = [n x , n y , n z ] T represents the unit normal vector of the plane, and d represents the distance from the coordinate origin of K i to the plane. Next, apply RANSAC [25] to extract planes for S i , but not all planes are reserved for building constraints, but only those planes that can represent the structure of the building (e.g., ground, walls, etc.) are selected. Here we adopt the following strategies for the extraction of SRP: • Keep all the planes with more than N points (Here, we set N to 400). • According to the normal vector of the extracted plane, three planes containing the most points and almost orthogonal are retained. • Use 80% of the points in S i to extract the plane, and the remaining points default to the unextractable points. Too many planes are extracted will increase the uncertainty of the RANSAC process and cause mismatches in the plane matching process. Here we only use three orthogonal planes to obtain the precise pose of the LiDAR with 6-DOF. At the same time, fewer edges will be constructed in the factor graph to reduce the calculation time. B. SRP Global Constraint To construct the global constraint, all SRP extracted from keyframe K i will be checked whether they have appeared in the previous keyframes. Here we denote all the planes added to the graph as Π = π K0 1 , · · · , π K0 k0 , · · · , π Ki−1 1 , · · · , π Ki−1 ki−1 , and the SRP under the K i frame as Π Ki = π Ki 1 , · · · , π Ki ki . First, according to the optimized results T W Km , m ∈ {1, · · · , i − 1} and the front-end odometry T Ki Ki−1 , the planes in Π are transformed to the frame of keyframe K i . T Ki Km = T W Km T Ki−1 W T Ki Ki−1 = R Ki Km p Ki Km 0 1 n Ki d Ki = R Ki Km 0 −p Ki Km T 1 n Km d Km(10) For all π Ki m (n Ki , d Ki ) ∈ Π Ki , m ∈ {1, · · · , k i }, calculate the angle δθ between its normal vector n Ki and n Ki and the distance δd between d Ki and d Ki . Once δθ and δd are lower than the preset threshold, add a plane edge to the factor graph. Otherwise, it's considered a new plane and added to Π. C. Graph optimization When the LiDAR-Inertial odometry and SRP construct the constraints between keyframes, the SLAM problem is expressed in a factor graph. The vertices of the graph represent states of being optimized, and the edges represent the constraints formed by the sensors' measurements, as shown in Fig. 3. Following [26], [27], the maximum likelihood estimation problem can be expressed as this nonlinear leastsquares problem: F (x) = i,j ∈C e (x i , x j , z ij ) T Ω ij e (x i , x j , z ij )(11) where x represents all states to be optimized and x i , x j ∈ x, z ij and Ω ij represent the mean and the information matrix of a constraint between x i and x j , C is the set of pairs of indices for which the constraint exist, and e (x i , x j , z ij ) is the error function between x i , x j and z ij . Eq. (11) can be minimized by Gauss-Newton or Levenberg-Marquardt algorithm. VI. EXPERIMENTS A. Experimental Settings To verify the versatility of the algorithm, we conducted experiments in different buildings. We used the Jueying Mini robot (Fig. 4) equipped with Velodyne VLP-16 and Xsens Mti-300 to collect data from multiple sets of multi-story scenes. The LiDAR is fixed on the head of Jueying, and the IMU is assembled at the center of mass. The LiDAR operates at a frequency of 10 Hz, and the IMU outputs orientation, angular velocity, and linear acceleration at 400 Hz. Since there are currently no publicly available datasets of LiDAR and IMU for indoor multi-story scenes, we used Jueying to collect actual data in two buildings and named them Building A and Building B, respectively. Building A is a five-story building in the shape of long corridor, and Building B is a six-story building with two long corridor-shaped scattered on the left and right. Our algorithm is tested on a NUC mini PC with Intel Core i7-7567U, 16G memory. B. Results and Analysis In this section, we provide the results of our experiments. We compared the state-of-the-art SLAM algorithms based on multi-sensor fusion, including LIO-Mapping [13], Fast-LIO2 [16], and LIO-SAM [5], and provided the mean running time of each part of our algorithm. These three are perfect LiDAR-Inertial SLAM algorithms that can apply to most scenarios. Still, in particular scenarios such as indoor multistory, they may not achieve the best results. Due to the unique experimental scene, we cannot obtain the ground truth of the robot motion. At the same time, we set the robot's starting point and ending point to be the same when collecting data to calculate the relative position and orientation deviation. In the case of not adding loop closure, we can use this criterion to judge the accuracy of all algorithms. Overview The performance of four algorithms on the Building A and Building B datasets are shown in Fig. 5. We can see that our algorithm is significantly better than the other three algorithms on both datasets because of plane constraints. When the 16-line LiDAR moves horizontally, the number of points scanned on the ground is relatively minor. Therefore, during the scan-to-scan registration, the height estimation will produce more significant deviations, especially in degraded scenarios such as corridors. Despite the aid of IMU, there will still be cumulative errors, which can be seen more clearly in FAST-LIO2 and LIO-SAM. LIO-Mapping optimizes each state in the sliding window, which consumes more time, so the effect of height estimation is better, but in the end, it does not return to the starting point as well. The other three algorithms did not return to the starting point in the end due to the lack of performing loop closure. Trajectory Fig. 6 shows the trajectories of three algorithms in two datasets. LIO-SAM failed in Building A and Building B, so we did not plot its trajectory. Although we do not have global ground truth, we can see in Fig. 6(b) and Fig. 6 front-end odometry can prevent mismatching in the plane matching process, it takes a little longer to estimate each state to ensure accuracy. VII. CONCLUSION AND FUTURE WORK This paper proposes a SLAM algorithm for indoor multistory scenes with a plane as the main feature. We use the tightly coupled LiDAR and IMU as the front-end and optimize the two simultaneously by constructing the error function of IMU pre-integration and scan registration to obtain more accurate odometry information. We search for SRP in keyframes instead of all planes at the back-end, which makes it construct fewer but more significant edges in the factor graph. According to the normal direction of the plane and the distance to the origin, it searches and matches with global SRP and constructs constraints. This allows the robot to build the same constraints on different floors, eliminates the cumulative error, and achieves an effect similar to "dimensionality reduction." Experiments show that our algorithm can significantly improve the state estimation. The constructed global map records the structural characteristics of the building well and is better than the state-of-the-art SLAM algorithm, LIO-Mapping [13], Fast-LIO2 [16], and LIO-SAM [5] in the indoor multi-story scenario. However, the current plane matching process relies heavily on front-end odometry. If the front-end odometry drifts a lot, the plane could be mismatched. Therefore, our algorithm takes a lot of time in the previous stage, which causes it to not run in real-time. In the future, it may be necessary to combine features unique to the plane to make the matching process more robust. * This work was supported by NSFC 62088101 Autonomous Intelligent Unmanned Systems and the National Key R&D Program of China (2020YFB1313300) and National Quality Infrastructure Research Program of Zhejiang Administration for Market Regulation (20190104). Jiashi Zhang, Chengyang Zhang, Jun Wu, Jianxiang Jin, Qiuguo Zhu are with the State Key Laboratory of Industrial Control and Technology, Zhejiang University, Institute of Cyber-System and Control, Zhejiang University, Hangzhou, P.R. China. Qiuguo Zhu is the corresponding author (e-mail: [email protected]). Fig. 1 . 1Schematic diagram of SRP. On the left are actual scenes from different stories. Different colors represent different SRPs, and the same color represents the same plane. On the right is the SRP extracted from the LiDAR point cloud. Using the same SRP to construct constraints on different stories can eliminate accumulated errors. Fig. 2 . 2System overview of our algorithm. . They will be downsampled to get the centroid points in each 3D voxel grid to remove duplicate points.2) Scan Matching: The relationship between the feature points and the local maps at time t i can be calculated by the point-line and the point-plane distances. First, transform the feature points F Li e and F Li p to frame L i−1 . The prediction transformation T i i−1 used here is obtained through IMU preintegration and extrinsic matrix T L I . Here we take the plane points as an example. For each transformed plane point x Fig. 4 . 4The Jueying Mini quadruped robot. Fig. 5 .Fig. 6 . 56Maps generated by Ours, LIO-Mapping, FAST-LIO2, and LIO-SAM. The other three algorithms drift on different stories, except ours. (a) Maps of Building A. (b) Maps of Building B. Comparison of trajectories estimated by different algorithms (LIO-SAM fails on both datasets, so we don't plot its trajectory.). (a) Front view of trajectories in Building A. (b) Front view of trajectories in Building B. (c) Top View of trajectories in Building A. (d) Top View of trajectories in Building B. (d) that in the staircase on the left of Building A and Building B, the other two algorithms drift a lot. Still, after adding plane constraints, ours can maintain the consistency of different floors.Table Iprovides the relative deviations of translation and rotation. Since accurate state estimation is achieved on other stories, our algorithm can return to the starting point without loop closure. Because of the same planes used to construct constraints, both translation and rotation are almost consistent with the starting point.Table IIlists the running times of different parts of our algorithm. Since an accurate TABLE I RELATIVE IDEVIATION OF DIFFERENT SLAM ALGORITHMS UNDER THE SAME STARTING POINT AND ENDING POINT.TABLE II MEAN RUNNING TIME (MS) OF DIFFERENT COMPONENTS.Dataset Distance System Translation (m) Rotation (rad) (m) ∆X ∆Y ∆Z ∆XYZ ∆Yaw ∆Pitch ∆Roll ∆Angle Building A 396 Ours 0.018 0.023 0.015 0.033 -0.021 -0.002 0.018 0.028 LIO-Mapping -0.073 3.001 -0.305 3.017 0.114 -0.019 -0.018 0.117 FAST-LIO2 -1.529 1.424 -1.442 2.539 -0.078 -0.023 -0.086 0.118 LIO-SAM -8.576 -35.110 -25.802 44.407 1.629 -0.629 -1.438 2.262 Building B 613 Ours 0.021 0.023 0.002 0.031 -0.006 -0.007 -0.006 0.011 LIO-Mapping 0.412 12.857 -2.588 13.121 -0.036 -0.008 -0.038 0.054 FAST-LIO2 2.943 21.479 -1.328 21.720 -0.356 -0.176 -0.096 0.409 LIO-SAM 4.296 12.792 12.109 18.131 2.409 0.030 -0.180 2.416 Dataset LiDAR-Inertial Odometry SRP Constraint Back-End Optimization SRP Extraction Find Global Constraint Building A 249.10 413.34 4.90 252.80 Building B 316.14 301.08 8.41 353.53 A Laser-Aided Inertial Navigation System (L-INS) for human localization in unknown indoor environments. 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[ "Integration of SOA and Cloud Computing in RM- ODP", "Integration of SOA and Cloud Computing in RM- ODP" ]	[ "Mostafa Jebbar [email protected] \nDepartement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco\n", "Abedrrahim Sekkaki [email protected] \nDepartement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco\n", "Othmane Benamar \nDepartement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco\n" ]	[ "Departement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco", "Departement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco", "Departement of Mathematics and Computer\nAin Chock\nFaculty of Sciences Casablanca\nScience University Hassan II\nMorocco" ]	[]	The objective of ODP is according to ITU-T Recommendation X.901 stated as follows: "The objective of ODP standardization is the development of standards that allow the benefits of distributing information processing services to be realized in an environment of heterogeneous IT resources and multiple organizational domains. These standards address constraints on system specification and the provision of a system infrastructure that accommodate difficulties inherent in the design and programming of distributed systems." This objective seems to cover cloud computing systems. Therefore, we in this paper discuss the concepts of cloud, and discuss the use of RM-ODP for specifying the solution. We indicate that the current RM-ODP may be too abstract for the purpose, and indicate how to adapt RM-ODP to fit the purpose.	10.1109/setit.2012.6481897	[ "https://arxiv.org/pdf/1303.2857v1.pdf" ]	14,556,942	1303.2857	34a6e0f2db8678dae2db59b63259c9ace2603360	Integration of SOA and Cloud Computing in RM- ODP Mostafa Jebbar [email protected] Departement of Mathematics and Computer Ain Chock Faculty of Sciences Casablanca Science University Hassan II Morocco Abedrrahim Sekkaki [email protected] Departement of Mathematics and Computer Ain Chock Faculty of Sciences Casablanca Science University Hassan II Morocco Othmane Benamar Departement of Mathematics and Computer Ain Chock Faculty of Sciences Casablanca Science University Hassan II Morocco Integration of SOA and Cloud Computing in RM- ODP RM-ODPSOACloud Computing The objective of ODP is according to ITU-T Recommendation X.901 stated as follows: "The objective of ODP standardization is the development of standards that allow the benefits of distributing information processing services to be realized in an environment of heterogeneous IT resources and multiple organizational domains. These standards address constraints on system specification and the provision of a system infrastructure that accommodate difficulties inherent in the design and programming of distributed systems." This objective seems to cover cloud computing systems. Therefore, we in this paper discuss the concepts of cloud, and discuss the use of RM-ODP for specifying the solution. We indicate that the current RM-ODP may be too abstract for the purpose, and indicate how to adapt RM-ODP to fit the purpose. INTRODUCTION Businesses have grown in complexity and have become increasingly reliant on information systems. The advent of Cloud Computing technologies opened up opportunities and business evolved different forms such as e-commerce, ebusiness, supply chains and virtual enterprises. This increased the complexity and challenges for businesses as they struggled to align their IT with their strategic intent. Hence a need arose for a holistic approach to handle this complexity. In this paper, we look at the concepts of cloud computing. We discuss the capabilities of RM-ODP in solving the complexity and challenges for specifying cloud computing systems. Then we introduce some existing standards in SOA to integrate them into the standard RM-ODP. II. CLOUD : DEFINITIONS AND TAXONOMY The following definitions and taxonomy are included to provide an overview of cloud computing concepts. A. Definitions of Cloud Computing Concepts Cloud Computing: Cloud computing is a model for enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction. (This definition is from the latest draft of the NIST Working Definition of Cloud Computing published by the U.S. Government's National Institute of Standards and Technology [2]. 1) Delivery Models The NIST definition of cloud computing defines three delivery models:  Software as a Service (SaaS).  Platform as a Service (PaaS).  Infrastructure as a Service (IaaS). 2) Deployment Models The NIST definition defines four deployment models:  Public Cloud.  Private Cloud.  Community Cloud.  Hybrid Cloud. In the diagram - Fig. 1‖, Service Consumers use the services provided through the cloud, Service Providers manage the cloud infrastructure and Service Developers create the services themselves. B. Taxonomy[1] (Notice that open standards are needed for the interactions between these roles.) Each role is discussed in more detail in [1]. 1) Service Consumer The service consumer is the end user or enterprise that actually uses the service, whether it is Software, Platform or Infrastructure as a Service. Depending on the type of service and their role, the consumer works with different user interfaces and programming interfaces. Some user interfaces look like any other application; the consumer does not need to know about cloud computing as they use the application. Other user interfaces provide administrative functions such as starting and stopping virtual machines or managing cloud storage. Consumers writing application code use different programming interfaces depending on the application they are writing. Consumers work with SLAs (Service Level Agreements) and contracts as well. Typically these are negotiated via human intervention between the consumer and the provider. The expectations of the consumer and the reputation of the provider are a key part of those negotiations. 2) Service Provider The service provider delivers the service to the consumer. The actual task of the provider varies depending on the type of service In the service provider diagram, the lowest layer of the stack is the firmware and hardware on which everything else is based. Above that is the software kernel, either the operating system or virtual machine manager that hosts the infrastructure beneath the cloud. The virtualized resources and images include the basic cloud computing services such as processing power, storage and middleware. The virtual images controlled by the VM manager include both the images themselves and the metadata required to manage them. Crucial to the service provider's operations is the management layer. At a low level, management requires metering to determine who uses the services and to what extent, provisioning to determine how resources are allocated to consumers, and monitoring to track the status of the system and its resources. At a higher level, management involves billing to recover costs, capacity planning to ensure that consumer demands will be met, SLA management to ensure that the terms of service agreed to by the provider and consumer are adhered to, and reporting for administrators. Security applies to all aspects of the service provider's operations. Open standards apply to the provider's operations as well. A well-rounded set of standards simplify operations within the provider and interoperability with other providers. 3) Service Developer The service developer creates, publishes and monitors the cloud service. These are typically "line-of-business" applications that are delivered directly to end users via the SaaS model. Applications written at the IaaS and PaaS levels will subsequently be used by SaaS developers and cloud providers. Development environments for service creation vary. If developers are creating a SaaS application, they are most likely writing code for an environment hosted by a cloud provider. In this case, publishing the service is deploying it to the cloud provider's infrastructure. During service creation, analytics involve remote debugging to test the service before it is published to consumers. Once the service is published, analytics allow developers to monitor the performance of their service and make changes as necessary. III. RM-ODP The Reference Model of Open Distributed Processing RM-ODP , is a joint standardization effort by ISO/IEC and ITU-T that creates an architecture within which support of distribution, interworking and portability can be integrated. Several years after its final adoption as ITU-T Recommendation and ISO/IEC International Standard, the RM-ODP is increasingly relevant, mainly because the size and complexity of current IT systems is challenging most of the current software engineering methods and tools. These methods and tools were not conceived for use with large, open and distributed systems, which are precisely the systems that the RM-ODP addresses. In addition, the use of international standards has become the most effective way to achieve the required interoperability between the different parties and the organizations involved in the design and development of complex systems. As a result, we are now witnessing many major companies and organizations investigating RM-ODP as a promising alternative for specifying their IT systems, and for structuring their large-scale distributed software designs especially new generation of Cloud applications. The RM-ODP provides five - Fig. 2‖ generic and complementary viewpoints on the system and its environment: enterprise, information, computational, engineering and technology. They allow different participants to observe a system from different perspectives [4]. These viewpoints are sufficiently independent to simplify reasoning about the complete specification of the system. The architecture defined by RM-ODP tries to ensure the mutual consistency among the viewpoints, and the use of a common object model and a common foundation defining concepts used in all of them (composition, type, subtype, actions, etc.) provide the glue that binds them all together. Although the ODP reference model provides abstract languages for the relevant concepts, it does not prescribe particular notations to be used in the individual viewpoints. UML4ODP [7] defines a set of UML profiles, one for each viewpoint language and one to express the correspondences between viewpoints, by which ODP modelers can use the UML notation for expressing their ODP specifications in a standard graphical way, and UML modelers could use the RM-ODP concepts and mechanisms to structure their large UML system specifications according to a mature and standard proposal. IV. CLOUD COMPUTING AND RM-ODP : CONVERGENCE OR DIVERGENCE By definition Cloud computing is a large distributed system, users in this account per million, working together on computer networks on different applications; The standard RM-ODP aims to large distributed system specification; share this observation we can say that RM-ODP and the cloud converges towards the same objectives (Applications for a large distributed systems and Specification of application of large distributed systems ). The question is: can RM-ODP support the specification of a cloud applications? The Reference Model for Open Distributed Processing (RM-ODP) has been published in 1995 and is currently under a revision process that includes better alignment with SOA concepts. Hence, a new clause -service concepts‖ has been added to Part 2 (Foundations) [4] and some adaptations have been included in the trading function of Part 3 (Architecture) [5]. For our part we did a study on various SOA standards and Cloud standards which already exists for the proposed extension as the standard RM-ODP. V. STANDARDS ANALYSIS UNDER STANDARDS DEVELOPMENT ORGANIZATION A number of standards consortia are working on the standardization of various aspects of SOA. Many of these standards define terms for SOA, Three of the consortia, The Open Group, OASIS, and OMG, collaborated to understand and position the architectural standards underway. The result of the collaboration is the jointly published positioning paper, -Navigating the SOA Open Standards Landscape Around SOA‖ [8] In addition to the standards on architecture, there are standards for implementation and infrastructure for SOA underway in OASIS and OMG as well. Here we present the standards that can help us make the extension of RM-ODP, The standards organized by the standards organization A. Open Group The Open Group's vision is Boundaryless Information Flow™ which will enable access to integrated information within and between enterprises based on open standards and global interoperability. The Open Group is known for its development of TOGAF and Unix. 1) SOA Reference Architecture The draft standard SOA reference architecture [12] uses a partially layered approach since one layer does not solely depend upon the adjacent layers. Layers are defined around sets of key architectural concerns and capabilities, the interaction protocols between layers, and the details within a layer using a set of architectural building blocks. There are five functional horizontal layers and four non-functional vertical layers that support various cross-cutting concerns of the SOA architectural style. Enterprise Business Aspect The purpose, scope and policies for the organization that will own the system The SOA Reference Architecture enumerates the fundamental elements of a SOA solution and provides the architectural foundation for the solution.. 2) Service Oriented Cloud Computing Infrastructure (SOCCI) The goal of The Service-Oriented Cloud Computing Infrastructure [13] project will provide recommendations and guidelines that enable the provisioning of infrastructure as a service in the SOA solutions and cloud computing environments. This draft will cover: 1. Definition of Service Oriented Cloud Computing Infrastructure, SOI and Infrastructure as a Service (IaaS) 2. Identify required components for enabling Service-Oriented Infrastructure as a Cloud Service and SOA service 3. Application of Enterprise Service Management concepts 4. Define relationship between SOA and XaaS (Business Process (BPaaS), Software (SaaS), Platform (PaaS), and Infrastructure (IaaS)) Define consumption models for IaaS This specification is important to SOA standards as it defines how to expose existing hardware, infrastructure, software, and virtualized versions of these as services that can be used equally by SOA solutions and Cloud computing. B. OASIS OASIS (Organization for the Advancement of Structured Information Standards) is well known for Web services standards along with standards for security, e-business, the public sector and application-specific markets 1) Reference Model for Service Oriented Architecture 1.0 The reference model [10] provides a normative reference that remains relevant for SOA as an abstract, powerful model, regardless of the inevitable technology changes that have influenced or will influence SOA deployment. The SOA RM is an abstract framework for understanding significant entities and relationships between them within a service-oriented environment, and for the development of consistent standards or specifications supporting that environment. It is based on unifying concepts of SOA and may be used by architects developing specific service oriented architectures or in training and explaining SOA. The OASIS SOA Reference Model - Fig. 4‖ applies directly to the Vocabulary category of RM-ODP. The goal of the OASIS Reference Architecture for SOA Foundation [9] is to define a view-based abstract reference architecture foundation that models SOA from an ecosystem/paradigm perspective. It specifies three viewpoints; specifically, the Service Ecosystem viewpoint, the Realizing SOAs viewpoint, and the Owning SOAs viewpoint. It is based on the concepts and relationships defined in the OASIS Reference Model for Service Oriented Architecture. Each of the associated views that are obtained from these three viewpoints is briefly described below. The Service Ecosystem view - Fig. 5‖ contains models that are intended to capture how SOA integrates with and supports the service model from the perspective of the people who perform their tasks and achieve their goals as mediated by SOAs. Since the Service Ecosystem viewpoint (on which this view is based) emphasizes the use of SOA to allow people to access and provide services that cross ownership boundaries, it is explicit about those boundaries and what it means to cross an ownership boundary. The Realizing SOAs view - Fig. 6‖ contains models for description of, visibility of, interaction with, and policies for services. The Owning SOAs view - Fig. 7‖ contains models for securing, managing, governing, and testing SOA-based systems. This Reference Architecture is principally targeted at Enterprise Architects; however, Business and IT Architects as well as CIOs and other senior executives involved in strategic business and IT planning should also find the architectural views and models described to be of value. C. OMG OMG has been an international, open membership, not-forprofit computer industry consortium since 1989. OMG Task Forces develop enterprise integration standards for a wide range of technologies. OMG's modeling standards, including the Unified Modeling Language™ (UML®) and Model Driven Architecture® (MDA®), enable powerful visual design, execution and maintenance of software and other processes, including IT Systems Modeling and Business Process Management. 1) Service oriented architecture Modeling Language (SoaML) The goals of SoaML [11] are to define extensions to UML for services modeling and provide functional, component, and service-oriented modeling capabilities. SoaML extends UML in order to provide additional capabilities for managing cohesion and coupling afforded by an SOA style. The standard is intended to be sufficiently detailed to define platformindependent SOA models (PIM) that can be transformed into platform-specific models (PSM) for particular technical architectures as described by the OMG MDA. The intent of SoaML was to provide a foundation for integration, interoperability, and extension. The fundamental element of SoaML - Fig. 8‖ is the participant, representing a service consumer and/or provider. Participants express their goals, needs, and expectations through requests for services as defined by service interfaces or service contracts. Other participants express their value propositions, capabilities, and commitments through services. Participants are then assembled into service value chains where participant requests are connected to the compatible services of other participants through service channels through which they interact. SoaML uses facilities of UML to define the services interfaces and method behaviours for carrying out and using services. SoaML also defines autonomous agents that can choreograph participants in a service value chain while adapting to the changing needs of the community of collaborating participants. SoaML provides a means of defining milestones that indicate the achievement of progress toward achieving the desired real-world effect of the services value chain, and for evaluating different approaches to achieving progress by different participants. The purpose of the dealer network architecture is to establish the financially related services between dealers and manufactures such that any dealer can do business with any manufacturer VI. EXTENTIONS OF RM-ODP From the previous paragraph it seems that the different standards are in line with RM-ODP, and to integrate the SOA into RM-ODP, we can take different concepts existing in these standards and reused in RM-ODP. A. Integration of OASIS SOA-RM in RM-ODP Foundation (ISO / IEC 10746-2): The Reference Model for Service Oriented Architecture is intended to capture the -essence‖ of SOA, as well as provide a vocabulary and common understanding of SOA. The goals of the reference model include a common conceptual framework that can be used consistently across and between different SOA implementations, common semantics that can be used unambiguously in modeling specific SOA solutions, unifying concepts to explain and underpin a generic design template supporting a specific SOA, and definitions that should apply to all SOA. While Rec. ITU-T X.902 \| ISO / IEC 10746-2 Foundations: it contains the definition of concepts and analytical framework to be used for the standardized description of distributed processing systems (arbitrary). She sticks to a level of detail sufficient to support Rec. ITU-T X.903 \| ISO / IEC 10746-3 and establish requirements for new specification techniques. It is noted that OASIS RM-SOA are in the same direction, so we take all the necessary definitions to SOA from SOA-RM and joined it in (Concepts services) of ISO / IEC 10746-2. B. Integration of RA-SOA of the Open Group and OASIS SOA-RA in RM-ODP Architecture (ISO / IEC 10746-3) The SOA reference architecture of the Open Group uses a layered approach as a partial layer does not only depend on adjacent layers. The layers are defined around a set of architectural concepts, protocols of interaction between the layers and details in a layer with a set of architectural building blocks. There are five functional layers and four horizontal layers non-functional vertical support issues cross the style of SOA. The reference architecture of OASIS SOA specifies three points of view, more precisely, the view through business service point of view of achievements SOA, and SOA perspective of the owner. It is based on the concepts and relationships defined in the model reference for the OASIS SOA. Each of the points of view represented in the form of UML diagram. On the other hand Rec. ITU-T X.903 \| ISO / IEC 10746-3: Architecture: it decomposes a system into five Viewpoint, business, information, computational, engineering, technology. It is found that RA-SOA from the Open Group and OASIS RA-SOA and RM-ODP Architecture (ISO / IEC 10746-3) are along the same lines, The Open Group RA-SOA decompose the system in five functional layers, four no-functional layers, all this is a point of view soa, while OASIS SOA-RA decompose the system in three views to a point of view soa, while RM-ODP Architecture (ISO / IEC 10746-3) decomposes the system in 5 viewpoints that includes all the system. From our point of view, the same concepts of decomposition to view that suggest the two RA-SOA are already existing in RM-ODP architecture, it is sufficient to include in each viewpoint of RM-ODP Architecture a sub view SOA, These subs views take their concepts from RA-SOA OASIS and RA-SOA OPEN GROUP. C. INTEGRATION OF SOAML IN UML4ODP: SoaML objectives are to define extensions to UML for modeling services and provide functional components and how service-oriented modeling. Each of these modeling approaches provide different and improved ways to deal with cohesion and coupling in complex systems. SoaML extends UML to provide additional features to manage cohesion and coupling offered by an SOA style. On the other hand UML4ODP fact the extension of UML for modeling of the five viewpoints RM-ODP, and the correspondence between views, and sets for each language point of view, a meta model. With the adoption of SOA by enterprises and the emergence of cloud computing, new needs appear and the specification of the new system based on SOA and cloud remains impossible, because the concept of service itself is not defined in RM-ODP. SoaML support the instantiation of the SOA reference model OASIS [11] and provides a concrete platform for modeling integrated with UML and supporting the OMG's MDA. This profile modeling can be used in conjunction with the normalization in the standard ODP precisely UML4ODP. Use language common to these various modeling systems and the integration of separate fields to enable business agility that can be represented by models of architectural art. We propose to take the new profiles UML defines SOAML and integrated into the UML4ODP to enable interoperability with already existing, and secondly to allow the modeling of system specifications based on SOA and Cloud Computing. VII. EXAMPLE USE CASE In contrast to most LBSs for tourism, which represents the user geo-referenced information about tourist attractions and other POIs, the presented application [14] tries to give the user a ranked list of tourist attractions. The ranking is calculated using a MCE, based on user preferences, e.g., personal interests and the actual location. The following scenario shall be solved efficiently with such a system: A tourist visits a city, where she or he is not familiar with. The tourist has huge interest in the culture and architecture of the city and moderate interest in nature and parks as well as shopping and events. She/he wants to see places other people recommend or many people think that these are worth visiting. The tourist wants to have a suggestion of attractive places, and how well they fit to her/his preferences. Based on such a list the tourist wants to make a decision which places are worth to visit for her/him and select these tourist attractions. Additional she/he needs to receive relevant information and the routing to the selected places. A. Enterprise viewpoint and requirements With the enterprise viewpoint tries to specify the scope of the tourist guide application. Figure 9 emphasises on the purpose and scope of the system and specifies which actors and use cases are involved in the system process. The diagram tries to show the roles of the actors and which basic activities have to be performed for the objective of the application The Use Case diagram of Figure 9 identifies actors like Tourist and Tourist Attraction. The Tourist has the role of the consumer of tourist guide application and wants to visit some tourist attractions within a city. Therefore the user asks the tourist guide for support. The tourist guide evaluates the characteristics of the tourist attractions against the tourist interests, the position and further settings to recommend a list of ranked attractions to the tourist. B. Information management and processing The information viewpoint is concerned with information modelling. An information specification defines the semantics of information and the semantics of information processing in an ODP system, without considerations about other system details, such as its implementation, or the technology used to implement the system. The selection of evaluation criteria and data sources for tourist places plays a role to understand the information management and processing. C. Computational Viewpoint The computational viewpoint describes the functionality of the tourist guide LBDS application and its environment through the decomposition of the system, in distributed transparent terms, into objects which interact at communication interfaces. In the computational viewpoint, application and distributed functions consist of configurations of interacting computational objects. The computational viewpoint is directly concerned with the distribution of processing but not with the interaction mechanisms that enable distribution to occur. The computational specification decomposes the system into objects performing individual functions and interacting at well-defined interfaces. D. Engineering perspective The engineering viewpoint focuses on the mechanisms and functions required to support distributed interactions between objects in the system. It describes the distribution of processing performed by the system to manage the information and provide the functionality. E. Technological perspective The technological perspective should give the idea which real-world software, hardware and network components are used. This is the starting point for the engineering process. Figure 12 shows the technological viewpoint of the LBDS application including their components. The data layer consists of content relevant geographic data, which are in the prototype application information about the tourist attractions to form decision alternatives. Additional data like community data is coming from third-party web services. The logic layer includes elements for data merging and conversion and MCE techniques. In the tourist guide application the concordance method is used as decision rule. The communication between data layer and logic layer is based on the Internet. Also the communication between the mobile client and the logic layer is also done via the Internet using SOAP web services. The client includes a part for the management and communication with the model base, the UI and positioning technology. VIII. CONCLUSIONS AND FUTURE WORK Our study focused on the development of RM-ODP, after a literature review on the field SOA and Cloud Computing, we found the following possible changes in RM-ODP  RM-ODP does not incorporate the concept of SOA  RM-ODP does not incorporate the notion of Cloud Computing Building on these, we proceeded to identify the various SOA standards developed or being developed by different international consortia; and we make a comparison with the concepts of RM-ODP, and see what is possible to integrate with it. We found four existing SOA standards that can be integrated with the RM-ODP, SOA-RM of the Open Group RA-SOA Open Group, SOA-RA OASIS, and SOAML the OMG. Our future work will focus on the integration of new concepts of these standards and we will give the various topics that must be integrated or update in RM-ODP. Fig. 1 . 1Diagram defining a taxonomy for cloud computing Fig. 2 . 2RM-ODP viewpoints Fig. 3 . 3Layers of the SOA Reference Architecture Three of the layers -Fig. 3‖ address the implementation and interface with a service (the Operational Systems Layer, the Service Components Layer and the Services Layer). Two of them support the consumption of services (the Business Process Layer and the Consumer Layer). Four of them support Fig. 4 . 4Principal concepts in the OASIS SOA Reference Model 2) OASIS Reference Architecture for SOA Foundation Version 1.1 Fig. 5 . 5Model elements described in the Business via Services Fig. 6 . 6Model Elements Described in the Realizing a Service Oriented Architecture View Fig. 7 . 7Model elements described in the Owning Service Oriented Architectures view Fig. 8 . 8Example Fig. 9 . 9Use Case diagram of the tourist guide application. Fig. 10 . 10Static schema of exchanged information. Fig. 11 . 11Computational viewpoint of the tourist guide application, including functional objects and important operations. Fig. 12 . 12Engineering viewpoint of the tourist guide application, we use SoaML modelisation Fig. 12 . 12LBDS architecture from the technology view. What for? Why? Who? When?Information Information System AspectsInformation handled by the system and constraints on the use and interpretation of that information What is it about ?Computational Application Design AspectsFunctional decomposition of the system into objects suitable for distribution Quality of Service, Integration and Governance Layers). The RA-SOA as a whole provides the framework for the support of all the elements of a SOA, including all the components that support services, and their interactions.This logical view of the SOA Reference Architecture addresses the question, -If I build a SOA, what would it look like and what abstractions should be present?‖How does each bit work ? Technology Implementation System hardware & software and actual distribution With what ? Engineering Solution Types & Distribution Infrastructure required to support distribution How do the bits work together ? ODP System cross-cutting concerns of a more -non-functional‖ nature (the Information Architecture, Cloud Computing Use Cases. //Opencloudmanifesto.Org/Cloud_Computing_Use_Cases_ Whitepaper-4_0.PdfWhitepaper 4.0Cloud Computing Use Cases, Whitepaper 4.0, July 2010 Http://Opencloudmanifesto.Org/Cloud_Computing_Use_Cases_ Whitepaper-4_0.Pdf Definition Of Cloud Computing, Draft Of The NIST Working. Definition Of Cloud Computing, Draft Of The NIST Working , July 2009, Http://Csrc.Nist.Gov/Groups/Sns/Cloud- Computing/Cloud-Def-V15.Doc 901 \| ISO/IEC 10746-1 Information Technology -Open Distributed Processing -Reference Model -Overview. Itu-T X, ITU-T X.901 \| ISO/IEC 10746-1 Information Technology -Open Distributed Processing -Reference Model -Overview, 1996 902 \| ISO/IEC 10746-2 Information Technology -Open Distributed Processing -Reference Model -Foundations. Itu-T X, FIDSITU-T X.902 \| ISO/IEC 10746-2 Information Technology -Open Distributed Processing -Reference Model -Foundations, FIDS, January 2010 903 \| ISO/IEC 10746-3 Information Technology -Open Distributed Processing -Reference Model -Architectural. Itu-T X, FIDSITU-T X.903 \| ISO/IEC 10746-3 Information Technology -Open Distributed Processing -Reference Model -Architectural, FIDS, January 2010 Itu-T X, 904 \| ISO/IEC 10746-4 Information Technology -Open Distributed Processing -Reference Model -Architectural Semantics. ITU-T X.904 \| ISO/IEC 10746-4 Information Technology -Open Distributed Processing -Reference Model -Architectural Semantics, 1996 906 \| ISO/IEC 19793 -Information Technology -Open Distributed Processing -Use of UML for ODP System Specifications. Itu-T X, ITU-T X.906 \| ISO/IEC 19793 -Information Technology - Open Distributed Processing -Use of UML for ODP System Specifications, October 2009 Navigating the SOA Open Standards Landscape Around Architecture, White Paper. Navigating the SOA Open Standards Landscape Around Architecture, White Paper, June 2009, http://www.oasis- open.org/committees/ download.php/32911/wp_soa_harmonize_d1.pdf Reference Architecture for SOA Foundation Version 1.1, OASIS, Committee Draft. Process. essReference Architecture for SOA Foundation Version 1.1, OASIS, Committee Draft, 2009, In Process, http://docs.oasis- open.org/soa-rm/soa--ra/v1.0/soa-ra-pr-01.pdf Reference Model for Service Oriented Architecture 1.0, OASIS, Standard. Reference Model for Service Oriented Architecture 1.0, OASIS, Standard, October 2006, http://docs.oasis-open.org/soa-rm/v1.0/ soa-rm.pdf Service Oriented Architecture Modeling Language (SoaML), Standard Recommendation. Service Oriented Architecture Modeling Language (SoaML), Standard Recommendation, November 2008, www.omg.org/cgi- bin/doc?ad /08-11-01 Draft standard. Process. essSOA Reference Architecture, Open GroupSOA Reference Architecture, Open Group, Draft standard, In Process, 2009, www.opengroup.org/projects/soa-ref-arch Service Oriented Infrastructure, Open Group, Draft. Process. essService Oriented Infrastructure, Open Group, Draft, In Process, 2009, www.opengroup.org/projects/soa-soi Development of a distributed Service Framework for Locationbased Decision. Johannes SornigMaster ThesisDevelopment of a distributed Service Framework for Location- based Decision; August 2008, Master Thesis, Johannes Sornig.	[]
[ "Time-dependent quantum graph", "Time-dependent quantum graph" ]	[ "D U Matrasulov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n\nInstitute for Applied Physics\nNational University of Uzbekistan\n100174TashkentUzbekistan\n", "J R Yusupov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n", "K K Sabirov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n", "Z A Sobirov \nTurin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan\n\nTashkent Financial Institute\nAmir Temur Str60A, 100000TashkentUzbekistan\n" ]	[ "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Institute for Applied Physics\nNational University of Uzbekistan\n100174TashkentUzbekistan", "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Turin Polytechnic University\n17. Niyazov Str100095Tashkent, TashkentUzbekistan", "Tashkent Financial Institute\nAmir Temur Str60A, 100000TashkentUzbekistan" ]	[]	In this paper we study quantum star graphs with time-dependent bond lengths. Quantum dynamics is treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. Time-dependence of the average kinetic energy is analyzed. Space-time evolution of the Gaussian wave packet is treated for harmonically breathing star graph.	10.17586/2220-8054-2015-6-2-173-181	[ "https://arxiv.org/pdf/1206.5890v1.pdf" ]	53,554,256	1206.5890	6c39d5e32b01e19eadb239becb84576974dc431f	Time-dependent quantum graph 26 Jun 2012 D U Matrasulov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan Institute for Applied Physics National University of Uzbekistan 100174TashkentUzbekistan J R Yusupov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan K K Sabirov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan Z A Sobirov Turin Polytechnic University 17. Niyazov Str100095Tashkent, TashkentUzbekistan Tashkent Financial Institute Amir Temur Str60A, 100000TashkentUzbekistan Time-dependent quantum graph 26 Jun 2012 In this paper we study quantum star graphs with time-dependent bond lengths. Quantum dynamics is treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. Time-dependence of the average kinetic energy is analyzed. Space-time evolution of the Gaussian wave packet is treated for harmonically breathing star graph. Quantum particle dynamics in nanoscale networks and discrete structures is of fundamental and practical importance. Usually such systems are modeled by so-called quantum graphs, systems attracting much attention in physics [1]- [3] and mathematics [5]- [7] during past two decades. In physics quantum graphs were introduced as a toy model for studies of quantum chaos by Kottos and Smilansky [1]. However, the idea for studying of a system confined to a graph dates back to Pauling [4] who suggested to use such systems for modeling free electron motion in organic molecules. During last two decades quantum graphs found numerous applications in modeling different discrete structures and networks in nanoscale and mesoscopic physics(e.g., see reviews [1]- [3] and references therein). Mathematical properties of the Schrodinger operators on graphs [5][6][7] inverse problems for quantum graphs [8,9] were also subject for extensive research recently. Also, an experimental realization of quantum graphs was discussed earlier in the Ref. [10]. Despite the certain progress made in the study of quantum graphs some of important aspects are still remaining as less-or not explored. Especially, this concerns the problems of driven graphs, i.e. graphs perturbed by time-dependent external forces. An important example of such a driving force is that caused by driven (moving) boundaries. Treatment of such system can be reduced to the Schrödinger equation with time-dependent boundary conditions. Earlier, the problem of time-dependent boundary conditions has attracted much attention in the context of quantum Fermi acceleration [12]- [14], though different aspects of the problem was treated by many authors [16]- [27]. Detailed study of the problem can be found in series of papers by Makowski and co-authors [21]- [23]. It was pointed out in the above Refs. that the problem of 1D box with the moving wall can be mapped onto that of time-dependent harmonic oscillator confined inside the static box [21]. In this paper we treat similar problem for quantum star graph, i.e. we study the problem of quantum graphs with time-dependent bonds. In particular, we consider harmonically breathing quantum star graphs, the cases of contracting and expanding graphs. The latter can be solved exactly analytically. Motivation for the study of timedependent graphs comes from such practically important problems as quantum Fermi acceleration in nanoscale network structures, tunable particle transport in quantum wire networks, molecular wires, different lattices and discrete structures. In particular, sites, vertices, nodes of such discrete structures can fluctuate that makes them timedependent. We will study time-dependence of the average kinetic energy and wave packet dynamics in harmonically breathing graphs. Graphs are the systems consisting of bonds which are connected at the vertices. The bonds are connected according to a rule which is called topology of a graph. Topology of a graph is given by in terms of so-called adjacency matrix [1,2]: C ij = C ji = 1 if i and j are connected 0 otherwise i, j = 1, 2, ..., V. Quantum dynamics of a particle in a graph is described in terms of one-dimensional Schrödinger equation [1,2] (in the units = 2m = 1): −i d 2 Ψ b (x) dx 2 = k 2 Ψ b (x), b = (i, j),(1) where bdenotes a bond connecting ith and jthe vertices, and for each bond b, the component Ψ b of the total wavefunction Ψ b is a solution of the eq.(1). The wavefunction, Ψ b , satisfies boundary conditions at the vertices, which ensure continuity and current conservation [1]. General scheme for finding of eigenfunctions and eigenvalues for such boundary conditions can be found in the Ref. [1]. Different types of boundary conditions for the Schrodinger equation on graphs are discussed in the Refs. [5][6][7]. In the following we restrict our consideration by most simplest graph, so-called star graph. The star graph consist of three or more bonds connected at the single vertex which is called branching point (see Fig.1). Other ones are called edge vertices. The eigenvalue problem for a star graph with N bonds is given by the following Schrödinger equation: − d 2 dx 2 φ j (y) = k 2 φ j (y), 0 ≤ y ≤ l j , j = 1, ..., N. Here we consider the following boundary conditions [11]:          φ 1 \| y=0 = φ 2 \| y=0 = ... = φ N \| y=0 , φ 1 \| y=l1 = φ 2 \| y=l2 = ... = φ N \| y=lN = 0, N j=1 ∂ ∂y φ j \| y=0 = 0.(2) The eigenvalues can be found by solving the following equation [11] N j=1 ctan(k n l j ) = 0 where corresponding eigenfunctions are given as [11] φ (n) j = B n sin(k n l j ) sin(k n (l j − y)) B n = 2 j lj +sin(2knlj ) sin 2 (knlj ) . Time-dependent graph implies that lengths of the bonds of a graph are time-varying, i.e., when L j is a function of time. In this case particle dynamics in graph is described by the following time-dependent Schrödinger equation: i ∂ ∂t Ψ j (x, t) = − ∂ 2 ∂x 2 Ψ j (x, t), 0 < x < L j (t), j = 1, ..., N,(3) with N being the number of bonds. In the following we will consider the boundary conditions given by          Ψ 1 \| x=0 = Ψ 2 \| x=0 = ... = Ψ N \| x=0 , Ψ 1 \| x=L1(t) = Ψ 2 \| x=L2(t) = ... = Ψ N \| x=LN (t) = 0, N j=1 ∂ ∂x Ψ j \| x=+0 = 0. These boundary conditions imply that only edge vertices of the graph are moving while center (branching point) is fixed. Furthermore, we assume that L j (t) is given as L j (t) = l j L(t), where L(t) is a continuous function and l j are the positive constants. Then using the coordinate transformation y = x L(t) , Eq. (3) can be rewritten as i ∂ ∂t Ψ j (y, t) = − 1 L 2 ∂ 2 ∂y 2 Ψ j (y, t) + iL L y ∂ ∂y Ψ j (y, t), 0 < y < l j , j = 1, ..., N.(4) It is clear that the Schrödinger operator in the right hand side of Eq. (4) is not Hermitian due to the presence of second term. Therefore using the transformation Ψ j (y, t) = 1 √ L e i LL 4 y 2 ϕ j (y, t), we can make it Hermitian as i ∂ ∂t ϕ j (y, t) = − 1 L 2 ∂ 2 ∂y 2 ϕ j (y, t) + LL 4 y 2 ϕ j (y, t), 0 < y < l j , j = 1, ..., N.(5) We note that the above transformations of the wave function remain the boundary conditions unchanged. Time and coordinate variables in Eq. (5) can be separated only in case when L(t) obeys the equation L 3L 4 = −C 2 = const,(6) In this case using the substitution ϕ j (y, t) = φ j (y) · exp −ik 2 t 0 ds L 2 (s) , we get d 2 φ j dy 2 + (λ − C 2 y 2 )φ j = 0, y ∈ (0, l j ).(7) For C = 0 from Eq. (6) we have L(t) = αt 2 + βt + γ, C 2 = 1 16 (β 2 − 4αγ),(8) and L(t) = βt + γ, C 2 = 1 16 β 2 .(9) In both cases exact solutions of Eq.(5) can be obtained in terms confluent hypergeometric functions. In particular, for the case when time-dependence of L(t) is given by Eq.(9) fundamental solutions of Eq.(5) can be written as φ j,1 = y exp C 2 y M 3 4 − k 4C , 3 2 , −Cy 2 , and φ j,1 = exp C 2 y M 1 4 − k 4C , 1 2 , −Cy 2 . Therefore the general solution of Eq.(5) is given as φ j (y) = A j φ j,1 + B j φ j,2 .(10) From the boundary conditions given by Eq. (2) we have B j = A, A j = A · α j (k), j = 1, 2, 3, ..., N, where B is an arbitrary constant and 3 2 , −Cl 2 j , j = 1, 2, ..., N. Taking to account the relations dφj,1(y) dy y=0 = 1, dφj,2(y) dy y=0 = C 2 , from Eq. (2) we obtain the following spectral equation for finding the eigenvalues, k n of Eq.(5): α j (k) = − M 1 4 − k 4C , 1 2 , −Cl 2 j l j M 3 4 − k 4C ,N j=1 1 l j M 1 4 − k 4C , 1 2 , −Cl 2 j M 3 4 − k 4C , 3 2 , −Cl 2 j = CN 2 .(11) Thus the eigenfunctions of Eq.(5) can be written as φ j (y, k n ) = A [α j (k n )φ j,1 (y) + φ j,2 (y)] , j = 1, 2, ..., N. Furthermore, we provide the solution of Eq.(5) for simplest case L(t) = at + b, which correspond to C = 0 in Eq. (6). In this case of the eigenvalues of Eq.(5) can be written as time-dependence of the wall is given as φ j (y, k n ) = A sin( √ k n l j ) sin( k n (l j − y)), j = 1, 2, ..., N,(13) where k n is the nth positive root of the equation N j=1 ctan(l 2 j √ k) = 0.(14) and L(t) > 0, B is the normalization constant. Now let us consider harmonically breathing graph, i.e. the case when time-dependence of L(t) is given as L(t) = b + a cos ωt with ω = 2π T being oscillation frequency and T is the oscillation period. It is clear that in this case time and coordinate variables in Eq. (5) cannot be separated. Expanding ϕ(y, t) in Eq. (5) in terms of static graphs wave functions as ϕ j (y, t) = n C n (t)φ (n) j (y),(15) and inserting this expansion into Eq.(5)we haveĊ n (t) = m M mn C m (t) where M mn = −i k 2 m L 2 (t) − i LL 4 K mn The quantity we are interested to compute is the average kinetic energy which is defined as E(t) = ψ\|H\|ψ = N j=1 Lj(t) 0 ∂ψ j (x, t) ∂x 2 dx.(16) In Fig. 1 time dependence of the average kinetic energy of the harmonically breathing star graph is presented for different values of the breathing frequency and amplitude. As it can be seen from these plots, < E(t) > is almost periodic for ω = 0.5 and a = 1, while for ω = 10 and a = 1 such a periodicity is completely broken and energy grows in time. For ω = 10 and a = 20 the behavior of < E(t) > demonstrates "quasiperiodic behavior". Appearing of periodic behavior in < E(t) > can be explained by synchronization of the motion of particle with the frequency. The lack of such synchronization causes breaking of the periodicity of the average energy in time. Furthermore, we consider wave packet evolution in harmonically breathing star graph by taking the wave function at t = 0 (for the first bond) as the following Gaussian wave packet: Correspondingly, the expansion coefficients at t = 0 can be written as Ψ 1 (x, 0) = Φ(x) = 1 √ 2πσ e − (x−µ) 2 2σ ,(17)C n (0) = j lj 0 ϕ (j) (y, 0)φ (j) n (y) * . In calculation of the wave packet evolution we will choose initial condition as the wave packet being on the first bond only, while for other two bonds the wave function at t = 0 is taken as zero. In Fig.2 the time evolution of the wave packet is plotted for harmonically breathing primary star graph whose bonds oscillate according to the law L(t) = 40 + a cos ωt . The oscillation parameters (frequency and amplitude) are chosen as follows: a) ω = 10, a = 20; b) ω = 10, a = 1; c) ω = 0.5, a = 1. Fig.2d presents wave packet evolution in static(time-independent) star graph. At t = 0 a Gaussian packet of the width σ and velocity v 0 is assumed being in the first bond. As it can be seen from these plots, for higher frequencies dispersion of the packet and its transition to other bonds occur more faster compared to that for smaller smaller frequencies. Again, an important role plays here possible synchronization between the bond edge and wave packets motions. Existence or absence of ssuch synchronization defines how the collision of the packet with the bond edges will occur and how extensively it gains or loses its energy. Therefore more detailed treatment of the wave packet dynamics in harmonically breathing graphs should be based on the analysis of the role of synchronization and its criterions. In this paper we have treated time-dependent quantum network by considering expanding and harmonically breathing quantum star graphs. Edge boundaries are considered to be time-dependent, while branching point is assumed to be fixed(static). Time-dependence of the average kinetic energy and space-time evolution of the Gaussian wave packet are studied by solving the Schrodinger equation with time-dependent boundary conditions. It is shown that for certain frequencies energy is a periodic function of time, while for others it gan be non-monotonically growing function of time. Such a feature can be caused by possible synchronization of of the particles motion and the motions of the moving edges of graph bonds. Similar feature can be seen also from the analysis of the wave packet evolution. The above study can be useful for the treatment of particle transport in different discrete structures, such as molecular and quantum wire networks, networks of carbon nanotubes, crystal lattices, and others nanoscale systems that can be modeled by quantum graphs. FIG. 1 : 1Time-dependence of the average kinetic energy for harmonically oscillating primary star graph. with σ being the width of the packet. For other bonds initial wave function is assumed to be zero, i.e. Ψ 2 (x, 0) = Ψ 3 (x, 0) = 0. Then for the initial values of the functions ϕ (j) (y, t) in Eq. (15) we have ϕ (j) (y, 0) = L(0)e −i L( FIG. 2 : 2Time evolution of the Gaussian wave packet given by Eq. (17) for the parameters: a) ω = 10, a = 20; b) ω = 10, a = 1; c) ω = 0.5, a = 1; d) Wave packet evolution in static star graph. . Tsampikos Kottos, Uzy Smilansky, Ann.Phys. 76274Tsampikos Kottos and Uzy Smilansky, Ann.Phys., 76 274 (1999). . Sven Gnutzmann, Uzy Smilansky, Adv.Phys. 55527Sven Gnutzmann and Uzy Smilansky, Adv.Phys. 55 527 (2006). . S P Gnutzmannj, F Keating B, Piotet, Ann.Phys. 3252595S. GnutzmannJ.P. Keating b, F. Piotet, Ann.Phys., 325 2595 (2010). . L Pauling, J. Chem. Phys. 4673L. Pauling, J. Chem. Phys. 4 673 (1936). . P Exner, P Seba, P Stovicek, J. Phys. 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[ "Neutrino effects in two-body electron-capture measurements at GSI", "Neutrino effects in two-body electron-capture measurements at GSI" ]	[ "Avraham Gal \nRacah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael\n" ]	[ "Racah Institute of Physics\nThe Hebrew University\n91904JerusalemIsrael" ]	[]	I conjecture that the time modulated decay rates reported in single ion measurements of two body electron capture decay of hydrogen like heavy ions at GSI may be related to neutrino spin precession in the static magnetic field of the storage ring. These 'GSI Oscillations' arise from interference between amplitudes of decay within and without the magnetic field, a scenario that requires a Dirac neutrino magnetic moment six times lower than the Borexino solar neutrino upper limit of 0.54 x 10E(-10) Bohr magneton. I also show in a way not discussed before that the time modulation associated with interference between massive neutrino amplitudes, if such interference could arise, is of a period at least four orders of magnitude shorter than reported and must average to zero given the time resolution of the GSI measurements.	10.1016/j.nuclphysa.2010.04.007	[ "https://arxiv.org/pdf/0809.1213v7.pdf" ]	12,714,304	0809.1213	a9282041676813cec904da29a672865b955e70c8	Neutrino effects in two-body electron-capture measurements at GSI 30 May 2010 Avraham Gal Racah Institute of Physics The Hebrew University 91904JerusalemIsrael Neutrino effects in two-body electron-capture measurements at GSI 30 May 2010neutrino interactionsmassmixing and moments; electron capture PACS: 1315+g1340Em1460Pq2340-s I conjecture that the time modulated decay rates reported in single ion measurements of two body electron capture decay of hydrogen like heavy ions at GSI may be related to neutrino spin precession in the static magnetic field of the storage ring. These 'GSI Oscillations' arise from interference between amplitudes of decay within and without the magnetic field, a scenario that requires a Dirac neutrino magnetic moment six times lower than the Borexino solar neutrino upper limit of 0.54 x 10E(-10) Bohr magneton. I also show in a way not discussed before that the time modulation associated with interference between massive neutrino amplitudes, if such interference could arise, is of a period at least four orders of magnitude shorter than reported and must average to zero given the time resolution of the GSI measurements. Introduction Measurements of weak interaction decay of multiply ionized heavy ions coasting in the ion storage-cooler ring ESR at the GSI laboratory, since the first report in 1992 [1], open up new vistas for dedicated studies of weak interactions. In particular, electron capture (EC) decay rates in hydrogen-like and helium-like 140 Pr ions have been recently measured for the first time [2] by following the motion of the decay ions (D) and the recoil ions (R). The overall decay rates λ EC of these two-body 140 Pr → 140 Ce + ν EC decays, in which no neutrino ν is detected, are well understood within standard weak interaction calculations of the underlying e − p → ν e n reaction [3,4]. However, a time-resolved decay spectroscopy applied subsequently to the two-body EC decay of H-like 140 Pr and 142 Pm single ions revealed an oscillatory behavior, or more specifically a time modulation of the two-body EC decay rate [5]: λ EC (t) = λ EC [1 + a EC cos(ω EC t + φ EC )],(1) with amplitude a EC ≈ 0.2, and angular frequency ω lab EC ≈ 0.89 s −1 (period T lab EC ≈ 7.1 s) in the laboratory system which is equivalent in the rest frame of the decay ion to a minute energyhω EC ≈ 0.84 × 10 −15 eV. Subsequent experiments on EC decays of neutral atoms in solid environment have found no evidence for oscillations with periodicities of this order of magnitude [6,7]. Thus, the oscillations observed in the GSI experiment could have their origin in some characteristics of the H-like ions, produced and isolated in the ESR, and in the electromagnetic fields specific to the ESR which are not operative in normal laboratory experiments. It is suggested here, in Sect. 3, that the 'GSI Oscillations' could indeed be due to the magnetic field which stabilizes and navigates the motion of the ions in the ESR. Several works, by Kienle and collaborators, relegated the 'GSI Oscillations' to interference between neutrino mass eigenstates that evolve coherently from the electron neutrino ν e [8,9,10,11,12]. This idea apparently also motivated the GSI experiment [5]. Such interferences, according to these works, lead to oscillatory behavior given by Eq. (1) with angular frequency ω νe where, again in the decay-ion rest frame, hω νe = ∆(m ν c 2 ) 2 2M D c 2 ≈ 0.29 × 10 −15 eV.(2) Here, ∆(m ν c 2 ) 2 = (0.76 ± 0.02) × 10 −4 eV 2 from accumulated solar ν plus KamLAND reactorν data [13] for the two mass-eigenstate neutrinos that almost exhaust the coupling to ν e , and M D ≈ 130 GeV/c 2 is the mass of the decay ion 140 Pr 58+ . Although the value ofhω νe on the r.h.s. of Eq. (2) is about three times smaller than the value ofhω EC required to resolve the 'GSI Oscillation' puzzle, getting down to this order of magnitude nevertheless presents a remarkable achievement if correct. 1 However, it is shown here in Sect. 2 by following the methodology of Ref. [12] that the correct energy scale under circumstances allowing oscillatory behavior is given bȳ hΩ νe = ∆(m ν c 2 ) 2 2E ν ≈ 0.95 × 10 −11 eV,(3) where E ν ≈ 4 MeV is a representative value for neutrino energy in the Hlike 140 Pr → 140 Ce + ν e and 142 Pm → 142 Nd + ν e EC decays [5]. The energȳ hΩ νe is larger by over four orders of magnitude thanhω EC orhω νe given by Eq. (2), and so it would lead to modulation period shorter by over four orders of magnitude than the 7 s period reported by the GSI experiment. Given a time measurement resolution of order 0.5 s [5], the effect of such oscillatory behavior would average out to zero. Other authors [16,17,18,19,20,21] have rejected any link between neutrino mass eigenstates and the EC decay rate oscillatory behavior reported by the GSI experiment [5], the underlying argument being that since no neutrino is detected, the EC decay rate sums incoherently over neutrino mass eigenstates, whereas any oscillatory behavior requires interference between amplitudes summed upon coherently. It is instructive, however, to demonstrate this assertion also by adopting the methodology of Ref. [12], but with a caveat explained below. To this end the time-dependent EC transition amplitude A νe (i → f ; t), from initial state i (D injected at time t = 0) to a final state f (R plus a coherent combination of neutrino mass eigenstates at time t), is written in terms of transition amplitudes A ν j (i → f ; t) that involve propagating mass-eigenstates neutrinos ν j as A νe (i → f ; t) = j U ej A ν j (i → f ; t),(4) where U ej is a neutrino mixing matrix element of the 3 × 3 unitary matrix U \|ν α = 3 j=1 U * αj \|ν j (α = e, µ, τ )(5) between the emitted electron-neutrino ν e and a mass-eigenstate neutrino ν j [22]. For times of order seconds, appropriate to the 'GSI Oscillations', the coherence implied by Eq. (4) is still in effect [20] and the flavor basis is of physical significance. If the GSI experiment were to detect neutrino ν β by a flavor measurement, the corresponding amplitude would have been generated by projecting Eq. (4) onto flavor β: A νe→ν β (i → f ; t) = j U ej A ν j (i → f ; t)U * βj ,(6) in close analogy with the discussion of neutrino oscillations in dedicated oscillation experiments (Eq. (13.4) in Ref. [22]). The probability associated with the amplitude (6) is then given by P νe→ν β (i → f ; t) = \|A νe→ν β (i → f ; t)\| 2 .(7) Interference terms A ν j A * ν j ′ will arise in P νe→ν β (i → f ; t), leading to oscillations as shown in Sect. 2. Since the GSI experiment does not detect any neutrino, the overall probability is the sum of probabilities P νe→ν β (i → f ; t) over all three flavors β. The probability of observing the transition i → f , in which D decays to R and a neutrino is emitted but remains undetected, is thus given by P νe (i → f ; t) = β \| j U ej A ν j (i → f ; t)U * βj \| 2 .(8) Note that the probability P νe (i → f ; t) cannot be written as a square of one amplitude, simply because it involves different-flavor final states which require measurement schemes differing from each other and therefore adding up incoherently. Using the unitarity of the mixing matrix U, the summation over β in Eq. (8) gets rid of the interference terms, leading to the final expression: P νe (i → f ; t) = j \|U ej \| 2 \|A ν j (i → f ; t)\| 2 ≈ \|A ν (i → f ; t)\| 2 ,(9) where the dependence of the absolute-squared terms \|A ν j (i → f ; t)\| 2 on the species ν j was neglected, 2 enabling repeated use of unitarity. The final result, Eq. (9), is that the probability P νe (i → f ; t) for the two-body EC decay to occur is what standard weak interaction theory yields for a massless electron neutrino, regardless of its coupling to the mass-eigenstate neutrinos. This holds true also for the total EC decay rate which is obtained by time differentiation of P νe (i → f ; t) and which is found identical with the time-independent decay rate λ EC derived ignoring neutrino mixing. Thus, although the masseigenstate components of the emitted neutrino oscillate against each other, the total EC decay rate does not exhibit any oscillatory behavior owing to the unitarity of the matrix U, Eq. (5), which transforms incoherence in one basis into incoherence in the other basis. If U is nonunitary, the above argumentation breaks down, but this does not spoil the more straightforward argumentation that mass-eigenstate neutrinos, as distinct mass particles, have to be summed upon incoherently; one then goes directly from the amplitude A νe , Eq. (4), into the probability P νe , Eq. (9), which indeed is an incoherent sum over the neutrino mass-eigenstates ν j . Detection of a flavor neutrino, neutrino oscillations Oscillatory behavior of EC decay rates is possible when a neutrino of a given flavor is detected. The relatively small energy of order few MeV released in EC limits the detected neutrino to ν e . Here I show within a straightforward 'gedanken' extension of the GSI experiment, in which an electron-neutrino ν e is detected, that the corresponding angular frequency of the oscillations is given byhΩ νe , Eq. (3). To this end, the specific time-dependent first-order perturbation theory amplitudes A ν j (i → f ; t) introduced by Ivanov and Kienle [12] are followed as much as possible: A ν j (i → f ; t) = −i t 0 f ( q )ν j ( k j )\|H eν j (τ )\|i( 0 ) dτ,(10) with a weak-interaction Hamiltonian for the leptonic transition e − → ν j given by H eν j (τ ) = G F √ 2 V ud d 3 x[ψ n γ λ (1 − g A γ 5 )ψ p ][ψ ν j γ λ (1 − γ 5 )ψ e − ].(11) Here, x = (τ, x ), G F is the Fermi constant, V ud is the CKM matrix element, g A is the axial coupling constant, and with ψ n (x), ψ p (x), ψ ν j (x) and ψ e − (x) denoting neutron, proton, mass-eigenstate neutrino ν j and electron field operators, respectively. EC decays occur at any time τ within [0, t], from time t ′ = 0 of injection of D into the ESR to time t ′ = t of order seconds and longer at which the EC decay rate is evaluated. In the single-ion GSI experiment [5] the heavy ions revolve in the ESR with a period of order 10 −6 s and their motion is monitored nondestructively once per revolution. The decay is defined experimentally by the correlated disappearance of D and appearance of R, but the appearance in the frequency spectrum is delayed by times of order 1 s needed to cool R. The order of magnitude of the experimental time resolution is similar, about 0.5 s, as reflected in the time intervals used to exhibit the experimental decay rates R(t) in Figs. 3,4,5 of Ref. [5]. The decay rates determined in the ESR appear to agree with those measured elswhere, e.g. for 142 Pm [6], and this consistency suggests that details of kinematics and motion of the heavy ions in the storage ring affect little the overall decay rates which are evaluated here in conventional time-dependent perturbation theory. Therefore, it is plausible to assume that the evolution of the final state in these single-ion EC measurements at GSI proceeds over times of order 1 s which is used here as a working hypothesis. To obtain the time dependence of the amplitude A ν j (i → f ; t) (similarly structured to Eq. (6) of Ref. [12]), recall that the time dependence of the integrand in Eq. (10) is given by exp (i∆ j τ ) where 3 ∆ j ( q ) = E R (− q ) + E j ( q ) − M D(12) with E R = M 2 R + (− q ) 2 , E j = m 2 j + q 2(13) for the recoil ion and neutrino ν j energies, respectively, in the decay-ion rest frame. Integrating on this time dependence results in a standard timedependent perturbation-theory energy-time dependence [23] A ν j (i → f ; t) ∼ 1 − exp(i∆ j t) ∆ j .(14) Finally, the EC partial decay rate R νe→νe (i → f ; t) is obtained from the probability P νe→νe (i → f ; t), Eq. (7), by differentiating: R = ∂ t P. The term 'partial' applied to the rate R νe→νe owes to its limitation to the detection of one particular kind of flavor neutrinos: depleted electron neutrinos. Using Eq. (14) for the time dependence of A ν j (i → f ; t), one gets for the contribution of any j ′ = j non oscillatory term to R νe→νe : R ν j = d dt \|A ν j (i → f ; t)\| 2 ∼ 2 sin(∆ j t) ∆ j → 2πδ(∆ j ),(15) where the last step requires a sufficiently long time t. The properly normalized contribution of these terms to R νe→νe (i → f ; t) is given by j R ν j = λ EC j \|U ej \| 4 δ(∆ j ).(16) Similarly, the contribution of the j ′ = j oscillatory terms to the EC partial decay rate R νe→νe (i → f ; t), again for sufficiently long times, is given by λ EC j>j ′ \|U ej \| 2 \|U ej ′ \| 2 [δ(∆ j ) + δ(∆ j ′ )] cos[(∆ j − ∆ j ′ )t].(17) The δ symbols in Eqs. (16) and (17) differ from Dirac δ functions in that no further integration on the implied c.m. momentum q has to be done. Their meaning is straightforward for the non oscillatory terms, but more delicate for the oscillatory terms in which the sum of δ symbols imply that ∆ j − ∆ j ′ be evaluated for momentum once derived from the constraint ∆ j ( q ) = 0 and once derived from ∆ j ′ ( q ) = 0. On each occasion, using a generic notation k for the momentum implied by each one of the δ symbols, one obtains to an excellent approximation ∆ j (k) − ∆ j ′ (k) = E j (k) − E j ′ (k) =hΩ jj ′ ,(18) where Ω jj ′ is related to Ω νe of Eq. (3): hΩ jj ′ = m 2 j − m 2 j ′ 2E ν ≈hΩ νe .(19) The requirement of sufficiently long times for Eq. (17) to hold translates in the present case to requiring t ≫ Ω −1 νe ∼ 7 × 10 −5 s, which is comfortably satisfied given the experimental time resolution scale of ∼ 0.5 s [5]. The final expression for the depleted ν e rate is obtained by integrating over the δ symbols in Eqs. (16) and (17), resulting in R νe→νe (i → f ; t) = λ EC { j \|U ej \| 4 + 2 j>j ′ \|U ej \| 2 \|U ej ′ \| 2 cos(Ω jj ′ t)}.(20) Using the unitarity of U, Eq. (20) may be simplified to the following form: R νe→νe (i → f ; t) = λ EC {1 − 4 j>j ′ \|U ej \| 2 \|U ej ′ \| 2 sin 2 ( Ω jj ′ 2 t)}.(21) This expression is identical with the probability for ν e → ν e oscillation in neutrino spatial oscillation experiments (Eq. (13.9) in Ref. [22]) upon making the identification t = L/c, where L is the distance traversed by the neutrino between its source and the detector. A more rigorous wave-packet treatment is required to justify this transition from t to L [24]. A further simplification of Eq. (21) occurs when only two of the mass-eigenstate neutrinos are coupled to ν e : R νe→νe (i → f ; t) = λ EC {1 − sin 2 2θ sin 2 ( Ω νe 2 t)},(22) where θ is the ν 1 ↔ ν 2 mixing angle (cf. Eq. (13.20) in Ref. [22]). Note that it is the neutrino energy E ν to which the period of oscillations is proportional, not to the mass M D of the decaying heavy ion in the GSI experiments. 4 Magnetic field effects The preceding discussion ignored a possible role of the electromagnetic fields surrounding the ESR for guidance and stabilization of the heavy-ion motion. The nuclei 140 Pr and 142 Pm in the GSI experiment [5] have spin-parity I π i = 1 + , and the electron-nucleus hyperfine interaction in the decay ion forms a doublet of levels F π i = ( 1 2 + , 3 2 + ), the 'sterile' 3 2 + level lying about 1 eV above the 'active' 1 2 + g.s. from which EC occurs to a F f = 1 2 final state of a fully ionized recoil ion with spin-parity I π f = 0 + plus a left-handed neutrino of spin 1 2 . 5 The lifetime of the F π i = 3 2 + excited level is of order 10 −2 s, so that it deexcites sufficiently rapidly to the F π i = 1 2 + g.s. [2,4]. Periodic excitations of this 'sterile' state cannot explain the reported time dependence and intensity pattern [25]. The static magnetic field which is perpendicular to the ESR, B = 1.19 T for 140 Pr [26], gives rise to precession of the F π i = 1 2 + initial-state spin with angular frequency ω i of orderhω i ∼ µ B B ≈ 0.7×10 −4 eV [27], where µ B is the Bohr magneton. The corresponding time scale of order 10 −11 s is substantially shorter than even the ESR revolution period t revol ≈ 0.5×10 −6 s, so any oscillation arising from this initial-state precession would average out to zero over 1 cm of the approximately 100 m long circumference. A nonstatic magnetic field could lead through its high harmonics to oscillations with the desired frequency between the magnetic substates of the F π i = 1 2 + g.s. [28], but the modulation amplitude a EC expected for such harmonics is below a 1% level, and hence negligible. Furthermore, the associated mixing between the two hyperfine levels F π i = ( 1 2 + , 3 2 + ) is negligible. In conclusion, no initial-state coherence effects are expected from internal or external electromagnetic fields in the GSI experiment. In the final configuration, interferences may arise from the precession of the neutrino spin in the static magnetic field of the ESR. The corresponding angular frequency ω µν is given byhω µν = µ ν γB < 0.5 × 10 −14 eV in the decay ion rest frame, due to the neutrino anomalous magnetic moment µ ν interacting with the static magnetic field B. Here, γ = 1.43 is the Lorentz factor relating the rest frame to the laboratory frame, and µ ν < 0.54 × 10 −10 µ B from the Borexino solar neutrino data [29]. Below I show how the total EC rate gets time-modulated with angular frequency ω µν . To agree with the reported GSI measurements, ω µν = ω EC , a value of the electron-neutrino magnetic moment µ ν ∼ 0.9 × 10 −11 µ B is required which is six times smaller than provided by the published Borexino solar neutrino upper limit [29]. δ(∆ j ′ ) respectively, and replacing ∆ j − ∆ j ′ in the oscillatory terms of Eq. (17) by E j (k j ) − E j ′ (k j ′ ) ≈hω νe , Eq. (2). A similar error was made by Kleinert and Kienle when evaluating Eq. (54) in Ref. [10]. 5 The subscript f in this section relates to both the recoil ion and the neutrino. Interference due to a Dirac neutrino magnetic moment For definiteness I first assume that neutrinos are Dirac fermions with only diagonal magnetic moments µ jk = µ j δ jk , and that these diagonal moments are the same for all 3 species: µ j = µ ν . The emitted electron-neutrino is a left-handed lepton. The amplitude for producing it right-handed, namely with a positive helicity is negligible, of order m ν /E ν < 10 −7 and thus may be safely ignored. A static magnetic field perpendicular to the ESR flips the neutrino spin. Each of the mass-eigenstate components of the emitted neutrino will then precess, with amplitude cos(ω µν t) for the depleted left handed components and with amplitude i sin(ω µν t) for the spin-flip right handed components [30]. Both are legitimate neutrino final states which are summed upon incoherently. The summed probability is of course time independent: cos 2 (ω µν t)+sin 2 (ω µν t) = 1. However, the magnetic field dipoles of the storage ring do not cover its full circumference, except for about 35% of it [26]. This results in interference between the decay amplitude A 0 ν j for events with no magnetic interaction and the decay amplitude A m ν j for events undergoing magnetic interaction (superscript m) with depleted left handed components, i.e. with a superimposed amplitude of cos(ω µν t): A 0 ν j ∼ −i t 0 exp(i∆ j τ )dτ, A m ν j ∼ −i t 0 exp(i∆ j τ ) cos[ω µν (t − τ )]dτ,(23) using the same normalization as in Eq. (14) for any of the left-handed masseigenstate neutrinos. This expression for A m ν j is a crude approximation, but has the merit of representing physically the sequential time dependence anticipated for magnetic interactions. For completeness, I also list the amplitude A R ν j for events undergoing magnetic interaction which have resulted in a right-handed neutrino (superscript R), with a superimposed amplitude of i sin(ω µν t): A R ν j ∼ −i t 0 exp(i∆ j τ )i sin[ω µν (t − τ )]dτ.(24) Repeating the same steps in going from amplitudes A ν j , Eq. (14), to decay rates R ν j , Eq. (15), and adopting the same normalization, the decay rates associated with each one of these three amplitudes are given by: R 0 ν j ∼ 2πδ(∆ j ),(25)R m ν j ∼ π 2 [δ(∆ j + ω µν ) + δ(∆ j − ω µν )](1 + cos(2ω µν t)),(26)R R ν j ∼ π 2 [δ(∆ j + ω µν ) + δ(∆ j − ω µν )](1 − cos(2ω µν t)).(27) Note that although the two latter expressions, for rates associated with the magnetic interaction, are time dependent, their sum is time independent as expected from summing incoherently over the two separate helicities. The only time dependence in this schematic model arises from interference of the two amplitudes A 0 ν j and A m ν j for a left-handed neutrino. The sum of these partial rates, all of which correspond to ν j , and incorporating this interference, is given by R ν j ∼ \|a 0 \| 2 2πδ(∆ j ) + \|a m \| 2 π[δ(∆ j + ω µν ) + δ(∆ j − ω µν )] + 2Re(a 0 a * m ) π 2 [δ(∆ j + ω µν ) + 2δ(∆ j ) + δ(∆ j − ω µν )] cos(ω µν t) −2Im(a 0 a * m ) π 2 [δ(∆ j + ω µν ) − δ(∆ j − ω µν )] sin(ω µν t),(28) where \|a m \| 2 ∼ 0.35 and \|a 0 \| 2 ∼ 0.65, with unknown relative phase between the probability amplitudes a m and a 0 for undergoing or not undergoing magnetic interaction, respectively. Working out the complete normalization of this expression, the final rate expression is given by R νe = λ EC [1 + 2Re(a 0 a * m ) cos(ω µν t)],(29) showing explicitly a time modulation of the kind Eq. (1) reported by the GSI experiment [5]. It is beyond the present schematic model to explain the magnitude of the modulation amplitude a EC and the phase shift φ EC , except that \|a EC \| < 1. In particular, a more realistic calculation is required in order to study effects of departures from the idealized kinematics implicitly considered above by which both the recoil ion and the neutrino go forward with respect to the decay-ion instantaneous laboratory forward direction. Whereas this is an excellent approximation for the recoil-ion motion, it is less so for the neutrino. 6 Nevertheless, for a rest-frame isotropic distribution, it is estimated that neutrino forward angles in the laboratory dominate over backward angles by more than a factor five. For distinct diagonal Dirac-neutrino magnetic moments, Eq. (29) gets generalized to R νe = λ EC [1 + 2Re(a 0 a * m ) j \|U ej \| 2 cos(ω µ j t)],(30) resulting in a more involved pattern of modulation. 6 I owe this observation to Eli Friedman. For vanishing diagonal magnetic moments, and nonzero values of transition magnetic moments, the discussion proceeds identically to that for Majorana neutrinos in the next subsection. Majorana neutrino magnetic moments Majorana neutrinos can have no diagonal electromagnetic moments, but are allowed to have nonzero transition moments connecting different mass-eigenstate neutrinos, or different flavor neutrinos. A static magnetic field perpendicular to the storage ring will induce spin-flavor precession [31]. However, the magnetic interaction effect is masked in this case by neutrino mass differences, such that the amplitudes cos(ω µν t ′ ) and sin(ω µν t ′ ) in Eqs. (23) and (24) are replaced, to leading order in ω µν /Ω νe << 1, by cos(ω µν t ′ ) → exp(−iΩ jj ′ t ′ ), sin(ω µν t ′ ) → ω µ jj ′ Ω jj ′ sin(Ω jj ′ t ′ ),(31) wherehω µ jj ′ = µ jj ′ γB, and Ω jj ′ is defined by Eq. (18). The period of any oscillation that might be induced by these amplitudes is of order Ω −1 νe ∼ 7 × 10 −5 s which is several orders of magnitude shorter than the time resolution scale of ∼ 0.5 s in the GSI experiment [5]. Therefore, such oscillations will completely average out to zero over realistic detection periods. Discussion and summary In conclusion, it was shown that interference terms between different propagating mass-eigenstate neutrino amplitudes in two-body EC reactions on nuclei cancel out to zero when no neutrinos are detected. Coherence between propagating mass-eigenstate neutrinos is evident within each one of the amplitudes for detecting a given flavor neutrino. It is only when all final flavor rates are summed upon incoherently, as motivated by the different flavor measurements required, that cancelations occur and the overall rate becomes independent of time and is reliably calculable from standard weak interaction theory for massless neutrinos. The underlying logic here is that summing on all possible phase space for flavor neutrinos is equivalent within quantum mechanics to not detecting any specific neutrino. Interference terms from different propagating mass-eigenstate neutrinos would survive and give rise to oscillatory behavior of the EC decay rate, if and only if neutrinos are detected. It was shown that the period of oscillations in such a case is T ∼ 4πE ν /∆(m 2 ν ) which for E ν ≈ 4 MeV as in the GSI experiments [5], and for ∆(m 2 ν ) ≈ 0.76 × 10 −4 eV 2 [13], assumes the value T ∼ 4.4×10 −4 s, shorter by over four orders of magnitude than the period reported in these experiments. The oscillation period cited here is in full agreement with the oscillation length tested in dedicated neutrino oscillation experiments, provided the time t is identified with L/c where L is the distance traversed by the neutrino. In particular, besides the ∆(m 2 ν ) neutrino input, it depends on the neutrino energy E ν , not on the mass M D of the decay ion. On the positive side, I have proposed a possible explanation of the 'GSI Oscillations' puzzle connected with the magnetic field that guides the heavy-ion motion in the ESR, requiring a Dirac neutrino magnetic moment µ ν about six times smaller than the laboratory upper limit value from the Borexino Collaboration [29]. The underlying mechanism is the interference between decay amplitudes not affected by the static magnetic field of the ESR and decay amplitudes affected by this field which induces spin precession of the emitted neutrino. The motion of the recoil ion in the ESR is constrained by the interference long after the neutrino has fled away. This mechanism does not work for Majorana neutrinos that can have no diagonal magnetic moments. For nonzero values of transition magnetic moments, the resulting spin-flavor precession is suppressed by neutrino mass differences, and it becomes impossible to relate then the GSI Oscillations puzzle to magnetic effects. It is not yet resolved experimentally whether neutrinos are Dirac or Majorana fermions, although the theoretical bias rests with Majorana fermions, in which case the present paper accomplished nothing towards providing a credible explanation of this puzzle. For experimental verification, note that the time-modulation period T lab EC is inversely proportional to B, so the effect proposed here may be checked by varying B, for example by varying β = v/c for the coasting decay ions. For a fixed value of β, B depends on the charge-to-mass ratio of the decay ion which varies only to a few percent with the decay-ion mass M D . Finally, the proposed effect is unique to two-body EC reactions, since three-body weak decays do not constrain the neutrino direction of motion with respect to the fixed direction of B. Indeed, preliminary data on the three-body β + decay of 142 Pm indicate no time modulation of the β + decay rate, limiting its modulation amplitude to a β + < 0.03(3) [32]. versität München, where Sect. 3 was conceived, is gratefully acknowledged. Eq.(2) was also obtained byLipkin [14] assuming interference between two unspecified components of the initial state with different momenta and energies that can both decay into the same final state, an electron neutrino and a recoil ion with definite energy and momentum. This scenario was criticized by Peshkin[15]. This neglect does not hold for interference terms A ν j A * ν j ′ , j = j ′ , which give rise to oscillatory behavior, as discussed in Sect. 2. From here onh = c = 1 units are almost exclusively used. Ivanov and Kienle[12] overlooked this distinction by using in Eq. 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[ "IDENTIFYING BID LEAKAGE IN PROCUREMENT AUCTIONS: MACHINE LEARNING APPROACH", "IDENTIFYING BID LEAKAGE IN PROCUREMENT AUCTIONS: MACHINE LEARNING APPROACH" ]	[ "Dmitry I Ivanov ", "Alexander S Nesterov " ]	[]	[]	We propose a novel machine-learning-based approach to detect bid leakage in first-price sealed-bid auctions. We extract and analyze the data on more than 1.4 million Russian procurement auctions between 2014 and 2018. As bid leakage in each particular auction is tacit, the direct classification is impossible. Instead, we reduce the problem of bid leakage detection to Positive-Unlabeled Classification. The key idea is to regard the losing participants as fair and the winners as possibly corrupted. This allows us to estimate the prior probability of bid leakage in the sample, as well as the posterior probability of bid leakage for each specific auction. We find that at least 16% of auctions are exposed to bid leakage. Bid leakage is more likely in auctions with a higher reserve price, lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline.	10.1145/3328526.3329642	[ "https://arxiv.org/pdf/1903.00261v1.pdf" ]	85,529,625	1903.00261	c95c3a8081e9c58926c577d9d8b626f0823f1ff5	IDENTIFYING BID LEAKAGE IN PROCUREMENT AUCTIONS: MACHINE LEARNING APPROACH Dmitry I Ivanov Alexander S Nesterov IDENTIFYING BID LEAKAGE IN PROCUREMENT AUCTIONS: MACHINE LEARNING APPROACH corruptionbid leakageprocurement auctionspositive-unlabeled learning JEL Classification: C38C57D82H57 We propose a novel machine-learning-based approach to detect bid leakage in first-price sealed-bid auctions. We extract and analyze the data on more than 1.4 million Russian procurement auctions between 2014 and 2018. As bid leakage in each particular auction is tacit, the direct classification is impossible. Instead, we reduce the problem of bid leakage detection to Positive-Unlabeled Classification. The key idea is to regard the losing participants as fair and the winners as possibly corrupted. This allows us to estimate the prior probability of bid leakage in the sample, as well as the posterior probability of bid leakage for each specific auction. We find that at least 16% of auctions are exposed to bid leakage. Bid leakage is more likely in auctions with a higher reserve price, lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline. Introduction In each country public procurement is an important and complex sector of the economy. In 2017 in Russia, the annual total volume of public procurement was 36.5 trillion rubles, which amounts to around a third of the annual GDP. A majority of contracts in Russian procurement are awarded through auctions, which in theory allocates the contract to the most efficient firm at the lowest possible price. In practice, however, certain tacit manipulations can corrupt the outcome both in terms of efficiency of allocation and the contract price. In this paper we study "requests for quotations" -small and frequent online first price sealed-bid procurement auctions. These auctions can suffer from bid leakage -the corruption scheme where procurer illegally provides his favored participant with the information about the bids of the other participants. Our goal is to estimate how widespread bid leakage is in general and to determine how likely it is that each particular auction has been affected by bid leakage. We analyze the dataset containing more than 1.4 mln Russian requests for quotations. The dataset covers all the auctions that took place from January 2014 to March 2018 and is extracted from the online database. 1 Figure 1. Example of typical request for quotations with leaked bids Notes: The auction lasts more than 7 days. The auction is suspicious for bid leakage as the winner bids near the deadline, after every other bid, and only slightly below the runner-up. Our work is inspired by Andreyanov et al. (2016) who observed the patterns that are likely to reflect rational behavior of the favoured participant that received leaked bids (see Figure 1): these participants are (1) bidding last, (2) bidding close to the deadline, and (3) winning by a small margin. The intuition behind these three patterns in auctions with bid leakage is straightforward. First, the only way for the unfair participant to know every other bid and ensure her win is to bid the last, hence pattern (1). Similarly, she delays bidding as much as possible to lower the risks of not being the last, hence pattern (2). As she aims for the highest profit, she slightly undercuts the current best bid, hence pattern (3). We use these three patterns to determine whether a particular auction has been corrupted by bid leakage. To do that we develop a two-stage identification strategy. In the first stage we build a classifier that distinguishes the winners from the runner-ups by using features associated with patterns (1), (2) and (3). For a given auction winner, the higher is the predicted probability of winning, the more suspicious the auction is. In a world without bid leakage, and assuming that these features are not related to the actual bid, such classifier would fail and if it does not then it has to be due to bid leakage. In practice however, the classifier might still be able to predict the winners well even in auctions without bid leakage, which leads to biased estimates. To correct our estimates we construct a synthetic placebo dataset of fair auctions and estimate the sign and the size of the bias. In the second stage we use the classifier's predictions and performance to estimate the prior probability that a random auction in the dataset is corrupted, and the posterior probability that a specific auction is corrupted -conditional on the probability of winning that the classifier has assigned to its winner. We estimate the prior probability of bid leakage as 16%. We also find that the bid leakage is more likely in auctions with a higher reserve price lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline. The rest of the paper is organized as follows. In Section 2 we present the background on requests for quotations and the relevant literature and sketch our identification strategy. In Section 3 we describe the dataset. In Section 4 we present the two stages of our bid leakage estimation: the classifier and the estimates for prior and posterior probability of bid leakage. In Section 5 we present the results of estimation, provide few robustness checks and economic implications. Section 6 concludes. 2. Problem setup and identification strategy 2.1. Requests for quotations and background on bid leakage detection. In Russia requests for quotations are used for distributing small contracts such as roof repair for a factory or products delivery to a school kitchen. Before each auction starts, the procurer makes an announcement with the relevant information about the contract and the auction. The announcement includes reserve price -the maximal price the contract can be assigned for, the reserve price is bounded by 500000 rubles (approximately $8000). The auction lasts at least one week. During this period potential participants can submit their bids. Each participant can submit only one bid, the bids are sealed. After the auction ends, all bids are revealed and the smallest bid wins, the final price equals the winning bid (first-price auction). Throughout the paper we only study successful bid leakage, that is when the honest winning bid has been leaked to and undercut by a favored bidder. The literature on manipulations in auctions is prolific but mostly concerned with collusion schemes such as bid rigging Porter and Zona (1993) and bid rotation Aoyagi (2003); a rather recent review of the literature on collusion detection is available in Harrington (2005). Crucial to our research question is the timing of bids, and the Russian procurement data is unique to contain this information. Previously, timing of bids has been studied in repeated Internet auctions such as eBay, where each bidder has a set of moments in time where he can submit a bid Song (2004). But, to the best of our knowledge, timing of bids has not been used before to detect corruption (except for Andreyanov et al. (2016) that we discuss below). Other papers studying bid leakage or other forms of corruption using Russian data do so on a local scale of specific market or during specific period of time Yakovlev et al. (2016); Mironov and Zhuravskaya (2016); Balsevich and Podkolzina (2014). The empirical literature on auctions is rich (see, e.g., the seminal works Athey and Haile (2007) and Krasnokutskaya and Seim (2011)), yet there are only few studies that use supervised machine learning (classification) in auction corruption detection. Typically they utilize small datasets of few hundreds of labeled auctions Ferwerda et al. (2017); Huber et al. (2018). 2.2. Identification strategy and placebo. The only closely related papers to ours are Andreyanov et al. (2016) and its recent version Korovkin et al. (2018): we study the same object using the same data. However, our identification and estimation strategy is different in three crucial ways: we use weaker assumptions, more advanced methods and larger set of characteristics (i.e., features), which additionally enables us to determine posterior probability of bid leakage for each specific auction. The identification in Andreyanov et al. (2016); Korovkin et al. (2018) relies on a crucial assumption that, in the auctions without bid leakage, bids and timing of the bids are independent. If independence holds, then the higher likelihood of the last bids to win compared to earlier bids is attributed exclusively to bid leakage. The independence assumption might fail for a number of reasons. The longer time it takes a risk-averse bidder to study the case, the lower will his bid be. For example, in our data we observe that 1,2% of participants behave like "snipers": they bid during the first day of the auction and bid slightly below the reserve price (up to 5%). Another reason comes from the honest bidders' attempts to resist bid leakage. On the one hand, the later the bid is submitted, the lower is the chance that it is going to be leaked and undercut by a corrupted bidder. On the other hand, submitting closer to the deadline requires attention and possibly costly logistics. 2 As a result, a bidder with a higher valuation (i.e., lower execution costs) submits a lower bid and, simultaneously, is ready to delay the submission more relative to a bidders with a lower valuation. Because bidding and timing are confounded through valuation, the independence assumption does not hold. A later bid has a higher chance of winning not only due to presence of bid leakage, but due to mere expectation of bid leakage. We present this argument formally in the Subsection 2.3. We do not rely on independence assumption. Instead, we correct our estimates using a synthetic placebo dataset of fair auctions. We remove the first-ranked bidders (the true winners) from all the auctions and recalculate the features accordingly. This way, we obtain a new dataset where the second places are treated as the winners and the third placesas the runner-ups ( Figure 2). We estimate the bias in these synthetic auctions and assume 2 Anecdotal evidence suggests that some bidders did not rely on post or courier services and delivered their bids personally to make sure to submit just before the deadline. that this bias is equal to the bias in fair auctions in the original dataset. We verify this assumption and compare it to independence in Section 4.3. Figure 2. Placebo auction example, transformed from the auction at Figure 1 Notes: The placebo auction is generated by removing the winning bid from the auction. In this hypothetical auction, bid 1 belongs to the winner, bid 2 -to the runner-up. The auction is no longer suspicious for bid leakage. Second, in contrast to reduced-form and structural statistical models we use machine learning techniques. This allows us to consider all the evidence on bid leakage patterns at once and without imposing restrictions on their interconnection. When we only include subset of the relevant features as it is done in Korovkin et al. (2018), we get significantly less accurate predictions. Finally, we descend from the population level and develop a method able to assign the posterior probability of the bid leakage presence to every auction in the dataset. Our method provides more precise and specific estimates of bid leakage and can be used for automatic ex-post bid leakage detection, which can be useful for regulation and auditing authorities. 2.3. Game-theoretic model of bid leakage. In this Section we formalize the intuition behind our identification strategy using a simple game-theoretic model. First consider the world without bid leakage. An auctioneer is selling a procurement contract with reserve price normalized to 1, and the lowest possible cost of executing the contract is normalized to zero. Each bidder is risk-neutral and is drawing his execution costs e from a uniform distribution on [0, 1], or, equivalently, each bidder has an iid valuation v = 1 − e drawn from a uniform distribution on [0, 1]; the cumulative density function is F (v) = v. Let the expected number of bidders participating in the auction be n. 3 For each bidder i with a valuation v his equilibrium bid is the expected bid of his runner-up conditional on i being the winner, b * (v) = 1 − v 0 xdF n−1 (x) v 0 dF n−1 (x) = 1 − v n − 1 n . Since the bid is monotonic in the valuation, in order to win the bidder needs to have the highest valuation. The probability of winning is thus F n−1 = v n−1 , and the expected profit is Eπ(v, b * (v)) = v n n . Now we add the time dimension to the problem. Each bidder chooses the submission time t ∈ [0, 1]. We assume that delaying submission is costly, submitting at time t costs the bidder c(t), where c is increasing and convex, c > 0, c > 0, and extremely high close to deadline, c(1) = ∞. These costs represent the stress and attention costs of not missing the deadline and also the costs of more precise bid delivery. In the world without bid leakage the timing of the bid is irrelevant for winning and each bidder submits at time t = 0. Now let bid leakage be possible. We assume that each bidder has the same prior belief regarding the possibility of bid leakage. Conditional on that the auction is corrupted, the probability that a specific bid is leaked and undercut decreases in time of submission: if you submit later, then the chances of leakage of your bid are lower. We assume that for each bidder the perceived probability that his bid submitted at time t is leaked and undercut is exogenously given by some function β(t) decreasing in time, β (t) < 0, down to zero at the time of deadline t = 1, β(1) = 0. Thus the expected profit of the bidder with valuation v and bid b * (v) is as follows: Eπ(v, b * (v), t) = v n n (1 − β(t)) − c(t). The optimal submission time t * (v) is given by the first order condition: c (t * (v)) = − v n n β (t * (v)). Observe that both b * (v) is decreasing and t * (v) is increasing in valuation v. Thus the optimal bid and the submission time of the bid are confounded by the valuation. The higher the valuation is, the more is the bidder ready to pay to get the contract: both in terms of submitting a lower bid and in terms of costly delay of the submission. Observe also that the correlation between the timing and the bid holds for each valuation, and thus will be true not only for the winners but also for the runner-ups. We uncover this correlation for runner-ups using a placebo dataset in Section 4.2 and use it to correct our biased estimates for the winners. 2.4. Positive-Unlabeled Classification. Our bid leakage estimation strategy is based on Positive-Unlabeled (PU) Classification. Generally, PU Classification is applied instead of Supervised Classification in the cases when the training data set is not fully labeled. Specifically, only a subsample of the Positive data needs to be labeled as such, while the remaining part of the Positive data and all the Negative data are mixed in the Unlabeled sample. This setting may be applied to our case if we regard runner-ups as fair (Positive) and the winners as possibly corrupted (Unlabeled). Numerous methods are proposed to solve PU Classification Elkan and Noto (2008) Auction Data We extracted data 4 on 1444718 requests for quotations that took place between January 2014 and March 2018. The data was preprocessed in the following way: • The auctions with missing data or with obvious coding mistakes were dropped. These obvious mistakes include: reserve price being negative or higher than the upper bound 500000 rubles; the starting date being after the ending date; the starting date, the ending date or the bidding date being in the future; the bid being negative or higher than the reserve price. 86% of the initial size is left. • Our identification methods cannot be applied to auctions with 1 participant, so these auctions are dropped. 44% of initial size is left. The data on 636866 auctions remains after the preprocessing. The main characteristics of this data set are shown in Table 1. Notes: The data set excludes auctions with 1 participant and auctions with missing data. We define Price fall as r−b1 r , where r is reserve price, b 1 is winner's bid. Bid Leakage Estimation Here we describe our two-stage bid leakage estimation strategy. Mainly, we reduce the problem to Positive-Unlabeled Classification by considering the runner-ups as fair (Positive) participants and the winners as a mixture of fair (Positive) and corrupted (Negative) participants. We follow the state of the art DEDPUL procedure proposed in Ivanov (2019) for general purposes. At the first stage of the procedure a supervised binary classifier is trained to distinguish the winners from the runner-ups. Using cross-validation technique, predictions of this classifier are obtained for all the winners in the data set. At the second stage these predictions are transformed into bid leakage probabilities by estimating the ratio between the densities of the predictions for the runner-ups and for the winners. There is a crucial distinction of our strategy compared to the original DEDPUL procedure. The original procedure would assume that bid leakage is the only reason why the winners and the runner-ups differ for the classifier. However, as we have already discussed in Sections 2.2 and 2.3, this might not be the case, and the difference may exist even in the fair auctions. To account for this we introduce into the analysis the synthetic placebo auctions defined in Section 2.2, which are assumed to be fair, and modify the procedure correspondingly. These modifications are discussed in details in Section 4.2. 4.1. First Stage: Winner vs Runner-up Classifier. In the first stage we train the classifier to distinguish the bids of the winners from the bids of the runner-ups. The features are presented in Table 2. These features are specifically designed to reflect possible bid leakage patterns, while uncovering only little information about fair auctions. Specifically, the features bid last? and bid timing reflect intention of a corrupted participant to gather information about all the other bids. Small values of relative bid reflect undercutting. The feature met before? reflects the possibility of repeated procurer-participant cooperation. Small values of relative bid timing might reflect fairness of participant, as bids are unlikely to be leaked instantly. Notes: relative bid is truncated at 0.1: values bigger than 0.1 are set to this threshold. Likewise, bid timing and relative bid timing are truncated at 1440 minutes (1 day). Auctions with 1 participant are excluded from the analysis. Observe that the information on whether some two participants are from the same auction is lost on purpose. The classifier does not choose the winner between the two participants in each auction. Instead, it determines the chances that each set of features in the data set belongs to a winner as opposed to a runner-up. As a classifier we use gradient boosting of decision trees -namely, xgboost. 5 We train an ensemble of 60 trees on the features described above with depth of each tree limited to 5 levels. With this classifier we obtain predicted probability of winning for each winner in the data set using cross-validation. At the second stage we establish the connection between these predictions and the probability of bid leakage. Second Stage: Transforming Classifier's Predictions into Bid Leakage Probability. We show how to use the discussed classifier to estimate both the prior and the posterior probabilities of bid leakage, neither of which are assumed to be known in advance for any of the auctions. First we introduce notations and formally define the problem. At the first stage the classifier estimates the probability that the participant wins based on corresponding vector of features x. Denote this probability of winning as y(x). Denote distributions of y(x) for winners and runner-ups as f w (y) and f w (y) respectively. As was previously discussed, we consider the runner-ups to be fair participants (as we aim to detect only successful bid leakage), while the winners may contain both fair and corrupted participants. Moreover, the distributions of y(x) for the winners and the runner-ups of the fair auctions might also differ. This may generally be expressed in the following mixture model: (1) f w (y) = f corr 2 (y) (2) f w (y) = αf corr (y) + (1 − α)(f corr 2 (y) + ∆ 12 ) (3) ∆ 12 = f corr 1 (y) − f corr 2 (y) where α denotes the prior probability of bid leakage; f corr (y), f corr 1 (y), and f corr 2 (y) denote the distributions of y(x) for the corrupted winners, the fair winners, and the fair runner-ups respectively; ∆ 12 denotes the difference between the distributions of the winners and the runner-ups in fair auctions. Introduction of ∆ 12 into (2) is exactly what distinguishes our case from the standard PU Classification problem setup. However, we will address the issue of estimating ∆ 12 later. For now, consider it exogenous. Our goal is to estimate the prior probability that a random winner is corrupted α and the posterior probability that a specific winner is corrupted f (corr \| y). The latter may be expressed using the Bayes rule: (4) f (corr \| y) = αf corr (y) f w (y) = 1 − (1 − α)(f w (y) + ∆ 12 ) f w (y) Following DEDPUL (Ivanov, 2019), the densities f w (y) and f w (y) may be estimated by applying Kernel Density Estimation to the classifier's predictions for the winners and the runner-ups respectively. Then, both priors α and posteriors f (corr \| y) may simultaneously be estimated by applying Expectation-Maximization algorithm to (4), thus reaching our goal. Now we address the issue of ∆ 12 estimation. The key step is to construct the synthetic data set of implicitly fair placebo auctions. As was previously discussed, placebo auctions are generated from the real auctions by dropping the winners and keeping the other participants which we know to be fair. In each hypothetical auction, the second-ranked bidder is assumed to be the winner, and the third-ranked bidder is assumed to be the runner-up. By applying the classifier that is trained on the real auctions to the placebo data set, we may obtain its predictions to later estimate the densities f wp (x) and f wp (x) for the winners and the runner-ups of placebo auctions respectively, where: (5) ∆ 23 = f wp (y) − f wp (y) Thus, we may estimate ∆ 23 by using placebo auctions. This becomes crucial as we make a major assumption regarding equality of ∆ 12 and ∆ 23 : PARITY: ∆ 12 = ∆ 23 . The difference between the winners and the runner-ups in the real fair auctions is equal to this difference in the placebo auctions. Using the parity assumption we may estimate ∆ 12 and thus the priors α and the posteriors f (corr \| y) of bid leakage. In the next subsection we verify applicability of the parity assumption. 4.3. Verifying parity and independence assumptions. First observe that independence implies parity. Namely, if timing is independent from bidding, then both differences ∆ 12 and ∆ 23 (when estimated using only time-related features) are equal to zero ∆ 12 = 0 = ∆ 23 implying parity. To test the independence assumption Korovkin et al. (2018) exclude all winning last bids and then test if being last predicts a lower bid. They find the opposite effect (in this subsample earlier bids are lower) of a much lower size. We replicate this observation with our dataset. However, if we exclude all winning bids (and not only those bids that were submitted last as in Korovkin et al. (2018)), we find that later bids are more likely to be smaller. That is, in the imaginary auction with only 2nd and 3rd lowest bids, the lower 2nd bid is on average submitted later. This is also true when we exclude winners and 2nd lowest bids, and compare 3rd and 4th latest bids, the lower 3rd bid is on average submitted later. Both independence and parity assumptions may be tested by training the classifier on the real auctions and applying it to the placebo auctions. In the case if independence holds, we expect the classifier's performance on the placebo data set to be on a level of fair coin (0.5 accuracy and ROC-AUC), which is clearly not the case (Column 2 in Table 3). In the case if parity holds, we expect the classifier to show the same performance on the placebo data sets regardless of how many participants are dropped. Observe that this is the case for two placebo data sets: when only the winners are dropped (Column 2 in Table 3) and when both winners and runner-ups are dropped (Column 3 in Table 3). At the same time, classifier's performance on the real data set (Column 1 in Table 3) is considerably higher than on the placebo data sets, which is expected in the presence of bid leakage. Thus the evidence suggests that the parity assumption holds, meaning that our bid leakage estimation strategy is applicable, while the stronger assumption of independence does not. Both results hold when the bid-related features are excluded. Empirical results The overall prior probability of bid leakage in our sample is estimated as 16%, which is slightly higher than 10-11% found previously in Korovkin et al. (2018). One of the reasons for this discrepancy is that our sample also includes auctions with only two participants, and bid leakage is significantly more likely there. The other reasons are due differences in identification and estimation. Table 3. Measures of classifier's performance on different data sets: on real auctions (first column), on placebo auctions with dropped winners (second column), on placebo auctions with dropped winners and runner-ups (third column) ∆ 12 , f corr ∆ 23 ∆ 34 Accuracy 0.5819 +-0.0005 0.5375 +-0.0008 0.5387 +-0.0012 ROC-AUC 0.6245 +-0.0004 0.5581 +-0.0008 0.5534 +-0.0011 Notes: mean and standard deviation statistics of scores are calculated on outputs of 3-fold cross-validation, repeated several times. Since classifier's purpose is to learn the patterns that reflect bid leakage, its suitability for our estimation strategy should be evaluated with the difference of the scores on the real (first column) and the placebo (second and third columns) data sets, rather than solely the score on the real data set. The probability α for specific subsamples provides insights into the mechanisms behind bid leakage. We present these results in Fig.3. The top left diagram in Fig.3 demonstrates that as the reserve price increases, the bid leakage is more likely to be observed. This is very natural and might have two explanations. First, a higher reserve price for a contract gives higher incentives to organize risky corruption schemes. Second, the better organized corruption schemes allow setting a higher reserve price in order to maximize the surplus. The top center diagram in Fig.3 demonstrates that the lower number of bidders on average corresponds to higher probability of bid leakage. The number of participants in an auction is endogenous: the entry is always costly and the expected benefit depends on the reserve price, description of the contract and other details posted in the announcement. If some of these details (such as required certification) deter entry or signal the experienced participants that bid leakage is likely, which results in fewer bidders entering the auction. If there is no such deterrence, then the auction is desirable and entry is high. And a desirable auction is more likely to be the one where bid leakage occurs. As the number of bidders grows above four, the relation to bid leakage becomes ambiguous. On the one hand, higher participation might mean that the auction was more competitive, which corresponds to low α. On the other hand, as demonstrated earlier bid leakage is correlated with reserve price, and an overly high reserve price might attract more bidders. The top right diagram in Fig.3 demonstrates that there is no particular pattern in seasonality of bid leakage. The only exception is December, where the procurement agencies might be distributing their accumulated reserves. The bottom left diagram in Fig.3 shows a stark relation of bid leakage and the difference between the final price and the reserve price. When surplus is high, price fall can be a measure of bidders' competition. In general, the larger is the price fall, the lower is the probability of bid leakage. However, if the price fall is below 5% then the predicted probability of bid leakage increases with the price fall. This can be due to that the reserve price has been already set very close to bidders' valuations, providing little opportunity for bid leakage. The bottom center diagram in Fig.3 demonstrates that commission size has very low influence on the probability of bid leakage. There is however a marginally significant difference between commissions with 3 members and commission with 7 members, which naturally suggests that larger commissions correspond to lower bid leakage. The bottom right diagram in Fig.3 shows that the (estimated) favored participants submit their bids at constant rates throughout the last 24 hours of the auction, except for the last hour, when the submission rate is 2-3 times higher. Finally, Fig.4 presents the cross-regional average probabilities of bid leakage. Perhaps surprisingly, the variance between different regions is rather large, up to 90%. However, these results are consistent with other measures of corruption available for Russian regions, such as electoral fraud (see, e.g., Mebane and Kalinin (2009),Bader and van Ham (2015)). Conclusions We study first-price sealed-bid auctions and identify auctions corrupted with bid leakage. The first stage of our strategy is to build a classifier that very well distinguishes the winners from the runner-ups in the corrupted auctions but not as well in the fair auctions. In the second stage we process the classifier's predicted probabilities of winning for the winner and the runner-up of each auction into the probability that the auction is corrupted. We apply our estimation strategy to the Russian procurement data between January 2014 and March 2018 containing 636866 auctions. We estimate the share of the corrupted auctions in the dataset as 16%. We believe that this estimate is conservative due to the assumptions we make. First, we are only concerned with effective bid leakage, that is if the bids are leaked to the favored participant, she inevitably wins. Consequently, when the classifier selects a runner-up, we assume it to be a mistake. Second, we assume that the classifier selects all the winners of the corrupted auctions. Consequently, when the classifier doesn't select the winner of an auction, we assume this auction to be fair. Several problems remain open. First, our current strategy can only be applied to the auctions with 2 and more participants. Second, and arguably the most important problem is that the validation of our strategy in its classical way of comparison with labeled data is not available. Yet the confidence in our strategy is reinforced by the economic interpretability of the results: the bid leakage is more likely in auctions with a higher reserve price, lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline. 1 ftp://ftp.zakupki.gov.ru/ ; Kiryo et al. (2017); Ivanov (2019). They are applied to various real-world problems, which include detection of fake texts Ren et al. (2014), time-series classification Nguyen et al. (2011), bioinformatics Yang et al. (2012), etc. Figure 3 . 3Bid leakage probability aggregated by auction characteristics Figure 4 . 4Estimated prevalence of bid leakage in regions of Russia Table 1 . 1Data set characteristicsCharacteristics Mean Median Std. Dev. Number of participants 3 2 1.9 Reserve price, rubles 182000 134000 150000 Winner's bid, rubles 142000 97000 128000 Runner-up's bid, rubles 154000 108000 134000 Price fall 0.23 0.18 0.21 Time from bid to deadline, hours 40 20 54 Time from winner's bid to deadline, hours 39 19 54 Time from runner-up's bid to deadline, hours 39 20 53 Duration, hours 195 169 71 Table 2 . 2Features descriptionName Type Range Description bid last? Binary {0,1} Did participant bid after other participants? met before? 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[ "A Born-Infeld Scalar and a Dynamical Violation of the Scale Invariance from the Modified Measure Action", "A Born-Infeld Scalar and a Dynamical Violation of the Scale Invariance from the Modified Measure Action" ]	[ "T O Vulfs [email protected] \nDepartment of Physics\nBen-Gurion University of the Negev\nBeer-ShevaIsrael\n", "E I Guendelman \nDepartment of Physics\nBen-Gurion University of the Negev\nBeer-ShevaIsrael\n\nFrankfurt Institute for Advanced Studies\nGiersch Science Center\nCampus RiedbergFrankfurt am MainGermany\n" ]	[ "Department of Physics\nBen-Gurion University of the Negev\nBeer-ShevaIsrael", "Department of Physics\nBen-Gurion University of the Negev\nBeer-ShevaIsrael", "Frankfurt Institute for Advanced Studies\nGiersch Science Center\nCampus RiedbergFrankfurt am MainGermany" ]	[]	Starting with a simple two scalar field system coupled to a modified measure that is independent of the metric, we, first, find a Born-Infeld dynamics sector of the theory for a scalar field and second, show that the initial scale invariance of the action is dynamically broken and leads to a scale charge nonconservation, although there is still a conserved dilatation current.	10.1142/s0217732320501989	[ "https://arxiv.org/pdf/1903.01792v1.pdf" ]	119,200,147	1903.01792	1a3c627fd982312af5d6c3c86aae33d0f6d21794	A Born-Infeld Scalar and a Dynamical Violation of the Scale Invariance from the Modified Measure Action 5 Mar 2019 T O Vulfs [email protected] Department of Physics Ben-Gurion University of the Negev Beer-ShevaIsrael E I Guendelman Department of Physics Ben-Gurion University of the Negev Beer-ShevaIsrael Frankfurt Institute for Advanced Studies Giersch Science Center Campus RiedbergFrankfurt am MainGermany A Born-Infeld Scalar and a Dynamical Violation of the Scale Invariance from the Modified Measure Action 5 Mar 20193 Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas Starting with a simple two scalar field system coupled to a modified measure that is independent of the metric, we, first, find a Born-Infeld dynamics sector of the theory for a scalar field and second, show that the initial scale invariance of the action is dynamically broken and leads to a scale charge nonconservation, although there is still a conserved dilatation current. Introduction Every physical quantity is defined by its transformation properties. A scalar field, φ, being a single real function of spacetime behaves as a scalar under Lorentz transformations: φ(x) → φ ′ (x ′ ). Its dynamics is determined by the kinetic and potential energy densities. In our paper we show, first, how this simplest scalar field can be transformed to a Born-Infeld scalar and second, how the dynamically broken scale invariance leads to the nonconservation of a scale charge. It all becomes possible when we use a modified measure in the action instead of a standard one. The notion of a measure is usually associated with the theories of gravity. There √ −g, where g is the determinant of the metric, is included to the action, i.e. S = L √ −gdx, to make the volume element invariant. However, such choice for the measure is not unique. The only requirement that it must be a density under diffeomorphic transformations can be fulfilled in other ways. Our new measure is Φ = ǫ µ 1 µ 2 ...µ D ǫ a 1 a 2 ...a D ∂ µ 1 ϕ a 1 . . . ∂ µ D ϕ a D ,(1) where ǫ µ 1 µ 2 ...µ D and ǫ a 1 a 2 ...a D are Levi-Civita symbols and ϕ a 1 . . . ϕ a D are additional scalar fields that have nothing to do with the original φ. This modified measure is already applied in gravity as first proposed in [1,2]. We assume that our spacetime is two-dimensional to avoid any unnecessary complications. Then Φ = ǫ µν ǫ ab ∂ µ ϕ a ∂ ν ϕ b .(2) We choose this particular realization for Φ because it is appropriate for our goals. We are able to do it because Φ → det( ∂x µ ′ ∂x µ ) −1 Φ, d 2 x → det( ∂x µ ′ ∂x µ )d 2 x.(3) Therefore, Φd 2 x → Φd 2 x.(4) The general form of the action is S = ΦLd D x.(5) To avoid a confusion in the terminology: the lagrangian is ΦL, let's call it L f ull and by L we mean the part of L f ull without the measure Φ. Notice that when the measure appears only linearly, as in (5) and the measure fields ϕ a do not enter in L, there is an infinite dimensional symmetry ϕ a → ϕ a + f a (L),(6) as has been discussed in [1]. As the background is clear, let's check what exactly the goals are. The essence of the Born-Infeld theory is the requirement of finitness of a physical quantities. Originated as a specific theory of nonlinear electrodynamics in [3], it put limitations on the self-energy of a point charge. Later it reappeared in string theory to describe the electromagnetic fields on the world-volumes of D-branes as it guarantees that the energy of the string is finite in [4,5]. Recently, to bring limits on scalar fields in cosmology, the Born-Infeld scalar was considered in [6,7]. This integration was developed later in [8,9,10,11,12,13,14,15]. Our first aim is a naturally arising restraints on our scalar field. The essence of the scale invariance is the requirement that physics must be the same at all scales, i.e. the system must be invariant under the global scale transformations (ω is a constant): g µν → ωg µν .(7) However, the physical universe definitely does not have such property, and different scales show different behavior. Then to approach reality the scale invariance must be broken. Our second aim is a naturally arising dynamical violation of the scale invariance. In this paper we are going to work with a fixed background metric, so the transformation (7) will not be used, instead the fields will transform and a volume element independent of the metric will be allowed to transform. Moreover, according to Noether's theorem, the symmetries and conservation laws are tightly connected. Then our third aim is to show that despite being a symmetry, the scale invariance does not lead to the conservation of the scale charge. The anomalous infrared behavior of the conserved chiral current in the presence of instantons was discussed in [16]. The conclusion was made that in this case there was no conserved U (1) charge and Goldstone's theorem therefore failed, solving the U (1) problem in QCD. The case of global scale invariance in the presence of a modified measure was considered in [17,18,19,20,21,22,23,24,25] and the dilatation currents were calculated in a special model in [26], where the current was shown to be singular in the infrared. Here also, the resulting scale current produces a nonzero flux of the dilatation current to infinity, so once again, although there is a conserved current, there is no conserved scalar charge. Section 2 is devoted to the preparations for the later sections: the guiding principles are considered in more details and the lagrangian is provided. In Section 3 we arrive at the Born-Infeld scalar dynamics. In Section 4 we show how the asymptotic behavior of the conserved current leads to the nonconservation of a scale charge. The conclusions are given in Section 5. General Considerations The modified measure results in the dynamical violation of the scale invariance. To see that, we consider the variation of (5) with respect to ϕ a : A µ 1 a 1 ∂ µ 1 L = 0,(8) where A µ 1 a 1 = ǫ µ 1 µ 2 ...µ D ǫ a 1 a 2 ...a D ∂ µ 2 ϕ a 2 . . . ∂ µ D ϕ a D and we assume that L is independent of ϕ a 's. If det(A µ i a j ) ∼ Φ is non-trivial, then the solution is L = M = constant.(9) The appearance of the constant in (9) breaks the scale invariance. To break the scale invariance in a consequence, the action must be scale invariant initially. Then S = 1 2 Φ∂ µ φ∂ ν φg µν d 2 x.(10) Without loss of generality, we assume that the scalar field possesses only the kinetic energy. The theory has the scale invariance with the following choice of the rescaling of the fields: φ → λ − 1 2 φ, ϕ a → λ 1 2 ϕ a ,(11) where λ is the rescaling parameter that applies to the scalar fields and the measure only (and the metric remains invariant). However, it turns out that this model is not able to bring the enviable results. We must add one more scalar to the lagrangian. The final lagrangian is L = 1 2 (∂ µ φ 1 ∂ ν φ 1 g µν + ∂ µ φ 2 ∂ ν φ 2 g µν ),(12) where φ 1 is the former scalar field φ and φ 2 is the supplemented one. So that the final action is S = 1 2 Φ(∂ µ φ 1 ∂ ν φ 1 g µν + ∂ µ φ 2 ∂ ν φ 2 g µν )d 2 x,(13) where for S to be scale invariant we choose the rescaling of the additional field as φ 2 → λ − 1 2 φ 2 .(14) So at that level the dynamics of the initial scalar field φ is defined by the action (13). To achieve our aims we added to that action three more scalar fields: φ 2 is physically equivalent to φ and enter the lagrangian in the same footing, ϕ a,a=1,2 are the base for the newly constructed modified measure in 2D. In the following sections we show how such complexity leads to the solutions. A step further is to obtain the equations of motion, i.e. the variations of S with respect to the dynamical variables. For simplicity we consider flat Minkowski 2D spacetime so that g µν = η µν ,(15) with the signature (−+). The Appearance of a Born-Infeld Scalar Sector. The dynamical variables of (13) are φ 1 , φ 2 , ϕ a . The variation with respect to ϕ a is ǫ µν ǫ ab ∂ ν ϕ b ∂ µ L = 0.(16) If Φ is non-trivial then ǫ µν ǫ ab ∂ ν φ b is non-trivial. Then we can obtain L = 1 2 (∂ µ φ 1 ∂ ν φ 1 g µν + ∂ µ φ 2 ∂ ν φ 2 g µν ) = M.(17) Note that we can see (even before studying the Born-Infeld scalar sector) that for static case the gradients of the two scalar fields are bounded by √ 2M . (17) is rewritten to give (∂ µ φ 1 ) 2 + (∂ µ φ 2 ) 2 = 2M.(18) The variation with respect to φ 2 is ∂ µ (Φ∂ µ φ 2 ) = 0.(19) We assume that φ 2 depends only on the spatial coordinate x, φ 2 = φ 2 (x). Then (19) becomes ∂ 1 (Φ∂ 1 φ 2 ) = 0,(20) which can be integrated to give Φ∂ 1 φ 2 = J = constant.(21) We observe that the action has the additional shift symmetry: φ 2 → φ 2 + c 2 ,(22) where c 2 is a constant. This symmetry leads to the conservation law. Then J has the interpretation of a constant current flowing in the x-direction. Then ∂ 1 φ 2 = J Φ .(23) Inserting this into (17), we get ∂ µ φ 1 ∂ µ φ 1 + J 2 Φ 2 = 2M,(24) which can be used to solve for the measure Φ, giving Φ = J 2M − ∂ µ φ 1 ∂ µ φ 1 .(25) The variation with respect to φ 1 is ∂ µ (Φ∂ µ φ 1 ) = 0.(26) Making the same assumptions as for the φ 2 , namely φ 1 = φ 1 (x) and φ 1 → φ 1 + c 1 ,(27) where c 1 is a constant, we obtain ∂ µ (Φ∂ µ φ 1 ) = 0.(28) Inserting (25) into (26), we get the Born-Infeld scalar equation (for M > 0): J∂ µ ( ∂ µ φ 1 2M − ∂ µ φ 1 ∂ µ φ 1 ) = 0,(29) which is also obtained from the effective Born-Infeld action for this kind of solutions. S ef f = 2M − ∂ µ φ 1 ∂ µ φ 1 d 2 x.(30) The dynamics of φ 1 defined by the equation (29) is the same as the dynamics of φ 1 derived from the variation of (30). This means that ∂ µ φ 1 ∂ µ φ 1 is bounded in this sector of the theory (the Born-Infeld scalar sector) Notice, however, that we are now considering only a sector of the theory. The Breaking of Charge Conservation. The action (13) is invariant under the scale transformations (11) and (14). By the Noether's theorem a conservation quantity must appear, namely, the scale charge, Q. Then the continuity equation must be satisfied: ∂ρ ∂t + ∇j = 0,(31) where ρ is a density of Q and j is the flux of Q. By the the integration x 2 x 1 ∂ρ ∂t d 2 x + x 2 x 1 ∂j 1 ∂x d 2 x = 0(32) we obtain dQ dt + j 1 (x 2 ) − j 1 (x 1 ) = 0.(33) Therefore, the conservation of the total charge requires j 1 (x 1 → −∞) − j 1 (x 2 → +∞) = 0.(34) However, it does not always happen. Our case is one of the exceptions. The conserved current is given by j µ = ∂L f ull ∂(∂ µ ϕ a ) δϕ a + ∂L f ull ∂(∂ µ φ 1 ) δφ a + ∂L f ull ∂(∂ µ φ 2 ) δφ 2 .(35) We consider a scale transformations infinitesimally closed to the identity: λ = (1 + θ), so that (11) and (14) turn into ϕ a → (1 + θ) 1 2 ϕ a ≃ ϕ a + θ 2 ϕ a ,(36)φ 1 → (1 + θ) − 1 2 φ 1 ≃ φ 1 − θ 2 φ 1 ,(37)φ 2 → (1 + θ) − 1 2 φ 2 ≃ φ 2 − θ 2 φ 2 .(38) Therefore, δϕ a = θ 2 ϕ a , δφ 1 = − θ 2 φ 1 , δφ 1 = − θ 2 φ 2 .(39) Then (35) becomes j µ = M θ 2 ǫ µν ǫ ab ϕ a ∂ ν ϕ b − θ 2 Φ∂ µ φ 1 − θ 2 Φ∂ µ φ 2 .(40) Let's go back to (29) and find the static solutions (∂ 0 φ 1 = 0): ∂ 1 ( ∂ 1 φ 1 2M − (∂ 1 φ 1 ) 2 ) = 0.(41) By integration we get ∂ 1 φ 1 2M − (∂ 1 φ 1 ) 2 = c 3 ,(42) where c 3 is a constant. Then φ 1 = √ 2M \|c 3 \| 1 + \|c 3 \| 2 (x 2 − x 1 ).(43) We have done all the calculations for φ 1 . However, the same is relevant for φ 2 . So that φ 2 = √ 2M \|c 4 \| 1 + \|c 4 \| 2 (x 2 − x 1 ),(44) where c 4 is a constant. Inserting this solution to (25), we obtain Φ = J 1 + c 2 3 (1 − 2M )c 2 3 + 1 .(45) Then we see Φ = Φ 0 = constant.(46) It is satisfied for ϕ 1 = c 5 t, ϕ 2 = c 6 x,(47) where c 5 and c 6 are constants. Then indeed Φ = c 5 c 6 .(48) Now by inserting the solutions for ϕ a , φ 1 and φ 2 into (35), we obtain for j 1 j 1 = θ 2 M c 5 c 6 x − θ 2 c 5 c 6 √ 2M \|c 3 \| 1 + \|c 3 \| 2 − θ 2 c 5 c 6 √ 2M \|c 4 \| 1 + \|c 4 \| 2 .(49) We see that j 1 is a constant plus a term proportional to x and therefore, j 1 (∞) − j 1 (−∞) = 0 and in fact diverges. Therefore, Q is not conserved. Let's calculate j 0 explicitly. j 0 = − θ 2 M c 5 c 6 t.(50) Then we see that Q = x 2 x 1 j 0 dx = −(x 2 − x 1 ) θ 2 M c 5 c 6 t.(51) We checked that Q is not conserved. Conclusions In this paper we start with the scalar field, surround it with three supplementary scalar fields and investigate the resulting action, (13). One scalar field is physically equivalent to the former scalar field. However, the new measure of integration is constructed from the other two scalars. The source of the following findings is this modified measure. First, we show that the gradient of this initial scalar field is finite and in particular there is a sector which can be presented in the form of the Born-Infeld scalar. Second, the initial action is scale invariant, however, the invariance gets spontaneously broken. In addition to having spontaneous symmetry breaking, our physical system serves as an example of a system with the symmetry that does not lead to the conserved charge. A scale invariance in cosmology was considered in [27,28]. In this case when the scale symmetry is spontaneously broken, there is a conserved current and since no singular behavior of the conserved current is obtained, so there is a conserved scale charge and the Goldstone theorem holds. 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[ "ORTHOGONALITY OF HOMOGENEOUS GEODESICS ON THE TANGENT BUNDLE", "ORTHOGONALITY OF HOMOGENEOUS GEODESICS ON THE TANGENT BUNDLE" ]	[ "R Chavosh Khatamy " ]	[]	[]	Let ξ = (G × K G/K, ρ ξ , G/K, G/K) be the associated bundle andbe the tangent bundle of special examples of odd dimension solvable Lie groups equipped with left invariant Riemannian metric. In this paper we prove some conditions about the existence of homogeneous geodesic on the base space of τ G/K and homogeneous (geodesic) vectors on the fiber space of ξ .	10.22080/cjms.2018.10527.1292	[ "https://arxiv.org/pdf/1101.1928v1.pdf" ]	117,944,411	1101.1928	82dc74ca3eb76bd1a59a65d8acaa6111286a4ec1	ORTHOGONALITY OF HOMOGENEOUS GEODESICS ON THE TANGENT BUNDLE 10 Jan 2011 R Chavosh Khatamy ORTHOGONALITY OF HOMOGENEOUS GEODESICS ON THE TANGENT BUNDLE 10 Jan 2011 Let ξ = (G × K G/K, ρ ξ , G/K, G/K) be the associated bundle andbe the tangent bundle of special examples of odd dimension solvable Lie groups equipped with left invariant Riemannian metric. In this paper we prove some conditions about the existence of homogeneous geodesic on the base space of τ G/K and homogeneous (geodesic) vectors on the fiber space of ξ . Introduction and preliminaries Let G be a connected Lie group and K be a closed subgroup of G. The set of left cosets of K in G is denoted by G/K and can be given a unique differentiable structure ( [6], vol.II, chap.2), and hence M = G/K is called a homogeneous manifold. When a Lie group G acts transitively isometric on a Riemannian manifold M , we can identify M with the set G/K of left cosets of the isotropy group K of a point Then a geodesic γ : I −→ M is called a homogeneous geodesic if, there exists a 1-parameter subgroup t −→ exp tX , t ∈ R, of G with X ∈ G= T e G such that γ(t) = T (exp tX, x 0 ). Where γ(0) = x 0 ∈ M , and exp : G → G is the exponential map [9]. Definition 1.1. A vector 0 = X ∈ G is called a homogeneous vector (or geodesic vector ), if the curve γ(t) = (exp tX)(x 0 ) is a geodesic on M = G/K [9]. Any homogeneous Riemannian manifold G/K has the reductive decomposition of the form G = M + K where M ⊂ G is a vector subspace, such that Ad(K)(M) ⊂ M. Let M = G/K be a Riemannian manifold and G = M + K, its reductive decomposition. Then the natural map φ : G −→ G/K = M induces a linear epimorphism (dφ) e : T e G −→ T x0 M , and the vector space M can be identified with T x0 M . If C is a scalar product on M induced by the scalar product on T x0 M , then the following lemma holds (see [9], proposition 2.1). Lemma 1.2. If X belongs to G, let [X, Y ] M and X M be the components of [X, Y ] and X in M with respect to reductive decomposition, then X is homogeneous vector Let ℘ = (P, π, B, G) be a smooth fiber bundle. A pair (℘, T ) is called a (smooth)principal bundle with structure group G, if T : P × G −→ P is a right action of G on P and ℘ admits a coordinate representation {(U α , ψ α )} such that (or geodesic vector ) iff C(X M , [X, Y ] M ) = 0 ∀Y ∈ G.ψ α (x, ab) = ψ α (x, a)b, x ∈ U α , a, b ∈ G, (see [6], vol.II, chap.V). Let ℘ = (P, π, B, G) be a principal bundle and F be a differentiable manifold. Consider the left action Q, of G on the product manifold P × F given by Q a (z, y) = (z, y)a = (za, a −1 y) z ∈ P, y ∈ F, a ∈ G. The set of orbits of this action is denoted by P × G F and q : P × F → P × G F will denote the corresponding projection, i.e., q(z, y) is the orbit through (z, y). The map q determines a map ρ ξ : P × G F → B such that, ρ ξ • q = π • π p . Where, π p : P × F → P is the canonical projection and π : P → B is the bundle map. There is a unique smooth structure on P × G F , such that ξ = (P × G F, ρ ξ , B, F ) is a smooth fiber bundle (see [6], vol.II, chap.V, sec.2). Definition 1.4. The fiber bundle ξ = (P × G F, ρ ξ , B, F ), is called the associated bundle with ℘ = (P, π, B, G). Let K be a closed subgroup of G. The principal fiber bundle ℑ = (G, π, G/K, K), is called homogeneous bundle, (See [3]). Let ℑ = (G, π, G/K, K) be a fiber bundle with group structure K, and let G be a connected Lie group and K a closed subgroup of G, (see [1], definition 2.2). We take the Lie algebras G and K of G and K respectively, in [1] and [2], we proved some relations between the homogeneous vector in the fiber space of the associated bundle, ξ = (G × K G/K, ρ ξ , G/K, G/K) and the homogeneous geodesic in the base space of a principal homogeneous bundle ℑ = (G, π, G/K, K).In [3], we consider the homogeneous bundle ℑ = (G, π, G/K, K) and the tangent bundle τ G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ G/K , for both cases of G semisimple and weakly semisimple Lie group. Now, we investigate the existence of mutually orthogonal linearly independent homogeneous geodesics in the base space of the tangent bundle τ G of homogeneous Riemannian manifold G given in theorem 2.2. Main results Let ℑ = (G, π, G/K, K) be a principal homogeneous bundle, with the associated bundle ξ = (G × K K, ρ ξ , G/K, G/K). Let G be the matrix group of all matrices of the form         e z0 0 . . . 0 x 0 0 e z1 . . . 0 x 1 . . . . . . . . . . . . . . . 0 0 . . . e zn x n 0 0 . . . 0 1         where, (x 0 , x 1 , · · · , x n , z 1 · · · , z n ) ∈ R 2n+1 . The Lie group G is unimodular and solvable (see [8], pp.134-136), with the left invariant Riemannian metric g = n i=0 e −2zi dx 2 i + λ 2 n k,j=0 dz k dz j . Where λ = 0 is a constant. Then G is a homogeneous Riemannian manifold with the origin at (0, 0, · · · , 0) ( [8], p.134). Let G = M + K be the reductive decomposition of G , then K= 0, and hence G = M. In [3], we prove the following lemma Lemma 2.1. Let ℑ = (G, π, G/K, K) , be a homogeneous bundle. Then ξ = (G × K G/K, ρ ξ , G/K, G/K), is the associated bundle of ℑ = (G, π, G/K, K). By lemma 2.1, we can take ξ = (G × M, ρ ξ , G, M), be the associated bundle of ℑ = (G, π, G/K, K). In [4], we let G be a 3-dimensional solvable Lie group, given in [8], pp.134, and prove some results about the existence of homogeneous vectors on the fiber space of τ G/K and ξ. In [5], we extend theorems 5.6 and 5.7 in [4], and give the following theorem, for the odd dimensional solvable Lie group. Theorem 2.2.( [5]). Let ℑ = (G, π, G/K, K), be a principal homogeneous bundle and ξ = (G× K G/K, ρ ξ , G/K, G/K), be the associated bundle of ℑ = (G, π, G/K, K). If G is the matrix group of all matrices of the form         e z0 0 . . . 0 x 0 0 e z1 . . . 0 x 1 . . . . . . . . . . . . . . . 0 0 . . . e zn x n 0 0 . . . 0 1         where (x 0 , x 1 , · · · , x n , z 1 · · · , z n ) ∈ R 2n+1 and z 0 = −(z 1 + z 2 + · · · + z n ). Then a vector V in the fiber space of ξ is a homogenous (geodesic ) if and only if its components (x 0 , x 1 , · · · , x n , z 1 , · · · , z n ) satisfy the following conditions x 0 (z 1 + z 2 + · · · + z n ) = 0 x 1 z 1 = 0, · · · , x n z 1 = 0 x 2 0 − x 2 1 = 0, · · · , x 2 0 − x 2 n = 0. In the proof of the Theorem 5.3 in [3], we give a strong isomorphism between the tangent bundle τ G/K = (T G/K , π G/K , G/K, R m ) and the associated bundle ξ = (G × K G/K, ρ ξ , G/K, G/K),τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian manifold G. Then a vector W in R 2n+1 is a homogeneous vector (under isomorphism), if and only if its component (x 0 , x 1 , · · · , x n , z 1 · · · , z n ) satisfy the following conditions x 0 (z 1 + z 2 + · · · + z n ) = 0 x 1 z 1 = 0, · · · , x n z 1 = 0 [3], theorem 5.4. we give a subspace of G ′ such that all member of this subspace are homogeneous vectors, and by strong isomorphism between τ G/K and ξ we can find a subspace of R m (under isomorphism) such that all members of this subspace are homogeneous vectors, In the following theorem, we consider the tangent bundle x 2 0 − x 2 1 = 0, · · · , x 2 0 − x 2 n = 0. Inτ G = (T G , π G , G, R 2n+1 ) of the homogeneous Riemannian manifold G in theorem 2.2, and give structure of all subspaces of R 2n+1 such that all their members are homogeneous vectors. Theorem 2.4.( [5]). Let τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian manifold G, (given in theorem 2.2). Then all homogeneous vectors are decomposed into an n-dimension vector subspace W in R 2n+1 and 2 n , one-dimension vector subspace in R 2n+1 generated by all vectors of the form X 0 ± X 1 ± · · · ± X n . τ G = (T G , π G , G, R 2n+1 ). In [3] we prove some conditions about existence and orthogonality of homogeneous vectors for both cases of G semisimple and weakly semisimple. For example in theorem 5.3 in [3], we prove that if G is a semisimple Lie group then there are m orthogonal homogeneous vectors on the fiber space of the tangent bundle, τ G/K = (T G/K , π G/K , G/K, R m ) In the follow, we want to get some conditions about linearly independent and orthogonality of homogeneous vectors on the fiber space and homogeneous geodesics on the base space of the tangent bundle of the homogeneous Riemannian manifold G (given in theorem 2.2). For this we need to considering to relations between orthogonality of homogeneous vectors and the Hadamard matrices. Definition 2.6. A Hadamard matrix of order k is k × k square matrix whose entries are all equal to ±1, and such that A.A t = kI k , where I k is the unit matrix. The condition A.A t = kI k , in definition 2.6, implies that the k rows or columns of a Hadamard matrix represent orthogonal k-tuples, with all entries equal to +1 or -1, we can use this fact for considering to structure of Hadamard matrices and orthogonality of homogeneous (geodesic) vectors. where (x 0 , x 1 , · · · , x n , z 1 · · · , z n ) ∈ R 2n+1 and z 0 = −(z 1 + z 2 + · · · + z n ) , then; (i) If (n + 1) is odd, then there are not any two mutually orthogonal (n + 1)-tuples with all entries equal to ±1. (ii)If (n + 1) is even and not divisible by 4, then there are exactly two mutually orthogonal (n + 1)-tuples with all entries equal to ±1. Proof. Let τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian manifold G (given in theorem 2.2). By Corollary 2.3, and theorem 2.4 a vector w in R 2n+1 is a homogeneous (geodesics) vector (under isomorphism), if and only if A) w ∈ W = span(Z 1 , Z 2 , · · · , Z n ) B) w = i=n i=0 x i X i and x 2 0 − x 2 1 = 0, · · · , x 2 0 − x 2 n = 0. As concerns homogeneous (geodesics) vectors of type (B), they are all generated by the vectors of the form X 0 + ǫ 1 X 1 + · · · + ǫ n X n , where ǫ i ∈ {1, −1}. Therefore, the problem of finding mutually orthogonal geodesics vectors of type (B) is equivalent to the algebraic problem of finding (n + 1)-tuples, with all entries equal to ±1, which are mutually orthogonal with respect to the standard scaler product in R n+1 . Let (n + 1) be odd number and W 1 and W 2 be two (n + 1)-tuples with all entries equal to ±1. The scaler product of W 1 and W 2 is the sum of the products of their entries and all such products are equal to ±1. By hypotheses, (n + 1) is odd, then sum of the products of their entries dose not vanish, so W 1 and W 2 can not be orthogonal, so we obtain (i). For the second statement of the lemma, we spouse that (n + 1) = 2m, where m is odd, let V 1 and V 2 be two (n + 1)-tuples with all entries equal to ±1, such that V 1 = (1, 1, · · · , 1) and V 2 = (−1, 1, · · · − 1, 1), then V 1 and V 2 are orthogonal. Now, we spouse that V , W , Z, are three mutually orthogonal (n + 1)-tuples with all entries equal to ±1. Then, we compute the scaler product of V , W and Z by V . In this way, we can obtain three mutually orthogonal (n + 1)-tuples vectors V ′ , W ′ , Z ′ such that all entries equal to ±1. If we take V ′ = (−1, −1, · · · , −1), then by orthogonality of V ′ and W ′ , W ′ has exactly m entries equal to −1 and exactly m entries equal to 1. We then multiply, component by component, and applying a fixed permutation of the all entries for mutually orthogonal (n + 1)-tuples vectors V ′ , W ′ , Z ′ , such that this applications will preserve the orthogonality of V ′ , W ′ , Z ′ . By this way, we can obtain W ′ = (1, 1, · · · , 1, −1, −1, · · · , −1), but m is odd and the orthogonality of V ′ , W ′ , Z ′ is imposable, this gives a contradiction, and the proof of the lemma is complete. Before starting some additional results, we recall the fact that A.A t = kI k , in definition 2.6 implies that the k rows or columns of a Hadamard matrix represent orthogonal k-tuples, with all entries equal to +1 or -1, for the case n + 1 be divisible by 4, the problem related to algebraic problem of the existence of Hadamard matrices of order n + 1. Therefore, we get at once the following proposition. Proposition 2.8.With hypothesis of lemma 2.7, let n + 1 be divisible by 4, then R 2n+1 admits n + 1 mutually orthogonal (n + 1)-tuples vectors with all entries equal to ±1, if and only if, there exists a Hadamard matrices of order n + 1. Now, we can prove the following theorem about the linearly independent and the maximum number of the orthogonal homogeneous (geodesic) vectors on the fiber space of τ G = (T G , π G , G, R 2n+1 ). Theorem 2.9. Let τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian Lie group G, (given in theorem 2.2 and lemma 2.7) then; (i)There are 2n + 1 linearly independent homogeneous (geodesics) vectors in the fiber space of through the τ G . (ii) The maximum number of the orthogonal homogeneous (geodesic) vectors on the fiber space of τ G is n + 1, in the case that n + 1 is odd. (ii)The maximum number of the orthogonal homogeneous (geodesic) vectors on the fiber space of τ G , is n + 2, in the case that n + 1 is even and not divisible by 4. (iv)The maximum number of the orthogonal homogeneous (geodesic) vectors on the fiber space of τ G , is 2n + 1, in the case that n + 1 is even and divisible by 4 and there exists a Hadamard matrices of order n + 1. Proof. Theorem 2.4 and corollary 2.3, conclude the fist part of theorem, it is easy to see that, there exist n + 1 linearly independent homogeneous (geodesics) vectors of type (B), ( see proof of lemma 2.7), then there are 2n + 1 linearly independent homogeneous (geodesics) vectors in the fiber space of τ G . The second and the third part of the theorem follows from (i) and (ii), in lemma 2.7. Finally, as an immediate consequence from proposition 2.8, we obtain (iv). By proposition 1.3 and theorem 2.9 we complete corollary 2.5 about the number of linearly independent homogeneous geodesics through origin of the base space of τ G . Corollary 2.10. Let τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian Lie group G, (given in theorem 2.2 and lemma 2.7) then; (i)There are 2n + 1 linearly independent homogeneous geodesics vectors through the origin {e} of the base space of τ G . (ii) The maximum number of the orthogonal homogeneous geodesic through the origin {e} of the base space of τ G , is n + 1, in the case that n + 1 is odd. (ii)The maximum number of the orthogonal homogeneous geodesic through the origin {e} of the base space of τ G , is n + 2, in the case that n + 1 is even and not divisible by 4. (iv)The maximum number of the orthogonal homogeneous geodesic through the origin {e} of the base space of τ G , is 2n + 1, in the case that n + 1 is even and divisible by 4 and there exists a Hadamard matrices of order n + 1. x 0 ∈ 0M . The point x 0 is called the origin of M . Let ▽ be an affine connection on M = G/K and let ▽ be invariant under the natural action of T : G × M −→ M . Proposition 1.3.([10]). A finite family {γ 1 , γ 2 , . . . , γ n } of homogeneous geodesics through x o ∈ M is orthogonal ( respectively, linearly independent) if the M-component of the corresponding homogeneous vectors are orthogonal (respectively, linearly independent). then under hypothesis of theorem 2.2 there is a strong isomorphism between,the associated bundleξ = (G × M, ρ ξ , G, M)and the tangent bundleτ G = (T G , π G , G, R 2n+1 ) so we have,Corollary 2.3.([5]). With hypothesis of theorem 2.2, let By proposition 1.3 and Theorem 2.4 we have, the following result about linearly independence of homogeneous geodesics on the base space of τ G Corollary 2.5.([5]). With hypothesis of theorem 2.2, the tangent bundle τ G = (T G , π G , G, R 2n+1 ) admits 2n + 1 linearly independent homogeneous geodesics through the origin {e} of the base space of τ G . Now, we investigate orthogonality of homogeneous vectors on the fiber space of tangent bundle, Lemma 2 . 7 . 27Let τ G = (T G , π G , G, R 2n+1 ) be the tangent bundle of the homogeneous Riemannian Lie group G of all matrices of the form z1 . . . 0 x 1 . . . . . . . . . . . . . . . 0 0 . . . e zn x AcknowledgementThe author would like to express his appreciation of professor O. Kowalski for his invaluable suggestions and also professor M. Toomanian for his constructive comments.The author was supported by the funds of the Islamic Azad University-Tabriz Branch, (IAUT). On the homogeneity of principal bundles. R , Chavosh Khatamy, M Toomanian, Izv. Nats. R.Chavosh Khatamy, and M.Toomanian, On the homogeneity of principal bundles, Izv. Nats. . Akad. Nauk Armenii Mat, J. Contemp. Math. Anal. 385Springer-VerlagAkad. Nauk Armenii Mat. 38, no. 5 (2003), 39-46; translation published by Springer-Verlag, NY, in J. Contemp. Math. Anal. 38, no. 5 (2004), 37-45. R , Chavosh Khatamy, M Toomanian, On the Associated Bundles of Homogeneous Principal Bundles, Proc. of 3 rd Sem. of Top. and Geo. Tabriz, IranR.Chavosh Khatamy, and M.Toomanian, On the Associated Bundles of Homogeneous Prin- cipal Bundles, Proc. of 3 rd Sem. of Top. and Geo. Tabriz, Iran (2004), 33-41. Existence of homogeneous vectors on the fiber space of the tangent bundle. R , Chavosh Khatamy, M Toomanian, Acta Mathematica Hungarica116R.Chavosh Khatamy, and M.Toomanian, Existence of homogeneous vectors on the fiber space of the tangent bundle , Acta Mathematica Hungarica, 116, (4), (2007), 285-294. Homogeneous geodesics on the solvable homogeneous principal bundles. R , Chavosh Khatamy, M Toomanian, Proc. of 4 th Sem. of Top. and Geo. of 4 th Sem. of Top. and GeoUromia, IranR.Chavosh Khatamy, and M.Toomanian,Homogeneous geodesics on the solvable homogeneous principal bundles, To appear in: Proc. of 4 th Sem. of Top. and Geo. Uromia, Iran (2007) Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles. R , Chavosh Khatamy, Izv. Nats. Akad. Nauk Armenii Mat. 451Springer-VerlagJ. Contemp. Math. Anal.R.Chavosh Khatamy, Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles, Izv. Nats. Akad. Nauk Armenii Mat. 45, no. 1 (2010), 53-60, trans- lation published by Springer-Verlag, NY, in J. Contemp. Math. Anal. 45, no. 1 (2010), 60-65. W Greub, S Halperin, R Vanstone, Connections, curvature, and cohomology. Academic PressIW. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, Academic Press, Vol. I, II, (1973). On the existence of homogeneous geodesics in homogeneous Riemannian manifolds Geometriae Dedicata. O Kowalski, J Szenthe, 84O.Kowalski, and J.Szenthe, On the existence of homogeneous geodesics in homogeneous Rie- mannian manifolds Geometriae Dedicata, 84,(2001), 331-332. Generalized symmetric spaces , Lecture notes in Math. O Kowalski, Springer-VerlagO.Kowalski, Generalized symmetric spaces , Lecture notes in Math. Springer-Verlag, no.805, 1980. . O Kowalski, L Vanhecke, Riemannian manifolds with homogeneous geodesics Bull. Un.Math Ital. 5O.Kowalski, and L.Vanhecke, Riemannian manifolds with homogeneous geodesics Bull. Un.Math Ital, 5, (1991), 184-246. O Kowaski, S Nikčević, Z Vlášek, Homogenueos geodesics in homogeneous Riemannian manifolds (examples), Geometry and topology of submanifolds. Beijing/Berlin; River Edge., NJWorld Sci. Publishing CoO. Kowaski, S. Nikčević,and Z. Vlášek, Homogenueos geodesics in homogeneous Riemannian manifolds (examples), Geometry and topology of submanifolds, Beijing/Berlin, World Sci. Publishing Co., 1999, River Edge., NJ (2000), 104-112.	[]
[ "Ping-pong configurations and circular orders on free groups", "Ping-pong configurations and circular orders on free groups" ]	[ "Dominique Malicet ", "Kathryn Mann ", "Cristóbal Rivas ", "Michele Triestino " ]	[]	[]	We discuss actions of free groups on the circle with "ping-pong" dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group F n admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from[21]. Inspired by work in [2], we also exhibit examples of "exotic" isolated points in the space of all circular orders on F 2 . Analogous results are obtained for linear orders on the groups F n × Z.	10.4171/ggd/519	[ "https://arxiv.org/pdf/1709.02348v2.pdf" ]	119,123,551	1709.02348	3280a17e3d0e5b0c9b6da775bffacb38c13b4389	Ping-pong configurations and circular orders on free groups Dominique Malicet Kathryn Mann Cristóbal Rivas Michele Triestino Ping-pong configurations and circular orders on free groups We discuss actions of free groups on the circle with "ping-pong" dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group F n admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from[21]. Inspired by work in [2], we also exhibit examples of "exotic" isolated points in the space of all circular orders on F 2 . Analogous results are obtained for linear orders on the groups F n × Z. Introduction Let G be a group. A (left-invariant) linear order, often called a left order on G is a total order invariant under left multiplication. Left-invariance directly implies that the order is determined by the set of elements greater than the identity, called the positive cone. It is often far from obvious whether a given order can be determined by only finitely many inequalities, or whether a given group admits such a finitely-determined order. This latter question turns out to be quite natural from an algebraic perspective, and can be traced back to Arora and McCleary [3] for the special case of free groups. McCleary answered this question for free groups shortly afterwards, showing that F n has no finitely determined orders [25]. The question of finite determination gained a topological interpretation following Sikora's definition of the space of linear orders on G in [29]. This space, denoted LO(G), is the set of all linear orders on G endowed with the topology generated by open sets U ( ,X) := { \| x y iff x y for all x, y ∈ X} as X ranges over all finite sets of G. Finitely determined linear orders on G are precisely the isolated points of LO(G); going forward, we will refer to these as isolated orders. This correspondence between isolated points and finitely determined orders is perhaps the simplest instance of the general theme that topological properties of LO(G) should reflect algebraic properties of G. Presently, several families of groups are known to either admit or fail to admit isolated orders, with proofs that use both purely algebraic and dynamical methods. Some examples of groups that do not admit isolated orders include free abelian groups [29], free groups [25,26], free products of arbitrary linearly orderable groups [28], and some amalgamated free products such as fundamental groups of orientable closed surfaces [1]. Large families of groups which do have isolated orders include braid groups [11,15], groups of the form x, y \| x n = y m (n, m ∈ Z) [19,27], and groups with triangular presentations [10]. (In fact, all of these latter examples have orders for which the positive cone is finitely generated as a semi-group, a strictly stronger condition.) As a consequence of our work here, we give a family of groups where, interestingly, both behaviors occur. Theorem 1.1. Let F n denote the free group of n generators. The group F n × Z has isolated linear orders if and only if n is even. This result appears to give the first examples of any group G with a finite index subgroup H (in this case F n × Z ⊂ F m × Z, for n odd and m even) such that LO(G) and LO(H) are both infinite, but only G contains isolated points. Theorem 1.1 also has an interesting consequence regarding the space of marked groups. As shown in [8,Prop. 2.13], the set of left-orderable groups is a closed subset of the space of marked groups on n generators. However, Theorem 1.1 implies that this is not the case either for the subset of groups admitting isolated linear orders (or its complement): one may take a sequence of markings of F 2 × Z so as to approach F 3 × Z, and similarly, a sequence of markings on F 3 × Z can be chosen to approach F 4 × Z (see [8, §2.4]). Thus, Theorem 1.1 immediately gives the following. Corollary 1.2. In the space of finitely generated marked groups, having an isolated linear order is neither a closed, nor an open property. The main tool for Theorem 1.1, and main focus of this work, is the study of circular orders on F n and the dynamics of their corresponding actions of F n on S 1 . It is well known that, for countable G, admitting a linear order is equivalent to acting faithfully by orientation-preserving homeomorphisms on the line. In the same vein, a circular order on G is an algebraic condition which, for countable groups, is equivalent to acting faithfully by orientation-preserving homeomorphisms on S 1 . We recall the definition and basic properties in Section 3. Any action of G on S 1 lifts to an action of a central extension of G by Z on the line, giving us a way to pass between circular and linear orders on these groups, and giving us many dynamical tools for their study. Analogous to LO(G), one can define a space of circular orders CO(G). In [21], the second and third authors showed that a circular order on F n is isolated if and only if the corresponding action on the circle has what they called ping-pong dynamics. They gave examples of isolated circular orders on free groups of even rank, but the odd rank case was left as an open problem. Here we answer this question in the negative: Theorem 1.3. F n admits an isolated circular order if and only if n is even. Similarly to Corollary 1.2, one can also prove that the set of groups admitting isolated circular orders is neither closed nor open in the space of marked groups. We prove Theorem 1.3 by developing a combinatorial tool for the study of actions on S 1 with ping-pong dynamics (similar to actions admitting Markov partitions), inspired by the work in [13] and [2]. We expect these to have applications beyond the study of linear and circular orders; one such statement is given in Theorem 3.9. The notion of ping-pong dynamics is defined and motivated in the next section. Sections 3 and 4 give the application to the study of circular and linear orders, respectively, and the proofs of Theorems 1.1 and 1.3. Ping-pong actions and configurations Definition 2.1. Let G = F n be the free group of rank n, freely generated by S = {a 1 , . . . , a n }. A ping-pong action of (G, S) on S 1 is a representation ρ : G → Homeo + (S 1 ) such that there exist pairwise disjoint open sets D(a) ⊂ S 1 , a ∈ S ∪ S −1 , each of which has finitely many connected components, and such that ρ(a) S 1 \ D(a −1 ) ⊂ D(a). We further assume that if I and J are any connected components of D(a), thenĪ ∩J = ∅. D(a) D(b) D(a −1 ) D(b −1 ) ρ(a) S 1 \ D(a −1 ) We call the sets D(a) the ping-pong domains for ρ. A similar definition is given in [21], with the additional requirement that ping-pong domains be closed. The above, more general definition is more natural for our purposes, although we will later introduce Convention 3.7 to reconcile the two. The reader may notice that, for a given ping-pong action ρ of (G, S), there can be many choices of sets D(a) satisfying the property in Definition 2.1. For instance, if ρ is a ping-pong action such that a∈S∪S −1 D(a) = S 1 , then one may choose an arbitrary open set I disjoint from a∈S∪S −1 D(a) and replace D(a 1 ) with D(a 1 ) ∪ I, leaving the other domains unchanged. These new domains still satisfy ρ(a) S 1 \ D(a −1 ) ⊂ D(a). Later we will adopt a convention to avoid this kind of ambiguity. Motivation: why ping-pong actions? The classical ping-pong lemma implies that ping-pong actions are always faithful, and a little more work shows that the action is determined up to semi-conjugacy by a finite amount of combinatorial data coming from the cyclic ordering and the images of the connected components of the sets D(a) (see Definition 2.4 and Lemma 3.4 below, or [23,Thm. 4.7]). In particular, one can think of ping-pong actions as the family of "simplest possible" faithful actions of F n on S 1 , and it is very easy to produce a diverse array of examples. Perhaps the best-known examples are the actions of discrete, free subgroups of PSL(2, R) on RP 1 . For these actions, one can choose domains D(a) with a single connected component. Despite their simplicity, ping-pong actions are quite useful. For instance, in [2] ping-pong actions were used to construct the first known examples of discrete groups of real-analytic circle diffeomorphisms acting minimally, but not conjugate to a subgroup of a finite central extension of PSL(2, R). This was a by-product of a series of papers concerning longstanding open conjectures of Hector, Ghys and Sullivan on the relationship between minimality and ergodicity of a codimensionone foliation (see for instance [12,13,16]). In general, it is quite tractable to study the dynamic and ergodic properties of a ping-pong action (or a Markov system), and this program has been carried out by many authors [6,7,18,20,22]. Basic properties Lemma 2.2. Given a ping-pong action of (G, S), there exists a choice of ping-pong domains D(a) such that ρ(a) S 1 \ D(a −1 ) = D(a) holds for all a ∈ S ∪ S −1 . Proof. Let ρ be a ping-pong action with sets D(a) given. We will modify these domains to satisfy the requirements of the lemma. For each generator a ∈ S (recall this is the free, not the symmetric, generating set), we shrink the domain D(a), setting D (a) := ρ(a) S 1 \ D(a −1 ) . Applying a −1 to both sides of the above expression gives ρ(a −1 )(D (a)) = S 1 \ D(a −1 ). Moreover, since the connected components of D(a) have disjoint closures, and the same holds for D(a −1 ), hence also for D (a), we also have ρ(a −1 ) D (a) = S 1 \ D(a −1 ); or equivalently ρ(a −1 ) S 1 \ D (a) = D(a −1 ). This is what we needed to show. Convention 2.3. From now on, we assume all choices of domains D(a) for every ping-pong action are as in Lemma 2.2. In particular, this means that, for each a ∈ S, the sets of connected components π 0 (D(a)) and π 0 (D(a −1 )) have the same cardinality, and ρ(a) induces a bijection between the connected components of S 1 \ D(a −1 ) and connected components of D(a). Definition 2.4. Let ρ be a ping-pong action of (G, S). The ping-pong configuration of ρ is the data consisting of 1. the cyclic order of the connected components of a∈S∪S −1 D(a) in S 1 , and 2. for each a ∈ S ∪ S −1 , the assignment of connected components λ a : π 0   b∈S∪S −1 \{a −1 } D(b)   → π 0 (D(a)) induced by the action. Note that not every abstract assignment λ a as in the definition above can be realized by an action F 2 → Homeo + (S 1 ). The following construction gives one way to produce some large families of examples. Example 2.5 (An easy construction of ping-pong actions). For a ∈ S, let X a and Y a ⊂ S 1 be disjoint sets each of cardinality k(a), for some integer k(a) ≥ 1, such that every two points of X a are separated by exactly one point of Y a . Choose these so that all the sets X a ∪ Y a are pairwise disjoint as a ranges over S. Let D(a) and D(a −1 ) be neighborhoods of X a and Y a , respectively, chosen small enough so that all these sets remain pairwise disjoint. Now one can easily construct a piecewise linear homeomorphism (or even a smooth diffeomorphism) ρ(a) with X a as its set of attracting periodic points, and Y a as the set of repelling periodic points such that ρ(a) S 1 \ D(a −1 ) = D(a). The assignments λ a are now dictated by the period of ρ(a) and the cyclic order of the sets X a and Y a . While the reader should keep the construction above in mind as a source of examples, we will show in Example 3.10 that not every ping-pong configuration can be obtained in this manner. However, the regularity (PL or smooth) in the construction is attainable in general. The following construction gives one possibility for a PL realization that will be useful later in the text. We leave the modifications for the smooth case as an easy exercise. The next definition and proposition give a further means of encoding the combinatorial data of a ping-pong action. This will be used later in the proof of Theorem 1.3. Definition 2.7. Let ρ be a ping-pong action of (G, S) with domains D(a). For each a ∈ S, we define an oriented bipartite graph Γ a with vertex set equal to π 0 (D(a)) ∪ π 0 (D(a −1 )), and edges defined as follows: • For I + ∈ π 0 (D(a)), let J + denote the connected component of S 1 \ D(a) adjacent to I + on the right. Put an oriented edge from I + to an interval I − ∈ π 0 (D(a −1 )) if and only if ρ(a −1 )(J + ) = I − . • Similarly, for I − ∈ π 0 (D(a)), with J − the adjacent interval of S 1 \ D(a −1 ) on the right, put an oriented edge from I − ∈ π 0 (D(a −1 )) to I + ∈ π 0 (D(a)) if and only if ρ(a)(J − ) = I + . Proposition 2.8. Let ρ : G → Homeo + (S 1 ) be a ping-pong action of a free group (G, S). Then, for each generator a ∈ S, there exists k(a) ∈ N such that the graph Γ a is an oriented 2k(a)-cycle. Proof. First, the construction of the graph ensures that it is bipartite, and that each vertex has at most one outgoing edge. As a consequence of . This proves that if I + 1 and I + 2 are consecutive connected components of D(a) in S 1 , then they belong to the same cycle in Γ a . Hence we easily deduce that all connected components of D(a) are in the same cycle in Γ a . The same also holds for the components of D(a −1 ), and the graph is connected. Left-invariant circular orders We begin by quickly recalling standard definitions and properties. A reader familiar with circular orders may skip to Section 3.1. Definition 3.1. Let G be a group. A left-invariant circular order is a function c : G × G × G → {0, ±1} such that 1. c is homogeneous: c(γg 0 , γg 1 , γg 2 ) = c(g 0 , g 1 , g 2 ) for any γ, g 0 , g 1 , g 2 ∈ G; 2. c is a 2-cocycle on G: c(g 1 , g 2 , g 3 ) − c(g 0 , g 2 , g 3 ) + c(g 0 , g 1 , g 3 ) − c(g 0 , g 1 , g 2 ) = 0 for any g 0 , g 1 , g 2 , g 3 ∈ G; 3. c is non-degenerate: c(g 0 , g 1 , g 2 ) = 0 if and only if g i = g j for some i = j. The space of all left-invariant circular orders on G, denoted CO(G), is the set of all such functions, endowed with the subset topology from {0, ±1} G×G×G (with the natural product topology). Although spaces of left-invariant linear orders have been well-studied, there are very few cases where the topology of CO(G) is completely understood. Other than a few sporadic examples, the only complete description of spaces of circular orders known to the authors comes from [9], which gives a classification of all groups such that CO(G) is finite, and also a proof that CO(A) is homeomorphic to a Cantor set for any Abelian group A. Given that left-orders on free groups are well understood, a natural next case of circular orders to study is CO(F n ). Our main tool for this purpose is the following classical relationship between circular orders and actions on S 1 (see [5,21]). Proposition 3.2. Given a left-invariant circular order c on a countable group G, there is an action ρ c : G → Homeo + (S 1 ) such that c(g 0 , g 1 , g 2 ) = ord (ρ c (g 0 )(x), ρ c (g 1 )(x), ρ c (g 2 )(x)) for some x ∈ S 1 , where ord denotes cyclic orientation. Moreover, there is a canonical procedure for producing ρ c which gives a well-defined conjugacy class of action. This conjugacy class is called the dynamical realization of c with basepoint x. A description of this procedure is given in [21], modeled on the analogous linear case (see e.g. [17]). Note that modifying a dynamical realization by blowing up the orbit of some point y / ∈ ρ(G)(x) may result in a non-conjugate action that still satisfies the property c(g 0 , g 1 , g 2 ) = ord (ρ c (g 0 )(x), ρ c (g 1 )(x), ρ c (g 2 )(x) ). However, this non-conjugate action cannot be obtained through the canonical procedure. Remark 3.3. The converse to the above proposition is also true: if G is a countable subgroup of Homeo + (S 1 ), then G admits a circular order. A proof is given in [5, Thm. 2.2.14] 2 . As a special case, if ρ : G → Homeo + (S 1 ) is such that some point x has trivial stabilizer, then we may define an induced order on G by c(g 1 , g 2 , g 3 ) := ord(ρ(g 1 )(x), ρ(g 2 )(x), ρ(g 3 )(x)). While one cannot expect in general to find a point with trivial stabilizer, this does hold for ping-pong actions by the following lemma. Isolated circular orders on free groups In this section we will use ping-pong actions to prove Theorem 1.3 from the introduction. As this builds on the framework of [21], we start by introducing two results obtained there. Let G be any group, and ρ : G → Homeo + (S 1 ). Recall that, if ρ(G) does not have a finite orbit, then there is a unique closed, ρ(G)-invariant set contained in the closure of every orbit, called the minimal set of ρ(G). We denote this set by Λ(ρ). If Λ(ρ) = S 1 , the action is called minimal. Otherwise, Λ(ρ) is homeomorphic to a Cantor set and ρ permutes the connected components of S 1 \ Λ(ρ). While, for many examples of actions, the permutation will have many disjoint cycles, the next lemma states that this is not the case for dynamical realizations. Since a ping-pong action of a free group of rank at least 2 cannot have finite orbits, invariance of the minimal set immediately implies that Λ(ρ) ⊂ a∈S∪S −1 D(a). If additionally, for each s = t ∈ S ∪ S −1 , one has D(s) ∩ D(t) = ∅, then invariance of Λ(ρ) and the definition of ping-pong implies that in fact Λ(ρ) ⊂ a∈S∪S −1 D(a). Going forward, it will be convenient to have the this stronger condition, which is given by following lemma. Since x has trivial stabilizer and G is free, we may extend the action of G to this new circle by allowing a ∈ S to act as any orientation-preserving map from I y to I a(y) . We now show that we may choose maps in such a way as to achieve a ping-pong action with the desired properties. For each inserted interval I = [p, q] that is adjacent to a set of the form D 1 (s) on the left and D 1 (t) on the right (where s, t ∈ S ∪ S −1 ), fix points p s < p t in the interior of I and extend D 1 (s) into I to include [p, p s ) and D 1 (t) to include (p t , q]. Having done this on each such interval, let D(s) denote the new extended domains, and note that these have disjoint closures. Now for a ∈ S, define the action of a on such an interval I = I y as follows. If ρ 0 (a)(y) ∈ D 0 (a), the restriction of ρ to I may be any orientation-preserving homomorphism between I y and I a(y) . Otherwise, I y is adjacent to D 1 (a −1 ) either on the right or the left, and we define ρ(a) on I y to map the chosen point p a −1 ∈ I y to the point p a in I a(y) . This ensures that ρ(a)(S 1 \ D(a −1 )) = D(a), so that these are indeed ping-pong domains for the action. Finally, note that by construction, the ping-pong configuration has not changed. Convention 3.7. In a ping-pong action of (G, S), we assume from now on that the domains D(s) satisfy D(s) ∩ D(t) = ∅ whenever s = t ∈ S ∪ S −1 . It follows easily from invariance of Λ(ρ) and the definition of ping-pong that, for actions as in Convention 3.7, we have the inclusion Λ(ρ) ⊆ a∈S∪S −1 D(a). The following theorem from [21] relates circular orders and ping-pong actions. With these tools, we proceed to the main goal of this section. Proof of Theorem 1.3. The case where n is even is covered in [21]. As explained there, the representation of G into PSL(2, R) coming from a hyperbolic structure on a genus n/2 surface with one boundary component gives an isolated circular order. (In fact, by taking lifts to cyclic covers, one can obtain infinitely many isolated circular orders in distinct equivalence classes under the action of Aut(F n ) on CO(G).) To show that F n does not admit an isolated circular order when n is odd, we need more work. We begin with some generalities, applicable to free groups of any rank (even or odd). Suppose that ρ : F n → Homeo + (S 1 ) is a dynamical realization of an isolated circular order, and fix a free generating set S = {a 1 , . . . , a n } for F n . By Theorem 3.8 and Lemma 3.5, ρ is a ping-pong action with domains satisfying Convention 3.7 and the connected components of S 1 \ Λ(ρ) form a unique orbit. Let c 0 , . . . , c r be the (finitely many) connected components of S 1 \ Λ(ρ) that are not contained in any domain D(s). Suppose that c i has endpoints in D(s) and D(t), for some s = t. Then, for any generator u / ∈ {s −1 , t −1 }, we have that ρ(u)(c i ) ∈ D(u). In addition, we have that ρ(s −1 )(c i ) and ρ(t −1 )(c i ) belong to {c 0 , . . . , c r }: indeed, the intersection c i ∩D(s) is nonempty; its image by ρ(s −1 ) is contained in S 1 \ (Λ ∪ D(s −1 )) and is adjacent to D(s −1 ) because of Convention 2.3; moreover, it has to intersect some c j because of Convention 3.7. Then we must have ρ(s −1 )(c i ) = c j . The same holds for t −1 . This implies that c i and c j are in the same orbit if and only if they are equivalent under the equivalence relation ∼ on {c 0 , . . . , c r } generated by c i ∼ c j if there exists t ∈ S ∪ S −1 such that c i = ρ(t)(c j ) and c i ∩ D(t) = ∅ We will now argue that the number of equivalence classes under this relation can be 1 only if n is even. This is done by using the combinatorial data of the graphs from Definition 2.7 to build a surface with boundary using the disc, and then making an Euler characteristic argument. For each generator a ∈ S, let k(a) be the integer given by Proposition 2.8. Let P a be a 4k(a)-gon (topologically a disc) with cyclically ordered vertices . Iterate this process until all edges v 2j−1 v 2j have been glued to S 1 = ∂D. Our convention to follow the orientation of S 1 implies that the resulting surface with boundary is orientable. Note that the remaining (unglued) edges of P a correspond exactly to the edges of the graph Γ a from Definition 2.7; precisely, collapsing each connected component of D(a) and of D(a −1 ) to a point representing a vertex recovers the cycle Γ a . Now repeat this procedure for each generator in S, to obtain an orientable surface with boundary, which we will denote by Σ. A cartoon of the result of this procedure for the ping-pong action of Example 3.10 is shown in Figure 3.1, and may be helpful to the reader. We claim that the number of boundary components of the surface Σ is exactly the number of equivalence classes of the relation ∼. To see this, we proceed as follows. By construction, the connected components of ∂Σ ∩ {c 0 , . . . , c r } are exactly the intervals c i . If some interval c i has endpoints in D(s) and D(t), then ∂Σ ∩ c i is joined to ρ(t −1 )c i ∩ ∂Σ and ρ(s −1 )c i ∩ ∂Σ by edges of P s and P t respectively. Thus, c i ∼ c j implies that c i and c j lie in the same boundary component of Σ, and the intersection of that boundary component with {c 0 , . . . , c r } defines an equivalence class. This proves the claim. We now compute the Euler characteristic of Σ and conclude the proof. Proposition 2.8 implies that the gluing of P a described in our procedure adds one face and 2k(a) edges to the existing surface. Therefore after all the polygons P a (as a ranges over elements of S) have been glued, the surface Σ obtained has χ(Σ) ≡ n + 1 mod 2. Since Σ is orientable, χ(Σ) agrees mod 2 with the number of boundary components of Σ, which, by our claim proved above, agrees with the number of equivalence classes of c i . As discussed above, if ρ is the dynamical realization of an isolated order then this number is equal to 1, hence n + 1 ≡ 1 mod 2, and n must be even. The proof above can be improved to give a statement about general ping-pong actions: Theorem 3.9. Let G = F n be the free group of rank n with free generating set S. Consider a ping-pong action ρ of (G, S) satisfying Conventions 2.3 and 3.7. Let Λ(ρ) be the minimal invariant Cantor set for the action. Then the number of orbits of connected components of the complement S 1 \ Λ(ρ) is congruent to n + 1 mod 2. Proof. As in the previous proof, let c 0 , c 1 , . . . , c r be the connected components of S 1 \ Λ(ρ) that are not contained in any domain D(s), and recall that G permutes the connected components of S 1 \ Λ(ρ). We claim that each cycle of this permutation contains at least one of the c i . Given this claim, we may construct an orientable surface Σ as in the proof of Theorem 1.3, whose boundary components count the number of cycles. Computing Euler characteristic as above shows that the number of cycles is congruent to n + 1 mod 2. We now prove the claim. Suppose that I is a connected component of S 1 \ Λ(ρ) contained in some D(s). By Lemma 2.6, we can take ρ(s) to be piecewise linear, and such that each ρ(s −1 ) expands D(s) uniformly, increasing the length of each connected component by a factor of some µ > 1, independent of s. Iteratively, assuming that ρ(s k s k−1 · · · s 1 )(I) ⊂ D(s −1 k+1 ), then the length of ρ(s k+1 s k · · · s 1 )(I) is at least µ k+1 length(I). This process cannot continue indefinitely, so some image of I is not contained in a ping-pong domain. Exotic examples To indicate some of the potential difficulty of the problem of classifying all isolated orders on F n , we give an example of a ping-pong configuration for F 2 that, even after applying an automorphism of F 2 , cannot arise from the construction in Example 2.5. Example 3.10. Let F 2 = a, b and consider a ping-pong action where ρ(b) is as defined by the graph in Figure 3.2, and ρ(a) is a hyperbolic element of SL(2, R) chosen so that the connected components D(b) D(b) D(b −1 ) D(b −1 ) ρ(b) S 1 \ D(b −1 ) ρ(b) S 1 \ D(b −1 ) D(b) D(b) D(b −1 ) D(b −1 ) D(b) D(b) D(b −1 ) D(b −1 ) Left-invariant linear orders on F n × Z The purpose of this section is to prove Theorem 1.1, stating that F n × Z admits an isolated linear order if and only if n is even. Preliminaries on linear orders Before proceeding to the proof of Theorem 1.1, we recall some standard tools. As for circular orders, linear orders on countable groups have a dynamical realization (see for instance [14, Prop. 1.1.8]). One quick way of seeing this given what we have already described, is by thinking of a linear order as a special case of a circular order. Indeed, given a linear order on a group G, one defines the cocycle c by setting, for distinct g 1 , g 2 , g 3 ∈ G c (g 1 , g 2 , g 3 ) = sign(σ), where σ is the permutation of the indices such that g σ(1) ≺ g σ(2) ≺ g σ (3) . Thus, the construction of the dynamical realization sketched in the proof of Proposition 3.2 may be performed also for a linear order. The result is an action on the circle with a single one global fixed point, which one can view as an action on the line with no global fixed point. Conversely, a faithful action on the real line ρ : G → Homeo + (R) can be viewed as a faithful action on the circle with a single fixed point, and the circular orders produced as in Remark 3.3 will be linear orders on G. Next, we recall the notion of convex subgroups, their dynamical interpretation, and their relationship to isolated orders. Definition 4.1. A subgroup C in a linearly-ordered group (G, ) is convex if for any two elements h, k ∈ C, and for any g ∈ G, the condition h g k implies g ∈ C. Lemma 4.2 (see [14], Prop. 2.1.3). Let G be a countable left-ordered group and consider a dynamical realization ρ of (G, ) with basepoint x such that C is a convex subgroup. Let I be the interval bounded by inf h∈C ρ(h)(x) and sup h∈C ρ(h)(x). Then I has the following property: for any g ∈ G, either ρ(g)(I) = I, or ρ(g)(I) ∩ I = ∅. (4.1) Moreover, the stabilizer of I is precisely C. Conversely, given a faithful action on the real line ρ : G → Homeo + (R), if an interval I has the property (4.1), then the stabilizer C = Stab G (I) is convex in any induced order with basepoint x ∈ I. It is easy to see that the family of convex subgroups of a linearly ordered group (G, ) forms a chain: if C 1 , C 2 are two convex subgroups of (G, ), then either C 1 ⊂ C 2 or C 2 ⊂ C 1 . Moreover, for any convex subgroup C ⊂ G, the group G acts on the ordered coset space (G/C, C ) by orderpreserving transformation. (The induced order on the coset space is given by f C < C gC if and only if f c < gc for every c, c ∈ C, which makes sense because C is convex.) In particular, this implies that if C is convex in (G, ), then any linear order C on C may be extended to a (new) order on G by declaring id g ⇔ C C gC if g / ∈ C, id g if g ∈ C. Elaborating on this, one can show the following lemma (see [14,Prop. 3.2.53] or [24, Thm. 2] for details). Lemma 4.3. If (G, ) has an infinite chain of convex subgroups, then is non-isolated in LO(G). Let us also introduce a dynamical property that implies that an order is non-isolated. Recall that two representations ρ 1 and ρ 2 : A representation ρ ∈ Rep # (G, Homeo + (R)) is said to be flexible if every open neighborhood of ρ in Rep # (G, Homeo + (R)) contains a representation that is not semi-conjugate to ρ. G → Homeo + (R) are semi-conjugate if there is a proper, non-decreasing map f : R → R such that f • ρ 1 = ρ 2 • f . The following lemma is implicit in work of Navas [26] as well as in [28]. An explicit proof can be found in [1, Prop. 2.8]. Lemma 4.5. Let G be a discrete, countable group and let ρ 0 be the dynamical realization of an order with basepoint x ∈ R. If ρ 0 is flexible, then is non-isolated in LO(G). Remark 4.6. Though not needed for our work here, we note that a precise characterization of isolated circular and linear orders in terms of a strong form of rigidity (i.e. strong non-flexibility) of their dynamical realizations is given in [21]. As mentioned in the introduction, in order to prove Theorem 1.1 we use the relationship between circular orders on groups and linear orders on their central extensions by Z. For this purpose, we need the notion of cofinal elements. Given a group G with a circular order c, there is a natural procedure to lift c to a linear order c on a central extension of G by Z such that any generator of the central Z subgroup is cofinal for c [21,30]. The following statement appears as Proposition 5.4 in [21]. Finally, we recall the definition of crossings. Definition 4.10. Let G be a group acting on a totally ordered space (Ω, ≤). The action has crossings if there exist f, g ∈ G and u, v, w ∈ Ω such that: 1. u < w < v. 2. g n u < v and f n v > u for every n ∈ N, and 3. there exist M, N in N such that f N v < w < g M u. In this case, we say that f and g are crossed. If f and g are crossed, then the graph of ρ(f ) and ρ(g) in the dynamical realization is locally given by the picture in Figure 4.1. Our application of the notion of crossings will be through the following lemma. Lemma 4.11 ([14] Cor. 3.2.28). Let C be a convex subgroup of (G, ) and suppose that the (natural) action of G on (G/C, C ) has no crossings. Then there exists a homomorphism τ : G → R with C in its kernel. Moreover, if C is the maximal convex subgroup of (G, ), then C agrees with the kernel of τ . Isolated linear orders on F n × Z We now turn to our main goal of describing isolated linear orders on F n × Z and proving Theorem 1.1. We begin by reducing the proof to the statement of Proposition 4.12 below. Since every central extension of F n by Z splits, Proposition 4.9 tell us that F 2n × Z admits isolated linear orders -more precisely, any lift of an isolated order on F 2n to F 2n × Z will be isolated. Furthermore, if 0 → Z →Ĝ π → G → 1 is a central extension of G by Z then any linear order onĜ in which Z is cofinal gives a canonical circular order on G as follows. Let z be the generator of Z such that z id. Since z is cofinal, for each g ∈ G, there exists a unique representativeĝ ∈ π −1 (g) such that id ĝ ≺ z. Given distinct elements g 1 , g 2 , g 3 ∈ G, let σ be the permutation such that id ĝ σ(1) ≺ĝ σ(2) ≺ĝ σ(3) ≺ z. Define π * ( )(g 1 , g 2 , g 3 ) := sign(σ). One checks that this is a well defined circular order on G. In the proof of Proposition 5.4 of [21], it is shown that π * is continuous and is locally injective when G is finitely generated, which implies that an isolated linear order of F 2n+1 × Z with cofinal center induces an isolated circular order of F 2n+1 by this procedure. Since F 2n+1 has no isolated circular orders, by Theorem 1.3, to finish the proof of Theorem 1.1 it is enough to show the following: Proposition 4.12. Let F be a free group, and a linear order on G = F × Z in which the central factor is not cofinal. Then is non-isolated. As a warm-up, as well as tool to be used in the proof, we start with a short proof of a special case. Proof. Let f 1 , f 2 , . . . be a set of free generators of the free factor F and g the generator of the central factor Z. Let be any order on G and ρ 0 a dynamical realization with basepoint x. For any fixed n ∈ N, we can define a representation ρ n : G → Homeo + (R) by setting ρ n (g) = ρ 0 (g), ρ n (f k ) = ρ 0 (f k ) if k = n, ρ 0 (f n ) −1 if k = n. It is easy to see that the orbit of x is free for all the actions ρ n , and that no two distinct representations ρ n and ρ m are semi-conjugate one to another. Thus, they determine distinct orders n ; and these orders converge to in LO(G) as n → ∞. Proof of Proposition 4.12. We have already eliminated the case where F has infinite rank. If F has rank one, then F × Z is abelian, and so admits no isolated orders (see [29]). So from now on we assume that the rank of F is finite and at least 2. Looking for a contradiction, suppose that is a linear order on G = F × Z which is isolated, and in which the center is not cofinal. Let ρ be its dynamical realization, and let z be a generator of the central Z subgroup. By Remark 4.8, ρ(z) acts with fixed points. Moreover since Z is central, the set of fixed points of ρ(z) is ρ(G)-invariant. Since ρ(G) has no global fixed point, this implies that ρ(z) has fixed points in every neighborhood of +∞ and of −∞. We now find a convex subgroup in which z is cofinal. Let I denote the connected component of R \ Fix(ρ(z)) that contains the basepoint x 0 of ρ, so in particular I is a bounded interval. Let C = Stab G (I). Claim 1. C is a convex subgroup of (G, ) and z is cofinal in (C, ). Proof of Claim. If h, k ∈ C and g ∈ G satisfy ρ(h)(x 0 ) < ρ(g)(x 0 ) < ρ(k)(x 0 ), then for any n we also have ρ(z n h)(x 0 ) < ρ(z n g)(x 0 ) < ρ(z n k)(x 0 ). Since z is central, this implies ρ(hz n )(x 0 ) < ρ(gz n )(x 0 ) < ρ(kz n )(x 0 ). (4.3) Up to replacing z with z −1 , without loss of generality we may assume that ρ(z)(x 0 ) > x 0 . Thus, as n → ∞, the sequence of points ρ(z n )(x 0 ) converges to the rightmost point of I, which is fixed by both ρ(h) and ρ(k). We deduce from (4.3) that ρ(g) also fixes this point. Similarly, considering the limit in (4.3) as n → −∞ shows that the leftmost point of I is fixed by ρ(g). Hence g ∈ C, which shows C is convex. Finally, by Remark 4.8, the fact that ρ(z) has no fixed points in I implies that z is cofinal in (C, ). Since (G, ) is isolated and C is convex, the restriction of to C is also an isolated order on C. Additionally, the fact that is isolated implies, by Lemma 4.3, that the chain of convex subgroups of G is finite. Let G denote the smallest convex subgroup properly containing C. Since Z ⊂ C, we have that G is also a direct product of Z and a free group (a subgroup of F ), and again, our assumptions on imply that (the restriction of) is an isolated left-order on G in which Z is not cofinal. Thus, we may work from now on with G instead of G. Equivalently -and, for notational convenience, this is how we will proceed -we may assume that C is the maximal convex subgroup of G. For our next claim observe that this maximal convex subgroup C also admits a decomposition of the form F * × Z, where F * is a subgroup of F . Claim 2. F * is a non trivial free group of even rank. Proof of Claim. Since the restriction of to C = F * × Z is isolated, as before, Lemma 4.13 implies that F * cannot have infinite rank. If F * were trivial, then the action of G would be semi-conjugate to an action of F , thus making very easy to perturb the action of F × Z and thus the order (recall that free groups have no isolated orders [25]). Thus, F * is a nontrivial free group of finite rank, and as z is cofinal in the ordering in C, its rank must be even (c.f. the remarks at the beginning of Section 4.2). Claim 3. C has infinite index in G. Proof of Claim. If C had finite index, then the G-orbit of the interval I would be bounded. This would imply that the dynamical realization has a global fixed point, which is absurd. Since every nontrivial normal, infinite index subgroup of F has infinite rank, we conclude from Claims 2 and 3 that F * (and thus C) is not a normal subgroup of G. Lemma 4.11 thus implies that the action of G on (G/C, C ) has crossings, as otherwise C would be normal. In particular, if we collapse I and its G-orbit, we obtain a semi-conjugate actionρ : G → Homeo + (R) which is minimal and has crossings. Using this observation, we now prove the following claim. Proof of Claim. Fix a compact set K. We will modify the action of F outside K to produce an action of G that is not semi-conjugate to ρ. Suppose as an initial case that there is a primitive element (i.e. a generator in some free generating set) a of F such that ρ(a) has a fixed point p / ∈ K. Without loss of generality, assume p is to the right of K, the other case is completely analogous. Since Fix(ρ(a)) is ρ(z)-invariant, and ρ(z n )(p) is bounded and accumulates at a fixed point of ρ(z), we may also assume without loss of generality that we have chosen p to be a common fixed point of ρ(z) and ρ(a). We now define a + and a − ∈ Homeo + (R) which commute with ρ(z), and have the property that a + (x) ≥ x for all x ≥ p, and a − • ρ(a)(x) ≤ x for all x ≥ p. For this, let J be any connected component of (R\Fix(ρ(z)))∩[p, ∞). Suppose first that J contains a point of Fix(ρ(a)). The fact that a and z commute means that the endpoints of J are preserved by ρ(a). Then define the restriction of a + to J to agree with ρ(z) if ρ(z)(x) > x on J, or with ρ(z −1 ) if ρ(z)(x) < x on J. If J contains no point of Fix(ρ(a)), then J ⊆ J where J is a connected component of R \ Fix(ρ(a)) ∩ [p, ∞), and we may define a + to agree with ρ(a) or ρ(a −1 ) there, so as to satisfy a + (x) > x for x ∈ J. Lastly, set a + (x) = x for any x ∈ Fix(ρ(z)) ∩ [p, ∞). The definition of a − is analogous. Let ρ ± be the actions obtained by replacing the action of ρ(a) by that of a ± on [p, ∞) and leaving the other generators unchanged. Since a + and a − commute with ρ(z), this defines representations of G, and clearly ρ + and ρ − are not semi-conjugate. We are left to deal with the case where no primitive element of F has a fixed point outside K. In this case, we will perturb the action ρ to obtain a primitive element with a fixed point outside K, and hence a non-semi-conjugate action. To do this, we use the fact that the semi-conjugate actionρ has crossings and is minimal. Minimality implies that crossings can be found outside any compact set and, thus, for any compact K ⊂ R there is g ∈ G such that R \ Fix(ρ(g)) has a component outside (and on the right of) K. Let J = (j 0 , j 1 ) denote one of those components. Notice that some primitive element a ∈ F has the property that ρ(a)(j 0 ) ∈ J, but ρ(a)(j 1 ) / ∈ J. For if this was not the case, J would satisfy property (4.1) and as observed in Remark 4.2, G would then have a convex subgroup properly containing a conjugate of C. Since C was assumed maximal, this is impossible. Fix a primitive element a with the property above, and letḡ be the homeomorphism defined as the identity outside J and agreeing with ρ(g) on J. Define ρḡ by ρḡ(a) =ḡρ(a), and ρḡ(b) = ρ(b) for any other generator of F , and ρḡ(z) = ρ(z). Sinceḡ commutes with ρ(z) = ρḡ(z), the new action ρḡ is a representation of G. Moreover, by changingḡ by some power if necessary, we have that ρḡ(a) has a fixed point in J. This ends the proof of Claim 4. To finish the proof of Proposition 4.12 (and thus that of Theorem 1.1), we note that the flexibility of ρ from Claim 4 together with the statement of Lemma 4.5 implies that the order is non-isolated, giving the desired contradiction. Figure 2 . 1 : 21Classical ping-pong on two generators Figure 2 . 21 shows an example of the dynamics of such an action of F 2 = a, b . Lemma 2. 6 . 6Given a ping-pong action ρ 0 of (G, S) with domains {D 0 (a)} a∈S∪S −1 following Convention 2.3, one can find another ping-pong action ρ of (G, S) with domains {D(a)} a∈S∪S −1 such that:1. the action ρ is piecewise linear,2. there exists µ > 1 such that for any a ∈ S ∪ S −1 , one has ρ(a) \| D(a −1 ) ≥ µ,and 3. the actions ρ 0 and ρ have the same ping-pong configuration. Proof. Let ρ 0 be as in the statement of the Lemma. For each a ∈ S ∪ S −1 , replace the original domains by smaller domains D(a) ⊂ D 0 (a), chosen small enough so that the largest connected component of D(a) is at most half the length of the smallest connected component of S 1 \ D(a −1 ). We require also that D(a) has exactly one connected component in each connected component of D 0 (a). Now define ρ(a) as a piecewise linear homeomorphism that maps connected components of S 1 \ D(a −1 ) onto connected components of D(a) linearly following the assignment λ a . Convention 2.3, for any s ∈ S ∪ S −1 and connected component J of S 1 \ D(s −1 ), there exists I ∈ π 0 (D(s)) such that ρ(s)(J) = I, so each vertex does indeed have an outgoing edge. Moreover, if J = J is a different connected component, then ρ(s)(J) ∩ ρ(s)(J ) = ∅, so each vertex I has a unique incoming edge. This shows that Γ a is a union of disjoint cycles, and it remains only to prove that the graph is connected. To show connectivity, let I − be a connected component of D(a −1 ) and consider the connected component J + of S 1 \ D(a) such that ρ(a −1 )(J + ) = I − . Let I + 1 and I + 2 be the connected components of D(a) (possibly the same) which are adjacent to J + on either side. By definition of the graph Γ a , the intervals I + 1 , I − , I + 2 are consecutive vertices in the same cycle of the graph. And vice versa: if three intervals I + 1 , I − , I + 2 are consecutive vertices, then J + := ρ(a)(I − ) is the connected component of S 1 \ D(a) adjacent to both I + 1 and I + 2 Lemma 3. 4 . 4Suppose that ρ is a ping-pong action of (G, S) with domains D(a). If x 0 ∈ S 1 \ a∈S∪S −1 D(a), then the orbit of x 0 is free and its cyclic order is completely determined by the cyclic order of the elements of {π 0 ( a∈S∪S −1 D(a)) , {x 0 }} and the assignments λ a .The proof is obtained by a careful reading of the standard proof of the classical ping-pong lemma. Details are given in[21, Lemma 4.2]. Lemma 3.5 ([21] Lemma 3.21 and Cor. 3.24). Let ρ : G → Homeo + (S 1 ) be a dynamical realization of a circular order c. Suppose that ρ has a minimal invariant Cantor set Λ(ρ). Then ρ acts transitively on the set of connected components of S 1 \ Λ(ρ). Lemma 3. 6 . 6Let ρ 0 be a ping-pong action of (G, S) with domains D 0 (a) for a ∈ S ∪ S −1 . Then there exists an action ρ with the same ping-pong configuration as ρ 0 and with domains D(a) satisfyingD(s) ∩ D(t) = ∅ whenever s = t.Proof. Let ρ 0 be a ping-pong action. There are only finitely many points x contained in sets of the form D 0 (s) ∩ D 0 (t) for t = s ∈ S ∪ S −1 . For each such point x, blow up its orbit, replacing each point y ∈ ρ 0 (G)(x) with an interval I y ; if lengths of the I y are chosen so their sum converges, then we obtain a new circle, sayŜ 1 , with a natural continuous, degree one map h :Ŝ 1 → S 1 given by collapsing each I y to the point y. For each s ∈ S ∪ S −1 , let D 1 (s) ⊂Ŝ 1 be the preimage of D 0 (S) under h. Theorem 3.8 ([21] Thm. 1.5). Let G = F n be a free group. A circular order c ∈ CO(G) is isolated if and only if its dynamical realization ρ c : G → Homeo + (S 1 ) is a ping-pong action satisfying Convention 3.7. v 1 , v 2 , . . . , v 4k(a) . Choose a connected component I = [x 1 , y 1 ] of D(a) and glue the oriented edge v 1 v 2 to I so as to agree with the orientation of I ⊂ S 1 . Then glue the edge v 3 v 4 to the connected component of D(a −1 ) containing ρ(a −1 )(x 1 ), according to the orientation in S 1 . Let y 2 denote the other endpoint of this connected component, and glue v 5 v 6 to the connected component of D(a) containing ρ(a)(y 2 ) Figure 3 . 1 : 31The surface associated with the exotic example (left), and its boundary component (right). Figure 3 . 2 : 32The ping-pong domains for ρ(b) (left) and its graph (right). The circle is oriented counterclockwise. of the domains for the ping-pong action are in cyclic order as follows:D(b −1 ), D(a −1 ), D(b −1 ), D(a), D(b), D(a −1 ), D(b), D(a)(we are abusing notation slightly here, using each appearance of D(s) to stand for a connected component of D(s)). SeeFigure 3.1 (left) for an illustration of the domains and the surface Σ constructed in the proof of Theorem 1.3.Since ρ(b) has two hyperbolic fixed points, and ρ(a) has four, this example is not realized by a ping-pong action in PSL(2, R), nor in any finite extension of it. In fact, the ping-pong configuration for ρ(b) alone is atypical, in the sense that it is not the classical ping-pong configuration for a hyperbolic element in PSL(2, R)ρ(b) has a "slow" contraction on the left half of the circle, as two iterations of ρ(b) are needed in order to bring the external gaps of D(b) ∪ D(b −1 ) into the component of D(b) with the attracting fixed point.However, the surface Σ from this construction has one boundary component, as shown inFigure 3.1 (right), so it corresponds to an isolated circular order in CO(F 2 ).Observe that one can create several examples of this kind, by choosing ρ(b) to have two hyperbolic fixed points, but with an arbitrarily slow contraction (i.e. with N connected components for D(b), N ∈ N arbitrary) and then choosing ρ(a) to be a N -fold lift of a hyperbolic element in PSL(2, R). Definition 4. 4 . 4Let G be a discrete group. Let Rep(G, Homeo + (R)) denote the space of representations (homomorphisms) G → Homeo + (R), endowed with the compact-open topology; let Rep # (G, Homeo + (R)) be the subspace of representations with no global fixed points. Definition 4. 7 . 7An element h in a linearly-ordered group (G, ) is called cofinal if for all g ∈ G, there exist m, n ∈ Z such that h m g h n . (4.2) Remark 4.8. Cofinal elements also have a characterization in terms of the dynamical realization: If ρ is a dynamical realization of with basepoint x, then h ∈ G is cofinal if and only if ρ(h) has no fixed point. Indeed, if h is not cofinal, then the point inf{ρ(g)(x) \| h n g for every n ∈ Z} is fixed by ρ(h). Conversely, if h satisfies (4.2), then the orbit of x under ρ(h) is clearly unbounded on both sides. Figure 4 . 41: A crossing in the dynamical realization. Proposition 4. 9 . 9Assume that G is finitely generated and c is an isolated circular order on G. If c is the lift of c to a central extension G of G by Z, then the induced linear order c is isolated in LO( G). Lemma 4 . 13 . 413Let F be a free group of infinite rank and G = F × Z. Then no order in LO(G) is isolated. Claim 4 . 4For any compact set K ⊂ R, there exists ρ agreeing with ρ on K, but not semi-conjugate to ρ. In[4, Prop. 2.4] the authors propose an alternative way of inducing an ordering of G, different from that in[5]. However their method is incorrect, as the following example shows: suppose to have three distinct homeomorphisms f, g, h, with f coinciding with g on one half circle and with h on the other half. Then for any point x ∈ S 1 , there are always two equal points in the triple (f (x), g(x), h(x)). AcknowledgmentsThe authors thank Yago Antolín for suggesting Corollary 1. Orderings and flexibility of some subgroups of Homeo+(R). J Alonso, J Brum, C Rivas, J. Lond. Math. Soc. 36645242J. Alonso, J. Brum, and C. 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[ "Full counting statistics of time of flight images", "Full counting statistics of time of flight images" ]	[ "Izabella Lovas \nDepartment of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary\n", "Balázs Dóra \nDepartment of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary\n", "Eugene Demler \nPhysics Department\nHarvard University\n02138CambridgeMassachusettsUSA\n", "Gergely Zaránd \nDepartment of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary\n" ]	[ "Department of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary", "Department of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary", "Physics Department\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Theoretical Physics\nMTA-BME Exotic Quantum Phases \"Momentum\" Research Group\nBudapest University of Technology and Economics\n1111BudapestHungary" ]	[]	Inspired by recent advances in cold atomic systems and non-equilibrium physics, we introduce a novel characterization scheme, the time of flight full counting statistics. We benchmark this method on an interacting one dimensional Bose gas, and show that there the time of flight image displays several universal regimes. Finite momentum fluctuations are observed at larger distances, where a crossover from exponential to Gamma distribution occurs upon decreasing momentum resolution. Zero momentum particles, on the other hand, obey a Gumbel distribution in the weakly interacting limit, characterizing the quantum fluctuations of the former quasi-condensate. Time of flight full counting statistics is demonstrated to capture (pre-)thermalization processes after a quantum quench, and can be useful for characterizing exotic quantum states such as many-body localized systems or models of holography.	10.1103/physreva.95.053621	[ "https://arxiv.org/pdf/1612.02837v2.pdf" ]	27,006,018	1612.02837	bbec27c8fcf1068aaddfb2aa57af4934023d1577	Full counting statistics of time of flight images Izabella Lovas Department of Theoretical Physics MTA-BME Exotic Quantum Phases "Momentum" Research Group Budapest University of Technology and Economics 1111BudapestHungary Balázs Dóra Department of Theoretical Physics MTA-BME Exotic Quantum Phases "Momentum" Research Group Budapest University of Technology and Economics 1111BudapestHungary Eugene Demler Physics Department Harvard University 02138CambridgeMassachusettsUSA Gergely Zaránd Department of Theoretical Physics MTA-BME Exotic Quantum Phases "Momentum" Research Group Budapest University of Technology and Economics 1111BudapestHungary Full counting statistics of time of flight images numbers: 6785-d4250Lc0530Jp6785Hj Inspired by recent advances in cold atomic systems and non-equilibrium physics, we introduce a novel characterization scheme, the time of flight full counting statistics. We benchmark this method on an interacting one dimensional Bose gas, and show that there the time of flight image displays several universal regimes. Finite momentum fluctuations are observed at larger distances, where a crossover from exponential to Gamma distribution occurs upon decreasing momentum resolution. Zero momentum particles, on the other hand, obey a Gumbel distribution in the weakly interacting limit, characterizing the quantum fluctuations of the former quasi-condensate. Time of flight full counting statistics is demonstrated to capture (pre-)thermalization processes after a quantum quench, and can be useful for characterizing exotic quantum states such as many-body localized systems or models of holography. Inspired by recent advances in cold atomic systems and non-equilibrium physics, we introduce a novel characterization scheme, the time of flight full counting statistics. We benchmark this method on an interacting one dimensional Bose gas, and show that there the time of flight image displays several universal regimes. Finite momentum fluctuations are observed at larger distances, where a crossover from exponential to Gamma distribution occurs upon decreasing momentum resolution. Zero momentum particles, on the other hand, obey a Gumbel distribution in the weakly interacting limit, characterizing the quantum fluctuations of the former quasi-condensate. Time of flight full counting statistics is demonstrated to capture (pre-)thermalization processes after a quantum quench, and can be useful for characterizing exotic quantum states such as many-body localized systems or models of holography. I. INTRODUCTION One of the fundamental principles of modern theory of strongly correlated many-body systems is emergent universal behavior. For example, in the vicinity of a thermal phase transition, one finds universal behavior of correlation functions determined by the nature of the transition but not the microscopic details [1][2][3][4]. Close to criticality, the behavior of correlation functions is just determined by the dimensionless ratio of the system size and one emergent lengthscale: the correlation length [2,4]. This statement is expected to hold beyond two point correlation functions. Higher order correlation functions and distribution functions should also obey hyper-scaling property: they are universal functions of the the system size to the correlation length. While hyperscaling has been well studied theoretically [2,4], it has not been observed in experiments so far. In quantum systems we expect manifestations of emergent universality to be even stronger. For example, we expect that a broad class of one dimensional quantum systems can be described by a universal Luttinger theory [5][6][7][8][9]. This powerful approach demonstrates that long distance correlation functions as well as low energy collective modes are described by a universal theory which is not sensitive to details of underlying microscopic Hamiltonians. This powerful paradigm of universality has been commonly discussed in the context of two point correlation functions, such as probed by scattering and tunneling experiments [10][11][12][13][14][15]. In principle, to fully characterize these in or out of equilibrium quantum states at every instant, one should reconstruct them by performing Quantum State Tomography. In practice, however, quantum state tomography is restricted to tiny quantum systems [67]. The most complete information on the many-body wave function can be obtained through investigating the full distribution of some properly chosen physical observables [16,[19][20][21][22][23][24][25]. Observing universality in these distribution func-tions would therefore be a direct and striking demonstration of the universal nature of the entire many-body state and emergent universality. Unfortunately, in traditional solid state systems, experimental studies of such distribution functions are extremely challenging. Most of experimental techniques rely either on averaging over many 1d systems, such as in a crystal containing many 1d systems [15,26,27], or on long time averaging such as in STM experiments [28][29][30]. As a result, no theoretical work has been done on understanding universality classes of distribution functions of observables in quantum systems. Recent progress with ultracold atoms, however, makes it possible to perform experiments that look like textbook classical measurements of quantum mechanical wavefunctions on individual 1d systems [31]. By collecting a histogram of single shot results one can obtain full distribution functions. In particular, quasi-one dimensional gases have provided an interesting test-ground to realize and test low dimensional quantum field theories [36]. In a peculiar setup, a pioneering series of sophisticated experiments was performed [16,17,37] to access the probability distribution function (PDF) [38,40,71] of matter-wave interference fringes of a coherently split one-dimensional Bose gas and to gain deeper insight into phase correlations. Here we propose that even the most wide-spread and extensively used standard Time of Flight (ToF) images contain a lot more precious information -never exploited so far, which can be extracted and used to characterize the quantum state observed. In particular, we propose to study the full distribution function of Time of Flight images, a procedure we dubbed Time of Flight Full Counting Statistics to parallel the method used in nanophysics [19][20][21][22][23]. ToF imaging is in fact probably the most widespread tool to investigate cold atomic systems [31][32][33][34], and a wide range of other, more sophisticated experimental techniques like Bragg spectroscopy or matter-wave interference are also based on ToF measurements. In a arXiv:1612.02837v2 [cond-mat.quant-gas] 15 Apr 2022 ToF experiment with quasi one and two dimensional systems, atoms quickly cease to interact after being released from a trap, and therefore their position after some time is directly proportional to their momenta in the initial quantum state. ToF images thus picture the momentum distribution of the atoms in the initial interacting state (see Fig. 1). They contain, however, a lot more information than just the average intensities or their correlations [75] they contain the full probability distribution function (PDF) of particles at each momentum, which is expected to reflect the universal behavior of low dimensional quantum systems or critical states. In this work, we concentrate on this so far unexploited information, accessible in a wide range of experimental settings for many experimental groups. To demonstrate this approach, we analyze the fingerprints of abundant quantum fluctuations on one dimensional interacting quasi-condensates, and determine the complete distribution of the time of flight image. The particular setup considered is sketched in Fig. 1: a one dimensional Bose gas, confined to a tube of length L, is suddenly released from the trap. Due to the rapid expansion in the tightly confined directions (not shown in Fig. 1), the interactions become quickly negligible, and it is enough to consider a free, one dimensional propagation along the longitudinal axis [61]. After expansion time t, the density profile is imaged by a laser beam at position R, which measures the integrated density of particles within the spotsize of the laser, ∆R, I R,∆R (t) ≡ ∞ −∞ dx e −(x−R) 2 /(2∆R 2 )ψ † (x, t)ψ(x, t). (1) t = 0 t > 0 R ΔR Laser beam L V(x) V(x)=0 FIG. 1. Sketch of ToF experiment with quasi-one-dimensional Bose gas. At t = 0 the atoms, initially confined to a tube of length L, are released form the trap. Interactions between the particles are typically short ranged, and become quickly negligible due to the rapid expansion in transverse direction (not shown here). After propagation time t, the density profile of the expanded cloud is investigated by taking an absorption image at position R with a laser beam of waist ∆R. Atoms expand freely after release from the trap, and the distribution of the measured intensity provides direct information on the structure of the initial quantum state. Hereψ(x, t) denotes the bosonic field operator, and we assumed [41] a Gaussian laser intensity profile [42]. Since bosons propagate freely during the ToF expansion, Eq. (1) provides information on the correlations in the initial state of the system at time t = 0. In particular, for R L and ∆R R, the measured intensity can be interpreted as the number of particlesN p with a given initial momentum, p = mR/t [32]. Let us note that the same information about momentum correlations can also be obtained by performing another experimental procedure, the focusing technique [60][61][62] (see Appendix C). As we discuss later, apart from minor corrections, our results apply for this type of measurement as well [64], which offers, however, a more accurate approach to measuring distributions in momentum space than the usual ToF technique. We determine the full distribution of the operator I R,∆R , and show that it contains important information on the quantum fluctuations of the condensate, leading to the emergence of several universal distribution functions. Analysing first the image of a T ≈ 0 temperature condensate, we show that intensity distributions at finite momenta follow Gamma distribution, and reflect squeezing. The signal of zero momentum particles is, on the other hand, shown to follow a Gumbel distribution in the weakly interacting limit, a characteristic universal distribution of extreme value statistics, and reflecting large correlated particle number fluctuations of the quasicondensate. We also extend our calculations to finite temperatures and show that the predicted Gumbel distribution should be observable at realistic temperatures for typical system parameters. Then we study the image of the condensate after a quench, and show how thermalization of the condensate manifests itself as a crossover to an -also universal -exponential distribution in the time of flight full counting statistics. II. THEORETICAL FRAMEWORK To reach our main goal and to determine the full distribution ofÎ R,∆R (t) for a one dimensional interacting Bose gas, we shall make use of Luttinger-liquid theory, [39,40] and compute all moments ofÎ R,∆R to show that for long times of flight and large enough distances Î n R,∆R (t) → ∞ 0 dI I n W p (I)(2) with n positive integer. The function W p (I) can be viewed as the probability distribution function (PDF) of the intensity, measuring the number of particles N p ∼ I with momentum p = mR/t. Notice that the function W p (I) depends implicitly on the time of flight as well as on the momentum resolution ∆p, suppressed for clarity in Eq. (2). Luttinger-liquid theory describes the low energy properties of quasi-one-dimensional bosons [7] as well as a wide range of one-dimensional systems [5]. Long wavelength excitations of a Luttinger liquid are collective bosonic modes, described in terms of a phase field,φ(x) [5]. For quasi-condensates, the field operatorψ(x) is directly related to this phase operator [5] ψ (x) ≈ √ ρ e iφ(x) ,(3) with ρ the average density of the quasi-condensate. Fluctuations of the density generate dynamical phase fluctuations, described by a simple Gaussian action [5,31], S = K 2π dt dx 1 c (∂ t φ) 2 − c (∇φ) 2 ,(4) that involves the sound velocity of bosonic excitations, c, and the Luttinger parameter, K. The dimensionless parameter K characterizes the strength of the interactions: for hard-core bosons K → 1, corresponding to the socalled Tonks-Girardeau limit [43,44], while for weaker repulsive interactions K > 1, with K → ∞ corresponding to the non-interacting limit [5]. The connection between the parameters c and K and the microscopic parameters is model dependent. For a weak repulsive Dirac-delta in- teraction, V (x − x ) = g δ(x − x ), both are determined by perturbative expressions [7] c ∼ = gρ m , K ∼ = πρ mc = π ρ mg ,(5) with the Planck constant. To evaluate the moments of the operatorÎ R,∆R (t), we first observe that the interactions between the atoms become quickly negligible once the confining potential is turned off and the atoms start to expand. Therefore the fieldsψ(x, t) evolve almost freely in time for times t > 0, with a time evolution described by the Feynman propa- gator, G(x, t) ∼ e imx 2 /(2 t) / √ i t, ψ(x, t) = dx G(x − x , t)ψ(x ) .(6) For large times, and points far away from the initial position of the condensate one finds thatψ(x, t) is approximately equal to the Fourier transform of the fieldψ p at a momentum p = mx/t. This relation becomes exact if, instead of a simple time of flight experiment, one uses the previously mentioned focusing technique (see Appendix C), allowing to reach much better resolutions [60][61][62]. Applying the representation Eq. (3) and the Gaussian action Eq. (4), we can evaluate Î n R,∆R (t) in any moment [39,40], and construct the intensity distribution W p (I). Using open boundary conditions for the phase operator we obtain, e.g. W p (I) = ∞ −∞ j dτ j e −τ 2 j /2 √ 2π δ I − N ∆ p √ 2π g ({τ j }) ,(7) with j = 1, 2, . . . labeling the auxiliary variables τ j and the function g ({τ j }) determined by the double integral g ({τ j }) = 1/2 −1/2 du dv e −∆ p 2 (u−v) 2 /2+i p(u−v)(1− u+v 2R/L ) · exp   i j τ j e −ξ h πj/(2L) √ K j cos π j u + j π 2 − cos π j v + j π 2 .(8) The derivation of Eqs. (7) and (8) is detailed in Appendix A. The healing length ξ h ≡ /(mc) here serves as a short distance cutoff [45], N = Lρ denotes the total number of particles, and we introduced the dimensionless time of flight momentum and its resolution p ≡ mR t L , ∆ p ≡ m∆R t L ,(9) both measured in units of /L. We note that the intensity measured in a focusing experiment also follows a distribution of the form of Eq. (7), apart from a small change in the function g ({τ j }). As discussed in Appendix C, in a focusing experiment Eq. (6) yields just the Fourier transform of the fieldψ, and the real space coordinates R and ∆R are directly proportional to the dimensionless momenta,p and ∆p. As a technical consequence, the term exp(−ip(u 2 −v 2 )L/(2R)) is absent from the integral giving g ({τ j }), but for a giveñ p and ∆p, the shape of distribution is hardly affected by this minor modification in the relevant limit, R L. Therefore all the results presented below apply also for intensities measured by the refocusing method. III. EQUILIBRIUM QUANTUM FLUCTUATIONS We evaluated Eqs. (7) and (8) by performing classical Monte Carlo simulations. Already the expectation values, Î R carry valuable information, since they account for the size of interaction induced quantum (or thermal) fluctuations of bosons with momentum p = mR/t. They are proportional to N p and to the corresponding momentum dependent effective temperatures. The momentum and temperature dependence of N p has been studied theoretically [5] and experimentally [46,47] in detail (see also the following subsections and Appendix D). In an infinitely long Luttinger liquid, in particular, N p falls of as ∼ 1/\|p\| 1−1/2K at T = 0 temperature, while at finite temperatures its value depends on p: For small momenta it saturates to a constant proportional to 1/T 1−1/2K ≈ 1/T , while at large momenta the power law behavior is recovered. For weak interactions, the cross-over between these two regimes occurs through a regime, where a power law behavior is observed with a modified exponent (see Appendix D). ∆p = 0.1 × 2π ∆p = 0.5 × 2π ∆p = 1 × 2π ∆p = 2 × 2π ∆p = 3 × 2π FIG. 2. Distribution of normalized intensity I (symbols) at T = 0 temperature, plotted for different momentum resolutions, ∆ p. We used K = 10, p = 15 × 2π and ξ h /L = 0.002. Solid lines are fits with the Gamma distribution from Eq. (10). The distribution smoothly evolves from exponential to Gamma as ∆p increases, and reflects the two-mode squeezed structure of the Bogoliubov ground state in momenta p and −p. Inset: parameter of the fitted Gamma distribution α as a function of ∆ p. The average being well understood, here we concentrate on the shape of the full intensity distribution. Therefore, we introduce the normalized intensity I =Î R,∆R / Î R,∆R , and determine the corresponding distribution function W p ( I). The intensity distributions for p = 0 and for typical p = 0 exhibit drastically different characters; the zero momentum intensity is just associated with particles in the quasi-condensate, while intensities corresponding to p = 0 reflect quantum fluctuations to states of momentum p. We shall therefore discuss these separately. A. Zero-temperature intensity distribution of finite momentum particles Let us first discuss the intensity distribution of finite momentum particles, p = 0, at T = 0 temperature, allowing us to take a glimpse at the structure of interactiongenerated quantum fluctuations. Fig. 2 shows the typical structure of the distribution function W p ( I) for a moderate Luttinger parameter K = 10 for various momentum resolutions ∆ p. The shape of W p ( I) has a strong dependence on the resolution ∆ p, and is well described by a Gamma distribution W p =0 ( I) ≈ α α Γ(α) I α−1 e −α I .(10) The parameter α here incorporates the momentum resolution, ∆ p, and increases linearly with it (see inset of Fig 2). For good resolutions α ≈ 1, an exponential distribution is recovered, W p =0 ( I) ≈ e − I , for ∆ p 2π. These observations can be understood in terms of the Bogoliubov approximation [48], valid for weak interactions and short system sizes. For small sizes of the laser spot, i. e. ∆ p 2π, the intensity, Eq. (1) can be interpreted as the number of particles with dimensionless wave number p = mR/t. The Bogoliubov ground state has a two-mode squeezed structure, i.e., particles with momenta p and −p are always created in pairs, implying perfect correlations at the operator level,N p =N −p . This two-mode squeezed structure gives rise to a geometric distribution for the particle numberN p [50], and the exponential intensity distribution observed is just the continuous version of this geometric distribution. Moreover, Bogoliubov theory predicts vanishing correlation between nonzero momenta \|p\| = \|p \| [48]. Therefore, the total number of particles in a given momentum window ∆p can be viewed as the sum of ∼ ∆ p/2π independent, exponentially distributed random variables, with approximately equal expectation values [49] N p ≈ ρ π 2K\|p\| .(11) The Gamma distribution with a parameter α ∝ ∆ p thus arises as the weighted sum of independent exponential variables. The precise prefactor here depends on the shape of the intensity profile in Eq. (1). For a Gaussian profile we find α ≈ 4.1 ∆ p/(2π), while other profiles amount in other numerical prefactors of O(1). Though the Bogoliubov approach has only a limited range of validity, a similar crossover from exponential to Gamma distribution persists even for strong interactions (see Appendix F). B. Quasicondensate distribution at T = 0 temperature Let us now turn to the zero-momentum distribution, corresponding to the number of particles in the quasicondensate, and exhibiting a completely different behavior, shown in Fig. 3. The distribution, plotted for different interaction strengths K, converges quickly to a so-called Gumbel distribution as K increases. This distribution, arising frequently in extreme value statistics [51], is expressed as W Gumbel ( I) = π √ 6 exp π √ 6 I − γ − exp π √ 6 I − γ ,(12) with γ ≈ 0.5772 the Euler constant. We can prove that the extreme value distribution (12) follows from particle number conservation combined with the fact thatN p =0 display exponential distributions with 3. PDF of the normalized variable (Î − Î )/δI for p = mR/t = 0, plotted for different Luttinger parameters K, with δI referring to standard deviation. We used periodic boundary conditions to compare to analytical results. For weak interactions (K 1) the PDF converges to Gumbel distribution, Eq. (12) (solid line), also predicted by a particle number preserving Bogoliubov approach. We used ∆ p = 0.1 × 2π and ξ h /L = 0.002. expectation values N p =0 ∼ 1/\|p\|. Particle number conservation relates the fluctuations of the number of particles in the condensate,N 0 with those of p = 0 particles, N 0 = N − p =0N p . This can be achieved within the particle number preserving Bogoliubov approach of Ref. [53] by performing a second order expansion in the bosonic fluctuations. As discussed above, all finite momentum particle numbersN p =0 exhibit exponential distributions with expectation values ∼ 1/\|p\|. Therefore, as we show in Appendix E, the distribution of the sum p =0N p can be rewritten analytically, and reexpressed as the maximum of a large number of independent, identically distributed exponential random variables, leading to the observed Gumbel distribution. For strong interactions K ∼ 1, the zero-momentum distribution starts to deviate form the Gumbel distribution, Eq. (12), considerably. However, the observed distribution is still universal in the sense that it does not depend on the momentum cutoff, and remains unchanged if we consider a Bogoliubov spectrum instead of the linear dispersion relation of a Luttinger liquid. I R2 , Î n1 R1Î n2 R2 (t) → ∞ 0 dI I n1 1 I n2 2 W (I 1 , I 2 ),(13) for any positive integers n 1 and n 2 . The previous calculations can be extended to compute these probability distributions with little effort (see Apendix B for details). Without analysing them in detail, here we just briefly discuss the joint distribution function of the of p = 0 and p = 0 modes, providing further evidence for the role of particle number conservation behind the emergent extreme value statistics. The distribution of the normalized variables I 0 and I 1 , corresponding to dimensionless momenta p 0 = 0 and p 1 = 2π is plotted in Fig. 4 for strong (K = 2) and weak (K = 10) interactions. The wave number resolution was chosen to be such that particles contributing to the signals I 0 and I 1 have well defined momenta. The joint PDFs reveal strong anticorrelation between the intensities I 0 and I 1 , interpreted as particle numbers N 0 and N 1 , for all interaction strengths, persisting for higher values of p 1 . Anticorrelations manifest in the fact that the joint PDF is sharply peaked around the line I 0 + I 1 = const., implying that a high intensity I 0 is typically accompanied by a low signal I 1 . The origin of these anticorrelations is particle number conservation: a particle with non-zero wave number p 1 , removed from the quasi-condensate, leaves a 'hole' behind, eventually appearing as anticorrelation in the joint PDF of I 1 and I 0 . D. Finite temperature effects and thermal depletion of the quasi-condensate So far we focused on the limit of T = 0 temperature. At finite temperatures, modes with energies E = p c < ∼ k B T get thermally excited and, at some point, destroy the quasi-condensate. As we show now, this thermal depletion of the quasi-condensate is controlled by the dimensionless temperature T = 1 K k B T ∆ ,(14) with ∆ = h c/L the 'level spacing', i.e. the typical separation of sound modes in a condensate of size L. Fig. 5 displays the intensity distribution of the zeromode, derived in Appendix A, as a function of T for experimentally relevant parameters [54,65]. The PDF retains the characteristic shape of a Gumbel distribution for realistic but small temperatures, T < ∼ 1, though the distribution broadens with increasing temperature. At temperatures T > ∼ 1, however, the PDF turns quickly into an exponential distribution. This behavior and the crossover scale in Eq. (14) are deeply related to the structure of correlations in a finite temperature Luttinger liquid. At T = 0 temperature, a The PDF crosses over from the zero temperature Gumbel distribution, Eq. (12), to an exponential distribution, as a signature of the thermal depletion of the quasi-condensate by the thermally populated p = 0 modes. The experimentally accessible temperature range, T ∼ 30 nK − 120 nK [54], corresponds to T ∼ 0.12 − 0.46. We used N = 3500, and L = 39 µm (density ρ = 90 µm −1 ), and a chemical potential µ/h = 1.6 kHz, implying K ≈ 77, c ≈ 2, 7 mm/s and ξ h /L ≈ 0.007 for 87 Rb atoms. We assumed ∆ p/(2π) = 0.1, corresponding to a time of flight t = 1 s, and a real space resolution ∆R = 12 µm, but shorter times of flight can also be applied using a focusing method, yielding similar images. bosonic Luttinger liquid exhibits power law correlations at distances larger than the healing length [7,40]. At finite temperatures, however, these power law correlations turn into an exponential decay beyond the thermal wavelength [7], where ψ † (x)ψ(0) ≈ ρ 2ξ h λ T 1/2K e −\|x\|/ξ T , for \|x\| > λ T .(15) Notice that the thermal correlation length ξ T appearing here (often denoted by λ T in the literature) is proportional to but not identical with the thermal wavelength of the sound modes, denoted here by λ T = c/(πk B T ); being influenced by the stiffness of the condensate, ξ T is larger by a factor of 2K [55], ξ T = 2 K λ T , implying that ξ T can be several orders of magnitude larger than λ T in a weakly interacting condensate. Notice that the product Kc ∼ ρ/m is independent of the interaction strength by Galilean invariance [58]. Thus the correlation length ξ T ∼ 2 ρ/(mk B T ) is independent of the strength of interaction. It is precisely this length scale that appears in Eq (14), which can be re-expressed asT = L/(ξ T π 2 ). The conditionT < ∼ 1 thus corresponds to the inequality L < ∼ ξ T π 2 ensuring that the phase of the condensate remains close to uniform for sizeable segments of gas. As shown in Appendix D, the number of particles in the p = 0 mode is also determined by this ratio, N 0 ≈ N 2ξ T /L. Thus T < ∼ 1 also implies that at least about 20 % of the particles remain in the homogeneous condensate. As stated earlier in this section, this condition is independent of the interaction strength. Indeed, although the discussion above focused on the weakly interacting limit, K 1, we observe a similar crossover to an exponential function even for strong interactions, for which λ T ∼ ξ T (see Appendix F). The exponential distribution emerging for T > ∼ 1 can be understood as a consequence of the thermal depletion of the condensate by low energy p = 0 modes. Considering the latter naively as particle reservoirs leads to Prob(N p=0 = n) ∝ e −βµ eff n , with some effective chemical potential µ eff , set by the population of low energy modes. IV. DISTRIBUTION AFTER INTERACTION QUENCHES So far we have focused on applying Time of Flight Full Counting Statistics to study equilibrium correlations. Even more interestingly, we can use it to study Distribution of the normalized zero momentum intensity Ip=0 after an interaction quench, for different holding times, τ h = 1.3µs − 177µs, measured in dimensionless units,τ h = τ h c f /L. Distributions are plotted for two different quenches of durations τ = 0.71µs (rapid), and τ = 71µs (slow). We have used N = 3684, L = 39µm and a chemical potential µ/h = 1.6 kHz, corresponding to K0 = 80, c0 = 2.7mm/s, and ξ 0 h = 0.27µm, and assumed an interaction quench to K f = 7, yielding c f = c0K0/K f = 30.8 mm/s. We assumed a modest momentum resolution, ∆p/(2π) = 0.1. Similar to the finite temperature thermalization plotted in Fig. 5, the PDF after a rapid quench crosses over from the equilibrium Gumbel distribution, Eq. (12), to an exponential distribution asτ h increases, even though the number of excitations in the system remains constant after the quench. For a slower quench, the PDF for short holding times τ h is much wider than the Gumbel distribution, showing that increasing interactions have time to deplete the quasi-condensate during the quench protocol, resulting in larger particle number fluctuations. As in the case of rapid quench, for larger holding times this PDF crosses over to a thermal distribution. non-equilibrium dynamics and to gain information about the non-equilibrium states and time evolution of a system after a quantum quench [56,57]. Here we demonstrate this perspective by focusing on interaction quenches, i.e., on changing g using a Feshbach resonance [31]. For the sake of simplicity, we consider linear quench procedures of g, where the product c(t)K(t) = πρ/m remains constant by Galilean invariance [58], while c/K changes approximately linearly over a quench time τ [63]. After the quench, the atoms are held in the trap for an additional holding time τ h , while the final parameters c f and K f remain constant, and the ToF experiment is performed only afterwards. For short enough quench times τ , the quench creates abundant excitations. Here we focus on these excitations and concentrate therefore on zero temperature quenches. The initial state is then simply the Gaussian ground state wave function corresponding to the initial parameters c 0 and K 0 . Moreover, the wave function remains Gaussian during the time evolution [59], and can be expressed as Ψ ({φ k }, t) ∼ k>0 exp (−σ k (t) φ k φ −k ) , with the parameters σ k (t) obeying simple differential equations [59]. This observation allows us to evaluate the full distribution of the intensityĨ p , by only slightly modifying the derivation outlined in Appendix A. Fig. 6 shows the intensity distribution of the zero mode,Ĩ 0 , for a large quench between Luttinger parameters K 0 = 80 and K f = 7, as a function of the holding time after the quench, τ h . The distributions are plotted for two different quench times τ . After a rapid quench, for short holding times the probability density function still resembles the Gumbel distribution, Eq.(12), valid for the equilibrium case. However, we observe a crossover to an exponential distribution upon increasing the holding time, τ h . The phenomenon observed is similar to the finite temperature thermalization plotted in Fig. 5, even though the number of excitations in each mode k is a conserved quantity for the Luttinger model considered here, and the final state is definitely not thermal. The structure of this non-thermal final state can be understood as follows. After long enough holding times, τ h , the distribution of the particle numberN p looks thermal for each momentum p. Based on this thermal, exponential distribution ofN p , one can define an effective inverse temperature β p [68], Prob(N p = n) ∝ e −βpεpn , with ε p denoting the quasiparticles' dispersion relation. The non-thermal nature of the state is reflected by the fact that in contrast to a thermal state characterized by a single inverse temperature, β p strongly depend on the momentum p [68]. Similar pre-thermalization phenomena are encountered in some quench experiments on closed, cold atomic systems, where the long-time expectation value of local observables can be well described by a thermal ensemble, despite the non-equilibrium state of the system [69,70]. For a slower quench, the distribution for short holding times τ h gets much wider compared to the distribution after a sudden quench. This widening can be understood by noting that the interactions increase during the quench protocol. For slower quenches these stronger interactions have time to deplete the quasi condensate while the quench is performed, manifesting in more pronounced particle number fluctuations for short holding times. For larger holding times, however, we observe a crossover to a thermal distribution, similarly to the case of a rapid quench. For both quench procedures, the time scale of thermalization of the zero-mode is very fast, and for realistic parameters it falls to the range of ∼ 0.1 ms. V. CONCLUSIONS In this work we have proposed a novel approach to analyse time of flight images, namely to measure the full probability distribution function (PDF) of the intensities in a series of images. Similar to full counting statistics [19,20], the PDF of the intensity contains information on the complete distribution of the number of particles N p with a given momentum p, beyond its expectation value and variance, and reveals the structure of the quantum state observed and its quantum fluctuations. This so far unexploited information in ToF images reflects the emergent universal behavior of strongly correlated low dimensional quantum systems. We have demonstrated the perspectives of this versatile method on the specific example of an interacting onedimensional condensate. We have first focused on the equilibrium signal, and have shown that the intensity distribution of the image for p = 0 has an exponential character (deformed into a Gamma distribution with decreasing resolution), reflecting the squeezed structure of the superfluid ground state. The p = 0 intensity distribution, on the other hand, reveals fluctuations of the quasi-condensate, and turns out to be a Gumbel distribution in the weakly interacting limit, a familiar universal distribution from extreme value statistics. We have shown that the Gumbel distribution derives from particle number conservation, combined with large, interaction induced quantum fluctuations of the small momentum modes. We have also shown that these intriguing fingerprints of quantum fluctuations remain observable in a finite system at small but finite temperatures within the experimentally accessible range, but the predicted Gumbel distribution is destroyed once the small momentum thermal modes thermalize the p = 0 quasi-condensate mode. ToF full counting statistics can be used in a versatile way to study non-equilibrium dynamics and thermalization. As an example, we considered an interaction quench, and have shown that the intensity statistics of the p = 0 mode displays clear signatures of (pre-)thermalization as a function of the holding time after the quench, whereby the original Gumbel distribution, discussed above turns into a quasi-thermal exponential (Gamma) distribution. This universal exponential distribution describes a condensate connected to a particle reservoir, formed by the p > 0 modes. One can also go beyond measuring the PDF of the intensity at a given point of the ToF image by measuring the complete joint distribution functions, W (I p , I p ), rather than measuring just intensity correlations, I p I p . As an example, we have determined this joint distribution for the p = 0 quasi-condensate intensity and the p = 0 intensities, and have shown that W (I 0 , I p ) exhibits strong negative correlations, induced by particle number conservation. Clearly, our analysis can be generalized to multipoint distributions, W ({I p }), still expected to reflect universality, though the experimental and theoret-ical accessibility becomes less obvious for these complex quantities. As demonstrated here through the simplest example, ToF full counting statistics is expected to give insight to the exotic quantum states of various interacting quantum systems. Besides investigating the emergent universal behavior of low dimensional quantum systems, Time of Flight Full Counting Statistics could also be applied to study exotic quantum states in higher dimensional, fermionic or even anyonic systems where it is supposed to reflect the quantum statistics of particles. Another interesting direction would be the analysis of ToF full counting statistics at quantum critical points, such as the quantum critical points of the transverse field Ising model or that of spinor condensates [66], e.g., where quantum fluctuations get stronger and bare particles cease to exist. It is also a completely open question, how ToF distributions reflect the structure of a many-body localized state, but the images of chaotic and integrable models are also expected to exhibit different ToF full counting statistics. Here we derive the probability density function of the intensity, W p (I), both for the zero temperature case and for finite temperatures. First we perform the calculations at T = 0, then we generalize the results to finite temperatures. In order to derive the PDF at T = 0, stated in Eqs. (7) and (8), we have to calculate the momenta Î n R,∆R (t) for all n. First we express the intensityÎ R,∆R (t) in terms of the field operators at t = 0 by substituting the free propagator G(x, t) = m 2πit exp(imx 2 /(2t)) into Eq. (1), and use the density-phase representation (3) to arrive at I R,∆R (t) =ρ m∆R √ 2π t L/2 −L/2 dx 1 L/2 −L/2 dx 2 e − m 2 ∆R 2 2 t 2 (x1−x2) 2 e i mR t (x1−x2)− im 2 t (x 2 1 −x 2 2 ) e −i(φ(x1,0)−φ(x2,0)) .(A1) The nth momentum ofÎ R,∆R (t) involves the 2n point correlator of the phase operator. This can be determined by using the Fourier expansion ofφ, for open boundary conditions given bŷ φ(x) = 1 √ Lφ 0 + k>0 π KL\|k\| e −ξ h \|k\|/2 cos(k(x + L/2)) b k +b † k ,(A2) with k = πj/L, j ∈ Z + . Hereb † k andb k are bosonic creation and annihilation operators, withb k annihilating the ground state of the system. The inverse of the healing length ξ h serves as a momentum cutoff. All ground state expectation values Î n R,∆R (t) can be calculated by using the normal ordering identity e D kbk +D * kb † k = e D * kb † k e D kbk e −\|D k \| 2 /2 with D k = π/(KL\|k\|)e −ξ h \|k\|/2 cos(k(x+L/2)), leading to Î n R,∆R (t) = Lρ ∆p √ 2π n ... 1/2 −1/2 n i=1 (du i dv i C(u i , v i ))· exp   − j>0 e −ξ h πj/L 2Kj n i=1 cos πju i + jπ 2 − cos πjv i + jπ 2 2 ,(A3) with C(u, v) = e − ∆p 2 2 (u−v) 2 +ip(u−v)(1− u+v 2R/L ) ,(A4) and dimensionless variablesp and ∆p given by Eq. (9). The quadratic sum appearing in the exponent of Eq. (A3) can be decoupled by applying the Hubbard-Stratonovich transformation, performed by introducing a new integration variable τ j for every index j, exp − e −ξ h πj/L 2Kj n i=1 cos πju i + jπ 2 − cos πjv i + jπ 2 2 = ∞ −∞ dτ j √ 2π e −τ 2 j /2 exp i τ j e −ξ h πj/(2L) √ Kj × n i=1 cos πju i + jπ 2 − cos πjv i + jπ 2 . By substituting this expression into Eq. (A3), the integrals over different pairs of variables {u i , v i } can be performed independently, and we arrive at Î n R,∆R (t) = Lρ ∆p √ 2π n ∞ −∞ j>0 dτ j √ 2π e −τ 2 j /2 g ({τ j }) n , with g ({τ j }) given by Eq. (8). Comparing this result with Eq. (2) shows, that the distribution ofÎ R,∆R (t) can indeed be described by a PDF, given by Eqs. (7) and (8). Now we generalize these results to T > 0 temperatures. The Fourier expansion of the phase operator, Eq. (A2), together with the thermal occupation of the modes, b † kb k = 1/(e βck − 1), implies e iφ(x)−iφ(y) = exp   − j>0 e −ξ h πj/L 2K tanh(βcπj/(2L)) × cos πjx + jπ 2 − cos πjy + jπ 2 2 . The only difference compared to the expectation value at T = 0 temperature is the appearance of the thermal occupation factor tanh(βcπj/(2L)). By repeating the derivation above, we find that the distribution function still takes the form Eq. (7), but with a modified function g T ({τ j }) given by g T ({τ j }) = 1/2 −1/2 du dv C(u, v)× exp   i j τ j e −ξ h πj/(2L) K j tanh( βcπj 2L ) cos π j u + j π 2 − cos π j v + j π 2 . Appendix B: Joint distribution function In this appendix we derive a numerically tractable expression for the joint PDF at T = 0 temperature, defined in Eq. (13), by calculating the momenta Î n1 1Î n2 2 (t) for all n 1 and n 2 . Here we introduced the shorthand nota-tionÎ 1 ≡Î R1,∆R1 . By using Eq. (A1) and the Fourier expansion of the phase operator, Eq. (A2), we arrive at Î n1 1Î n2 2 (t) = Lρ √ 2π n1+n2 ∆p n1 1 ∆p n2 2 ... 1/2 −1/2 n1 i=1 (du i dv i C 1 (u i , v i )) n2 l=1 (dũ l dṽ l C 2 (ũ l ,ṽ l ))· exp   − j>0 e −ξ h πj/L 2Kj n1 i=1 cos πju i + jπ 2 − cos πjv i + jπ 2 + n2 l=1 cos πjũ l + jπ 2 − cos πjṽ l + jπ 2 2 ,(B1) with C i (u, v) given by Eq. (A4) with parameters R i and ∆R i for i = 1, 2. Similarly to the calculation presented in Appendix A, the quadratic sum appearing in the exponent of Eq. (B1) can be decoupled by applying a Hubbard-Stratonovich transformation. By introducing a new integration variable τ j for every index j, and performing the integrals over different pairs of variables {u i , v i } and {ũ l ,ṽ l } independently, we arrive at Î n1 1Î n2 2 (t) = Lρ √ 2π n1+n2 ∆p n1 1 ∆p n2 2 ∞ −∞ j>0 dτ j √ 2π e −τ 2 j /2 g 1 ({τ j }) n1 g 2 ({τ j }) n2 , with g i ({τ j }) given by Eq. (8) with parameters R i and ∆R i for i = 1, 2. Comparing this expression to the definition of the joint distribution, Eq. (13), we find that W (I 1 , I 2 ) = ∞ −∞ j dτ j e −τ 2 j /2 √ 2π × δ I 1 − N ∆ p 1 √ 2π g 1 ({τ j }) δ I 2 − N ∆ p 2 √ 2π g 2 ({τ j }) . The distribution can be evaluated by performing a Monte Carlo simulation for the normal random variables τ j , and calculating the two dimensional histogram for I 1 and I 2 . As T increases, the expectation value develops a wide flat region for small momenta p /ξT . For larger momenta /ξT p /λT , the intensity shows a power law decay ∼ 1/p 2 . This behaviour can be explained by the exponential decay of two-point correlations in finite temperature Luttinger-liquids, Eq. (15), with correlation length ξT . For even larger momenta p /λT , we get back the zero temperature results, resulting in a crossover to the different power law behavior Î R,∆R(t) ∼ 1/p. Here the last approximation is valid for p 2k B T /c ≈ /λ T . As already mentioned, this ∼ 1/p 2 decay is consistent with the numerical results plotted in Fig. 7. Appendix E: Gumbel distribution In this appendix we show that the Gumbel distribution (12), arising for weak interactions, can be derived from the structure of the Bogoliubov ground state, by taking into account particle number conservation. In this perturbative approach, the PDF (12) emerges as the distribution of the normalized operator giving the number of particles with zero momentum, N 0 =N p=0 − N p=0 δN p=0 . Here N p=0 denotes the expectation value, and δN p=0 is the standard deviation. For simplicity, we perform the calculations using periodic boundary conditions. As already noted in the main text, particle number conservation implieŝ N p=0 = N − p =0N p ,(E1) with N denoting the total number of particles. Moreover, the two mode squeezed structure of the ground state in non-zero momenta p and −p, resulting in a perfect cor-relationN p =N −p , leads to an exponential distribution for the random variable (N p +N −p )/N , with expectation value π/(KL\|p\|) [48] (see Eq. (11)). For PBC the momentum can only take values p = 2πn /L, so the PDF of the sum p =0N p /N can be written as P   p =0N p /N = x   = nc i=1 (2Kn) ∞ 0 dx 1 e −2K x1 × ∞ 0 dx 2 e −2K 2 x2 ... ∞ 0 dx nc e −2K nc xn c δ x − nc i=1 x i .(E2) Here n c ∼ L/ξ h denotes a cutoff in momentum space, restricting the momentum p to the low energy region, described by linear dispersion relation. The PDF (E2) can be rewritten by introducing new integration variables z 1 = x nc , z 2 = x nc + x nc−1 , ..., and z nc = nc i=1 x i as P   p =0N p /N = x   = (2K) nc n c ! × ∞ 0 dz 1 ∞ z1 dz 2 ... ∞ zn c −1 dz nc e −2K nc i=1 zi δ (x − z nc ) . This result shows, that the PDF associated to the operator p =0N p /N is equivalent to the distribution of the maximum of n c independent, exponentially distributed random variables, with equal expectation values 1/(2K). This observation follows from noting, that the integrand describes independent exponential random variables, subject to the constraint z 1 < z 2 < ... < z nc , with the factor n c ! taking into account all possible orderings of these n c variables. This interpretation explains the emergence of the extreme value distribution W Gumbel . The cumulative distribution function of the maximum of independent random variables can be easily calculated, leading to the probability Prob Ñ 0 < x = Prob   p =0N p N > p =0 N p N − x δN p=0 N   = 1 −   1 − exp    −2K   p =0 N p N − x δN p=0 N        nc ≈ 1 − exp   −n c exp    −2K   p =0 N p N − x δN p=0 N        ,(E3) with the approximation in the third line valid for large K. Here the expectation value p =0 N p /N is given by , plotted for different momentum resolutions ∆ p. We used K = 2, p = 15 × 2π and ξ h /L = 0.002. Similarly to the limit of weak interactions, the distribution smoothly evolves from exponential to Gamma as ∆p increases. Inset: parameter of the fitted Gamma distribution α as a function of ∆ p, increasing approximately linearly with the same slope as in the weakly interacting limit. Moreover, using the particle number conservation (E1), and the variances the variablesN p =0 , the standard deviation δN p=0 /N can be calculated as δN p=0 N = nc n=1 1 2Kn 2 ≈ π 2 √ 6K , taking the limit of large cutoff n c in the last step. Substituting these results into (E3) allows us to take the n c → ∞ limit, resulting in the cumulative distribution function Prob Ñ 0 < x ≈ 1 − exp − exp π √ 6 x − γ , with γ denoting the Euler constant, defined by the relation γ = lim nc→∞ nc n=1 1 n − log n c . By taking the derivative of this cumulative distribution function, we arrive at the PDF of the Gumbel distribution, Eq. (12). Appendix F: Numerical results for strong interactions In the figures of the main text we concentrated mostly on the limit of weak interactions. Here we present additional numerical results, corresponding to stronger interactions. . Finite temperature distribution of the normalized zero momentum intensity Ip=0 for strong interactions, using different dimensionless temperatures T = kBT /(K∆). The PDF crosses over from the zero temperature limit (deviating from Gumbel distribution due to strong interactions) to an exponential distribution, as a signature of the depletion of the zero mode by the thermally populated p = 0 modes. As in the limit of weak interactions, the crossover is governed by the dimensionless temperatureT . We used K = 1.5, ∆ p/(2π) = 0.1 and ξ h /L ≈ 0.002. By analyzing the equilibrium quantum fluctuations at T = 0 temperature, we have shown in Sec. III A that the distribution of the intensity at finite momentum crosses over from exponential to Gamma distribution with increasing momentum resolution ∆p. We plotted this crossover for weak interactions in Fig. 2. In Fig. 8 we show the same crossover for stronger interactions K = 2. We find that the parameter of the fitted Gamma distribution, Eq. (10), increases approximately linearly with ∆p, with the same slope as in the limit of weak interactions. We considered the finite temperature distribution of the zero mode in Sec. III D. In the limit of weak interactions, plotted in Fig. 5 of the main text, we found a crossover from the zero temperature Gumbel distribution to an exponential distribution, as the temperature is increased and thermal fluctuations deplete the quasicondensate. We observe a similar crossover for strong interactions K = 1.5, by plotting the zero-momentum distributions for different dimensionless temperaturesT in Fig. 9. For such strong interactions, the distribution at T = 0 deviates from the Gumbel distribution considerably (see also Fig. 3 in the main text), but a clear crossover from the T = 0 limit to an exponential distribution, governed by the dimensionless temperatureT , still persists. PACS numbers: 67.85.-d, 42.50.Lc, 05.30.Jp, 67.85.Hj FIG. 3. PDF of the normalized variable (Î − Î )/δI for p = mR/t = 0, plotted for different Luttinger parameters K, with δI referring to standard deviation. We used periodic boundary conditions to compare to analytical results. For weak interactions (K 1) the PDF converges to Gumbel distribution, Eq. (12) (solid line), also predicted by a particle number preserving Bogoliubov approach. We used ∆ p = 0.1 × 2π and ξ h /L = 0.002. FIG. 4 . 4the full distribution function, W p (I), we can generalize usual multipoint correlation functions and define the joint distribution W p1,p2,... (I 1 , I 2 , . . . ), corresponding to measuring the intensities Î R1 ,Î R2 , . . . at positions R i = p i t/m. More formally, in analogy with Eq. (2), the joint distribution function W (I R1 , I R2 ) can be defined through the moments of the variablesÎ R1 and Joint PDF of signals I0 and I1, evaluated for dimensionless momenta p0 = 0 and p1 = 2π, for strong (K = 2) and weak (K = 10) interactions at T = 0 temperature. The anticorrelation, observable for any interaction strength, reflects particle number conservation. Particles with non-zero wave numbers p1, leave holes behind in the quasi-condensate. FIG. 5 . 5Finite temperature distribution of the normalized zero momentum intensity Ip=0, for different dimensionless temperatures T = kBT /(K∆). FIG. 6. Distribution of the normalized zero momentum intensity Ip=0 after an interaction quench, for different holding times, τ h = 1.3µs − 177µs, measured in dimensionless units,τ h = τ h c f /L. Distributions are plotted for two different quenches of durations τ = 0.71µs (rapid), and τ = 71µs (slow). We have used N = 3684, L = 39µm and a chemical potential µ/h = 1.6 kHz, corresponding to K0 = 80, c0 = 2.7mm/s, and ξ 0 h = 0.27µm, and assumed an interaction quench to K f = 7, yielding c f = c0K0/K f = 30.8 mm/s. We assumed a modest momentum resolution, ∆p/(2π) = 0.1. Similar to the finite temperature thermalization plotted in Fig. 5, the PDF after a rapid quench crosses over from the equilibrium Gumbel distribution, Eq. (12), to an exponential distribution asτ h increases, even though the number of excitations in the system remains constant after the quench. For a slower quench, the PDF for short holding times τ h is much wider than the Gumbel distribution, showing that increasing interactions have time to deplete the quasi-condensate during the quench protocol, resulting in larger particle number fluctuations. As in the case of rapid quench, for larger holding times this PDF crosses over to a thermal distribution. ACKNOWLEDGMENTS This research has been supported by the Hungarian Scientific Research Funds Nos. K101244, K105149, SNN118028, K119442. ED acknowledges support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR Quantum Simulation MURI, AFOSR MURI Photonic Quantum Matter, the Humboldt Foundation, and the Max Planck Institute for Quantum Optics. Appendix A: Probability density function FIG. 7 . 7Expectation value Î R,∆R(t) /N plotted as a function of dimensionless wave numberp/(2π), for different dimensionless temperaturesT , with parameters K = 10, ξ h /L = 0.002 and ∆p = 0.1 × 2π, using logarithmic scale on both axis. FIG. 8 . 8Distribution of normalized intensity I (symbols) at T = 0 for stronger interactions, and fits with the Gamma distribution from Eq. (10) (solid lines) FIG. 9. Finite temperature distribution of the normalized zero momentum intensity Ip=0 for strong interactions, using different dimensionless temperatures T = kBT /(K∆). The PDF crosses over from the zero temperature limit (deviating from Gumbel distribution due to strong interactions) to an exponential distribution, as a signature of the depletion of the zero mode by the thermally populated p = 0 modes. As in the limit of weak interactions, the crossover is governed by the dimensionless temperatureT . We used K = 1.5, ∆ p/(2π) = 0.1 and ξ h /L ≈ 0.002. N p ∼ coth(βc\|p\|/2) \|p\| ∼ 1/p 2 . Appendix C: Focusing techniqueBesides the ToF measurements, the focusing technique provides an alternative way to access the momentum distribution[60][61][62]. The strong transverse confinement of the quasi one dimensional system is abruptly switched off, while the weak longitudinal confinement is replaced by a strong harmonic trap of frequency ω[63], and the gas is imaged after a quarter time period, t = T /4 = π/(2ω).To express the intensity (1) in this case with the field operators at time t = 0, we have to replace the free propagator in Eq. (6) by that of the harmonic oscillator G osc (x, y, t = T /4) = e −i x y/l 2 0 /(l 0 √ 2π i), with l 0 = /(mω) the oscillator length of the strong trapping potential. In this case, Eqs. (6) thus simply yields the Fourier transform of the field at t = 0, ψ(R, t = T /4) ∼ψ p at a momentum p = R/l 2 0 . Thus the intensity measured at R is directly proportional to the number of par-ticlesN p in this case. Performing calculations similar to those sketched in Appendix A, we arrive at Eqs. (7) and Eq. (8), with the weight function (A4) replaced byand the dimensionless momentum and momentum resolution expressed asp osc = R L/l 2 0 and ∆p osc = ∆R L/l 2 0 .Apart from these minor corrections, all our calculations can be performed for focusing experiments, and while this method allows to use shorter measurement times, the conclusions in the main text remain unaltered.Appendix D: The expectation value ofÎR,∆R(t)In this appendix we investigate the expectation value of the intensityÎ R,∆R (t), scaled out from the distribution functions calculated in the main text.In order to investigate the temperature dependence of the expectation value, we plotted Î R,∆R (t) /N as a function of dimensionless momentump/(2π) for different dimensionless temperaturesT inFig. 7. We concentrated on the weakly interacting regime, keeping the Luttinger-parameter, K = 10, constant. The low temperature results show pronounced oscillations, originating from the presence of the quasi-condensate due to finite size effects. For higher temperatures, the intensity Î R,∆R (t) increases for non-zero momentap = O(2π), while the zero-momentum expectation value decreases due to the depletion of the condensate. Moreover, we can distinguish two momentum regions, corresponding to different behavior of the expectation value. For momenta much smaller than the thermal wavelength, p /λ T (orp/(2π) πKT in dimensionless variables), the expectation value of the intensity is well approximated by the Fourier transform of Eq.(15), yieldingThis expression predicts a power law decay Î R,∆R (t) ∼ 1/p 2 for momenta /ξ T p /λ T . However, for even larger momenta, p /λ T , the short distance behaviour of the correlation function ψ † (x)ψ(0) becomes important, and it is not appropriate to approximate it by the simple exponential function Eq.(15). 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S Fölling, F Gerbier, A Widera, O Mandel, T Gericke, I Bloch, Nature. 434481S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke and I. Bloch, Nature 434, 481 (2005). . T Rom, Th, D Best, U Van Oosten, S Schneider, B Fölling, I Paredes, Bloch, Nature. 444733T. Rom, Th. Best, D. van Oosten, U. Schneider, S. Fölling, B. Paredes and I. Bloch, Nature 444, 733 (2006). Noise correlations, extracted from time of flight images, have already been used as a versatile tool to study strongly correlated many-body systems, see Refs. 72-74Noise correlations, extracted from time of flight images, have already been used as a versatile tool to study strongly correlated many-body systems, see Refs. [72- 74].	[]
[ "arXiv:0807.4398v1 [hep-th] Euclidean Methods and the entropy function", "arXiv:0807.4398v1 [hep-th] Euclidean Methods and the entropy function" ]	[ "Pedro J Silva [email protected] \nInstitut de Ciències de l'Espai (IEEC-CSIC) and Institut de Física d'Altes Energies (IFAE)\nE-08193Bellaterra, Barcelona)Spain\n" ]	[ "Institut de Ciències de l'Espai (IEEC-CSIC) and Institut de Física d'Altes Energies (IFAE)\nE-08193Bellaterra, Barcelona)Spain" ]	[ "Published in Fortschr. Phys" ]	We review results of articles hep-th/0607056, hep-th/0610163 and 0704.1405 [hep-th]. Here we focus on establish the connection between the entropy functional formalism of Sen and the standard Euclidean formalism taken at zero temperature. We find that Sen's entropy function f (on-shell) matches the zero temperature limit of the Euclidean action. Moreover, Sen's near horizon angular and electric fields agree with the chemical potentials that are defined from the zero-temperature limit of the Euclidean formalism. Connection with the Dual CFT thermodynamics is briefly discussed.	10.1002/prop.200810558	[ "https://arxiv.org/pdf/0807.4398v1.pdf" ]	15,557,059	0807.4398	0279f84bc78ace4248d5be3a4f3239f56602baac	arXiv:0807.4398v1 [hep-th] Euclidean Methods and the entropy function 2008 Pedro J Silva [email protected] Institut de Ciències de l'Espai (IEEC-CSIC) and Institut de Física d'Altes Energies (IFAE) E-08193Bellaterra, Barcelona)Spain arXiv:0807.4398v1 [hep-th] Euclidean Methods and the entropy function Published in Fortschr. Phys 567 -92008Received 15 April 2008 We review results of articles hep-th/0607056, hep-th/0610163 and 0704.1405 [hep-th]. Here we focus on establish the connection between the entropy functional formalism of Sen and the standard Euclidean formalism taken at zero temperature. We find that Sen's entropy function f (on-shell) matches the zero temperature limit of the Euclidean action. Moreover, Sen's near horizon angular and electric fields agree with the chemical potentials that are defined from the zero-temperature limit of the Euclidean formalism. Connection with the Dual CFT thermodynamics is briefly discussed. Introduction This article contains the talk based on the articles [1,2,3]. Here we only show the main results and general lessons that steam from our work. More References and more details should be found on our original papers. On Sen's entropy functional formalism Black holes (BH) are one of most interesting laboratories we have to investigate quantum gravity effects. Due to their thermodynamic behavior these objects have been associated to ensembles of microstates in the fundamental quantum gravity theory where ideally, quantum statistical analysis should account for all the BH coarse-grained thermodynamical behavior. In particular, many important insights in the classical and quantum structure of BH have been obtained studying supersymmetric configurations in string theory. In this context we have the so called attractor mechanism. It was originally thought in the context of four dimensional N = 2 supergravity, where we have that the values of the scalar fields at the horizon are given by the values of the BH conserved charges and are independent of the asymptotic values of the scalars at infinity. Importantly, the attractor mechanism has provided a new way to calculate the BH entropy. In a series of articles [4,5,6], Sen recovered the entropy of D-dimensional BPS BH using only the near horizon part of the geometry. Basically, in this regime the solution adopts the form AdS 2 ⊗ S D−2 1 plus some electric and magnetics fields. The entropy S is obtained by introducing a function f as the integral of the corresponding supergravity Lagrangian over the S D−2 . More concretely, an entropy function is defined as 2π times the Legendre transform of f with respect to the electric fields e i . Then, an extremization procedure fixes the on-shell BPS values of the different fields of the solution and in particular determines the BPS value of the entropy S, Sen's entropy functional formalism assumes that: (i) we start with a Lagrangian L with gravity plus some field strengths and uncharge massless scalar fields; and (ii) due to the attractor mechanism the near horizon geometry of a D-dimensional BH is set to be of the form AdS 2 ⊗ S D−2 . From the above input data, the general form of the near horizon BH solution is ds 2 = v 1 −ρ 2 dτ + dρ 2 ρ 2 + v 2 dΩ 2 D−2 , F (i) ρτ = e i , H (a) = p a ǫ D−2 , φ s = u s ,(1) where ǫ D−2 is the unit-volume form of S D−2 , and (e i , p a ) are respectively the electric fields and the magnetic charges of the BH. Note that ( u, v, e, p) are arbitrary constants up to now and therefore the solution is off-shell. Next, it is defined the following function f ( u, v, e, p) = S D−2 √ −gL ,(2) where L is the string frame Lagrangian of the theory. After minimizing f ( u, v, e, p) with respect to ( u, v) we obtain the exact supersymmetric near horizon BH solution in terms of ( e, p). In fact, the field equations are reproduced by this minimization procedure. Furhermore, minimization with respect to e gives the electric charges q. Explicitly, the on-shell values of u, v, e that specify (1) for a given theory described by (2) are found through the relations, ∂f ∂u s = 0 , ∂f ∂v j = 0 , ∂f ∂e i = q i .(3) Then, using Wald formalism [7], Sen derived that the entropy S of the corresponding BH is given by 2π times the Legendre transform of f , S = 2π e i ∂f ∂e i − f .(4) Finally notice that the minimization procedure, can be taken only after S is defined. In this form S is really an entropy function of ( u, v, q, p), that after minimization equals the BH entropy as a function of ( q, p) only. The above formalism fixes the form of the NH geometry and the entropy S in terms of the conserved charges but what is the geometric origin or motivation for the above definitions? and how is connected to the usual Bh thermodynamics? To answer these questions we revisit the Bh thermodynamics and the limit of zero temperature in next section. On GR thermodynamics and zero temperature limit In [1,2] the "thermodynamics" or better "the statistical mechanics" of supersymmetric solitons in gauged supergravity was studied in detail using an extension of standard Euclidean thermodynamical methods to zero temperature systems. We call this approach the Euclidean zero-temperature formalism. BPS BH can be studied as dual configurations of supersymmetric ensembles at zero temperature but non-zero chemical potentials in the dual CFT. These potentials control the expectation value of the conjugated conserved charges carried by the BH, like e.g., angular momenta and electric charge. In these articles, the two main ideas are: First, there is a supersymmetric field theory dual to the supergravity theory. Second, in this dual field theory the grand canonical partition function over a given supersymmetric sector can be obtained as the zero temperature limit of the general grand canonical partition function at finite temperature. This limit also fixes the values of several chemical potentials of the system. To make things more clear, recall that all supersymmetric states in a field theory saturate a BPS inequality that translates into a series of constraints between the different physical charges. For definiteness, let us consider a simple case where the BPS bound corresponds to the constraint: E = J. Then, defining the left and right variables E ± = 1 2 (E ν ± J ν ), β ± = β(1 ± Ω) the grand canonical partition function is given by Z (β,Ω) = ν e −(β + E + +β − E − ) .(5) At this point, it is clear that taking the limit β − → ∞ while β + → ω (constant), gives the correct supersymmetric partition function. The above limiting procedure takes T to zero, but also scales Ω in such a way that the new supersymmetric conjugated variable ω is finite and arbitrary. Note that among all available states, only those that satisfy the BPS bound are not suppress in the sum, resulting in the supersymmetric partition function Z(ω) = bps e −ωJ ,(6) where the sum is over all supersymmetric states (bps) with E = J. The above manipulations are easy to implement in more complicated supersymmetric field theories like, e.g., N = 4 SYM theory in four dimensions. What is less trivial is that amazingly it could also be implemented in the dual supersymmetric configurations of gauged supergravity, since it means that these extreme BPS solutions are somehow protected from higher string theory corrections. To apply the Euclidean zero-temperature formalism to concrete black hole systems, it is profitable to highlight its key steps. To study the statistical mechanics of supersymmetric black holes we take the off-BPS BH solution and we send T → 0. In this limiting procedure, the angular velocities and electric potentials at the horizon can be written as an expansion in powers of the temperature. More concretely one has when T → 0, β → ∞ , Ω → Ω bps − ω β + O(β −2 ) , Φ → Φ bps − φ β + O(β −2 ) ,(7) where β is the inverse temperature; (Ω, Φ) are the angular velocities and electric potentials at the horizon; the subscript bps stands for the values of these quantities in the on-shell BPS solution; and (ω, φ) are what we call the supersymmetric conjugated potentials, i.e., the next to leading order terms in the expansion. For all the systems studied, we find that the charges have the off-BPS expansion, E = E bps + O β −2 , Q = Q bps + O β −2 , J = J bps φ + O β −2 ,(8) where (E, Q, J) are the energy, charges and angular momenta of the BH. In supergravity, the grand canonical partition function in the saddle point approximation is related to so called quantum statistical relation (QSR) I (β,Φ,Ω) = βE − ΦQ − ΩJ − S ,(9) where S is the entropy, and (β, Φ, Ω) are interpreted as conjugated potentials to E, Q, J, respectively. I is the Euclidean action (evaluated on the off-BPS BH solution) that, in this ensemble, depends only on (β, Φ, Ω). It plays the role of free energy divided by the temperature. Inserting (7) and (8) into (9) yields I (β,Φ,Ω) = β(E bps − Φ bps Q bps − Ω bps J bps ) + φQ bps + ωJ bps − S bps + O β −1 .(10) Here, we observe that this action is still being evaluated off-BPS. Moreover, the term multiplying β boils down to the BPS relation between the charges of the system and thus vanishes (this will become explicitly clear in the several examples we will consider). This is an important feature, since now we can finally take the β → ∞ limit yielding relation I bps = φQ bps + ωJ bps − S bps .(11) It is important to stress that this zero temperature limiting procedure yields a finite, not diverging, supersymmetric version of QSR, or shortly SQSR. Note that if we had evaluated the Euclidean action (9) directly on-shell it would not be well defined, as is well-known. As a concrete realization, we picked (and will do so along the paper) the SQSR to exemplify that the T → 0 limit yields well-behaved supersymmetric relations. The reason being that this SQSR relation is the one that will provide direct contact with Sen's entropy functional formalism, which is the main aim of our study. However, it also provides a suitable framework that extends to the study of the full statistical mechanics of supersymmetric black holes. On entropy functional and zero temperature thermodynamics In previous sections we have described two apparently unrelated procedures to obtain the entropy of supersymmetric BH that naturally contain the definitions of pairs of conjugated variables, related to the BH charges. In this section we show that both procedures produce basically the same body of final definitions, even though conceptually both approaches are rather different. That both approaches produce the same final chemical potentials and definitions can be seen in any of the examples at hand. As usual, the best way to illustrate our point is to pick a system that captures the fundamental ingredients. Comparing the zero temperature thermodynamic relations with the corresponding Sen's definitions, in any BH like e.g., the D1-D5-P system, we can indeed confirm that all the key quantities agree in the two formalisms. Explicitly we have that φ i = 2πe i , Q i = q i , I bps = 2πf .(12) Nevertheless, that both frameworks are equivalent is a priori not at all obvious since they have important differences. Sen's approach relies completely on the structure of the near horizon geometry. In particular, the entropy is constructed analyzing Wald's prescription and Einstein equations in these spacetimes and all the analysis is carried on at the BPS bound i.e., when the solution is extremal. In contrast, the zero temperature limit approach relies on the thermodynamical properties of BH and, in principle, uses the whole spacetime, not only the near horizon region. The resulting thermodynamic definitions come as a limiting behavior of non-extremal BH and have a nice straightforward interpretation in terms of the dual CFT thermodynamics. Near-horizon and asymptotic contributions to the Euclidean action To understand why the above close relations between the two formalisms hold, let us go back to the calculation of the Euclidean action for general BH in the off-BPS regime. Inspired in ten dimensional type II supergravity, we start with the general action 2 I = 1 16πG Σ √ −g R − 1 2 (∂Ψ) 2 − 1 2n! e αΨ F 2 (n) + 1 8πG ∂Σ K ,(13) where Σ is the spacetime manifold, ∂Σ the boundary of that manifold and K is the extrinsic curvature. In the BH case, once we have switched to Euclidean regime, it is necessary to compactify the time direction to avoid a conical singularity. This compactification defines the Hawking temperature as the inverse of the corresponding compactification radius. To evaluate the Euclidean action on the BH solution, one of the methods to obtain a finite result, i.e., to regularize and renormalize the action, consists of putting the BH in a box and subtract the action of a background vacuum solution (g 0 , Ψ 0 , F 0 ). This procedures also defines the "zero" of all the conserved charges. For asymptotic flat solutions we use Minkowski, while for asymptotic AdS solutions we use AdS. Once in the box, the radial coordinate is restricted to the interval (r + , r b ), where r + is the position of the horizon and r b corresponds to an arbitrary point which limits the box and that at the end is sent to infinity. Another important ingredient is the boundary conditions on the box. Basically, depending on which conditions we impose on the different fields, we will have fixed charges or fixed potentials. If we do not add any boundary term to the above action, we will be working with fixed potentials, i.e., we will work in the grand canonical ensemble. At this point we are ready to rewrite the Euclidean action in two pieces, one evaluated in the first boundary at r = r + , and the other in the second boundary at r = ∞, I = r=r + c 8πG e aΨ F (n) C (n−1) + 1 8πG K + r=∞ c 8πG e aΨ F (n) C (n−1) + 1 8πG K − K 0 .(14) where the field equations have been used and c is a proportionality constant. Then we can rewrite the above expression as, I = β(Φ bps − Φ)Q − S + β(E − Φ bps Q) .(15) r = r + r = ∞ Therefore we can always find a gauge in which the Euclidean action splits in two contributions, one at the horizon and the other in the asymptotic region. In fact, from our discussion it is easy to see that the first term exactly reproduces the SQSR, i.e., lim BP S limit β(Φ bps − Φ)Q − S = φQ bps − S bps .(16) while the asymptotic term vanishes due to fact that Φ bps = 1, and thus the leading term in the expansion is nothing else than the BPS relation E bps = Q bps characteristic of supersymmetric regimes, i.e., lim BP S limit β(E − Φ bps Q) = lim BP S limit β(E − Q) = 0 .(17) The above results are trivially generalized to the case of rotating charged BH, see [3], for more detail. Hence, due to the above equations we have verified that I bps = 2πf .(18) Relation between chemical potentials in the two formalisms At this point only reminds to understand the relation between the conjugated potentials in both pictures. In Sen's approach, the information about them is contained in the electric fields of the near horizon geometry, while in the Euclidean zero temperature formalism this information is encoded in the next to leading order term in an off-BPS expansion of the full geometry. Although these definitions seem to be rather different at first sight, notice that in Sen's approach the field strength is just the radial derivative of the potential evaluated at the horizon. In the Euclidean zero temperature case, the off-BPS expansion can be rewritten as an expansion in the radial position of the horizon ρ + . Therefore, the next to leading order term in the off-BPS expansion of the gauge potential at ρ + is proportional to its derivative with respect to the radial position of the horizon. Hence it is proportional to the field strength at the horizon. It is not dificult to check that the above reasoning produces exactly the result φ i = 2πe i .(19) therefore we have been able to relate all the different terms in the entropy functional approach with the emergent chemical potentials and thermodynamic functions of the zero temperature Euclidean approach. Conclusions As stated above, the main goal of this article is to provide a bridge between Sen's entropy functional formalism and standard Euclidean analysis of the thermodynamics of a black hole system. While doing so, we also find that the supergravity conjugated potentials defined in Sen's formalism map into chemical potentials of the dual CFT. We obtain a unifying picture where: 1)We are able to recover the entropy function of Sen from the zero temperature limit of the usual BH thermodynamics and the statistical mechanics definitions of the dual CFT theory. The supergravity and their dual CFT chemical potentials are identified with the surviving Sen's near horizon electric and angular fields. The Euclidean action is identified with Sen's function 2πf . 2)As a byproduct of the above analysis we have understood how to calculate the BPS chemical potentials that control the statistical properties of the BH using only the BPS regime, i.e., without needing the knowledge of the non-BPS geometry. The CFT chemical potentials are dual to the supergravity ones. Traditionally, to compute the latter we have to start with the non-BPS solution and send the temperature to zero to find the next to leading order terms in the horizon angular velocities and electric potentials expansions that give the chemical potentials. This requires the knowledge of the non-BPS geometry. Unfortunately, sometimes this is not available and we only know the BPS solution. But, from item 1) we know that the near horizon fields, that Sen computes with the single knowledge of the BPS near horizon solution, give us the supergravity chemical potentials. So now we can compute the supergravity chemical potentials of any BPS BH solution, regardless of its embedding into a family of non-BPS solutions, while still keeping the relation with the dual CFT. The analysis of the near horizon geometry has been applied to more general BH that define squashed AdS2 ⊗S D−2 geometries. For simplicity, the reasoning is done at the level of two derivative Lagrangian. Nevertheless, following Wald's approach for higher derivative actions, we notice that the BH action can always be recast as surface integrals. Moreover, for definiteness, we anchor our discussion to type II action, but whenever needed we make comments to extend our arguments to more general theories. AcknowledgmentsThe author would like to thanks the organizers of the 3rd RTN Workshop for hospitality. This work was partially funded by the Ministerio de Educacion y Ciencia under grant FPA2005-02211 and CSIC via the I3P programme Thermodynamics at the BPS bound for black holes in AdS. P J Silva, arXiv:hep-th/0607056JHEP. 061022P. J. Silva, "Thermodynamics at the BPS bound for black holes in AdS," JHEP 0610, 022 (2006) [arXiv:hep-th/0607056]. Phase transitions and statistical mechanics for BPS black holes in AdS/CFT. P J Silva, arXiv:hep-th/0610163JHEP. 070315P. J. Silva, "Phase transitions and statistical mechanics for BPS black holes in AdS/CFT," JHEP 0703, 015 (2007) [arXiv:hep-th/0610163]. Euclidean analysis of the entropy functional formalism. O J C Dias, P J Silva, arXiv:0704.1405hep-thO. J. C. Dias and P. J. Silva, "Euclidean analysis of the entropy functional formalism," arXiv:0704.1405 [hep-th]. Black hole entropy function and the attractor mechanism in higher derivative gravity. A Sen, arXiv:hep-th/0506177JHEP. 050938A. Sen, "Black hole entropy function and the attractor mechanism in higher derivative gravity," JHEP 0509 (2005) 038 [arXiv:hep-th/0506177]. Entropy function for heterotic black holes. A Sen, arXiv:hep-th/0508042JHEP. 06038A. Sen, "Entropy function for heterotic black holes," JHEP 0603 (2006) 008 [arXiv:hep-th/0508042]. Rotating attractors. D Astefanesei, K Goldstein, R P Jena, A Sen, S P Trivedi, arXiv:hep-th/0606244JHEP. 061058D. Astefanesei, K. Goldstein, R. P. Jena, A. Sen and S. P. Trivedi, "Rotating attractors," JHEP 0610 (2006) 058 [arXiv:hep-th/0606244]. Black hole entropy in the Noether charge. R M Wald, arXiv:gr-qc/9307038Phys. Rev. D. 483427R. M. Wald, "Black hole entropy in the Noether charge," Phys. Rev. D 48 (1993) 3427 [arXiv:gr-qc/9307038]. On Black Hole Entropy. T Jacobson, G Kang, R C Myers, arXiv:gr-qc/9312023Phys. Rev. D. 496587T. Jacobson, G. Kang and R. C. Myers, "On Black Hole Entropy," Phys. Rev. D 49 (1994) 6587 [arXiv:gr-qc/9312023]. Some properties of Noether charge and a proposal for dynamical black hole entropy. V Iyer, R M Wald, arXiv:gr-qc/9403028Phys. Rev. D. 50846V. Iyer and R. M. Wald, "Some properties of Noether charge and a proposal for dynamical black hole entropy," Phys. Rev. D 50 (1994) 846 [arXiv:gr-qc/9403028].	[]
[ "Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity", "Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity" ]	[ "Kai Flathmann ", "Manuel Hohmann ", "\nInstitut für Physik\nInstitute of Physics\nLaboratory of Theoretical Physics\nUniversität Oldenburg\n26111OldenburgGermany\n", "\nUniversity of Tartu\nW. Ostwaldi 150411TartuEstonia\n" ]	[ "Institut für Physik\nInstitute of Physics\nLaboratory of Theoretical Physics\nUniversität Oldenburg\n26111OldenburgGermany", "University of Tartu\nW. Ostwaldi 150411TartuEstonia" ]	[]	In this article we calculate the post-Newtonian limit of a general class of scalar-nonmetricity theories of gravity. The action is assumed to be a free function of the nonmetricity scalar, the kinetic term of the scalar field, two derivative couplings and the scalar field itself. We use the parametrized post-Newtonian formalism to solve the arising field equations for the case of a massless scalar field in order to compare several subclasses of this theory to solar system observations. In particular, we find several classes of theories which are indistinguishable from general relativity on the post-Newtonian level and therefore, should be studied further. Most remarkably, we find that this is the generic case, while a post-Newtonian limit that deviates from general relativity occurs only for a particular coupling between the scalar field and nonmetricity.	10.1103/physrevd.105.044002	[ "https://arxiv.org/pdf/2111.02806v2.pdf" ]	242,757,576	2111.02806	40807a2fd2637e7a4f4873e853d32b389c3b67c9	Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity 1 Mar 2022 Kai Flathmann Manuel Hohmann Institut für Physik Institute of Physics Laboratory of Theoretical Physics Universität Oldenburg 26111OldenburgGermany University of Tartu W. Ostwaldi 150411TartuEstonia Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity 1 Mar 2022 In this article we calculate the post-Newtonian limit of a general class of scalar-nonmetricity theories of gravity. The action is assumed to be a free function of the nonmetricity scalar, the kinetic term of the scalar field, two derivative couplings and the scalar field itself. We use the parametrized post-Newtonian formalism to solve the arising field equations for the case of a massless scalar field in order to compare several subclasses of this theory to solar system observations. In particular, we find several classes of theories which are indistinguishable from general relativity on the post-Newtonian level and therefore, should be studied further. Most remarkably, we find that this is the generic case, while a post-Newtonian limit that deviates from general relativity occurs only for a particular coupling between the scalar field and nonmetricity. I. INTRODUCTION Being contested in numerous experiments during the past century, general relativity is still the best gravitational theory describing observations in our universe. However, by fixing the mediator of gravity through the Ricci scalar as curvature two equivalent possibilities are overlooked [1]. The first alternative ascribes gravity to the dynamics of the tetrad via the torsion scalar. This equivalent formulation of general relativity is called the Teleparallel Equivalent of General Relativity (TEGR) [2,3]. If we assume curvature and torsion as being zero and simultaneously nonmetricity as nonvanishing, we can construct the Symmetric Teleparallel Equivalent of General Relativity (STEGR) [4][5][6][7][8][9][10][11]. Another possibility invokes both torsion and nonmetricity [12,13]. However, even though all of these formulations are equivalent, generalizations thereof differ from each other. A vast number of modifications of general relativity and its alternative formulations in terms of torsion and nonmetricity mentioned above has been developed [14]. The main motivation for studying such theories comes from tensions between general relativity and current observations in cosmology, such as different measurements of the Hubble expansion rate [15]. These observations hint towards physics beyond the so-called ΛCDM model, which aims to describe the universe using general relativity, a cosmological constant Λ and cold dark matter (CDM). Among the most common modifications studied to address these tensions are generalizations of the action functional to a free function of the aforementioned scalar invariants of curvature, torsion or nonmetricity, giving rise to the so-called f (R), f (T ) and f (Q) classes of gravity theories [16][17][18][19][20]. Another, related type of modifications is obtained by including an additional scalar field in the theory, which couples non-minimally to the geometric quantities which mediate the gravitational interaction, and can thus itself be regarded as a mediator of gravity. This type of modifications gives rise to scalar-curvature, scalar-torsion and scalar-nonmetricity theories of gravity [21][22][23][24][25][26][27][28]. In order to be considered as a viable theory of gravity, any of the aforementioned modifications must not only address the observational tensions in cosmology, but also be in agreement with numerous precision observations of gravitational waves [29][30][31] and gravity on stellar or solar system scales [32]. The latter can comprehensively be studied using the parametrized post-Newtonian (PPN) formalism [33,34], which characterizes any metric theory of gravity by ten (usually constant) parameters. Their values predicted by any given theory of gravity can then be compared to their experimentally measured values, giving constraints on the considered theory or its parameters. In this article we make use of the PPN formalism in order to derive the post-Newtonian limit of a general class of scalar-nonmetricity theories of gravity, which generalizes the originally proposed class [27], following a similar idea as applied in scalar-torsion gravity [25], and allowing for a gravitational action defined by an arbitrary function of the nonmetricity scalar, two non-minimal coupling terms, the scalar field and its kinetic energy, and which we will therefore denote L(Q, X, Y, Z, φ) theories of gravity. For this purpose, we make use of the previously developed post-Newtonian expansion of symmetric teleparallel gravity theories [35,36], which we enhance by including a post-Newtonian expansion for the scalar field and a Taylor expansion for the free function defining the action, in full analogy to the case of scalar-torsion gravity [37]. Our conventions and notation follow the textbook [33]. The outline of the article is as follows. In section II, we briefly review the field variables of scalar-nonmetricity gravity and introduce the class of theories we study in the remaining section. We briefly discuss the post-Newtonian expansion of the field equations in section III. In section IV, we solve these field equations up to the required perturbation order. The resulting PPN parameters are shown and interpreted in section V. We end with a conclusion in section VI. II. FIELD VARIABLES AND THEIR DYNAMICS Before defining the action and the resulting field equations of the class of L(Q, X, Y, Z, φ) scalar-nonmetricity theories, we intend to review the underlying dynamical fields. As usual in theories where nonmetricity is the mediator of gravity, the dynamical fields are a Lorentzian metric g µν and an affine connection Γ ρ µν . In addition we couple a dynamical scalar field φ. We specify the properties of the connection by demanding vanishing torsion T ρ µν = −2Γ ρ [µν] = 0(1) and curvature R ρ σµν = 2∂ [µ Γ ρ \|σ\|ν] + 2Γ ρ λ[µ Γ λ \|σ\|ν] = 0 ,(2) whereas the covariant derivative of the metric with respect to the dynamical connection (i.e., nonmetricity) is nonzero Q ρµν = ∇ ρ g µν = 0 .(3) The combination of Eqns. (1) and (2) leads to the form of the connection Γ ρ µν = Λ −1 ρ λ ∂ ν Λ λ µ ,(4) with ∂ [µ Λ λ ν] = 0. We consider an action of the form S[g µν , Γ ρ µν , φ, χ] = S g [g µν , Γ ρ µν , φ] + S m [g µν , χ] ,(5) where χ is an arbitrary set of matter field fields. The gravitational part of the action S g [g µν , Γ ρ µν , φ] = 1 2κ 2 M d 4 x √ −gL(Q, X, Y, Z, φ) ,(6) is a free function of the scalar field φ, the nonmetricity scalar Q = − 1 4 Q µνρ Q µνρ + 1 2 Q µνρ Q ρνµ + 1 4 Q µ Q µ − 1 2 Q µQ µ ,(7) the kinetic term of the scalar field X = − 1 2 g µν ∂ µ φ∂ ν φ ,(8) and the derivative couplings Y = Q µ ∂ µ φ(9) and Z =Q µ ∂ µ φ ,(10) which couple the scalar field to the two independent contractions of the nonmetricity tensor Q µ = Q µρ ρ ,Q µ = Q ρ ρµ .(11) By varying the matter action S m with respect to the metric δS m = − 1 2 M Θ µν δg µν √ −gd 4 x(12) we obtain the energy momentum tensor Θ µν . The full variation of the action S with respect to the metric then leads to the field equations 0 = E µν = − Lg µν + • ∇ ρ (L Q P ρ µν ) + 1 2 g µν g ρσ • ∇ ρ (L Y ∂ ρ φ) + 1 2 • ∇ (µ L Z ∂ ν) φ − 1 2 L Q 2Q ρ µσ [Q ρν σ − Q σ ρν ] − Q µ ρσ Q νρσ + Q ρ 2Q (µν) ρ − Q ρ µν − L X ∂ µ φ∂ ν φ + 2L Y Q (µ ∂ ν) φ − L Z Q ρ µν ∂ ρ φ − 2Q (µν) ρ ∂ ρ φ − Q (µ ∂ ν) φ − κ 2 Θ µν(13) and similarly a variation with respect to the scalar field leads to the scalar field equation 0 = E φ = • ∇ µ L Y Q µ + L ZQµ − L Y ∂ µ φ − L φ .(14) Note that L Q,X,Y,Z,φ is the derivative of the free function L with respect to Q , X , Y , Z and φ, respectively. We finally remark that another field equation can be obtained by variation of the action with respect to the flat, symmetric affine connection; however this field equation is fully determined from the previously displayed equations through the Bianchi identities, and thus redundant [38]. We therefore omit it here for brevity and show only the independent equations, which we will solve in the following sections. III. POST-NEWTONIAN APPROXIMATION In order to compare this family of theories with observations, we employ the parametrized post-Newtonian (PPN) formalism in its form detailed in the textbook [33]. 1 First, we give some general remarks on the PPN formalism and then, we review how to perform the post-Newtonian expansion of the dynamical connection. We start with the description of the matter part of the field equation. As usual we assume a perfect fluid of the form Θ µν = (ρ + ρΠ + p)u µ u ν + pg µν ,(15) with ρ, Π, p and u µ being the rest energy density, specific internal energy, pressure and four velocity, respectively. We further assume a normalization of u µ u ν g µν = −1 for the four velocity and compared to the speed of light c = 1 the velocity of the matter v i = u i /u 0 in a given reference frame is assumed to be small. Next, we perform a perturbative expansion in orders of the velocity O(n) ∝ \| v\| n . This has to be done for the metric g µν , the matter fields Θ µν , the dynamical connection Γ ρ µν , the scalar field φ and the free function L. The metric g µν will be expanded around the flat Minkowski metric η µν = diag(−1, 1, 1, 1) g µν = η µν + h µν = η µν + 2 h µν + 3 h µν + 4 h µν + O(5) .(16) As a consequence, the energy-momentum tensor reads as Θ 00 = ρ 1 + Π + v 2 − 2 h 00 + O(6) ,(17a)Θ 0j = −ρv j + O(5) , (17b) Θ ij = ρv i v j + pδ ij + O(6) .(17c) Here, we used the standard assumptions for the orders of the matter fields. Next, we make use of the form of the connection in Eqn. (4). As developed in [35], we expand the coordinates around the coordinates of the coincident gauge up to quadratic orders of the generators of a "knight diffeomorphism" x ′µ = x µ + ξ µ + 1 2 ξ ν ∂ ν ξ µ ,(18) with which the connection can be written as Γ ρ µν = ∂ µ ∂ ν ξ ρ + 1 2 ξ σ ∂ µ ∂ ν ∂ σ ξ ρ + 2∂ (µ ξ σ ∂ ν) ∂ σ ξ ρ − ∂ µ ∂ ν ξ σ ∂ σ ξ ρ .(19) Now, we expand ξ µ similar to the metric in post-Newtonian orders ξ α = 2 ξ α + 3 ξ α + 4 ξ α + O (5) .(20) Furthermore, we expand the scalar field φ around its cosmological background value Φ, which we assume to be constant φ = Φ + ψ = Φ + 1 ψ + 2 ψ + 3 ψ + 4 ψ .(21) The components of the dynamical fields, we have to calculate are 2 h 00 , 2 h ij , 3 h i0 , 4 h 00 , 2 ξ i , 3 ξ 0 , 4 ξ i , 2 ψ .(22) Lastly, we have to deal with the free function L in the action. For this we perform a Taylor expansion, where we assume the Taylor coefficients to be of velocity order O(0): L = l 0 + l φ ψ + 1 2 l φφ ψ 2 + l Q Q + l X X + l Y Y + l Z Z , L Q = l Q + l T φ ψ + 1 2 l Qφφ ψ 2 + l QX X + l QY Y + l QZ Z + l QQ Q , L X = l X + l Xφ ψ + 1 2 l Xφφ ψ 2 + l QX Q + l XY Y + l XZ Z + l XX X , L Y = l Y + l Y φ ψ + 1 2 l Y φφ ψ 2 + l QY Q + l XY X + l Y Z Z + l Y Y Y , L Z = l Y + l Zφ ψ + 1 2 l Zφφ ψ 2 + l QZ Q + l XZ X + l Y Z Z + l ZZ Z , L φ = l φ + l φφ ψ + l Qφ T + l Xφ X + l Y φ Y + l Zφ Z + 1 2 l φφφ ψ 2 .(23) By combining all perturbative expansions of this section, we can now calculate and solve the field equations in the next section. IV. SOLVING THE FIELD EQUATIONS We now apply the post-Newtonian expansion displayed in the previous section to the class of symmetric teleparallel gravity theories outlined in section II, in order to derive and solve the post-Newtonian field equations. We proceed in ascending velocity orders; the zeroth, second, third and fourth velocity order is discussed in sections IV A, IV B, IV C and IV D, respectively. Calculations have been performed using xPPN [39]. Alternatively, one could make use of the gauge-invariant approach to the PPN formalism [36,40]; the resulting equations for the constant coefficients we obtain are identical. A. Zeroth order and assumption First of all for simplicity we assume a massless scalar field in order to avoid solutions in terms of Yukawa type potentials. This can be achieved by assuming both l φφ and l φφφ = 0. The zeroth order equations (calculated by inserting g µν = η µν and φ = Φ) read 0 = l 0 η µν , 0 = l φ .(24) Therefore, the perturbed metric is given in the standard PPN form if and only if l 0 = l φ = 0. For the remainder of this article, we will use these assumptions. B. Second order With the assumptions and the solutions of the zeroth order field equations, we can now calculate the second order field and scalar field equations. The only nonvanishing components read 2 E 00 = κ 2 ρ − 1 2 l Q ∂ j ∂ i 2 h ij − △ 2 h i i + l Y △ 2 φ = 0 , 2 E ij = 1 2 l Q −∂ j ∂ i 2 h 00 + ∂ j ∂ i 2 h k k + 2∂ k ∂ (i 2 h k j) + △ 2 h ij + δ ij △ 2 h 00 + ∂ k ∂ l 2 h kl − △ 2 h k k − δ ij l Y △ 2 φ − l Z ∂ j ∂ i 2 φ = 0 , 2 E φ = l X △ 2 φ + l Y △ 2 h 00 − 2 h k k + 2∂ k 2 ξ k + l Z −∂ k ∂ i 2 h ij + 2△∂ k 2 ξ k = 0 .(25) These three equations can be solved with the ansatz 2 h 00 = a 1 U , 2 h ij = a 2 δ ij U , 2 ξ i = a 3 ∂ i χ , 2 φ = a 4 U .(26) Here, U and χ are the usual PPN potentials, which are defined by the relations △χ = −2U and △U = −4πρ. Inserting this ansatz into the field equations leads to a system of algebraic equations for the coefficients a i . In order to determine the most general solution to this system, one must distinguish different cases. The solution for the generic case is given by a 1 = κ 2 4πl Q , a 2 = κ 2 4πl Q , a 3 = − κ 2 (l Z + 2l Y ) 16π (l Z + l Y ) l Q , a 4 = 0 ,(27) and is valid if and only if the denominator (l Y + l Z )l Q is non-vanishing. Otherwise, the system is degenerate and one must further distinguish between two cases. For l Q = 0, one cannot solve for a 1 . Since this component is required for the Newtonian limit of the theory, as it governs the contribution of the Newtonian potential, we conclude that this case is not physically viable, and henceforth assume l Q = 0. Further assuming l Y + l Z = 0, one cannot solve for a 3 , as it cancels from the algebraic equations. For the remaining coefficients one obtains the solution a 1 = κ 2 4πl Q 4l 2 Y + l X l Q 3l 2 Y + l X l Q , a 2 = κ 2 4πl Q 2l 2 Y + l X l Q 3l 2 Y + l X l Q , a 4 = κ 2 4π l Y 3l 2 Y + l X l Q ,(28) provided that 3l 2 Y + l X l Q = 0. Otherwise, if 3l 2 Y + l X l Q = 0 with l Y = 0, one finds that the system does not possess any solution with non-vanishing matter content. Finally, if l X = l Y = 0, one obtains the solution a 1 = a 2 = κ 2 4πl Q .(29) This contains general relativity as a special case. C. Third order At the third velocity order, the only non-trivial field equation is given by 3 E 0i = −κ 2 ρv i − l Z ∂ 0 ∂ i 2 φ + l Q ∂ 0 ∂ [i 2 h j] j + ∂ j ∂ [j 2 h i]0 .(30) Note in particular that the third order connection component 3 ξ 0 does not enter these equations and thus remains undetermined, so that we can solve for the metric perturbation 3 h 0i . This can be done by using the ansatz 3 h 0i = a 5 V i + a 6 W i ,(31) where V i and W i denote the standard PPN potentials defined in [33]. Further, we need to substitute the second order perturbations found in the previous section for the different non-degenerate and degenerate cases. It is remarkable that in all cases this procedure leads to the same coefficient equation κ 2 + 2πl Q (a 5 + a 6 ) = 0 ,(32) and hence we have the solution a 5 + a 6 = − κ 2 2πl Q ,(33) while their difference is not determined by the third order field equations. The latter is an expected result, as it reflects the invariance of the theory and its post-Newtonian limit under (infinitesimal) diffeomorphisms. D. Fourth order We finally come to the fourth velocity order. We will not display the full perturbative expansion of the field equations here, as it turns out to be lengthy, and restrict ourselves to presenting the steps which are necessary to obtain the solution, starting with the non-degenerate case l Q (l Y + l Z ) = 0. In this case we find that the fourth order field equations contain besides the component 4 h 00 , which we need to solve for in order to determine the PPN parameters, also the fourth order components φ. It turns out that these can be eliminated from the fourth order equations by considering the linear combination (2l Y + l Z )∂ i ∂ j 4 E ij − (l Y + l Z )△( 4 E 00 + 4 E i i ) = 0 .(34) To solve this equation, we make an ansatz of the form 4 h 00 = a 7 U 2 + a 8 Φ 1 + a 9 Φ 2 + a 10 Φ 3 + a 11 Φ 4 + a 12 Φ W + a 13 A ,(35) once again referring to [33] for the definition of the appearing PPN potentials. In addition to the coefficients a 7 , . . . , a 13 in this ansatz, we also need to determine the linear combination a 5 − a 6 from the third order ansatz, which is left undetermined in the third order equations. By extracting the coefficients of the independent matter terms in the fourth order equations, we find that they indeed possess a unique solution for these coefficients, which reads a 5 − a 6 = − 3κ 2 8πl Q , a 7 = − κ 4 32π 2 l 2 Q , a 8 = κ 2 2πl Q , a 9 = κ 4 16π 2 l 2 Q , a 10 = κ 2 4πl Q , a 11 = 3κ 2 4πl Q , a 12 = a 13 = 0 .(36) In the degenerate case l Y + l Z = 0 and 3l 2 Y + l X l Q = 0, the fourth order connection component (4l 2 Y + l X l Q ) 4 E 00 + (2l 2 Y + l X l Q ) 4 E i i + l Y l Q 4 E φ = 0 .(37) Using again the ansatz (35), we now find the solution a 5 − a 6 = − κ 2 8πl Q 8l 2 Y + 3l X l Q 3l 2 Y + l X l Q , a 10 = κ 2 4πl Q 4l 2 Y + l X l Q 3l 2 Y + l X l Q , a 11 = 3κ 2 4πl Q 2l 2 Y + l X l Q 3l 2 Y + l X l Q , a 8 = κ 2 2πl Q , a 12 = a 13 = 0 , a 9 − 2a 7 = κ 4 16π 2 l 2 Q 20l 4 Y + 13l X l 2 Y l Q + 2l 2 X l 2 Q (3l 2 Y + l X l Q ) 2 , a 9 + 2a 7 = − κ 4 l Y 16π 2 l 2 Q (38) × 24l 5 Y + 2l 3 Y l Q [7l X − 2(l Y φ + l Zφ )] + l Y l 2 Q [2l X (l X − 2l Y φ − l Zφ ) + l Y l Xφ ] + (2l 2 Y + l X l Q )(6l 2 Y + l X l Q )l Qφ (3l 2 Y + l X l Q ) 3 . Finally, for l X = l Y = l Z = 0 we find again the same solution (36) as for the non-degenerate case. Hence, we have determined all possible solutions for the fourth order, without introducing any further distinction between different cases beyond the one introduced at the second order. V. PPN PARAMETERS By comparing the solution for the metric perturbation components we have derived in the preceding section to their standard PPN form [32,33], we are now able to obtain the values of the PPN parameters for the different subclasses of scalar-nonmetricity theories we considered. We start by recalling that the theories we study here are restricted by the conditions l 0 = l φ = 0 in order to possess a Minkowski background solution, l φφ = l φφφ = 0 for a massless scalar field and l Q = 0 to obtain a well-defined Newtonian limit. We find that after imposing these conditions, the most generic class of theories satisfying l Y + l Z = 0 exhibits the PPN parameters β = γ = 1 , α 1 = α 2 = α 3 = ζ 1 = ζ 2 = ζ 3 = ζ 4 = ξ = 0 ,(39) and thus fully agrees with the PPN parameters of general relativity. Potential deviations from these values are encountered only in the subclass l Y + l Z = 0. Within this subclass, we found that theories which in addition satisfy l X = l Y = 0, so that the scalar field is minimally coupled to nonmetricity at the linear order, again yield the same PPN parameters (39). For theories with 3l 2 Y + l X l Q = 0 we obtain the PPN parameters γ = 1 − 2l 2 Y 4l 2 Y + l X l Q , (40a) β = 1 − l Y {12l 5 Y + l 3 Y l Q [7l X + 4(l Y φ + l Zφ )] + l Y l 2 Q [l X (l X + 4l Y φ + 2l Zφ ) − l Y l Xφ ] − (2l 2 Y + l X l Q )(6l 2 Y + l X l Q )l Qφ } 2(3l 2 Y + l X l Q )(4l 2 Y + l X l Q ) 2 ,(40b) while the remaining parameters vanish again, indicating that the theory is fully conservative, i.e., there are no preferred-frame or preferred-location effects or violation of total energy-momentum conservation. Taking a closer look at this result, one finds that even in this class there is another subclass given by l Y = 0 corresponding to a minimally coupled scalar field at the linear perturbation order which leads to the general relativity values (39). Further, we find that for 4l 2 Y + l X l Q = 0 the PPN parameters β and γ diverge. This is due to the fact that in this case the contribution of the Newtonian potential U to the perturbation 2 h 00 vanishes. Hence, also in these theories no physically meaningful Newtonian limit is obtained. We summarize our findings by listing all cases we studied and their corresponding results in diagram 1. We conclude from our findings that the generic class of scalar-nonmetricity theories of gravity with a massless scalar field possesses the same PPN parameters as general relativity, and is therefore indistinguishable from the latter, and passes all solar system tests. Deviations for the parameters β and γ are found only for a particular subclass, which contains the scalar-nonmetricity equivalent of scalar-curvature gravity as a special case [27]. In this case, solar system observations give bounds on the Taylor coefficients of the Lagrangian function, and so also this subclass contains theories passing the solar system tests. Another possibility, which remains to be studied, is theories in which the scalar field is massive, and in which its contribution to the post-Newtonian limit depends on the distance to the gravitating body. However, this study exceeds the scope of this article. VI. CONCLUSION We have studied the post-Newtonian limit of a general class of scalar-nonmetricity theories of gravity and calculated their PPN parameters for the case of a massless scalar field. Our results show that generically a scalar field which is non-minimally coupled to nonmetricity is suppressed in the post-Newtonian limit and does not contribute to the post-Newtonian dynamics, so that the PPN parameters agree with those of general relativity. Deviations for the PPN parameters β and γ are found only for a specific subclass of theories which are distinguished by their coupling between the scalar field and nonmetricity. Further, we find that also in this case all other PPN parameters agree with those of general relativity. Hence, we find that the theories are fully conservative and do not possess any violation of local position invariance, local Lorentz invariance or total energy-momentum conservation. The most remarkable result of our work is the suppression of the scalar field in the post-Newtonian limit if it is non-minimally coupled to the nonmetricity via any other linear combination than the mixed trace Q ν νµ − Q µν ν at the linear order. This particular coupling term also appears in other results. Coupling to this term restores the conformal invariance of scalar-nonmetricity theories when this is broken through a non-minimal coupling to the nonmetricity scalar Q [27]. Also in cosmology the behavior of theories coupled to this term only differs qualitatively from the generic coupling [41]. The fact that their PPN parameters are exactly identical to those of general relativity, while allowing for a richer cosmological dynamics, motivates further studies of this class of theories and their implications for observations in cosmology and gravitational waves, where higher order effects beyond the PPN parameters become relevant due to the strong gravity present at the gravitational wave source. Another possible line of future investigation is to allow for a massive scalar field and study the resulting PPN parameters, in analogy to previous works on scalar-curvature [42][43][44] and scalar-torsion theories of gravity [45]. A natural question is whether the aforementioned suppression of the scalar field is present also in this case, which would significantly simplify the post-Newtonian limit compared to the case of a non-vanishing scalar field contribution. In the latter case, the PPN parameters are no longer constant, but attain a dependence on the distance to the gravitating source, which is in general highly non-trivial. ACKNOWLEDGMENTS KF gratefully acknowledges support by the DFG within the Research Training Group Models of Gravity. MH gratefully acknowledges the full financial support by the Estonian Research Council through the Personal Research Funding project PRG356 and by the European Regional Development Fund through the Center of Excellence TK133 "The Dark Side of the Universe". This article is based upon work from COST Action QGMM (CA18108), supported by COST (European Cooperation in Science and Technology). from the remaining fourth order components by taking the linear combination FIG. 1 . 1Full classification of L(Q, X, Y, Z, φ) theories. The path highlighted by thick arrows corresponds to STEGR. Theories with β = γ = 1 are in full agreement with observations. 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[ "Particle Production in the Interiors of Acoustic Black Holes", "Particle Production in the Interiors of Acoustic Black Holes" ]	[ "Roberto Balbinot ", "Alessandro Fabbri ", "Richard A Dudley ", "Paul R Anderson ", "\nDipartimento di Fisica dell'Università di Bologna and INFN sezione di Bologna\nDepartamento de Física Teórica and IFIC\nCentro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi\nCentro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi\nVia Irnerio 46, Piazza del Viminale 1, Universidad de Valencia-CSIC, C. Dr. Moliner 50, Piazza del Viminale 140126, 00184, 46100, 00184Bologna, Roma, Burjassot, RomaItaly, Italy, Spain, Italy\n", "\nDepartment of Physics\nLaboratoire de Physique Théorique, CNRS UMR 8627\nUniversité Paris\nBât. 210, Sud 1191405Orsay CedexFrance\n", "\nWake Forest University\n27109Winston-SalemNorth CarolinaUSA\n" ]	[ "Dipartimento di Fisica dell'Università di Bologna and INFN sezione di Bologna\nDepartamento de Física Teórica and IFIC\nCentro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi\nCentro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi\nVia Irnerio 46, Piazza del Viminale 1, Universidad de Valencia-CSIC, C. Dr. Moliner 50, Piazza del Viminale 140126, 00184, 46100, 00184Bologna, Roma, Burjassot, RomaItaly, Italy, Spain, Italy", "Department of Physics\nLaboratoire de Physique Théorique, CNRS UMR 8627\nUniversité Paris\nBât. 210, Sud 1191405Orsay CedexFrance", "Wake Forest University\n27109Winston-SalemNorth CarolinaUSA" ]	[]	Phonon creation inside the horizons of acoustic black holes is investigated using two simple toy models. It is shown that, unlike what occurs in the exterior regions, the spectrum is not thermal.This non-thermality is due to the anomalous scattering that occurs in the interior regions.	10.1103/physrevd.100.105021	[ "https://arxiv.org/pdf/1910.04532v1.pdf" ]	204,008,957	1910.04532	a3909f5e0a09985b5ae1e4c24af360b31003c2da	Particle Production in the Interiors of Acoustic Black Holes arXiv:1910.04532v1 [gr-qc] 10 Oct 2019 Roberto Balbinot Alessandro Fabbri Richard A Dudley Paul R Anderson Dipartimento di Fisica dell'Università di Bologna and INFN sezione di Bologna Departamento de Física Teórica and IFIC Centro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi Centro Fermi -Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi Via Irnerio 46, Piazza del Viminale 1, Universidad de Valencia-CSIC, C. Dr. Moliner 50, Piazza del Viminale 140126, 00184, 46100, 00184Bologna, Roma, Burjassot, RomaItaly, Italy, Spain, Italy Department of Physics Laboratoire de Physique Théorique, CNRS UMR 8627 Université Paris Bât. 210, Sud 1191405Orsay CedexFrance Wake Forest University 27109Winston-SalemNorth CarolinaUSA Particle Production in the Interiors of Acoustic Black Holes arXiv:1910.04532v1 [gr-qc] 10 Oct 20191PACS numbers: * Electronic address: balbinot@boinfnit † Electronic address: afabbri@ificuves ‡ Electronic address: dudlra13@wfuedu § Electronic address: anderson@wfuedu Phonon creation inside the horizons of acoustic black holes is investigated using two simple toy models. It is shown that, unlike what occurs in the exterior regions, the spectrum is not thermal.This non-thermality is due to the anomalous scattering that occurs in the interior regions. I. INTRODUCTION Among the many spectacular celestial objects that populate our universe, black holes (BHs) are perhaps the most intriguing. The presence of a causal horizon prevents any direct observation of the interior. According to theoretical studies based on General Relativity, the interior is a place where peculiar physical effects occur, which cannot be confirmed by astronomical observations. The situation has positively changed in recent times with the advent of so called "analog BHs" [1, 2]. These are condensed matter systems that are realizable in the laboratory, which mimic some of the essential features of gravitational BHs. A typical example is a Bose-Einstein Condensate (BEC) fluid (see for example [3]) whose flow becomes supersonic [4][5][6]. The supersonic region, trapping sound waves inside it, is the analog of the BH interior. The sonic surface where the speed of the flow equals the local speed of sound, plays the role of the horizon. This sonic horizon however has no causal significance at all: there is nothing to prevent one from directly observing the interior region. Indeed the first experimental observations of the analog of Hawking radiation [7] in BECs by Steinhauer et. al. [8,9] were made by performing simultaneous measurements of the density outside and inside the sonic horizon. A peak was observed in the resulting in-out density-density correlation function that was predicted in [10,11] and which is the 'smoking gun' signaling the presence of Hawking radiation. In the same spirit, one can imagine that other processes that are predicted to take place in the interior of a BH can be experimentally verified by looking at appropriate analog models. With this as motivation, in this paper we discuss the unusual features of scattering by a potential inside the horizon of a stationary BEC analog BH and its consequences. The calculations are done in the analog spacetime using quantum field theory in curved space techniques. These are the same types of calculations that one would do to explore similar effects in the interior of a real black hole. In Quantum Mechanics in the presence of a potential, an incident flux is split into a transmitted and a reflected part (see Fig.1). Reflection (R) and Transmission (T ) coefficients satisfy the unitary relation \|R\| 2 + \|T \| 2 = 1, which is the conservation of probability. Note that the previous relation implies that \|R\| 2 and \|T \| 2 ≤ 1. Inside the horizon of a BH both the transmitted and the "would be reflected" part of the field are forced to propagate in the same direction, namely towards the center of the black Incident T R FIG. 1: Illustration of a plane wave incident onto a potential from the right, which then is partially transmitted to the left and also partial reflected back to the right of the barrier. Incident T R Direction of Flow FIG. 2: Illustration of a plane wave incident onto a potential from the right in the interior of a BH, which then is partially transmitted to the left and also partial reflected, however the reflected portion is also moving to the left of the barrier since in the interior the wave is forced to travel further into the BH. hole (see Fig. 2). The scattering is 'anomalous' and R and T no longer satisfy the previous unitary relation. Instead they satisfy \|T \| 2 − \|R\| 2 = 1 which implies particle creation since \|T \| 2 ≥ 1. Another way to think about this is that, while the outside region of a nonrotating black hole is static, the interior can be thought of as a dynamical cosmology in which particle creation occurs. We shall deal with both massless and massive quantum fields. For the latter case there is usually a mass gap, namely E ≥ m where E is the conserved (Killing) energy and m is the mass of the particle. Inside a BH the former inequality no longer holds, E can take any value, even negative ones. In Sec. II a brief review is given of the set-up for BEC analog black holes. In Sec. III particle production is investigated in the case of massless phonons with a double delta function potential. In Sec. IV particle production is investigated for massive phonons when the effective potential is zero and the mass term in the mode equation is approximated by two step functions. Sec. V contains a discussion of our results and comparisons with some previous work. II. THE SETTING Under the hydrodynamic approximation the phase fluctuation operatorφ in a BEC satisfies a covariant version of D'Alembert's wave equation (see for instance [2]) φ = 0 , (2.1) whereˆ =∇ µ∇ ν is evaluated on a fictitious curved spacetime metric, called the acoustic metric, which in our case we write as follows ds 2 = n m a c −c 2 (x)dT 2 + (dx + v 0 dT ) 2 + dy 2 + dz 2 (2.2) where n is the density of the condensate (here assumed to be constant), m a is the mass of a single atom of the BEC, and c(x) is the sound speed. The flow is assumed to be stationary and one dimensional along the x axis with the velocity v = −v ox constant and directed from right to left. For a typical profile used in BEC analog models c(x) becomes constant in both asymptotic regions (x → ±∞) so that lim x→+∞ c(x) = c r and lim x→−∞ c(x) = c l with c r > v 0 and c l < v 0 . Thus the asymptotic regions are homogeneous and the profile monotonically decreases from right to left. The profile c(x) is chosen so that the horizon c(x) = v 0 is at x = 0. In the region x < 0, where c(x) < v 0 the metric describes the interior region of the acoustic BH while for x > 0, where c(x) > v 0 , the metric describes the exterior region of the acoustic BH. We call the exterior the r region and the interior the l region. Performing a dimensional reduction along the transverse direction and passing from the Gullstrand-Painlevé coordinates (T, x) to the Schwarzschild like ones (t, x * ) via the transformation t = T − dx v 0 c(x) 2 − v 2 0 and x * = dx c(x) c(x) 2 − v 2 0 . (2.3) the wave equation (2.1) can be reduced to [−∂ 2 t + ∂ 2 x * − k 2 ⊥ (c 2 − v 2 0 ) + V eff ]φ (2) = 0 , (2.4) where the effective potential is given by V eff ≡ c 2 − v 2 0 c 1 2 d 2 c dx 2 1 − v 2 0 c 2 − 1 4c dc dx 2 + 5v 2 0 4c 3 dc dx 2 . (2.5) The coefficient k 2 ⊥ is related to the transverse momentum andφ (2) is the dimensionally reduced field operator (see the appendix of Ref. [12] for details). The last two terms in Eq. (2.4), the mass-like term and V eff , cause scattering of the modes. Note that both of these terms vanish at the horizon. There the modes are effectively massless and propagate freely. φ (2) = ∞ 0 dω â ω f ω +â † ω f * ω (2.6) The creation and annihilation operators,â ω andâ † ω , satisfy the usual commutation relations. The modes f ω are normalized using the conserved scalar product (f ω , f ω ) = −i dΣ µ f ω ← → ∂ µ f * ω [g Σ (x)] 1 2 (2.7) with dΣ µ = n µ dΣ, where Σ is a Cauchy surface, n µ a future directed unit vector perpendicular to Σ, and g Σ the determinant of the induced metric. Writing f ω = e ±iωt χ ω (x * ) (2.8) and substituting into (2.4) gives d 2 χ ω dx * 2 + ω 2 − k 2 ⊥ (c 2 − v 2 0 ) + V eff χ ω = 0 . (2.9) In this paper we consider two toy models for the terms in Eq. (2.9) responsible for the scattering which have the advantage of being exactly solvable while, despite their crudeness, encode all of the basic features of the process we wish to discuss. r l v I l + u I l + H − i + i − i 0 H + I r + I r − in f r III. DIRAC DELTA FUNCTION POTENTIALS In the first toy model, the transverse excitations are neglected (i.e., k ⊥ = 0) and V eff is approximated by two Dirac delta functions, one in region r and one in region l. For simplicity we choose them at x * r = 0 in r and at x * l = 0 in l leading to 1 V eff =    V l δ(x * − x * l ) , x < 0 , V r δ(x * − x * r ) , x > 0 . (3.1) The Penrose diagram for the BH metric given in Eq. (2.2) is shown in Fig 3, where the modes representing our 'in' basis are schematically indicated. The asymptotic behaviors of these modes are in f r I = e −iωt e −iωx * √ 4πω = e −iωv √ 4πω (3.2) on past null infinity, I r − ; in f r H = e −iωt e iωx * √ 4πω = e −iωu √ 4πω (3.3) on the portion of the past horizon in region r, H r − ; in f l H = e iωt e −iωx * √ 4πω = e iωu √ 4πω (3.4) on the portion of the past horizon in region l, H l − . These are positive norm modes on I r − or H − which together form a Cauchy surface for the spacetime. These modes are associated with annihilation operators in the expansion of the fieldφ (2) in Eq. (2.6). In Eqs (3.2-3.4), u = t − x * and v = t + x * are the Eddington-Finkelstein retarded and advanced null coordinates respectively. Note the + sign in the exponent of Eq. (3.4). The conserved (Killing) energy associated with it is negative and corresponds to excitations called "partners". We need to find the explicit forms of the modes throughout the spacetime. Let us begin with the in f r I mode whose evolution is represented schematically in Fig 4. The incoming v mode of the form Eq. (3.2) coming from I r − is partially transmitted (T r I ) towards the horizon as a v mode and partially reflected (R r I ) back to infinity I r + as a u mode by the delta potential located at x * r = 0 (see Fig. 5). The transmitted part crosses the horizon, enters the black hole and is split by the second delta function potential located the r region H − i + i − i 0 I r − R r I T r I R l I T l Iin f r I = e −iωt √ 4πω e −iωx * + R r I e iωx * , x * > x * r = 0 , = e −iωt √ 4πω T r I e −iωx * , x * < x * r = 0 , (3.5) and in the region in f r I = e −iωt √ 4πω T r I e −iωx * , x * < x * = 0 , = e −iωt √ 4πω T I e −iωx * + R I e iωx * , x * > x * l = 0 ,(3.6) The transmission and reflection coefficients are found by matching these solutions across the delta function potentials. In general for a potential of the form V = λδ(x * ) we require that χ(x * ) satisfies χ\| − = χ\| + (3.7) χ \| + − χ \| − = −λ χ\| − ,(3.8) where χ\| ± = lim x * →0 ± χ, and χ represents the derivative with respect to x * . The results for χ r I are T r I = 2iω Vr 2iω Vr − 1 , (3.9a) R r I = 1 2iω Vr − 1 , (3.9b) R l I = V l 2iω T r I , (3.9c) T l I = 1 − V l 2iω T r I . (3.9d) These satisfy the relations \|R r I \| 2 + \|T r I \| 2 = 1, (3.10a) \|T l I \| 2 − \|R l I \| 2 + \|R r I \| 2 = 1 . (3.10b) The negative sign in front of the R l I term in (3.10b) comes from the fact that the "reflected" modes R l I e −iωu inside the BH have a negative norm (see Eq. (2.7)). The asymptotic form of the in f r I mode as x → +∞ is in f r I = e −iωt √ 4πω e −iωx * + R r I e iωx * (3.11) and in f l I = e −iωt √ 4πω T l I e −iωx * + R l I e iωx * (3.12) for x → −∞. Following the same procedure for the in f r H modes coming out from the part of the past horizon in the r region (see Fig 7), we have FIG. 7: Illustration of a plane wave incident onto a potential from the left, which then is partially transmitted to the right and also partial reflected back to the left of the barrier. The reflected portion then travels into the interior of the BH where it encounters the potential in the interior. There the "reflected" and "transmitted" portions travel away from the potential to the left, see inside the horizon (as shown in Fig. 8) T r H = 1 1 − Vr 2iω , (3.13) R r H = Vr 2iω 1 − Vr 2iω ,(3.T l H = 1 − V l 2iω R r H , (3.15) R l H = V l 2iω R r H , (3.16) again with \|T l H \| 2 − \|R l H \| 2 + \|T r H \| 2 = 1 leading to the asymptotic form in f r H = e −iωt √ 4πω T r H e iωx * (3.17) for x → +∞ and in f r H = e −iωt √ 4πω R l H e −iωx * + T l H e iωx * (3.18) for x → −∞. Finally, for the modes in f l H coming from the part of the past horizon in region l, see 9, the effective transmission and reflection coefficients arẽ T l H = 1 − V l 2iω , (3.19a) R l H = V l 2iω , (3.19b) satisfying \|T l H \| 2 − \|R l H \| 2 = 1. The asymptotic (x → −∞ ) form of in f l H is in f l H = e iωt √ 4πω T l H e −iωx * +R l H e iωx * . (3.20) Having defined the "in" basis, the field operatorφ (2) can be expanded aŝ φ (2) = dω in râ I ( in f r I ) + in râ H ( in f r H ) + in lâ H ( in f l H ) + h.c. (3.21) where theâ's are the annihilation operators for the respective modes. Alternatively one can construct another basis called the "out" basis formed by modes having the asymptotic form and then obtain the following expressions of the field operator out f r u = e −iωt √ 4πω e iωx * = e −iωu √ 4πω (3.22) for x → +∞ and out f l u = e iωt √ 4πω e −iωx * = e iωu √ 4πω , (3.23) out f l v = e −iωt √ 4πω e −iωx * = e −iωv √ 4πωφ (2) = dω out râ u ( out f r u ) + out lâ u ( out f l u ) + out râ v ( out f r v ) + h.c. (3.25) where the outâ 's are the associated annihilation operators. The "in" and "out" basis are Note that there is no contribution to in f l H from the out f r u modes. Using the scattering S-matrix formalism we can write the relation between the two basis as      in f r I in f r H in f l * H      = S T      out f r u out f l v out f l * u      (3.29) where S T =      S ur,vr S v l ,vr S u l ,vr S ur,ur S v l ,ur S u l ,ur 0 S v l ,u l S u l ,u l      (3.30) is the transpose of the scattering matrix S. The notation used is borrowed from Ref [13] and is quite intuitive. For example S ur,vr indicates an incoming v mode from r leading to an outgoing u mode in r. The corresponding Bogoliubov transformation for the annihilation operators of the two bases is      outâr u outâl v outâl † u      = S      inâr I inâr H inâ l † H      (3.31) where S =      S ur,vr S ur,ur 0 S v l ,vr S v l ,ur S v l ,u l S u l ,vr S u l ,ur S u l ,u l      (3.32) For the two delta functions potential the S-matrix elements can be found by inspection of S v l ,vr = T l I , S v l ,ur = T l H , (3.34) S u l ,vr = R l I , S u l ,ur = R l H , (3.35) S v l ,u l =R l * H , S u l ,u l =T l * H . (3.36) We are interested in the numbers of outgoing particles in the various channels, namely for all values of ω. This is the most "natural" quantum state one can define on the extended manifold described by the Penrose diagram of Fig 3. Physically \|B describes a state in which there are no incoming particles either from past right infinity I r − or from the past horizon H − . Although "natural", this does not correctly describe the quantum state of the fieldφ (2) if the BH is formed by a dynamic gravitational collapse. The collapse in fact induces the conversion of quantum vacuum fluctuations to real on shell particles, the so called Hawking radiation [7]. The state which correctly describes this process, at least at late times, is called the Unruh vacuum \|U [15]. The difference between the two states can be schematically summarized as follows. For the Unruh vacuum the modes coming out from the past horizon are chosen to be positive and negative frequency, not with respect to the Schwarzschild time t, but with respect to Kruskal time. Thus instead of the mode in f r u and in f l u , the modes are chosen as f K H = e −iω K U √ 4πω , (3.39) U = ∓ e −κu κ (3.40) where the − and + refer to the r and l regions respectively, and κ is the surface gravity of the BH horizon, which for our metric is κ = 1 2c d dx c 2 − v 2 x=0 . (3.41) The modes coming from past null infinity for the Unruh vacuum are chosen as in f r I . The field can then be expanded in terms of a complete set of these modeŝ φ (2) = dω k (â ω K f K H +â † ω K f K * H + dω inâr I ( in f r I ) + inâr † I ( in f r * I ) . (3.42) The Unruh state is therefore defined aŝ a ω K \|U = 0 , inâr I \|U = 0 ,(3.43) for every ω and ω K . The relation between the two sets of operators is given by the following Bogoliubov transformations inâr H = dω k α r ω K ωâ ω K + β r * ω K ωâ † ω K , inâl H = dω k α l ω K ωâ ω K + β l * ω K ωâ † ω K ,(3.44) where the Bogoliubov coefficients are given by (see for example [16]) α r ω K ω = 1 2πκ ω ω K (−iω K ) iω κ Γ −iω κ , β r ω K ω = 1 2πκ ω ω K (−iω K ) − iω κ Γ iω κ , α l ω K ω = 1 2πκ ω ω K (iω K ) − iω κ Γ iω κ , β l ω K ω = 1 2πκ ω ω K (iω K ) iω κ Γ −iω κ . (3.45) Using the Bogoliubov transformations, Eq. (3.31) and Eq. (3.44), we obtain n r u ≡ U \| out râ † u out râ u \|U = dω K \|S ur,ur \| 2 β r ω K ω 2 , (3.46) n l v ≡ U \| out lâ † v out lâ v \|U = dω K \|S v l ,ur \| 2 β r ω K ω 2 + S * v l ,ur β r ω K ω S * v l ,u l α l * ω K ω +S v l ,u l α l ω K ω S v l ,ur β r * ω K ω + \|S v l ,u l \| 2 α l ω K ω 2 , (3.47) n l u ≡ U \| out lâ † u out lâ u \|U = dω K \|S u l ,ur \| 2 α r ω K ω 2 + S * u l ,ur α r * ω K ω S u l ,u l β l ω K ω +S u l ,ur α r ω K ω S * u l ,u l β l * ω K ω + \|S u l ,u l \| β l ω K ω 2 + \|S u l ,vr \| 2 . (3.48) One can see the combined effect of the near horizon mixing (the α and β) encoded in the Bogoliubov transformation (3.44), which engenders Hawking thermal radiation, and the scattering caused by the potential (the S matrix element). After some calculation we obtain n r u = 4ω 2 4ω 2 + V 2 R δ(0) e 2πω κ − 1 , (3.49) n l v = 1 4ω 2 V r 2iω − V l 2iω − V r − V l e πω κ 2 δ(0) e 2πω κ − 1 , (3.50) n l u = 1 4ω 2 e πω κ V r V l 2iω + V r + (2iω − V l ) 2 δ(0) e 2πω κ − 1 + V 2 l 4ω 2 + V 2 r . (3.51) The latter expression represents the numbers of the negative energy excitations created inside the BH. Here we see the usual problem of the normalization of plane waves leading to the δ(0). Using wave packets we can set δ(0) → 1 and verify that n r u + n l v = n l u . Thus the number of positive energy excitations created equals the number of negative ones as energy conservation requires. One notices immediately the striking difference between the emission in the exterior region compared to that of the interior region. In the exterior region the scattering is the standard one, n r u describes, as expected, a thermal emission at the temperature T H = κ 2πk B modulated by the gray body factor 4ω 2 4ω 2 +V 2 r which regulates the infrared divergence associated with the Planckian distribution. The gray body factor goes to one for ω V r . In the interior region the scattering is anomalous resulting in particle production; as a result we see that both n l u and n l v do not decay exponentially for large ω but as a power law as seen in Fig. 11. The emission in the interior is not thermal. It is infrared divergent, i.e, the spectrum is dominated by soft phonons. The low frequency behaviors of n r u and n l u for various values of V r > 0 and V l < 0 are shown in Fig .12. The qualitative behaviors of n v for the same cases are identical to that of n l u and thus are not depicted. In addition to the non thermal behaviors of the high frequency modes for n l u seen in Fig. 11, the plots in Fig.12 show another nontrivial feature, a peak, that arises in the quantity ωn l u ( and also occurs for ωn l v ). It appears the peak is most pronounced when \|V l \| >> V r ∼ κ/(2π). In this regime, the position, in ω, of the peak is proportional to V r so it moves to the right on a plot of ωn l u versus ω as V r increases. For V r >> κ/(2π) it disappears because it becomes lost in the power law decay that occurs at high frequencies. In contrast, as V r gets smaller and moves to the left on the plot, the height of the peak decreases relative to its base, which for small values of ω is the limit lim ω→0 ωn l u . If for fixed V r , \|V l \| decreases, but is still larger than V r , then the height of the peak also decreases. However, its location stays about the same. When V r = \|V l \| the peak no longer The quantity C ω n r u shows how the thermal nature of the exterior modes is modified by the gray-body factor. Here C is a scaling factor so that maximum value of C ω n r u = 1. Right: The quantity C ω n l u , where C is a different scaling constant so that C ω n l u = 1 for ω = 10 −6 . An unexpected peak in C ω n l u is visible, and exists in all 3 cases shown. The qualitative behavior of n l v is identical to n l u and so is not depicted here. exists. This can be shown analytically by looking at the derivative of ωn l v , d(ωn l v ) dω = V r 2 κ 1 − e 2πω κ + 2πωe πω κ 4κω 2 e πω κ + 1 2 . (3.52) As the denominator is positive for all ω we can just focus at the numerator. Making the substitution ω = πω κ it can be shown that 2V r 2 κe ω 2 (ω − sinh (ω )) (3.53) which is less than zero for all ω > 0. Thus there is no peak like the one seen in Fig. 12 in the V r < \|V l \| case. The same can also be shown for n u but the expressions are more complicated. This peak is also present in other, more realistic, configurations for the effective potential. This will be shown elsewhere. A final remark concerning the Boulware vacuum \|B . This state is characterized by being a vacuum state at infinity I r ± (no incoming and no outgoing particles for x → +∞), that is singular however at H ± . Indeed if we calculate the number of particles created in the r region one finds This is not true in the BH interior region because of the particle production that occurs there resulting in N l v ≡ B\| out lâ † v out lâ v \|B = \|S v l ,u l \| 2 = R l * H 2 = V 2 l 4ω 2 (3.55) and N l u ≡ B\| out lâ † u out lâ u \|B = \|S u l ,v l \| 2 + \|S u l ,v l \| 2 = R l I 2 + R l H 2 = V 2 l 4ω 2 \|R r H \| 2 + \|T r H \| 2 = V 2 l 4ω 2 . IV. THE MASSIVE MODEL The second toy model we want to investigate is the one introduced in Ref [12], where in the field equation (2.4) V eff is neglected and the mass term k 2 ⊥ (c 2 − v 2 0 ) is approximated as two step functions (see Fig 13 ) FIG. 13: Plot from Ref [12] of the coefficient of m 2 (dashed) and the approximation to that coefficient(solid). This result is based on the sound speed profile used in [11] and [16]. k 2 ⊥ (c 2 − v 2 0 ) →      m 2 r Θ(x * − x * 0r ) , x > 0 , −m 2 l Θ(x * − x * 0l ) , x < 0 , (4.1) where m 2 r = m 2 (c 2 r − v 2 0 ) and m 2 l = m 2 (v 2 0 − c 2 l ). Again c r and c l are the asymptotic values of c(x) as x → +∞ and x → −∞ respectively. The − sign in front of m 2 l comes from the fact that inside the BH c 2 < v 2 0 . We also choose x * 0l = 0 = x * 0r for simplicity. The field equation (2.4) simplifies to [−∂ 2 t + ∂ 2 x * − m 2 r Θ(x * )]φ (2) = 0 , x > 0 , (4.2) [−∂ 2 t + ∂ 2 x * + m 2 l Θ(x * )]φ (2) = 0 , x < 0 ,(4.3) Since the construction of the "in" basis for this model has been performed in Ref [12], here we briefly sketch the basic features. The asymptotic form of the incoming v mode coming from x → +∞ is in f r I = 1 √ 4πω e −iωt e −ikrx * (4.4) with k r ≡ ω 2 − m 2 r . This is a massive mode and it exists only if ω > m r i.e., there is, as usual, a mass gap. On the other hand on H − where these modes are massless in f r H = 1 √ 4πω e −iωu (4.5) in f l H = 1 √ 4πω e iωu . (4.6) The form of these modes throughout the spacetime can be found by enforcing continuity of the spatial part χ of the modes and their derivatives at the boundaries of the step functions with the result in f r I = e −iωt √ 4πω k r − ω k r + ω e ikrx * + e −ikrx * , for x → +∞ , (4.7) in f r I = e −iωt √ 4πk l √ k l k r k r + ω k l + ω 2k l e ik l x * + k l − ω 2k l e −ik l x * , for x → −∞ ,(4.8) where k l ≡ ω 2 + m 2 l . Note that unlike k r , k l is real for any value of omega and k l ≥ m l . These modes can be illustrated schematically in the same way as the previous toy model of Note that for ω < m r , n r u = 0 because no modes reach future null infinity in the r region. Thus there is a discontinuity in n r u at ω = m r . Right: Plot of n l u vs ω which shows the non-thermal nature of the interior particle number as the low ω behavior is shown to approach a constant and for large ω it decays as a power law. The qualitative behavior of n l v is very similar to n l u , thus it is not shown. For completeness we can also work out the numbers of created particles in the Boulware state \|B N r u = 0 (4.21) N l u = N l v = (k l − ω) 2 4ωk l . (4.22) One can see that, just like in the previous case, \|B is no longer a vacuum state in the interior of the BH. Moreover, unlike in the Unruh state the number of created particles diverges as ω → 0. V. CONCLUSIONS We have investigated scattering in the exterior of the acoustic horizon of a BEC analog BH and anomalous scattering or particle production in its interior in a simple model with massless phonons and a different one for massive phonons. We have considered both the Unruh and Boulware states. The latter is the natural vacuum state for a static star while the former gives a good approximation in the exterior region to the late time radiation produced by the black hole. As expected we find for the region outside the horizon that the spectrum at infinity is thermal modulo the graybody factor for the Unruh state and there are no particles for the Boulware state. In the massive case we find that, as expected, the emitted thermal radiation in the exterior is gapped. In the interior anomalous scattering produces additional particle production for both massless and massive phonons and this destroys the thermal nature of the spectrum for the Unruh state. At small frequency the emission is dominated by soft phonons but only in the massless case. At high frequency one finds that, for the considered models, the particle number falls off like an inverse power of the frequency rather than exponentially. Not surprisingly particle production also occurs for the Boulware state in the interior. So the Boulware state remains a vacuum state in the exterior and can only be considered to be an initial vacuum state in the interior. For massless phonons an unexpected peak was found in the quantities ωn u and ωn v when they are plotted as functions of ω, with n u and n v the number of right moving and left moving particles found at future null infinity in the interior. This peak represents a clear deviation from a thermal spectrum. It occurs for a limited range of the factors V r and V in the delta function potentials (3.1). The presence of particle creation even for the Boulware state inside a BH can be understood by the fact that the Killing vector ∂ ∂t , of which the Boulware modes "in" are eigenfunctions, is spacelike inside the horizon. The symmetry associated with it is homogeneity rather than stationarity. This is clearly seen by the switch of roles of the coordinates t and x * inside the BH; x * is timelike and t is spacelike so a potential depending on x * is a time dependent potential which as such causes particle creation. Particle production associated with anomalous scattering induced by curvature and consequent deviation from thermality of Hawking radiation was first noticed by Corley and Jacobson [17] in a different context in the region exterior to the event horizon. Specifically, they introduced an ad hoc modification of the two dimensional wave equation for the modes propagating in a BH metric which results in a nonlinear dispersion relation, subluminal in their case, i.e. ω − vk = ± k 2 − k 4 k 2 0 . Then they analyzed the influence of the induced anomalous scattering on the spectrum of the particles radiated by the BH in the region exterior to the horizon. The fact that the anomalous scattering occurs outside the horizon is a peculiar effect of the dispersion they chose. In a genuine General Relativity framework, like the one we use, anomalous scattering and related particle production can occur only inside the horizon, outside the scattering is always the standard one giving just a gray body factor and no extra particle production. Deviation of thermality of Hawking radiation in the context of BEC analog BHs, where the modification of the relativistic dispersion relation is superluminal, i.e. ω − vk = ± k 2 + k 4 k 2 0 , was first analyzed numerically by Macher and Parentani [5]. Our results are in the context of quantum field theory in curved space, as such they involve a strictly linear dispersion relation for which there are no superluminal/subluminal modes. The connection to actual analog BHs is that our results should be valid for long wavelength phonons for which the mode equation is approximately the same as that for a massless minimally coupled scalar field in the analog spacetime [20]. The connection of our results to real black holes is that, in the interior (where the Killing vector is spacelike) the spacetime is dynamic and there is also an effective potential, this time due to the spacetime curvature, and so nonthermal particle production should also occur. The advantage of analog gravity is that, unlike what happens in the gravitational context, one has direct experimental access to the region inside the horizon and so the spectrum of the phonons emitted there will be observable. Our results predict that this spectrum will be completely different from the one emitted outside the horizon. In particular, it will not be thermal. The second coordinate transformation in Eq. (2.3) maps the (0, +∞) interval in x in the rregion to (−∞, +∞) in x * while in l the interval (−∞, 0) in x is mapped to (+∞, −∞) in x * .According to the standard procedure of quantum field theory in curved space-time, the field operatorφ (2) is expanded in terms of a complete set of basis functions {f ω , f * ω }, which are solutions of the classical counterpart of the operator equation (2.4) with the result FIG. 3 : 3Penrose diagram with the "in" mode basis schematically illustrated in the l and r regions. FIG. 4 :FIG. 5 :FIG. 6 : 456Penrose diagram illustrating the scattering of an in f r I mode in the l and r regions. Illustration of a plane wave incident onto a potential from the right, which then is partially transmitted to the left and also partial reflected back to the right of the barrier.inside the black hole at x * l = 0 (seeFig. 6) into a transmitted (T l I ) v mode and a "reflected" (R l I ) u mode both traveling inside along the flow toward left future infinity (I l + )Illustration of a plane wave incident onto a negative potential from the right. As this is in the interior of the BH both the transmitted and reflected portions of the mode are forced to travel further into the BH. FIG. 8 : 8Scattering inside the horizon of the mode in f r H . FIG. 9 : 9Scattering of the mode in f l H . FIG. 10 : 10x → −∞. These modes are represented in the Penrose diagram inFig.10.Proceeding in the same manner we can construct the out f modes throughout the spacetime Penrose diagram illustrating the modes forming the "out" basis. related by a Bogoliubov transformation. Looking at the asymptotic form of the in modes Eqs.(3.11, 3.12, 3.17, 3.18) and(3.20), one can rewrite the modes on I + as follows this calculation we have first to specify the quantum state of theφ (2) operator in which the expectation values in Eq. (3.37) have to be taken. The "in" modes used in the expression of the field operatorφ (2) have a temporal part e ±iωt . These are the eigenfunctions of the Killing vector ∂ ∂t associated with the stationarity of the metric and are positive or negative (Killing) energy modes with respect to Schwarzschild time t. The quantum state associated with this expansion is annihilated by all the inâ operators and is called the Boulware vacuum [14], i.e., FIG. 11 : 11Plots for ω >> κ with −V l = V r /10 = κ/100 (Blue, Dashed) and V l = V r = 0 (Red, Solid). The quantity C ω n l u , where C is a different scaling constant so that C ω n l u = 1 for ω = 10 −1 . FIG. 12 : 12Plots for −V l = V r /10 = κ/100 (Blue, Dashed), −V l = V r /100 = κ/100 (Green, Dotted) and −V l = 3V r /200 = κ/100 (Orange, Solid). Left: u \|B = 0 . (3.54) BH \|B is no longer an out vacuum state. Instead there is a net flux of particles (of positive and negative energy) directed towards x → −∞ with N l v = N l u . 14: Illustration of a plane wave incident from x = +∞, which then is partially transmitted and partial reflected back. 15: Illustration of a plane wave incident onto the negative step function potential from the right. As this is in the interior of the BH both the transmitted and reflected portions of the mode are forced to travel further into the BH. FIG. 17 : 17Plots for m = 4 × 10 −2 . Left: Plot of n r u vs ω. 14 ) 14satisfying \|R r H \| 2 + \|T r H \| 2 = 1.Similarly, the ingoing R r H part gets scattered by the δ potentialIncident R r H T r H For typical flows discussed in the literature which mimic the experimental set up in Ref.[8,9] the effective potential in the interior is dominated by a negative peak. Thus, while our analytic results are valid for arbitrary values of V l , when plotting the results we restrict our attention to the case V l < 0. [1] W. Unruh, Phys. Rev. Lett. 46, 1351Lett. 46, (1981 [2] C. Barcelo, S. Liberati and M. Visser, Living Rev. Relativity 8, 12 (2005).Sec III. The scattering of these modes in the exterior is illustrated inFig. 14, whileFig. 15illustrates the interior scattering.The R r I part is the coefficient of the first exponential in Eq. (4.7), while T l I and R l I are the coefficients of the first and second exponentials respectively in Eq.(4.8).For the in f r H modes one findsSchematically the exterior scattering is described inFig. 16and the inner one is similar to the one represented inFig. 15.The T r H term is the coefficient of the first exponential in Eq. (4.9), while T l H and R l H are the coefficients of the first and second exponential respectively in Eq. (4.10). Note that for ω < m r the in f r H mode coming from H r − is completely reflected at x * 0r . This is the boomerang effect as seen in Ref[18].16: Illustration of a plane wave incident onto a step function potential from the left, which then is partially transmitted to the right and also partial reflected back to the left of the barrier. There, the reflected portion then travels into the interior of the BH where it encounters the step function potential in the interior. There the "reflected" and "transmitted" portions travel away from the potential to the left, seeFig. 15.The final set of modes in this basis are the in f l H modes which areSchematically this is the same as seen inFig. 15.T l H is the coefficient of first exponential in Eq. (4.11) andR l H is the coefficient of the second one. The "out" basis is constructed by a similar procedure to that described in the previous section starting from the asymptotic form of the modesas x → +∞, andfor x → −∞. From Eq (4.7 -4.11) we can express the "in" modes in terms of the "out"23 modes asNote that there is no contribution to in f l H = 0 from the out f r u modes. From these the Bogoliubov transformations between the "in" and "out" creation and annihilation operators can be found as in the previous section. The following expression is found for the number of outgoing created particles in the Unruh state,One can verify (again, if wave packets are used then one can set δ(0) = 1) that n r u + n l v = n l u above the threshold ω > m r while for 0 < ω < m r we have n l v = n l u . In this toy model we also find that, unlike the exterior region, the emission inside is not thermal. Furthermore, n l u and n l v are finite in the infrared ω → 0 limit (see also Ref.[19]). In the asymptotic interior region, x → −∞, the dispersion relation for the massive modes is ω 2 − k 2 = −m 2 l so there is no threshold for the conserved energy, one can have phonons whose energy is below m l , even a zero frequency mode with \|k\| = m l exists. There is a threshold in momentum \|k\| > m l for the outgoing x → −∞ particles. These features are a consequence of the switching roles between t and x * inside the BH as we have discussed previously. Note, however, that unlike the energy, momentum is not conserved along the trajectory of the created particle. Finally, the energy of (u, l) particles is negative. All of these unusual features exist only inside the BH. The deviation from a thermal spectrum is easily seen inFig. 17. Note that the spectrum in the exterior is truncated for modes where ω < m R . L Pitaevskii, S Stringari, Bose-Einstein Condensation. OxfordOxford University PressL. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford (2003) . L J Garay, J R Anglin, J I Cirac, P Zoller, Phys. Rev. Lett. 854643L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000) . J Macher, R Parentani, Phys. Rev. A. 8043601J. Macher and R. Parentani, Phys. Rev. A 80, 043601 (2009). . A Recati, N Pavloff, I Carusotto, Phys. Rev. A. 8043603A. Recati, N. Pavloff and I. Carusotto, Phys. Rev. A 80, 043603 (2009). . S W Hawking, Nature. 24830S.W. 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[ "The Bernstein mechanism: Function release under differential privacy", "The Bernstein mechanism: Function release under differential privacy" ]	[ "Francesco Aldà [email protected] \nGörtz Institute for IT Security and Faculty of Mathematics Ruhr\nDept. Computing and Information Systems\nUniversität Bochum\nHorstGermany\n", "Benjamin I P Rubinstein [email protected] \nThe University of Melbourne\nAustralia\n" ]	[ "Görtz Institute for IT Security and Faculty of Mathematics Ruhr\nDept. Computing and Information Systems\nUniversität Bochum\nHorstGermany", "The University of Melbourne\nAustralia" ]	[]	We investigate the general problem of function release under differential privacy. First we develop a mechanism leveraging the iterated Bernstein operator for polynomial approximation of the target function, followed by coefficient perturbation. We then prove ε-differential privacy using access to the target's sensitivity and function evaluation only-admitting treatment of functions described explicitly or implicitly. Moreover, under weak regularity conditions-either Hölder continuity or bounded derivatives-we establish fast rates on utility measured by high-probability uniform approximation. We provide lower bounds on the utility achievable for any functional mechanism that is ε-differentially private. The generality of our mechanism is demonstrated by the analysis of a number of example learners, including nonparametric estimators and regularized empirical risk minimization. Competitive rates are demonstrated for kernel density estimation; and ε-differential privacy is achieved for a broader class of support vector machines than known previously.	10.1609/aaai.v31i1.10884	[ "https://arxiv.org/pdf/1507.04499v2.pdf" ]	14,697,535	1507.04499	b0ec74b5fbd10539aae6aff7180afd0797b52b90	The Bernstein mechanism: Function release under differential privacy Francesco Aldà [email protected] Görtz Institute for IT Security and Faculty of Mathematics Ruhr Dept. Computing and Information Systems Universität Bochum HorstGermany Benjamin I P Rubinstein [email protected] The University of Melbourne Australia The Bernstein mechanism: Function release under differential privacy We investigate the general problem of function release under differential privacy. First we develop a mechanism leveraging the iterated Bernstein operator for polynomial approximation of the target function, followed by coefficient perturbation. We then prove ε-differential privacy using access to the target's sensitivity and function evaluation only-admitting treatment of functions described explicitly or implicitly. Moreover, under weak regularity conditions-either Hölder continuity or bounded derivatives-we establish fast rates on utility measured by high-probability uniform approximation. We provide lower bounds on the utility achievable for any functional mechanism that is ε-differentially private. The generality of our mechanism is demonstrated by the analysis of a number of example learners, including nonparametric estimators and regularized empirical risk minimization. Competitive rates are demonstrated for kernel density estimation; and ε-differential privacy is achieved for a broader class of support vector machines than known previously. Introduction A major challenge in statistics and machine learning is balancing the conflicting objectives of releasing accurate data analyses while protecting data privacy. A U.S. hospital running an effective clinical trial must still guarantee patient privacy as demanded by HIPAA laws; while a media-streaming service's recommendations that drive user engagement, should respect user privacy for risk of litigation. In recent years, differential privacy [10] has emerged as the standard paradigm for privacy-preserving statistical analysis. It provides formal guarantees that aggregate statistics output by a randomized mechanism are not significantly influenced by the presence or absence of an individual input datum. In this paper, we aim to privately release functions that depend on privacy-sensitive training data, that can be subsequently evaluated on arbitrary test points. This non-interactive setting matches a wide variety of machine learning tasks such as non-parametric methods (kernel density estimation and regression) where the function of train and test data is explicit, to generalized linear models and support vector machines where the function is only implicitly defined by an algorithm. We develop a mechanism that addresses this goal, and is based on functional approximation by Bernstein basis polynomials, specifically via an iterated Bernstein operator. Privacy is guaranteed by sanitizing the coefficients of approximation, which requires only function evaluation. As a result, our mechanism applies to releasing explicitly and implicitly defined functions. Our technique can be regarded as a function-valued version of the popular Laplace mechanism [10], which generically privatizes vector-valued mappings assuming the same oracle access to function evaluation and sensitivity. Polynomial approximation has proven useful in differential privacy settings outside function release [22,4]. Towards the goal of private function release, few previous attempts have been made [24,11]. Hall et al. [11] add Gaussian process noise which only yields a weaker form of privacy, namely (ε, δ)-differential privacy, and does not admit general utility rates. Zhang et al. [24] introduce a functional mechanism for the more specific task of perturbing the objective in private optimization, but they assume separability in the training data and do not obtain rates on utility. The Bernstein polynomials central to our mechanism are used in the Stone-Weierstrass theorem to uniformly approximate any continuous function on a closed interval; moreover, the Bernstein operator yields approximations that are pointwise convex combinations of the function evaluations on a cover. As a result, performing privacy-preserving perturbations to the approximation's coefficients, permit us to control utility and achieve fast convergence rates. Wang et al. [23] propose a mechanism that releases a summary of data in a trigonometric basis, able to respond to subsequent queries that are smooth as in our setting, but are in addition required to be separable in the training dataset as in [24]. A natural application is kernel density estimation, which would achieve a rate of O (log(1/β)/(nε)) h/( +h) as does our approach. Private KDE has also been explored in various other settings [9] and under weaker notions of utility [11]. As an example of an implicitly defined function, we consider regularized empirical risk minimization such as logistic regression, ridge regression, and the SVM. Previous mechanisms for private SVM release and ERM more generally [21,7,13,12,1] require finite-dimensional feature mapping or translation-invariant kernels. Hall et al. [11] consider more general mappings but provide (ε, δ)differential privacy. Our treatment of regularized ERM extends to kernels that may be translationvariant with infinite-dimensional mappings, while providing stronger privacy guarantees. Finally, we provide a lower bound that fundamentally limits utility under private function release, partly resolving a question posed by Hall et al. [11]. This matches (up to logarithmic factors) our upper bound in the linear case. Preliminaries Problem setting. In this work, we consider X an arbitrary (possibly infinite) domain and D ∈ X n a database consisting of n points in X . We refer to n as the size of the database D. For a positive integer , let Y = [0, 1] be a set of query points and F : X n × Y → R the function we aim to release. Once the database D is fixed, we denote by F D = F (D, ·) the function parametrized by D. For example, D might represent a training set-over X a product space of feature vectors and labels-with Y representing test points from the same feature space. In Section 3, we show how to privately release the function F D and we provide alternative error bounds depending on regularity of F . Definition 1. Let 0 ≤ γ ≤ 1 and L > 0. A function f : [0, 1] → R is (γ, L)-Hölder continuous if, for every x, y ∈ [0, 1] , \|f (x) − f (y)\| ≤ L x − y γ ∞ . When γ = 1, we refer to f as L-Lipschitz. Definition 2. Let h be a positive integer and T > 0. A function f : 1] ) and its partial derivatives up to order h are all bounded by T . [0, 1] → R is (h, T )-smooth if it is C h ([0, Our goal is to develop a private release mechanism for the function F D in the non-interactive setting. A non-interactive mechanism takes a function F and a database D as inputs and outputs a synopsis A which can be used to evaluate the function F D on Y without accessing the database D further. Differential privacy. To provide strong privacy guarantees on the release of F D , we adopt the well-established notion of differential privacy. Definition 3 ([10]). Let R be a (possibly infinite) set of responses. A mechanism M : X → R (meaning that, for every D ∈ X = n>0 X n , M(D) is an R-valued random variable) is said to provide ε-differential privacy for ε > 0 if, for every n ∈ N, for every pair (D, D ) ∈ X n × X n of databases differing in one entry only (henceforth denoted by D ∼ D ), and for every measurable S ⊆ R, we have Pr[M(D) ∈ S] ≤ e ε Pr[M(D ) ∈ S]. By limiting the influence of data on the induced response distribution, a powerful adversary (with knowledge of all but one input datum, the mechanism up to random source, and unbounded computation) cannot effectively identify an unknown input datum from mechanism responses. The Laplace mechanism [10] is a generic tool for differential privacy: adding zero-mean Laplace noise 1 to a vector-valued function provides privacy if the noise is calibrated to the function's sensitivity. Definition 4 ([10] ). The sensitivity of a function f : X n → R d is given by S(f ) = sup D∼D f (D)− f (D ) 1 , where the supremum is taken over all D, D ∈ X n that differ in one entry only. The sensitivity of a function F : X n × Y → R d is defined as S(F ) = sup y∈Y S(F (·, y)). Lemma 1 ([10] ). Let f : X n → R d be a non-private function of finite sensitivity, and let Y ∼ Lap(S(f )/ε) d . Then, the random functionf (D) = f (D) + Y provides ε-differential privacy. Given a mechanism, we measure its accuracy as follows. Definition 5. Let F : X n × Y → R. A mechanism M is (α, β)-accurate with respect to F D if for any database D ∈ X n and A = M(D), with probability at least 1 − β over the randomness of M, sup y∈Y \|A(y) − F D (y)\| ≤ α. The Bernstein mechanism In this section, we assume to be a constant and consider Y = [0, 1] . In Algorithm 1, we introduce a non-interactive and private mechanism for releasing F D : Y → R, a family of (γ, L)-Hölder continuous or (h, T )-smooth functions, parametrized by D ∈ X n . Algorithm 1 The Bernstein mechanism 1: Sanitization -Inputs: private dataset D ∈ X n ; sensitivity S(F ) and oracle access to target F 2: Parameters: cover size k, Bernstein order h positive integers; privacy budget ε > 0 3: P ← ({0, 1/k, 2/k, . . . , 1}) Lattice cover of Y 4: λ ← S(F )(k + 1) /ε Perturbation scale 5: For each p = (p 1 , . . . , p ) ∈ P : 6: F D (p) ← F D (p) + Y , where Y iid ∼ Lap(λ) 7: Return: F D (p) \| p ∈ P 8: 9: Evaluation -Inputs: query y ∈ Y; private response F D (p) \| p ∈ P 10: b (h) νi,k ← Compute data-independent basis See Definition 8 11: Return: j=1 k νj =0 F D (ν 1 /k, · · · , ν /k) i=1 b (h) νi,k (y i ) Main Theorem. Let , h, positive integers, 0 < γ ≤ 1, L > 0 and T > 0 be constants. Let X be an arbitrary space and Y = [0, 1] . Let furthermore F : X n × Y → R. For ε > 0, the Bernstein sanitization mechanism M provides ε-differential privacy. Moreover, for 0 < β < 1 the mechanism M is (α, β)-accurate if any of the following hold, where hidden constants depend on , L, γ, T, h. (i) If F D is (γ, L)-Hölder continuous for every D ∈ X n , with error rate scaling as α = O S(F ) ε log(1/β) γ 2 +γ ; or (ii) If F D is (2h, T )-smooth for every D ∈ X n , error scaling as α = O S(F ) ε log(1/β) h +h ; or (iii) If F D is linear for every D ∈ X n , error scaling as α = O S(F ) ε log(1/β) . Moreover, if 1/S(F ) ≤ poly(n, 1/ε), then the running-time of the mechanism and the running-time per evaluation are both polynomial in n and 1/ε. The proof of this result is detailed in Section 4. The mechanism makes use of the iterated Bernstein polynomial of F D , which we introduce next (for a comprehensive survey refer to [16,18]). This approximation consists of a linear combination of so-called Bernstein basis polynomials, whose coefficients are evaluations of target F D on a cover. We briefly introduce the univariate Bernstein basis polynomials and state some of their properties. Definition 6. Let k be a positive integer. The Bernstein basis polynomials of degree k are defined as b ν,k (x) = k ν x ν (1 − x) k−ν for ν = 0, . . . , k. Proposition 2 ([16]). For every x ∈ [0, 1], any positive integer k and 0 ≤ ν ≤ k, we have b ν,k (x) ≥ 0 and k ν=0 b ν,k (x) = 1. In order to introduce the iterated Bernstein polynomials, we first need to recall the Bernstein operator. Definition 7. Let f : [0, 1] → R and k a positive integer. The Bernstein polynomial of f of degree k is defined as B k (f ; x) = k ν=0 f (ν/k) b ν,k (x). We call B k the Bernstein operator. It maps a function f , defined on [0, 1], to B k f , where the function B k f evaluated at x is B k (f ; x). Note that the Bernstein operator is linear and if f (x) ∈ [a 1 , a 2 ] for every x ∈ [0, 1], then from Proposition 2 it follows that B k (f ; x) ∈ [a 1 ,k = I − (I − B n ) h = h i=1 h i (−1) i−1 B i k , where I = B 0 k denotes the identity operator. The iterated Bernstein polynomial of order h can then be computed as: B (h) k (f ; x) = k ν=0 f ν k b (h) ν,k (x), where b (h) ν,k (x) = h i=1 h i (−1) i−1 B i−1 k (b ν,k ; x). We observe that B (1) k = B k . Although the bases b (h) ν,k are not always positive for h ≥ 2, we still have k ν=0 b (h) ν,k (x) = 1 for every x ∈ [0, 1]. Proof of the Main Theorem To prove privacy we note that only the coefficients of the Bernstein polynomial of F D are sensitive and need to be protected. In order to provide ε-differential privacy, these coefficients-evaluations of target F D on a cover-are perturbed by means of Lemma 1. In this way, we can release the sanitized coefficients and use them for unlimited, efficient evaluation of the approximation of F D over Y, without further access to the data D. To establish utility, we separately analyze error due to the polynomial approximation of F D and error due to perturbation. Unidimensional case ( = 1) Let us fix k, a positive integer. As described in Algorithm 1, the Bernstein mechanism perturbs the evaluation of the function F D on a cover of the interval [0, 1]. Lemma 3. Let ε > 0. Then the Bernstein mechanism M provides ε-differential privacy. The proof of Lemma 3 follows from an application of Lemma 1. We provide the full argument in Appendix B. In order to analyze the accuracy of our mechanism, we denote by B (h) k (F D ; x) = k ν=0 [F D (ν/k) + Y ν ] b (h) ν,k (x) the iterated Bernstein polynomial of order h constructed using the coefficients output by the mechanism M. The error α introduced by the mechanism can be expressed as follows: α = max x∈[0,1] F D (x) − B (h) k (F D ; x) (1) ≤ max x∈[0,1] B (h) k (F D ; x) − B (h) k (F D ; x) + max x∈[0,1] F D (x) − B (h) k (F D ; x) .(2) For every x ∈ [0, 1], the first summand in Equation (2) consists of the absolute value of an affine combination of independent Laplace-distributed random variables. Proposition 4. Let Y 0 , . . . , Y k iid ∼ Lap(λ) , δ ≥ 0, and C h be a constant depending on h only. Then: Pr max x∈[0,1] k ν=0 Y ν b (h) ν,k (x) ≥ δ ≤ exp − δ C h λ . The proof of Proposition 4 follows from a result of Proschan [20] on the concentration of convex combinations of random variables drawn i.i.d from a log-concave symmetric distribution. For completeness, we give the full proof in Appendix C. Proposition 4 implies that with probability at least 1 − β the first summand in Equation (2) is bounded by O (S(F )k log(1/β)/ε). In order to bound the second summand we make use of the following convergence rates. , f (x) − B (h) k (f ; x) ≤ T D h k −h , where D h is a constant independent of k, f and x ∈ [0, 1]. According to the regularity of F D , the second summand in Equation (2) can then be bounded by a decreasing function g(k). All in all, the error in Equation (1) can be bounded as follows: α = O g(k) + S(F )k ε log(1/β) .(3) Since the second summand in Equation (3) is an increasing function in k, the optimal value for k (up to a constant factor) is achieved when k satisfies g(k) = S(F )k ε log(1/β).(4) Solving Equation (4) with the bounds for g(k) provided in Theorems 5 and 6 and substituting the thus obtained value of k into (3) prove the first two statements. The bound when F D is linear follows from the fact that for h = 1 and k = 1 the approximation error in Equation (2) is zero, since linear functions are fixed points of B 1 . The error is thus bounded by O (S(F ) log(1/β)/ε). The running time of the mechanism and the running time for answering a query is linear in k and hence upper bounded by a polynomial in n and 1/ε, if 1/S(F ) ≤ poly(n, 1/ε). Multidimensional case ( > 1) In order to extend the proof of the Main Theorem to > 1, we need to introduce the iterated Bernstein polynomial of a multivariate function f : [0, 1] → R. B (h) k1,...,k (f ; x 1 , . . . , x ) = j=1 kj νj =0 f ν 1 k 1 , · · · , ν k i=1 b (h) νi,ki (x i ). For ease of exposition, we consider k 1 = . . . = k = k. By induction, it is possible to show that the approximation error of B (h) k can be bounded by g(k), if the error of the corresponding univariate polynomial is bounded by g(k). For completeness, we provide a proof in Appendix E. Moreover, note that the mechanism M now outputs a (k + 1) -dimensional vector, where each component is perturbed by Laplace-distributed noise with parameter λ = S(F )(k + 1) /ε. Privacy then follows immediately from Lemma 1. In order to conclude the error analysis, we observe that the error bound provided in Proposition 4 can be extended, too. Pr   max x∈[0,1] j=1 k νj =0 Y νj i=1 b (h) νi,k (x i ) ≥ δ   ≤ exp − δ C h, λ . The proof of Proposition 7 is provided in Appendix D. Following the same steps as in Section 4.1, the error α introduced by the mechanism can thus be bounded by α = O g(k) + S(F )k ε log(1/β) .(5) Solving g(k) = S(F )k log(1/β)/ε with the bounds for g(k) provided in Theorems 5 and 6 and substituting the thus obtained value of k into (5) give the first two statements of the Main Theorem. The bound for linear functions follows from the fact that the approximation error is zero for h = 1 and k = 1. The analysis of the running time of the mechanism and the running time for answering a query is straightforward and hence omitted. Lower bound In this section we present a lower bound on the error that any ε-differentially private mechanism approximating a function F : X n × Y → R must introduce. Theorem 8. For ε > 0, there exists a function F : X n × Y → R such that the error that any ε-differentially private mechanism approximating F introduces is Ω (S(F )/ε), with high probability. Proof. In order to prove Theorem 8, we consider X ⊂ [0, 1] to be a finite set and without loss of generality we view the database D as an element of X n or as an element of (Z + ) \|X \| , i.e. a histogram over the elements of X , interchangeably. We can then make use of a general result provided by De [8]. Proposition 9 ([8]). Assume D 1 , D 2 , . . . , D 2 s ∈ (Z + ) N such that, for every i, D i 1 ≤ n and, for i = j, D i − D j 1 ≤ ∆. Moreover, let f : (Z + ) N → R t be such that for any i = j, f (D i ) − f (D j ) ∞ ≥ η. If ∆ ≤ (s − 1)/ε, then any mechanism which is ε-differentially private for the query f on databases of size n introduces an error which is Ω(η), with high probability. Therefore, we only need to show that there exist a suitable sequence of databases D 1 , D 2 , . . . , D 2 s , a function F : X n × Y → R and a y ∈ Y such that F (·, y) satisfies the assumptions of Proposition 9. We actually show that this holds for every y ∈ Y. Let ε > 0 and V be a non-negative integer. We define X = ({0, 1/(V + 8), 2/(V + 8), . . . , 1}) . Note that N = \|X \| = (V + 9) . Let furthermore c = 1/ε and n = V + c. The function F : X n × [0, 1] → R we consider is defined as follows: F (D, y) = η(d 0 + d 1 + . . . + d N −7 + 2d N −6 + . . . + 8d N + y, 1 ), where d i corresponds to the number of entries in D whose value is x i , for every x i ∈ X . For s = 3, we consider the sequence of databases D 1 , D 2 , . . . , D 8 , where, for j ∈ {1, 2, . . . , 8}, d i ∈ D j is such that d i =    1 for i ∈ {0, 1, . . . , V − 1} c for i = N − j + 8 0 otherwise. We first observe that, for every j ∈ {1, 2, . . . , 8}, D j 1 = n. Moreover, for i = j, D i − D j 1 = 2c ≤ 2/ε. Finally, for i = j, \|F (D i , y) − F (D j , y)\| ≥ cη for every y ∈ [0, 1] . Since S(F ) = 7η, Proposition 9 implies that, with high probability, any ε-differentially private mechanism approximating F must introduce error Ω (S(F )/ε). Examples In this section, we demonstrate the versatility of the Bernstein mechanism through the analysis of a number of example learners. O 1 nε det(H) log(1/β) h +h , with probability at least 1 − β. In Figure 1 we display the utility (averaged over 1000 repeats) of the Bernstein mechanism on 5000 points drawn from a mixture of two normal distributions N (0.5, 0.02) and N (0.75, 0.005) with weights 0.4, 0.6, respectively. Accuracy improves for increasing h, except for sufficiently large perturbations (very small ε) which affect approximations with larger derivatives (larger h) greater. Private cross validation [7,6] can be used to tune h. We conclude noting that the same error bounds can be provided by the mechanism of Wang et al. [23], since the function F H (D, ·) is separable in the training set D, i.e. F H (D, ·) = d∈D f H (d, ·). However, this assumption is overly restrictive for many applications. In the following, we discuss a few such cases and show how the Bernstein mechanism can still be successfully applied. l 2 ), . . . , (d n , l n )) ∈ X n , where d 1 ≤ d 2 ≤ . . . ≤ d n , and for every i ∈ {1, . . . , n} there exists j = i such that \|d i − d j \| ≤ c/n, for a given (and publicly known) 0 < c ≤ n. Small values of c restrict the data space under consideration, whereas c = n corresponds to the general case D ∈ X n . For kernel K and bandwidth δ > 0, the Priestley-Chao kernel estimator [19,2] is defined Priestley as F δ (D, y) = 1 δ n i=2 (d i − d i−1 )K ((y − d i )/δ) l i . This function is not separable in D and S(F δ ) = sup y∈Y S(F δ (·, y)) ≤ 4Bc nδ sup x∈[−1,1] K x δ . If K is the Gaussian kernel, then with probability at least 1 − β the error introduced by the mechanism can be bounded by In order to compute S(F ), we first observe that the sigmoid function is 1/4-Lipschitz. Denoting by w ∼ w the minimizers obtained from input databases D ∼ D , we have S(F ) ≤ sup y∈Y,w∼w 1 4 \| w − w , y \| ≤ sup y∈Y,w∼w 1 4 w − w 2 y 2 , where the last inequality follows from an application of the Cauchy-Schwarz inequality. Chaudhuri et al. [5] showed sup w∼w w − w 2 ≤ 2C/n. Since y 2 ≤ √ for every y ∈ Y, we have S(F ) ≤ C √ /(2n). Since F D is an (h, T )-smooth function for any positive integer h, with probability at least 1 − β the error introduced by the mechanism is bounded by O C nε log(1/β) h +h . We note that defining G D (y) = w , y the previous bound can be improved to l 2 ), . . . , (d n , l n )) ∈ X n , a regularized empirical risk minimization program with loss function L is defined as O C nε log(1/β) ,w ∈ arg min w∈R C n n i=1 L(l i , f w (d i )) + 1 2 w 2 2 , where f w (x) = φ(x), w for a chosen feature mapping φ : X → R F taking points from X to some (possibly infinite) F -dimensional feature space and a hyperplane normal w ∈ R F . Let K(x, y) = φ(x), φ(y) be the kernel function induced by the feature mapping φ. The Representer Theorem [15] implies that the minimizer w lies in the span of the functions K(·, d i ) ∈ H, where H is a reproducing kernel Hilbert space (RKHS). Therefore, we consider F : X n × Y → R such that F D (y) = f w (y) = n i=1 α i l i K(y, d i ), for some α i ∈ R. An upper bound on the sensitivity of this function follows from an argument provided by Hall et al. [11] based on a technique of Bousquet and Elisseeff [3]. In particular, we have S(F ) = sup y∈Y,w∼w \|f w (y) − f w (y)\| ≤ M C n sup y∈Y K(y, y). If K is (2h, T )-smooth, the error introduced is bounded, with probability at least 1 − β, by O M C sup y∈Y K(y, y) nε log(1/β) h +h , Note that this result holds with very mild assumptions, namely for any convex and locally M -Lipschitz loss function (e.g. square-loss, log-loss, hinge-loss) and any bounded kernel K. Conclusions In this paper we have considered the setting of releasing functions of test data and privacy-sensitive training data. We have presented a simple yet effective mechanism for this general setting, that makes use of iterated Bernstein polynomials to approximate any regular function with perturbations applied to the resulting coefficients. Both ε-differential privacy and utility rates are proved in general for the mechanism, with corresponding lower bounds provided, and a number of example learners analyzed, demonstrating the Bernstein mechanism's versatility. Although this is a classical result, we show for completeness that linear functions are fixed points of the Bernstein operator B k = B (1) k , for k ≥ 1. Let f (x) = mx + q, for m, q ∈ R and x ∈ [0, 1]. We have B k (f ; x) = k ν=0 f ν k b ν,k (x) = m k k ν=0 νb ν,k (x) + q k ν=0 b ν,k (x) = mx + q, since k ν=0 b ν,k (x) = 1 and k ν=0 νb ν,k (x) = kx. B Proof of Lemma 3 Let D ∈ X n be a second database differing from D in one entry only. Let furthermore ψ : X n → R k+1 be the map defined by ψ(D) = F D 0 k , F D 1 k , . . . , F D k k . Then S(ψ) = sup D∼D ψ(D) − ψ(D ) 1 ≤ k ν=0 sup D∼D F D ν k − F D ν k ≤ S(F )(k + 1). According to Lemma 1 (applied with k + 1 in place of d), the mechanism M provides ε-differential privacy. C Proof of Proposition 4 In order to prove the proposition, we make use of the following result. 1. a 1 ≥ a 2 ≥ . . . ≥ a m , b 1 ≥ b 2 ≥ . . . ≥ b m ; 2. k i=1 b i ≤ k i=1 a i for k = 1, . . . , m − 1; 3. m i=1 a i = m i=1 b i = 1. Then, for all δ ≥ 0 Pr m i=1 b i Y i ≥ δ < Pr m i=1 a i Y i ≥ δ . Choosing a 1 = 1 and a j = 0 for j = 2, . . . , m, Theorem 10 implies Pr m i=1 b i Y i ≥ δ < Pr [\|Y 1 \| ≥ δ] .(6) for every (b 1 , . . . , b m ) ∈ [0, 1] m which satisfies m i=1 b i = 1. We then observe that the density function h(y) = exp(−\|y\|/λ)/(2λ) of the Laplace distribution is symmetric and log-concave. If Y i ∼ Lap(λ) are i.i.d random variables for i = 1, . . . , m, the right-hand side of Equation (6) satisfies Pr [\|Y 1 \| ≥ δ] = exp − δ λ .(7) Although the bases b (h) ν,k are not always positive for h ≥ 2, we observe that, for x ∈ [0, 1], Z(x) = k ν=0 Y ν b (h) ν,k (x) and Z (x) = k ν=0 Y ν \|b (h) ν,k (x)\| have the same distribution, since the random variables Y ν are i.i.d. and symmetric around zero. We can thus restrict our analysis to Z (x). For x ∈ [0, 1], let U (x) = k ν=0 \|b (h) ν,k (x)\|. We first note that U (x) = k ν=0 h i=1 h i (−1) i−1 B i−1 k (b ν,k ; x) ≤ k ν=0 h i=1 h i B i−1 k (b ν,k ; x) = k ν=0 h i=1 h i B i−1 k (b ν,k ; x) = h i=1 h i k ν=0 B i−1 k (b ν,k ; x) = h i=1 h i B i−1 k k ν=0 b ν,k ; x = h i=1 h i = 2 h − 1.(8) According to Equations (6) and (7), for every x ∈ [0, 1] and δ ≥ 0 we have Pr 1 U (x) k ν=0 Y ν \|b (h) ν,k (x)\| ≥ δ ≤ exp − δ λ . Choosing δ = U (x)δ , we get Pr k ν=0 Y ν \|b (h) ν,k (x)\| ≥ δ ≤ exp − δ U (x)λ ≤ exp − δ (2 h − 1)λ , for every x ∈ [0, 1], concluding the proof. D Proof of Proposition 7 The proof of the proposition follows from the same argument provided in Appendix C, with some minor changes. In particular, it suffices to provide a tail bound for max x∈[0,1] j=1 k νj =0 Y νj i=1 b (h) νi,k (x i ) , since, as observed in Appendix C, the random variables Y νj are i.i.d. and symmetric around zero. In order to apply Theorem 10 and conclude the proof, we need to upper bound U (x) = j=1 k νj =0 i=1 b (h) νi,k (x i ) , for every x ∈ [0, 1] . We have U (x) = j=1 k νj =0 i=1 b (h) νi,k (x i ) = k ν1=0 b (h) ν1,k (x 1 ) k ν2=0 b (h) ν2,k (x 2 ) · · · k ν =0 b (h) ν ,k (x ) ≤ (2 h − 1) , since, according to Equation (8), k νj =0 b (h) νj ,k (x j ) ≤ (2 h − 1) for every j ∈ {1, . . . , }. The rest of the proof follows from the same computations done at the end of Appendix C. E Approximation error of multivariate Bernstein polynomials In what follows, we assume that f : [0, 1] → R is a (γ, L)-Hölder continuous function. The proof for (h, T )-smooth functions follows the same argument, with minor changes. The argument we present here is by induction on . The base case ( = 1) follows from the fact that the Bernstein polynomial a 2 ] 2for every positive integer k and x ∈ [0, 1]. Moreover, it is not hard to show that any linear function is a fixed point for B k . For completeness, we provide a short proof in Appendix A. Definition 8 ([18]). Let h be a positive integer. The iterated Bernstein operator of order h is defined as the sequence of linear operators B (h) Theorem 5 ( 5[14,17]). Let 0 < γ ≤ 1 and L > 0.If f : [0, 1] → R is a (γ, L)-Hölder continuous function, then f (x) − B (1) k (f ; x) ≤ L (4k) −γ/2 for all positive integers k and x ∈ [0, 1]. Theorem 6 ([18]). Let h be a positive integer and T > 0. If f : [0, 1] → R is a (2h, T )-smooth function, then, for all positive integers k and x ∈ [0, 1] Definition 9 . 9Assume f : [0, 1] → R and let k 1 , . . . , k , h be positive integers. The (multivariate) iterated Bernstein polynomial of f (of order h) is defined as Proposition 7 . 7For j ∈ {1, . . . , } and ν j = 0, . . . , k, let Y νj iid ∼ Lap(λ), let δ ≥ 0, and constant C h, depend only on h, . Then: Figure 1 :Figure 2 : 12Private Private SVM with Gaussian kernel Kernel density estimation. Let X = Y = [0, 1] and D = (d 1 , d 2 , . . . , d n ) ∈ X n . For a given kernel K H , with bandwidth H (a symmetric and positive definite × matrix), the kernel density estimator F H : X n × Y → R is defined as F H (D, y) = 1 n n i=1 K H (y − d i ). It is easy to see that S(F H ) ≤ sup x∈[−1,1] K H (x)/n. For instance, if K H is the Gaussian kernel with covariance matrix H, then S(F H ) ≤ 1/(n (2π) det(H)). Moreover, observe that F H (D, ·) is an (h, T )-smooth function for any positive integer h. Hence the error introduced by the mechanism is -Chao kernel regression. For ease of exposition, consider = 1. For constant B > 0, let X = [0, 1] × [−B, B] and Y = [0, 1]. Without loss of generality, consider datasets D = ((d 1 , l 1 ), (d 2 , Logistic regression. In the next two examples, the functions we aim to release are implicitly defined by an algorithm. Let X = {x ∈ [0, 1] : x 2 ≤ 1}. Let furthermore X = X × [0, 1] and Y = [0, 1] . The logistic regressor can be seen as a function F : X n × Y → [0, 1] such that, for D = ((d 1 , l 1 ), (d 2 , l 2 ), . . . , (d n , l n )) ∈ X n , F D (y) = 1/(1 + exp(− w , y )), where w is such that √ /n and G D (y) is a linear function. The prediction with the sigmoid function achieves the same error bound, being 1/4-Lipschitz. Regularized empirical risk minimization. Let now X = [0, 1] , X = X × [0, 1] and Y = X. Let L be a convex and locally M -Lipschitz (in the first argument) loss function. For D = ((d 1 , l 1 ), (d 2 , Figure 2 depicts 2SVM learning with RBF kernel (C = σ = 1) on 1000 each of positive (negative) Gaussian synthetic data with mean [0.3, 0.5] ([0.6, 0.4]), covariance [0.01, 0; 0, 0.01] (0.01 * [1, 0.8; 0.8, 1.5]). On a 200 point i.i.d. test, non-private vs. private SVM both achieve 0.01 misclassification rate. Theorem 10 ([20]). Suppose that f :R → [0, 1] is a log-concave density function such that f (x) = f (−x) for every x ∈ R. LetY 1 , . . . , Y m be i.i.d random variables with density f , and suppose that (a 1 , . . . , a m ), (b 1 , . . . , b m ) ∈ [0, 1] m satisfy . B k (f ; x 1 ) converge uniformly to f in the interval [0, 1], as shown in Theorem 5. Assume now \|B k (f ; x 1 , . . . , x ) − f (x 1 , . . . , x )\| ≤ every (x 1 , . . . , x ) ∈ [0, 1] . Let f : [0, 1] +1 → R be a (γ, L)-Hölder continuous function and let B k (f ; x 1 , . . . , x +1 ) be the corresponding Bernstein polynomial. For every(x 1 , . . . , x +1 ) ∈ [0, 1] +1 , the error \|B k (f ; x 1 , . . . , x +1 ) − f (x 1 , . . . , x +1 )\|can be bounded by≤ \|B k (f ; x 1 , . . . , x +1 ) − B k (f ; x 1 , . . . , x )\| + \|B k (f ; x 1 , . . . , x ) − f (x 1 , . . . , x +1 )\|In fact, the second term of Equation(9)is the error of the Bernstein polynomial of f seen as a function of x 1 , . . . , x only. The corresponding bound then follows from the inductive step. On the other hand, the first summand corresponds to the approximation error of the (univariate) Bernstein polynomial of B k (f ; x 1 , . . . , x ), as a function of the remaining variable x +1 . The proof for (h, T )-smooth functions is obtained by replacing B k with B (h) k and using the bound of Theorem 6. A Lap(λ)-distributed real random variable Y has probability density proportional to exp(−\|y\|/λ). AcknowledgmentsThis work was partially completed while F. Aldà was visiting the University of Melbourne. Moreover, he acknowledges support of the DFG Research Training Group GRK 1817/1. The work of B. Rubinstein was supported by the Australian Research Council (DE160100584). Private empirical risk minimization: Efficient algorithms and tight error bounds. R Bassily, A Smith, A Thakurta, Proceedings of the 55th Annual Symposium on Foundations of Computer Science. the 55th Annual Symposium on Foundations of Computer ScienceIEEER. Bassily, A. Smith, and A. Thakurta. Private empirical risk minimization: Efficient algorithms and tight error bounds. In Proceedings of the 55th Annual Symposium on Foundations of Computer Science, pages 464-473. IEEE, 2014. On the nonparametric estimation of regression functions. J K Benedetti, Journal of the Royal Statistical Society. Series B (Methodological). J. K. Benedetti. On the nonparametric estimation of regression functions. Journal of the Royal Statistical Society. Series B (Methodological), pages 248-253, 1977. Stability and generalization. O Bousquet, A Elisseeff, The Journal of Machine Learning Research. 2O. Bousquet and A. Elisseeff. Stability and generalization. The Journal of Machine Learning Research, 2: 499-526, 2002. Faster private release of marginals on small databases. K Chandrasekaran, J Thaler, J Ullman, A Wan, Proceedings of the 5th Conference on Innovations in Theoretical Computer Science. the 5th Conference on Innovations in Theoretical Computer ScienceACMK. Chandrasekaran, J. Thaler, J. Ullman, and A. Wan. Faster private release of marginals on small databases. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, pages 387-402. ACM, 2014. Privacy-preserving logistic regression. K Chaudhuri, C Monteleoni, Advances in Neural Information Processing Systems. K. Chaudhuri and C. Monteleoni. Privacy-preserving logistic regression. In Advances in Neural Information Processing Systems, pages 289-296, 2008. A stability-based validation procedure for differentially private machine learning. K Chaudhuri, S A Vinterbo, Advances in Neural Information Processing Systems. K. Chaudhuri and S. A. Vinterbo. A stability-based validation procedure for differentially private machine learning. In Advances in Neural Information Processing Systems, pages 2652-2660, 2013. Differentially private empirical risk minimization. K Chaudhuri, C Monteleoni, A D Sarwate, The Journal of Machine Learning Research. 12K. Chaudhuri, C. Monteleoni, and A. D. Sarwate. Differentially private empirical risk minimization. The Journal of Machine Learning Research, 12:1069-1109, 2011. Lower bounds in differential privacy. A De, Proceedings of the 9th Theory of Cryptography Conference. the 9th Theory of Cryptography ConferenceA. De. Lower bounds in differential privacy. In Proceedings of the 9th Theory of Cryptography Conference, pages 321-338, 2012. Local privacy and minimax bounds: Sharp rates for probability estimation. J Duchi, M J Wainwright, M I Jordan, Advances in Neural Information Processing Systems. J. Duchi, M. J. Wainwright, and M. I. Jordan. Local privacy and minimax bounds: Sharp rates for probability estimation. In Advances in Neural Information Processing Systems, pages 1529-1537, 2013. Calibrating noise to sensitivity in private data analysis. C Dwork, F Mcsherry, K Nissim, A Smith, Proceedings of the 3rd Theory of Cryptography Conference. the 3rd Theory of Cryptography ConferenceC. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In Proceedings of the 3rd Theory of Cryptography Conference, pages 265-284, 2006. Differential privacy for functions and functional data. R Hall, A Rinaldo, L Wasserman, The Journal of Machine Learning Research. 141R. Hall, A. Rinaldo, and L. Wasserman. Differential privacy for functions and functional data. The Journal of Machine Learning Research, 14(1):703-727, 2013. Differentially private learning with kernels. P Jain, A Thakurta, Proceedings of the 30th International Conference on Machine Learning. the 30th International Conference on Machine LearningP. Jain and A. Thakurta. Differentially private learning with kernels. In Proceedings of the 30th International Conference on Machine Learning, pages 118-126, 2013. Near) dimension independent risk bounds for differentially private learning. P Jain, A G Thakurta, Proceedings of the 31st International Conference on Machine Learning. the 31st International Conference on Machine LearningP. Jain and A. G. Thakurta. (Near) dimension independent risk bounds for differentially private learning. In Proceedings of the 31st International Conference on Machine Learning, pages 476-484, 2014. M Kac, Une remarque sur les polynomes de M. S. Bernstein. Studia Mathematica. 7M. Kac. Une remarque sur les polynomes de M. S. Bernstein. Studia Mathematica, 7(1):49-51, 1938. Some results on Tchebycheffian spline functions. G Kimeldorf, G Wahba, Journal of Mathematical Analysis and Applications. 331G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and Applications, 33(1):82-95, 1971. Bernstein polynomials. G G Lorentz, University of Toronto PressG. G. Lorentz. Bernstein polynomials. University of Toronto Press, 1953. Approximation of Hölder continuous functions by Bernstein polynomials. 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Learning in a large function space: Privacy-preserving mechanisms for SVM learning. B I Rubinstein, P L Bartlett, L Huang, N Taft, Journal of Privacy and Confidentiality. 414B. I. Rubinstein, P. L. Bartlett, L. Huang, and N. Taft. Learning in a large function space: Privacy-preserving mechanisms for SVM learning. Journal of Privacy and Confidentiality, 4(1):4, 2012. Faster algorithms for privately releasing marginals. J Thaler, J Ullman, S Vadhan, Automata, Languages, and Programming. SpringerJ. Thaler, J. Ullman, and S. Vadhan. Faster algorithms for privately releasing marginals. In Automata, Languages, and Programming, pages 810-821. Springer, 2012. Efficient algorithm for privately releasing smooth queries. Z Wang, K Fan, J Zhang, L Wang, Advances in Neural Information Processing Systems. Z. Wang, K. Fan, J. Zhang, and L. Wang. Efficient algorithm for privately releasing smooth queries. In Advances in Neural Information Processing Systems, pages 782-790, 2013. 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[ "Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity", "Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity" ]	[ "Nathaniel Rodriguez \nSchool of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America\n", "Johan Bollen \nSchool of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America\n", "Yong-Yeol Ahn *[email protected] \nSchool of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America\n" ]	[ "School of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America", "School of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America", "School of Informatics and Computing\nThe Center for Complex Networks and Systems Research\nIndiana University\nBloomingtonIndianaUnited States of America" ]	[]	Human history has been marked by social instability and conflict, often driven by the irreconcilability of opposing sets of beliefs, ideologies, and religious dogmas. The dynamics of belief systems has been studied mainly from two distinct perspectives, namely how cognitive biases lead to individual belief rigidity and how social influence leads to social conformity. Here we propose a unifying framework that connects cognitive and social forces together in order to study the dynamics of societal belief evolution. Each individual is endowed with a network of interacting beliefs that evolves through interaction with other individuals in a social network. The adoption of beliefs is affected by both internal coherence and social conformity. Our framework may offer explanations for how social transitions can arise in otherwise homogeneous populations, how small numbers of zealots with highly coherent beliefs can overturn societal consensus, and how belief rigidity protects fringe groups and cults against invasion from mainstream beliefs, allowing them to persist and even thrive in larger societies. Our results suggest that strong consensus may be insufficient to guarantee social stability, that the cognitive coherence of belief-systems is vital in determining their ability to spread, and that coherent belief-systems may pose a serious problem for resolving social polarization, due to their ability to prevent consensus even under high levels of social exposure. We argue that the inclusion of cognitive factors into a social model could provide a more complete picture of collective human dynamics. Data Availability Statement: Code used to generate the simulations are available on an online public repository at https://github.com/Cognitive-Social-Belief-Project/cognitive-social_belief_model.Funding: Microsoft Research and DefenseAdvanced Research Projects Agency NGS2 program (grant #D17AC00005) contributed funding to the project, but had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Competing Interests:The authors have declared that no competing interests exist. and accompanying widespread conflicts, such as the Thirty Years' War[5]. The abundance of such ideological transitions in history raises questions: are they driven by common psychosocial mechanisms? Do specific peculiarities of human psychology play a role in ideological dynamics?Although these questions have been tackled by numerous studies, most existing models of belief system dynamics focus on either social or cognitive factors rather than integrating both aspects. Social models focus on how social interactions transmit and shape beliefs. For instance, Axelrod's cultural dissemination model considers social influence and homophily as key drivers of cultural polarization (seeFig 1A)[6,7]. In this model each agent is represented as a vector of independent traits that can be modified through social influence. The study of spin systems in physics have inspired a number of opinion models, such as the voter model [8-10], Sznajd model[11][12][13], and Ising-like models[14]. Other approaches have drawn upon reaction-diffusion systems[15], or may use continuous opinions[16]or bounded-confidence[17].Cognitive modeling approaches mainly focus on information processing and decision making (seeFig 1B)[18]. Psychological research has revealed that individuals strive for internal consistency, which leads to cognitive mechanisms such as confirmation bias[19]and cognitive dissonance[20]. Our understanding of these cognitive forces and biases in belief formation allow us to move beyond the underlying assumptions-a set of independent beliefs [21, 22]of spin models or opinion vector-based models[23,24].Recent attempts to combine these forces have used mechanisms like social-and anti-conformity [13], foundational beliefs [25], confirmation bias [26], attitudes [27], and social network formation[28][29][30]. However, modeling belief-systems that consist of interacting beliefs, and the study of such systems under social influence, has not been fully investigated. Here we introduce a novel framework that can incorporate both social and cognitive factors in a coherent way (seeFig 1C). We show that the integration of social and cognitive factors produces elementary features of collective social phenomena-societal transitions, upheavals, and existence of fringe groups.Our framework represents a society as a network of individuals-a social network-where each individual possesses a network of concepts and beliefs. Social influence takes place via the social ties[8,[31][32][33]and each individual carries a belief network of interconnected relationships that represents the individual's belief system[20,24,[34][35][36][37][38]. We evaluate the internal coherence of each individual's belief network by applying the balance theory[39,40]. The coherence shifts an individual's willingness to integrate new beliefs, thus simulating cognitive traits such as confirmation bias and cognitive dissonance[18,19,23,[31][32][33]. At the same time, an individual's belief network is affected by social influence; repeated exposure to new ideas through social ties[41][42][43][44]increases their likelihood of adopting even incongruent ideas. Individuals thus experience both cognitive and social forces: (i) they prefer to have coherent belief networks and prefer beliefs that will increase their internal coherence; but (ii) may accept conflicting beliefs under the influence of strong social pressure.MethodsOur approach considers a network of concepts and beliefs where the nodes represent concepts and signed edges between them represents binary associative beliefs that capture the relation between two concepts (cf. Social Knowledge Structure (SKS) model[34]). This formulation allows us to define internal coherence through the principle of triad stability in social balance theory[45][46][47][48].For instance, consider the beliefs of Alice, who is a devoted spectator of soccer. The Eagles are her favorite soccer team, but it has been charged with match fixing. In Alice's belief network Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity PLOS ONE \| 3 / 15Fig 2. Phase space. (a-c) Phase diagrams of various combinations of the three parameters J, I, and T.Along with corresponding slices through the phase space as indicated by the dashed red lines (bottom row). (a,d) peer-influence I and susceptibility T conflict creating a regime where multiple belief systems with various coherences can coexist. We see a similar regime appear in (b, e) where peer-influence and coherentism contend for dominance. More traditional disorder-to-order transitions as in other opinion models also take place when I is small and fixed (c, f). S/N is the fractional size of the largest group. ER graphs with N = 10 4 nodes and average degree of 5 were used. The density was calculated from a 160 trails per point. The belief network was fully connected with M = 5.	10.1371/journal.pone.0165910	null	14,345,579	1509.01502	c31104ecad632895f970b64fcd0a40adf87a570e	Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity Nathaniel Rodriguez School of Informatics and Computing The Center for Complex Networks and Systems Research Indiana University BloomingtonIndianaUnited States of America Johan Bollen School of Informatics and Computing The Center for Complex Networks and Systems Research Indiana University BloomingtonIndianaUnited States of America Yong-Yeol Ahn *[email protected] School of Informatics and Computing The Center for Complex Networks and Systems Research Indiana University BloomingtonIndianaUnited States of America Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity RESEARCH ARTICLE Human history has been marked by social instability and conflict, often driven by the irreconcilability of opposing sets of beliefs, ideologies, and religious dogmas. The dynamics of belief systems has been studied mainly from two distinct perspectives, namely how cognitive biases lead to individual belief rigidity and how social influence leads to social conformity. Here we propose a unifying framework that connects cognitive and social forces together in order to study the dynamics of societal belief evolution. Each individual is endowed with a network of interacting beliefs that evolves through interaction with other individuals in a social network. The adoption of beliefs is affected by both internal coherence and social conformity. Our framework may offer explanations for how social transitions can arise in otherwise homogeneous populations, how small numbers of zealots with highly coherent beliefs can overturn societal consensus, and how belief rigidity protects fringe groups and cults against invasion from mainstream beliefs, allowing them to persist and even thrive in larger societies. Our results suggest that strong consensus may be insufficient to guarantee social stability, that the cognitive coherence of belief-systems is vital in determining their ability to spread, and that coherent belief-systems may pose a serious problem for resolving social polarization, due to their ability to prevent consensus even under high levels of social exposure. We argue that the inclusion of cognitive factors into a social model could provide a more complete picture of collective human dynamics. Data Availability Statement: Code used to generate the simulations are available on an online public repository at https://github.com/Cognitive-Social-Belief-Project/cognitive-social_belief_model.Funding: Microsoft Research and DefenseAdvanced Research Projects Agency NGS2 program (grant #D17AC00005) contributed funding to the project, but had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Competing Interests:The authors have declared that no competing interests exist. and accompanying widespread conflicts, such as the Thirty Years' War[5]. The abundance of such ideological transitions in history raises questions: are they driven by common psychosocial mechanisms? Do specific peculiarities of human psychology play a role in ideological dynamics?Although these questions have been tackled by numerous studies, most existing models of belief system dynamics focus on either social or cognitive factors rather than integrating both aspects. Social models focus on how social interactions transmit and shape beliefs. For instance, Axelrod's cultural dissemination model considers social influence and homophily as key drivers of cultural polarization (seeFig 1A)[6,7]. In this model each agent is represented as a vector of independent traits that can be modified through social influence. The study of spin systems in physics have inspired a number of opinion models, such as the voter model [8-10], Sznajd model[11][12][13], and Ising-like models[14]. Other approaches have drawn upon reaction-diffusion systems[15], or may use continuous opinions[16]or bounded-confidence[17].Cognitive modeling approaches mainly focus on information processing and decision making (seeFig 1B)[18]. Psychological research has revealed that individuals strive for internal consistency, which leads to cognitive mechanisms such as confirmation bias[19]and cognitive dissonance[20]. Our understanding of these cognitive forces and biases in belief formation allow us to move beyond the underlying assumptions-a set of independent beliefs [21, 22]of spin models or opinion vector-based models[23,24].Recent attempts to combine these forces have used mechanisms like social-and anti-conformity [13], foundational beliefs [25], confirmation bias [26], attitudes [27], and social network formation[28][29][30]. However, modeling belief-systems that consist of interacting beliefs, and the study of such systems under social influence, has not been fully investigated. Here we introduce a novel framework that can incorporate both social and cognitive factors in a coherent way (seeFig 1C). We show that the integration of social and cognitive factors produces elementary features of collective social phenomena-societal transitions, upheavals, and existence of fringe groups.Our framework represents a society as a network of individuals-a social network-where each individual possesses a network of concepts and beliefs. Social influence takes place via the social ties[8,[31][32][33]and each individual carries a belief network of interconnected relationships that represents the individual's belief system[20,24,[34][35][36][37][38]. We evaluate the internal coherence of each individual's belief network by applying the balance theory[39,40]. The coherence shifts an individual's willingness to integrate new beliefs, thus simulating cognitive traits such as confirmation bias and cognitive dissonance[18,19,23,[31][32][33]. At the same time, an individual's belief network is affected by social influence; repeated exposure to new ideas through social ties[41][42][43][44]increases their likelihood of adopting even incongruent ideas. Individuals thus experience both cognitive and social forces: (i) they prefer to have coherent belief networks and prefer beliefs that will increase their internal coherence; but (ii) may accept conflicting beliefs under the influence of strong social pressure.MethodsOur approach considers a network of concepts and beliefs where the nodes represent concepts and signed edges between them represents binary associative beliefs that capture the relation between two concepts (cf. Social Knowledge Structure (SKS) model[34]). This formulation allows us to define internal coherence through the principle of triad stability in social balance theory[45][46][47][48].For instance, consider the beliefs of Alice, who is a devoted spectator of soccer. The Eagles are her favorite soccer team, but it has been charged with match fixing. In Alice's belief network Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity PLOS ONE \| 3 / 15Fig 2. Phase space. (a-c) Phase diagrams of various combinations of the three parameters J, I, and T.Along with corresponding slices through the phase space as indicated by the dashed red lines (bottom row). (a,d) peer-influence I and susceptibility T conflict creating a regime where multiple belief systems with various coherences can coexist. We see a similar regime appear in (b, e) where peer-influence and coherentism contend for dominance. More traditional disorder-to-order transitions as in other opinion models also take place when I is small and fixed (c, f). S/N is the fractional size of the largest group. ER graphs with N = 10 4 nodes and average degree of 5 were used. The density was calculated from a 160 trails per point. The belief network was fully connected with M = 5. Introduction Ideological conflict has been a major challenge for human societies [1]. For instance, when post World-War I Germany was marked by economic depression and social trauma, ideological fringe groups like the National-Socialist Party and the Communist Party of Germany made their way into mainstream politics to eventually dominate the political landscape [2][3][4]. This process of ideological upheaval eventually led to World War II, one of the deadliest conflicts in human history. Similarly, 14th and 15th century Europe was torn by sharp religious transitions (c) Our model incorporates both forces, recognizing not only social pressures but also the connected nature of human beliefs. The social network acts as a conduit for belief transmission between individuals. We model a belief as a signed relationship between there is a positive link between the Eagles and soccer, and between the Eagles and match fixing, while there is a negative link between soccer and match fixing. Such pressured configurations are considered unstable (incoherent), and are analogous to frustrated states in spin-systems or unstable social triads. To resolve the frustration Alice may dissociate the Eagles from the allegations of match fixing, drop the Eagles' association with the sport, or change the relationship between soccer and match fixing. It has been shown that people tend to quickly resolve such inconsistency when provided the opportunity to choose dissonance reduction strategies [49]. Yet, in the presence of social pressure or more complicated concept associations, a concept may remain pressured. Each triad in a belief network can be either stable or unstable, as shown in Fig 1C. The incoherence of an entire belief system (of individual n) can thus be captured by an internal energy function [50] on the belief network M: E ðiÞ n ¼ À 1 M 3 X j;k;l a jk a kl a jl ;ð1Þ where M is the number of nodes in the belief network and a jk is the association connecting nodes j and k, which can be +1 (positive association) or −1 (negative association). The sum is taken over all triads in the belief network and normalized by the total number of triads. For simplicity, in our simulations we choose this network to be complete, meaning that all concepts have a positive or negative association with every other concept. A single association's contribution to the energy depends on the state of adjacent associations, providing interdependence and rigidity to the belief system. Beliefs do not necessarily reflect reality. They may be fabricated or completely false. It is, however, the interaction between beliefs that gives them their strength, reflective of psychological factors like confirmation bias and cognitive dissonance. The evolution of belief systems is also driven by social interactions, through which people communicate their beliefs to others. We represent this society as a social network, N , where N ¼ jN j, and whose nodes are individuals and edges represent social relationships through which ideas are communicated. We define a second, "social" energy term, inspired by energy in the spin-based models which captures the degree of alignment between connected individuals. The 'local' social energy that an individual n 2 N feels can be defined by: E ðsÞ n ¼ À 1 k max M 2 X q2GðnÞS n ÁS q ;ð2Þ where the sum is taken over the set of n's neighbors in N , denoted by Γ(n).S is a belief state vector where each element corresponds to an edge in the belief network, so jSj ¼ M 2 À Á . k max is a normalization constant that bounds the strength of peer-influence and is equal to the maximum degree of N . Alternatively, it could be replaced by a function that specifies the scaling relationship between exposure and the individual's energy. For our simulations everyone possesses the same set of concepts (nodes) so M is the same for each person. two concepts. We express the internal coherence of a network of such beliefs in terms of social balance theory where relationship triads can be either stable or unstable. The belief networks evolve over time as individuals decide whether to accept new beliefs transmitted by their peers. doi:10.1371/journal.pone.0165910.g001 We combine the internal energy with the social energy for all individuals to define the total energy as follows: H ¼ X n 2 N JE ðiÞ n þ IE ðsÞ n Â Ãð3Þ where the last sum is taken over all nodes on the network. The parameters J and I, which we refer to as the coherentism and peer-influence respectively, control the relative contribution of the internal energy and the social energy to the total. The dynamics is dominated by internal belief coherence if J ) I and by social consensus when I ) J. Each individual is endowed with their own internal belief network and may transfer some of their beliefs to their social contacts. A receiver of a belief either accepts the incoming belief or not based on the context of their own belief system (internal coherency) and similarity to their neighbors (social conformity). A belief is more likely to be accepted if it increases the coherence of an individual's own belief system, social pressure will also increase the odds of a belief being accepted, even if it conflicts with their belief system. We implement these ideas by creating the following rules: at each time step t, a random pair of connected individuals is chosen and one of the individuals (sender) randomly chooses a belief (association) from its internal belief system and sends it to the other individual (receiver), as illustrated in Fig 1C. We assume that each individual has an identical set of concept nodes. Fig 1C shows the selection and emission process on a graph. The receiver accepts the association if it decreases their individual energy: H n ¼ JE ðiÞ n þ IE ðsÞ n . Even if the change in energy is less than zero, ΔH n > 0, the receiver may still accept with the probability of e À DH n T . This term is analogous to the Boltzmann factor [51]. T, which we refer to as susceptibility, serves a similar purpose as temperature in physical systems for the belief network. As T increases, an individual is more likely to accept their neighbor's opinions that conflict with their own. We characterize the status of the whole society by defining two global energy functions. First, the mean individual energy hE (i) i measures the average internal coherence of individuals. It is expressed by the following equation: hE ðiÞ i ¼ 1 N X n2N E ðiÞ n :ð4Þ The average is taken over the energies of all individuals and it can take values between +1 and −1. hE (i) i = −1 means that every individual in the society possesses a completely coherent belief system, with no pressured beliefs. The other extreme (+1) represents a society where every individual has completely incoherent beliefs. Yet, this measure does not give us any indication of how homogeneous a society is, as belief systems can vary widely while still being coherent. We have a second energy measure inspired from spin systems: hE ðsÞ i ¼ 1 N X n2N E ðsÞ n ;ð5Þ which is minimized if the society is in consensus. For each simulation we use Erdös-Rènyi graphs with N = 10 4 nodes and average degree of 5, though similar results are found for 2D lattices. The belief network was fully connected with M = 5. Results Most opinion models exhibit a phase transition from a disordered state to an ordered one [8], where the ordered state represents consensus. As our model includes the two conflicting forcespersonal belief rigidity and social influence-we first ask how the relative strength of these two forces governs consensus dynamics. Through social interaction, the society may or may not reach a consensus, depending upon the relative strength of peer-influence (I) and coherentism (J). Fig 2 shows a set of phase diagrams for various combinations of J, I, and T. S/N is defined as the belief system with the largest number of constituents normalized by the size of the social network. Density in Fig 2D-2F is the probability density of S/N over 160 trials at a given parameter configuration. Since belief systems evolve locally, individuals may have to pass through incoherent states before fully converting. As the coherentism (J) increases individuals tend to cling to their own beliefs rather than make such incoherent transitions. This can prevent consensus, but small local consensus can still occur. As the susceptibility (T) increases individuals readily accept incoherent beliefs, allowing individuals to traverse through the belief space. It facilitates spreading of ideas and consensus. By making individuals more susceptible to belief spreading rather than to random switching, T plays the opposite role to temperature in standard spin models [8]. T acts as a temperature for individual's beliefs, and as an inverse-temperature for the whole system. As the peer-influence (I) increases individuals again become more prone to consensus. Novel dynamics occur when the effect of coherentism and social influence become comparable. Fig 2B shows that complete consensus is not guaranteed when the forces exerted by J and I remain comparable in size. The competition leads to a situation where belief systems can coexist and where less coherent systems can dominate. In Fig 2D and 2E, we see examples of these competing configurations for when T or J is decreased and I held constant, as denoted by the higher density of largest belief-systems around 0.5. In these cases there are usually two (sometimes more) larger competing belief systems that cannot completely dominate the system even after a very long time (hundreds of billions of time-steps). As noted in [50] there are local minima distributed throughout belief space where an individual's belief system can get stuck. At lower temperatures and when the drive for internal coherence (J) is much smaller than I, then groups of individuals (sometimes the whole population) can collectively become stuck in these "jammed" states. Under these conditions convergence to a completely coherent belief system is not guaranteed. In our model the energy contribution of belief networks can have a major impact on system stability as individuals seek more coherent beliefs and resolve dissonance. What happens when key beliefs are upset through an external shock or perturbation? Imagine two independent systems, both homogeneous with the same social energy, E (s) . In traditional social models, with the absence of the individual belief system (cognitive factor), these systems are identical. By contrast, the internal system of interconnected beliefs introduces a new force that drives people to seek coherence in the structure of their own belief systems. Given a homogeneous population of people with highly coherent belief systems, society remains stable. However, given a homogeneous population of incoherent belief systems, society will become unstable and following a small perturbation, breaks down (see Fig 3). In our simulation, the society is initialized at consensus with an incoherent belief system. Then 1% of the population are given a random belief system. Individuals attempt to reduce the energy of their own belief systems and leave consensus. This society eventually re-converges at a more coherent belief-system that is different from the original consensus (Fig 3C). In the model, consensus does not guarantee stability. These social instabilities may help explain why new ideologies can arise in the presence of belief systems that dominate most of the population. Traumatic events such as war or depression could make previously coherent beliefs less coherent, thereby destabilizing the belief system as a whole. Since these beliefs are shared by the overwhelming majority, this perturbation has the potential to have widespread impact. Such changes may reduce the internal coherence of individuals, which may make the whole society more prone to paradigm shifts. Throughout history, major external perturbations in the form of war, depression, and crippling inflation has frequently disrupted the world-views of a society's citizens, inducing subsequent social upheavals. More coherent belief systems, which otherwise fail to gain adherents in the presence of a dominant stable societal values, could then gain the upper-hand by recruiting among a population of disturbed citizens as Hoffer suggests [2]. Given that the coherence of personal beliefs fundamentally impacts collective behavior, we investigate the impact of "true believers" or zealots in a population. They can play an important role in shaping collective social dynamics [2,8]. Zealots were introduced in [52] and [53] in the context of the voter model. Since, there has been continued work on the impact of zealots in the voter model [54][55][56] as well as binary adoption [57], naming game [58,59], and other social models [60,61]. The related topic of minority spreading has also been explored in various opinion models [62][63][64]. Here we explore this aspect of opinion dynamics from the perspective of cognitive forces acting on the agents. In the context of our framework we define zealots as individuals who will never alter their own belief systems, but will continue to attempt to convert others to their own. To study the impact of zealots, we prepare a homogeneous society with highly coherent beliefs and introduce zealots with varying internal coherence. As seen in studies of previous social models, there is a tipping-point density above which the minority opinion takes over the population . Fig 4 shows that the internal coherence of zealots has a strong impact on their effectiveness in converting society. Low coherence zealots require much higher densities in order to convert the whole population (see squares in Fig 4) because converted individuals revert back to more coherent belief-systems at a higher rate, making it difficult for the zealot's belief-system to retain converts. Highly coherent zealots pull the whole population out of consensus and convert it to their belief-systems more easily. Coherent zealots require almost half fractional group size. As society is upset, the original dominant but incoherent belief system S o (solid black) is replaced by an emerging coherent alternative S f (dashed red). doi:10.1371/journal.pone.0165910.g003 Fig 4. Impact of the internal consistency of zealot beliefs. We define zealots as a group of individuals who share an identical, immutable belief system. Such belief systems can, however, vary in terms of their coherency. The dynamics of hS/Ni, the fractional size of the zealot population, over ρ o , the density of zealots introduced into the population, reveals that zealots with more coherent beliefs can convert a population much more efficiently. In converting the whole population, the coherent set of beliefs (circles) require only less than half the density of zealots compared with incoherent beliefs (squares). Bars show standard error and E z is the energy of the zealot's belief-system. The simulations were run using J = 2.0, T = 2.0 and I = 90.0. doi:10.1371/journal.pone.0165910.g004 the density, in this particular case, to convert the entire population. This suggests that coherence of a set of beliefs plays a vital role in determining how well the set of beliefs spread through society. Our model may offer a possible explanation for the co-existence of many seemingly invalid or impractical cults and fringe groups in our society. The beliefs of cults and other fringe groups may frequently contradict reality, yet they continue to thrive in spite of being surrounded by large majorities of other belief systems. In our model, this is explained by the degree to which the coherence of belief systems can out-balance social pressure in addition to social isolation. To investigate the influence of belief rigidity and social isolation on social dynamics we create a society with two communities, a large mainstream community and a smaller cult community (Fig 5A). The mainstream community acts like a reservoir of mainstream beliefs. The other community consists of people following a different coherent belief system. By controlling social exposure and the strength of belief coherence, we investigate what effect belief coherence has on the capacity for mainstream society to invade the cult. The parameter μ controls the fraction of edges in the cult community that are shared with the mainstream community [65]. When socially isolated (low μ) the cult can resist invasion regardless of mainstream coherence (Fig 5B). Groups of like-minded individuals can resist outside influence by reducing social contact with non-members (decreasing μ) and enhancing internal social interactions, both of which are common in many fringe groups and even in religious communities. On the other hand, the internal coherence also plays a key role. Less coherent mainstream beliefs have greater difficulty in converting the cult, even at high levels of exposure (high μ). Compared with the dogma of cults, the truth or the reality can be more complex and less coherent. In such We create a network with two communities with parameters T = 2.0, I = 0.09, and J = 2.0-putting the system in a regime where it will seek consensus. We vary the fraction of links that connect the cult community to the mainstream community, denoted μ. e o is the number of social links between communities and ∑k i is the total number of links in the cult (both shared and internal). The mainstream community attempts to convert the smaller cult. (b) At low μ the lack of exposure allows the cult to resist mainstream conversion. At higher μ there is sufficient exposure to the mainstream community to overcome the rigidity of the cult's belief system. However, the process of conversion becomes more difficult as the cult's beliefs become more coherent than mainstream beliefs. Cults are easily converted with highly coherent mainstream beliefs even at low exposure levels (black circles), while cults maintain their beliefs even at high exposure given low coherence of mainstream beliefs (red squares). Bars show standard deviation. doi:10.1371/journal.pone.0165910.g005 case, even a belief-system firmly grounded on truth may struggle to convert cults that possess a highly coherent set of beliefs. Interlocked beliefs turns tightly-knit communities into bastions of resistance to ideological invasion from outside. Fig 6A shows a phase diagram of the case when I ) J, where conversion is determined solely by social exposure (μ). This is expected by most spin-based social models. However, if the strength of social influence is comparable to belief strength, social exposure is not the sole determinant anymore (see Fig 6B). Notably, by increasing the coherentism (J) of its members and maintaining a coherent set of beliefs, cults can continue to thrive even with complete mixing with society. This contradicts, while underlining common intuition, the traditional exposure models where enough exposure is sufficient to convert populations. These results also hint at common characteristics of surviving cults. We do not expect to find successful cults that have low belief coherence and high mixing because they would be quickly converted by the mainstream society. We expect to see more cults that utilize a combination of policies that minimize their member's social contacts with outsiders, emphasize the importance of their dogma (increasing J), and maximizing the coherence of their beliefs. The latter could mean incorporating explanations for beliefs that contradict empirical evidence or belittling mainstream methods of reasoning. Coherence can be, but is not necessarily aligned with logical consistency, rather coherence is based on the strength of associations between concepts and valence concepts (see SKS model [34]), that is derived from a connectionist cognitive framework. Discussion We have shown that our model exhibits a disorder-to-order transition similar to other opinion models, while also exhibiting unique dynamics that could help explain common processes observed in the real world, such as the breakdown of a homogeneous society driven by shocks to individuals' beliefs, the dependence of zealotry on belief coherence, and the successful entrenchment of fringe groups. Each of these results arises from the fact that our framework integrates belief interdependence with social influence. The breakdown of homogeneous societies can manifest through changes in the connectivity of concepts at the concept-network level (cf. SKS model). Abrupt shocks to these conceptual connections can impact the coherence of beliefs. That impact is reflected in our model as an unstable belief system. When people share a common shock, one could expect that many of those people's belief-systems will enter a frustrated state. In order for this socially frustrated state to be resolved, members of the population partake in societal-dissonance resolution strategies by collectively transitioning toward more coherent belief systems. From this view we could interpret paradigm shifts in culture as something akin to a societal-wide coping strategy. Our model is a minimalistic approach toward introducing individualist concept-networks into a systematic social dynamics model. Though simple, the framework can be expanded to include more realistic features. For instance, communicated beliefs are chosen randomly, but this feature can be replaced with other models of belief communication, such as imitation, where someone will be more likely to communicate recently accepted beliefs. Additionally, our mapping from the SKS concept model could be expanded to included weighted relationships that emphasize the uncertainty that an individual has in their beliefs (conviction) and its relations to other beliefs. While we used the same parameter values (J, I, and T) for all individuals, we expect them to vary within the population from person to person. Variations of these parameters could produce the natural occurrence of zealot-like behavior. Finally, the formation and maintenance of social ties is an important facet of social dynamics [66,67] and could be implemented by allowing agents to choose their neighbors [68]. Our model expands upon previous social models by implementing an internal belief system that assumes that the interdependence among beliefs shapes their dynamics. In this framework we are able to incorporate known psychological forces into the model and show that a bottomup approach can successfully link micro-level behaviors to global dynamics. By making this small jump to using internal belief-systems, while preserving key features of standard social models, such as percolation and global consensus, it extends their dynamics and explanatory value, and could be a contributing factor for explaining the entrenchment of belief systems and social upheaval. Future work will be directed towards the development of new opinion models for specific applications that more closely integrate our behavioral and cognitive understanding of humanity. Fig 1 . 1Cognitive-Social Belief Model. (a) Social models, such as the voter or Sznajd models, focus on the assimilation process through social pressure. Beliefs are usually simplified as independent states. (b) Cognitive models, such as the SKS model, focus on the interaction and coherence of beliefs of a single individual and how individuals make decisions and change their minds. The effect of social networks is often unaddressed. Fig 3 . 3Belief driven social instability. Strong societal consensus does not guarantee a stable society in our model. If major paradigm shifts occur and make individual belief systems incoherent, then society may become unstable. (a) The plot shows the evolution of social energy E (s) over time. The system starts at consensus but with incoherent beliefs. After introducing a small perturbation, individuals leave consensus, searching for more coherent sets of beliefs, until society re-converges at a stable configuration. (b) Decreasing mean individual energies hE (i) i over time illustrates individual stabilization during societal transition. (c) hS/Ni is the Fig 5 . 5Belief Invasion. (a) The survival of cults and fringe groups depends on the coherence and strength of beliefs. Fig 6 . 6Community and belief rigidity. (a) Exposure determines conversion resistance when peer-influence (I) is strong. 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[ "Decomposition of Differential Games", "Decomposition of Differential Games" ]	[ "Adriano Festa \nENSTA ParisTech\nBoulevard des Marchaux828, 91120PalaiseauFR\n", "Richard Vinter \nEEE Department\nImperial College\nExhibition RoadSW7 2BTLondonUK\n" ]	[ "ENSTA ParisTech\nBoulevard des Marchaux828, 91120PalaiseauFR", "EEE Department\nImperial College\nExhibition RoadSW7 2BTLondonUK" ]	[]	This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets; the decomposition consists of replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games, where the decomposition arises from considering the interaction of individual pursuer/evader pairs.	null	[ "https://arxiv.org/pdf/1409.4624v1.pdf" ]	119,050,023	1409.4624	31b5df3647ed49bb36301d51218889be8c244aa1	Decomposition of Differential Games Adriano Festa ENSTA ParisTech Boulevard des Marchaux828, 91120PalaiseauFR Richard Vinter EEE Department Imperial College Exhibition RoadSW7 2BTLondonUK Decomposition of Differential Games Differential gamesviscosity solutionsdecomposition techniques 1991 MSC: [2010] 49N7035D4049M27 This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets; the decomposition consists of replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games, where the decomposition arises from considering the interaction of individual pursuer/evader pairs. Introduction We propose a decomposition technique to simplify the solution of zero-sum differential games that involve two players (the a-player and the b-player), whose actions govern the evolution of the state x. The state trajectory associated with open loop policies a(.) and b(.) ('open loop policies' are defined below), for a specified initial state x 0 , is given by the (absolutely continuous) solution of the differential equation ẋ(t) = f (x(t), a(t), b(t)), a.e. x(0) = x 0 . Here, f (., ., .) : R n × R m1 × R m2 → R n is a given function. Open loop policies a(.) and b(.) of the two players take values in specified sets A ⊂ R m1 and B ⊂ R m2 respectively. We write the solution x(t; x 0 , a(.), b(.)). It is assumed that hypotheses are imposed on the data ensuring that a solution exists and it is unique. We also specify a closed set T ⊂ R n called the 'target'. The first entry time τ for x(t; x 0 , a(.), b(.)) is. τ := sup{t \| x(t; x 0 , a(.), b(.)) / ∈ T } . For a(.) ∈ A and b(.) ∈ B the pay-off is J(x 0 , a(.), b(.)) = τ 0 e −λt l(x(t; 0, x 0 , a(.), b(.)), a(t), b(t))dt , in which λ ≥ 0 (the discount factor) is a given number and l(., ., .) : R n × R m1 × R m2 → R (the payoff integrand) is a given function. Here, τ is the first entry time for x(t; x 0 , a(.), b(.)). 'for any t ≥ 0, and b 1 (.), b 2 (.) ∈ B, b 1 (t) = b 2 (t) a.e. t ∈ [0, t ] =⇒ φ(b 1 (.))(t) = φ(b 2 (.))(t) for a.e. t ∈ [0, t ] . 'ψ(.) is non-anticipative' in the second defining relation is analogously defined. Using these interpretations, we define the upper and lower values u(x) and v(x) of the game, for a given starting start x ∈ R n , to be u(x) = sup There is an extensive literature on precise conditions on the data, target, etc., under which u(.) coincides with v(.), when u(.) can be characterized as the unique continuous viscosity solution of the HJI (Hamilton Jacobi Isaacs) equation: F (x, u, Du) = 0 for x ∈ R n \T , u(x) = 0 for x ∈ T ,(1) and when maximizing closed loop policies for the aplayer can be obtained from knowledge of u(.). See [1], [3], [12] for expository material on these topics, and [2] for numerical aspects. In this paper, attention focuses on the upper value functon u(.) and the associated HJI equation (1). We consider situations in which the target T can be represented as the union of a finite number of closed sets T j , j = 1, . . . , m: T = ∪ m j=1 T j . Here, the b-player, responding to the closed loop policy of the a-player, has a choice over which component T j , j = 1, . . . m, to exit into, to minimize the payoff. Consider the family of 'reduced' value functions u j (.), j = 1, . . . , m, that result when the target T is replaced by the subset T i . Of interest are cases in which the value functions u j (.), j = 1, . . . , m, for the target subsets are easier to calculate than the value function u(.) for the full target T and when u(.) can constructed as the lower envelope of the u j (.)'s, thus: u(x) = min{u j (x) \| j = 1, . . . , m}.(2) The motivation for seeking a decomposition of this nature is as follows. Optimal control problems are special cases of differential games in which the constraint set A for the a-player is a single point; there is then only one possible open loop policy for the a-player, which can therefore be effectively ignored. For optimal control problems, the decomposition (2) is always valid, since replacing T by one particular T j amounts to a strengthening of the problem constraints, and cannot therefore reduce the value. So, for any x and any j, u(x) ≤ u j (x). On the other hand, an optimal policy, for the given initial state x, must result in the state trajectory exiting into Tk for somek. But then u(x) ≥ u j (x). These inequalities validate the decomposition (2). When the presence of the a-player is restored and we are dealing with a true differential game, decomposition is a much more complicated issue. There are nontheless interesting cases when the decomposition can be achieved. The goal of this paper is to give criteria for decomposition, and to illustrate their application. We shall assume that the value functions involved are unique viscosity solutions of the HJI equation with appropriate boundary conditions. This means that checking the validity of the decomposition reduces to answering the question: when is the lower envelope of a family of viscosity solutions to a particular HJI equation also a viscosity solution? In Section 2 we give two criteria ((E) and (C)) under which the answer is affirmative. (E) is more general, but (C) is often easier to verify. (C) is satisfied, in particular, when F (x, u, .) is convex. This is a well-known fact: the viscosity solution property is preserved under the operation of taking the lower envelopes, for convex Hamiltonians. Notice that, for optimal control problems F (x, u, .) is always convex, so this fact is consistent with the earlier observation that, for optimal control problems, regarded as special cases of differential games, the decomposition is possible. However (C) is weaker than 'full' convexity of F (x, u, .), because it requires us to check, for each x ∈ R n \T , the convexity inequality only w.r.t. gradient vectors of the minimizing u j (.)'s at x. In the examples, this (restricted sense) convexity condition is satisfied while full convexity fails. We provide examples from pursuit/evasion games in which the decomposition simplifies computations by reducing the state dimensionality. Some examples of the decomposition, without detailed accompanying analysis were presented in [7]. Properties of the Lower Envelope of a Family of Viscosity Solutions Take a function F (., ., .) : R n ×R×R n → R and consider the partial differential equation F (x, u(x), Du(x)) = 0 .(3)F (x, u(x), p) ≤ 0, ∀p ∈ D + u(x) .(4) u(.) is a continuous viscosity supersolution of (3) on Ω if it is continuous and, for each x ∈ Ω, F (x, u(x), p) ≥ 0, ∀p ∈ D − u(x).(5) u(.) is a continuous viscosity solution of (3) on Ω if it is both a continuous subsolution and supersolution of (3) on Ω. Here, D + u(x) and D − u(x) denote, respectively, the Fréchet superdifferential and subdifferential of the continuous function u(.) defined on an open subset of R n containing the point x: D + u(x) := p ∈ R N : lim sup y→x u(y) − u(x) − p · (y − x) \|x − y\| ≤ 0 , D − u(x) := p ∈ R N : lim inf y→x u(y) − u(x) − p · (y − x) \|x − y\| ≥ 0 . (For the analysis of this paper it is helpful to define continuous viscosity solutions in terms of one-sided Fréchet differentials which is equivalent to the standard definition in terms of gradients of smooth majorizing and minoring functions [4]. The following proposition gives conditions under which the lower envelope of a collection of continuous viscosity solutions of (3) is also a continuous viscosity solution, expressed in terms of the limiting superdifferential ∂ L (x) of the continuous function u(.) at x: ∂ L u(x) := {p \| ∃ sequences p i → p and x i → x s.t. p i ∈ D + u(x i ) for each i} . Proposition 2.2 Take a collection of closed sets T j ⊂ R n , j = 1, . . . , m. For each j, let u j (.) be a scalar valued function with domain R n \T j . Define I(x) = {j ∈ {1, . . . , m} \| u j (x) = Min j u j (x)} for each x ∈ R n \(∪ m j=1 T j ) and Σ = {x ∈ R n \(∪ m j=1 T j ) \| Cardinality{I(x)} > 1} . Takeū(.) : R n \(∪ m j=1 T j ) → R to be the lower envelope functionū (x) = Min j {u j (x)} . (a): Suppose that u j (.) is a continuous viscosity supersolution of (3) on R n \T j for each j. Thenū(.) is a continuous viscosity supersolution of (3) on R n \(∪ m j=1 T j ). (b): Suppose that u j (.) is a continuous viscosity subsolution of (3) on R n \T j for each j, that H(., ., .) is continuous and that, for each x ∈ Σ, u j (.) is Lipschitz continuous on a neighbourhood of x. Consider the hypotheses: (C): for any x ∈ Σ, any set of vectors {p j \| j ∈ I(x)} such that p j ∈ ∂ L u j (x) for each j ∈ I(x), and any convex combination {λ j \| j ∈ I(x)}, F (x,ū(x), j∈I(x) λ j p j ) ≤ j∈I(x) λ j F (x, u j (x), p j ) . (E): for any x ∈ Σ, any set of vectors {p j \| j ∈ I(x)} such that p j ∈ ∂ L u j (x) for each j ∈ I(x), and any convex combination {λ j \| j ∈ I(x)}, F (x,ū(x), j∈I(x) λ j p j ) ≤ 0 . (i): (E) =⇒ 'ū(.) is a continuous viscosity subso- lution to (3) on R n \(∪ m j=1 T j )'. (ii): If, additionally, u j (.) is C 1 on a neighborhood of x for each j, then u(.) is a continuous viscosity subsolution to (3) on R n \(∪ m j=1 T j ) =⇒ (E). (iii): (C) =⇒ (E). Comments. (i): The proof of the proposition is based on a wellknown estimate for one-sided differentials to lower envelope functions, in terms of the one-sided differentials to the constituent functions (the 'Max Rule'). Such estimates are studied in depth in [13]. (ii): The proposition treats separately the preservation of the supersolution and subsolution properties of viscosity solutions under the operation of taking the lower envelope, because much weaker hypotheses need be imposed in connection wth supersolutions. (iii): We give two sufficient conditions for the lower envelope of a famility of continuous viscosity solutions also to be a continuous viscosity solution, namely (E) and (C). (C) is a more restrictive condition, but it is useful because, as illustrated in the following examples, it can be easier to verify. (iv): The proposition is an analytical tool for decomposing a differential game (associated with the value functionū(.)) into a collection of simpler problems. The critical hypothesis in this proposition is (E) (or (C)). (C) is automatically satisfied when F (x, u, .) is convex. This special case of the proposition is well-known [4]. However (C) imposes a convexity type condition on F (x, u, .), only with respect to selected vectors in its domain. In some cases, examples of which given below, the restricted sense convexity hypothesis is satisfied but the full convexity hypothesis is violated; the proposition thereby identifies a new class of differential games for which the decomposition is possible. Proof of Prop. 2.2. (a): Suppose that u j (.) is a continuous viscosity supersolution of (3) on R n \T j for each j. Take any x ∈ R n \(∪ m j=1 T j ) and p ∈ D −ū (x). Then u(x ) −ū(x) ≥ p · (x − x) − o(\|x − x\|) , for all x ∈ R n \(∪ m j=1 T j ). (Here, o(.) : R + → R + is some function such that lim s↓0 o(s)/s → 0.) Choose any j ∈ I(x). We know that u j (x) =ū(x) and u j (x ) ≥ū(x ) . It follows that, for all x ∈ R n \(∪ m j=1 T j ), u j (x ) − u j (x) ≥ p · (x − x) − o(\|x − x\|) . But then p ∈ D − u j (x) and, since u j is a continuous viscosity supersolution, we have F (x, u j (x), p) ≥ 0. It follows that F (x,ū(x), p) ≥ 0. Sinceū(.) is continuous, we have established thatū(.) is a continuous viscosity subsolution of (3) on R n \(∪ m j=1 T j ). (b)(i): Suppose that u j (.) is a continuous viscosity subsolution of (3) on R n \T j for each j. Take any x ∈ R n \(∪ m j=1 T j ) and p ∈ D +ū (x). We must show that F (x,ū(x), p) ≤ 0 .(6) Suppose first that x / ∈ Σ, i.e. I(x) contains a single index value j. Then, since the u i (.)'s are continuous,ū(x ) = u j (x ) for all x in some neighbourhood of x. It follows that p ∈ D + u j and so F (x, u j (x), p)(= F (x,ū(x), p)) ≤ 0. We have confirmed (6) in this case. It may be assumed then that x ∈ Σ. Now, u j (.) is Lipschitz continuous on a neighbourhood of x for each j ∈ I(x). Since p ∈ D +ū (x), it is certainly the case that p ∈ ∂ Lū (x). Using the property thatū(x ) coincides with max{u j (x ) \| j ∈ I(x )} for x in some neighbourhood of x, we deduce from the Max Rule for limiting subdifferentials of Lipschitz continuous functions (see, e.g., [16,Thm. 5.5.2]) applied to −ū(.) the following representation for p: p = j∈I(x) λ j p j , for some convex combination {λ j \| j ∈ I(x)} and vectors p j ∈ ∂ L u j (x), j ∈ I(x). But then, by hypothesis (E), F (x,ū(x), p) = F (x,ū(x), j∈I(x) λ j p j ) ≤ 0 . We have confirmed (6) and so (b)(i) is true. (b)(ii): Take any x ∈ Σ. Suppose that the u j 's are continously differentiable of a neighbourhood of x and that u(.) is a viscosity solution. Take any convex combination {λ i } on I(x). Then, for all x in some neighborhood of x, u(x ) −ū(x) ≤ i∈I(x) λ i (u i (x ) − u i (x)) ≤ i∈I(x) λ i ∇u i (x) · (x − x) + o(\|x − x\|) . This last inequality tells us that i∈I(x) λ i ∇u i (x) is a limiting superdifferential ofū(.) at x. But then, sincē u(.) is a viscosity subsolution, F (x,ū(x), j∈I(x) λ j p j ) ≤ 0 . We have confirmed that (E) is true. (b)(iii): Take any convex combination {λ i } on I(x) and vectors p i ∈ ∂ L u i (x) for i ∈ I(x). It follows from the definition of the limiting supergradient that, for each i, there exist sequences x i j → x and p i j → p i such that p i j ∈ D + u i (x i j ) for i = 1, 2, . . . But then, for each i ∈ I(x), F (x i j , u j (x i j ), p i j ) ≤ 0, i = 1, 2, . . . , since the u i (.)'s are viscosity subsolutions. It follows that j∈I(x) λ j F (x i j , u j (x i j ), p i j ) ≤ 0, i = 1, 2, . . .. Noting the continuity of F (., , ., ) and also the u j (.)'s, we may pass to the limit as i → ∞ to obtain j∈I(x) λ j F (x, u j (x), p j ) ≤ 0 . Assume (C). Then F (x,ū(x), j∈I(x) λ j p j ) ≤ j∈I(x) λ j F (x, u j , p j ) ≤ 0 , which is (E). Pursuit Evasion Games Pursuer/evader games are examples of the game posed in the Introduction. There is an extensive literature on such games, going back to Rufus Isaacs' work in the 1960's, and his monograph [10] contains many examples. Expository material is to be found in [8], [12]. We note also [5], [11], [9], [11], [14], and [15]. But none of these references systematically address decomposions of the game, each element of which is generated by a target subset. Pursuer/evader games is an application area for the methods proposed in this paper; they provide exemplar problems, both where decomposion is possible, and where it is not. We consider zero sum differential games which terminate when one of the pursuers is sufficently close to one of the evaders, where 'closeness' is understood in the sense of a specified target. The pay-off is the time until the target is attained. We analyse a number of examples, involving different numbers of pursuers and evaders, and different targets. dx 1 dt = f 1 (x 1 , a 1 ) , . . . , dx m1 dt = f m1 (x m1 , a m1 )) dx m1+1 dt = f m1+1 (x m1+1 , b 1 ), . . . , dx m1+m2 dt = f m1+m2 (x m1+m2 , b m2 ) . The variables a 1 , . . . , a m1 and b 1 , . . . , , b m2 are interpreted as controls for the evaders and the pursuers, respectively, which are subject to the constraints Write Φ for the space of non-anticipative mappings φ : B → A. The game fits the formulation Section 1, with λ = 0, and may be summarized as: 1 (t), ..., a m1 , b 1 (t), ..., b m2 (t)) a i ∈ A i , i = 1,(P )                                          Maximize φ∈Φ Minimize {bi}∈A τ 0 1 dt       ẋ 1 (t) = f 1 (x 1 (t), a 1 (t)) ... x m1+m2 (t) = f m1+m2 (x m1+m2 , b m2 ), a.e. (a∈ A 1 × ... × A m1 × B 1 × ... × B m2 , a.e. in which (a 1 (.), . . . , a m1 (.)) = φ(b 1 (.), . . . , b m2 (.)) and τ is first entry time into T (x 1 (0), . . . , x m1+m2 (0)) = (x 1 , . . . ,x m1+m2 ) for some given (x 1 , . . . ,x m1+m2 ) ∈ R n ×. . .×R n . Here T is a given closed subset of R n × . . . × R n . The Hamilton-Jacobi-Isaacs equation is F (x 1 , . . . , x m1+m2 , D x1 u, . . . , D xm 1 +m2 u) = 0 , (7) in which F (x 1 , . . . , x m1+m2 , p 1 , . . . , p m1+m2 ) = − m1 i=1 H i (x i , p i ) + m1+m2 i=m1+1 H i (x i , −p i ) − 1. Here H i (x i , p i ) :=      sup ai∈Ai p i · f (x i , a i ) for i = 1, . . . m 1 sup bi−m 1 ∈Bi−m 1 p i · f (x i , b i−m1 ) for i = m 1 + 1, . . . m 1 + m 2 .(8) A Single Pursuer/Multiple Evaders Game Consider first a case of the pursuit/evasion game, written (P 1 ), in which m 1 = m > 1, m 2 = 1 and n = 1 (a single pursuer/multiple evaders game in 1D space). The states of the m evaders, labeled 1, . . . , m and of the one pursuer, labeled m+1, are interpreted as the positions of the evaders and pursuer. The game terminates when the pursuer is first at a distance r from one of the evaders, where r ≥ 0 is a given constant. Accordingly, we take T = T 1 ∪ . . . ∪ T m , in which, for i = 1, . . . , m, T i := {(x 1 , . . . , x m+1 ) \| \|x m+1 − x i \| ≤ r} . The Hamilton-Jacobi-Isaacs equation is F 1 (x 1 , . . . , x m+1 , D x1 u, . . . , D xm+1 u) = 0 , (9) in which F 1 (x 1 , . . . , x m+1 , p 1 , . . . , p m+1 ) = − m i=1 H i (x i , p i ) + H m+1 (x m+1 , −p m+1 ) − 1 , where H i (x i , p i ) := sup ai∈Ai p i · f (x i , a i ) i = 1, . . . m, H m+1 (x m+1 , p m+1 ) = sup b1∈B1 p m+1 · f (x m+1 , b 1 ) .(10) Now take (P 1 i ) to be the modification of (P 1 ), when T i replaces T , i = 1, . . . , m. Let us assume that, for each i, the value function u i (.) for (P 1 i ) is a continuous viscosity solution of (7). The following proposition tells us that we can construct a viscosity solution to (9) from the u i (.)'s, by taking the pointwise infimum. Comment. Suppose hypotheses are imposed, ensuring that (1): for each i, the HJI equation for (P 1 i ) has a continuous viscosity solution on (R n × . . . R n )\T i with a continous extension to T i , on which set the solution vanishes, and (2): the value function (P 1 ) is the unique continuous viscosity solution on (R n × . . . R n )\T that has a continous extension to T i , on which set the solution vanishes. The proposition tells us that, under these circumstances, the upper value u(.) for (P 1 ) can be calculated as the lower envelope of the continuous viscosiy solutions for the (P 1 i )'s. (Notice that, since all upper values concerned are non-negative, and each u i (.) is assumed to have a continuous extension to T i , on which set it vanishes, the lower envelope has a continuous extension to T , on which set it vanishes.) Proof of Prop. 3.1. Note that, for any i, u i (x 1 , ..., x m+1 ) depends only on the two variables (x i , x m+1 ). This is because the first entry time into T i only concerns the state trajectories associated the i'th evader and the pursuer (labelled m + 1). In view of the hypotheses imposed on the u i (.)'s, the fact thatū(.) is a viscosity solution of (7) will follow from Prop. 2.2, if we can confirm hypothesis (C) of this proposition. Take any z = (x 1 , . . . , x m+1 ) ∈ R m+1 \T , any index set I(z) (of cardinality l > 1) such that the values u i (z), i ∈ I(z), coincide, and any convex combination {λ i } from I(z). To simplify, assume index values have been re-ordered so that I(z) = {1, . . . , , l}. Take alsop i ∈ R n , i = 1, . . . , l such that p i := (0, . . . , 0, p i i , 0, . . . , 0, p i m+1 ) ∈ ∂ L u i (z) . (11) (The possibly non-zero components p i i and p i m+1 ofp i appear at the i'th and (m + 1)'th locations. We must show η(λ 1 , . . . , λ l ) ≥ 0, where η(λ 1 , . . . , λ l ) := l i=1 λ i F (z,p i ) − F (z, l i=1 λ ipi ) . Noting the special structure (11) of thep i 's and the fact that H i (x i , p i ) = 0 when p 1 = 0, for each i, we see that η(λ 1 , . . . , λ l ) = l i=1 λ i H m+1 (x m+1 , −p i m+1 ) − H i (x i , p i i ) − H m+1 (x m+1 , − l i=1 λ i p i m+1 ) − l i=1 H i (x i , λ i p i i )) . We achieve a further simplification from the fact that H i (x i , .) is positively homogeneous, so H i (x i , λ i p i m+1 ) = λ i H i (x i , p i m+1 ). This gives η(λ 1 , . . . , λ l ) = l i=1 λ i H m+1 (x m+1 , −p 1 m+1 ) − H m+1 (x m+1 , − l i=1 λ i p i m+1 ) . But then η(λ 1 , . . . , λ l ) is non-negative, because the term H m+1 (x m+1 , .), defined by (8), is convex. The proof is complete. A Multiple Pursuers/Single Evader Game Consider next a case of the pursuit/evader game, written (P 2 ), in which m 1 = 1, m 2 ≥ 1 and n = 2 (single pursuer/multiple evaders). The dynamic behavior of each player is modelled as a thrust acting on a mass, in 1D space, with saturating damping. The state equations, governing the position and velocity of each player, are taken to be, for i = 2, . . . , m + 1,  ẋ 1 1 x 1 2   =   x 1 2 −d1(x i 2 ) + a1   and  ẋ i 1 x i 2   =   x i 2 −di(x i 2 ) + bi−1   . Here, d i (.) : R → R, i = 1, . . . , m+1 are given functions satisfying \|d i (y) − d i (y )\| ≤ k d \|y − y \|, d i (y) ≤ c d ,(12) for all y, y ∈ R and i = 1, . . . , m + 1 for some constants k d > 0 and c d > 0. The control actions the players are required to satisfy \|a\| ≤ α and \|b\| ≤ β i for i = 1, . . . .m . for positive constants α, β 1 , . . . , β m . We assume that β i > α + 2 × c d for i=1,. . . , m.(14) The game terminates when one of the pursuers overtakes the evader. Thus, we take the target to be T = T 1 ∪ . . . ∪ T m , in which, for i = 1, . . . , m, T i := {(x 1 = (x 1 1 , x 1 2 ), . . . , x m+1 = (x m+1 1 , x m+1 2 ) \| x i 1 ≥ x 1 1 } . The HJI equation is F 2 (x 1 , . . . , x m+1 , D x 1 u, . . . , D x m+1 u) = 0 ,(15) in which F 2 (x 1 , . . . , x m+1 , p 1 , . . . , p m+1 ) = m+1 i=1 −p i 1 x i 2 − p i 2 d(x i 2 ) −α×\|p 1 2 \|+ m+1 i=2 (β i ×\|p i 2 \|) . Let (P 2 i ) to be the modification of (P 2 ), when the target T i replaces T , i = 2, . . . , m + 1. Comment. When, for each i, the HJI equation for (P 2 i ) has a continuous viscosity solution u i (.) on (R 2 ) m+1 \T i (with appropriate boundary values) and the value function u(.) for (P 2 ) is the unique continuous viscosity solution on (R 2 ) m+1 \T (with appropriate boundary values), the proposition describes how the value function for (P 2 ) can be obtained, as the pointwise infimum of the u i (.)'s. Proof. Note that, for i = 2, . . . , m + 1, u i (x 1 , . . . , x m+1 ) depends only on the two variables (x 1 , x i ), since the first entry time into T i only concerns the state trajectories associated with the i'th pursuer and the evader. We write u i (x 1 , x i ), suppressing irrelevant arguments in the notation. Note that by assumptions (12) and (14) (which tell us that all evaders can accelerate at a faster rate than the evader), u i (x 1 , x i ) is finite when x 1 1 ≥ x i 1 . The left side of the HJI equation F 2 = 0 can be decomposed as F 2 = F 21 + F 22 ,(16) where 2 )), are: F 21 = − m+1 i=1 (p i 1 x i 2 − p i 2 d(x i 2 )) ,(17) and F 22 = −α × \|p 1 2 \| + m+1 i=2 β i × \|p i 2 \| .(18) We shall make use of the following Lemma, whose proof appears in the appendix. and z = (z 1 , z 2 ) such that z 1 ≥ z 1 and z 2 ≥ z 2 . Then x 1 1 (t;ā(.), z ) ≥ x 1 1 (t; a(.), z) for all t ≥ 0 ,(19) where t → (x 1 1 , x 1 2 )(t; a(.), z) is the state trajectory for the evader, under the open loop strategy a(.) and for initial state z. Fix i, and consider (P 2 i ). We deduce from the lemma that the optimal closed loop strategy for the a-player (the evader) isφ(b i (.)) ≡ +1, for arbitrary initial state ( x 1 = (x 1 1 , x 1 2 ), x i 1 = (x i 1 , x i 2 )) such that x 1 1 > x i 1 . Furthermore, if the a-player applies this optimal strategy then, for any open loop strategy b i (.), the effect of increasing the x 1 2 component of the initial state is to increase the first interception time. We conclude that x 1 2 → u i ((x 1 1 , x 1 2 ), (x i 1 , x i 2 )) is monotone increasing (20) for arbitrary ((x 1 1 , (x i 1 , x i 2 )) ∈ R 3 such that x 1 1 > x i 1 . Once again, we shall deduce that the lower envelopeū(.) of the u i 's is a continuous viscosity solution (15) from Prop. 2.2, by verifying hypothesis (C). Take any z = (x 1 , . . . , x m+1 ) ∈ (R 2 × . . . R 2 )\T , any index set I(z) (of cardinality l > 1) such that the values u i (z), i ∈ I(z) coincide, and any convex combination {λ i } from I(z). We may assume that index values have been re-ordered so that I(z) = {2, . . . , , l + 1}. For i = 2, . . . , l + 1, take anyp i ∈ R n , such that p i := ((p i,1 1 , p i,1 2 ), (0, 0), . . . , (0, 0), (p i,1 1 , p i,1 2 ), (0, 0), . . . (0, 0)) ∈ ∂ L u i (z) . (21) (We have used the fact that u i depends only on ( x 1 = (x 1 1 , x 1 2 ), x i = (x i 1 , x i 2 )) .) The possibly non-zero components (p i, 1 1 , p i,1 2 ) and (p i 1 , p i 2 ) ofp i appear at the first and i'th locations. Note that, by (20), p i,1 2 ≥ 0 for i = 2, . . . , m + 1 .(22) Verification of hypothesis (C) requires us to show that η(λ 2 , . . . , λ l+1 ) ≥ 0, where η(λ 2 , . . . , λ l+1 ) := l+2 i=2 λ i F 21 (z,p i ) − F 21 (z, l+2 i=2 λ ipi ) + l+2 i=2 λ i F 22 (z,p i ) − F 22 (z, l+2 i=2 λ ipi ) . Because F 21 (z, .) is linear, we have η(λ 2 , . . . , λ l+1 ) := l+2 i=2 λ i F 22 (z,p i ) − F 22 (z, l+2 i=2 λ ipi ) = c 1 + c 2 , where c 1 := l+1 i=2 −αλ i \|p i,1 2 \| + α\|λ i p 1 2 \| and But c 1 = 0 since, by (22), the p i,1 2 's all have the same sign. Also, c 2 ≥ 0, by convexity of f 22 (.). We have confirmed η(λ 2 , . . . , λ l+1 ) ≥ 0, and the proof of the proposition is complete. c 2 := l+1 i=2 λ i f 22 (p i ) − f 22 (z, l+1 i=2 λ i p i ) , For the special case when m = 2, d(x) = x, α = 1 and β 1 = β 2 = 0.5, (Figure 1) shows computations of the value function with respect to the reduced coordinates (y 1 , y 2 ) = (x 1 Figure 2 shows an example of the evolution of the positions of the players over time, with respect to the original coordinates. Capture occurs at the point marked X, when pursuer P 1 overtakes the evader, despite starting farther from the evader than pursuer P 2 . 1 − x 2 1 , x 1 2 − x 2 2 ) in R 2 . A Pursuit/Evasion Game With No Decomposition We now provide a simple example illustrating that, for a multiple pursuers/single evader game, with target a union of target subsets, each associated with the evader and just one of the pursuers, may fail to have a decomposition. In this example, it is possible to derive formulae for the value functions involved, and to test the conditions for decomposition directly. We denote by (P 3 ) the special case of (P ) in which m 1 = 1, m 2 = 2 and n = 1. f 1 (x 1 , a) = a, f 2 (x 2 , b 1 ) = b 1 and f 3 (x 3 , b 2 ) = b 2 . The controls actions of the players are constrained as follows: a ∈ A := [−α, +α], b 1 ∈ B 1 := [−1, +1] and b 2 ∈ B 2 := [−1, +1], for some α ∈ (0, 1). We take the target to be T = T 2 ∪ T 3 , where T 2 = {(x 1 , x 2 , x 3 ) \| x 1 = x 2 }, T 3 = {(x 1 , x 2 , x 3 ) \| x 1 = x 3 } . (In this version of the game, two pursuers chase a single evader in 1D space. The game terminates when either pursuer meets the evader.) Denote by (P 3 2 ) and (P 3 3 ) the modified games in which the target T is replaced by the subsets T 2 and T 3 respectively. The HJI equation is F 3 (D x1 u, D x2 u, D x3 u) = 0 ,(23) in which F 3 (p 1 , p 2 , p 3 ) = \|p 2 \| + \|p 3 \| − α\|p 1 \| − 1 . Optimal strategies for both games (P 3 2 ) and (P 3 3 ) are: the evader moves away from the pursuer, and the pursuer moves towards the evader, as quickly as possible. A simple calculation based on these observations yields upper values for (P 3 2 ) and (P 3 3 ), namely: u 2 (x 1 , x 2 , x 3 ) = (1 − α) −1 \|x 2 − x 1 \| , u 3 (x 1 , x 2 , x 3 ) = (1 − α) −1 \|x 3 − x 1 \| , for all x = (x 1 , x 2 , x 3 ) ∈ R 3 . Defineū(.) : R 3 → R to bē u(x) = min{u 1 (x), u 2 (x)} for x ∈ R 3 . Proposition 3.4ū(.) is not a continuous viscosity solution for (23) on R 3 \T . Since the upper value for (P 3 ) is a viscosity solution on R 3 \T , vanishing on T , we may conclude thatū(.) is not the value function for (P 3 ). Proof. Take any z > 0 and letx = (0, z, −z). Thenx ∈ R 3 \(T 2 ∪ T 3 ). Also, u 2 (x) = u 3 (x), and u 2 (.) and u 3 (.) are continuously differentiable atx. From the formulae for the value functions we have Then, for any λ ∈ (0, 1), F 3 (λ∇ x u 2 (x) + (1 − λ)∇ x u 3 (x)) = λ (1 − β) + 1 − λ (1 − β) − β (1 − β) (−λ + (1 − λ)) − 1 = 2λβ (1 − β) > 0 . So condition (E) is violated. Then,ū(.) cannot be a continous viscosity solution, by Prop 2.2, part (b)(iii). The true value function u(.) for (P 3 ) is expressed in terms of the subset: D = {(x 1 , x 2 , x 3 ) ∈ R 3 \| sgn{x 2 − x 1 } = −sgn{x 3 − x 1 } and 1 − α 1 + α \|x 3 − x 1 \| < \|x 2 − x 1 \| < 1 + α 1 − α \|x 3 − x 1 \|} . It is u(x 1 , x 2 , x 3 ) =              1 1−α min{\|x 2 − x 1 \|, \|x 3 − x 1 \|} for (x 1 , x 2 , x 3 ) ∈ R 3 \D 1 2 (\|x 2 − x 1 \| + \|x 3 − x 1 \|) for (x 1 , x 2 , x 3 ) ∈ D . We see that u(.) coincides with min{u 1 (x), u 2 (x)}, for x ∈ R 3 \D. But u(x) < min{u 1 (x), u 2 (x)}, for x ∈ D . (The value function is constructed according to the heuristic: each of the pursuers always travels at maximum speed towards the evader. if both pursuers are on the same side of the evader, the evader travels at maximum speed in the opposite direction until the evader is hit. If, on the other hand, the evader is between the two pursuers, the evader travels at maximum speed away from the closest pursuer until the two pursuers are equidistant. The evader then stops until the evader is reached. A check is then carried out that the value function is a continuous viscosity solution of (7), has a continuous extension to T on which it vanishes, and which is therefore the upper value of the game.) addresses: [email protected] (Adriano Festa), [email protected] ( Richard Vinter). Let A and B be the spaces of open loop policies for the a-player and b-player respectively, namely A := {a(.) : [0, ∞) → R m1 \| a(.) meas. and a(t) ∈ A a.e. }, B := {a(.) : [0, ∞) → R m2 \| b(.) meas. and b(t) ∈ B a.e. } . x(.; x, a(.), ψ(a(.))) .Define the real valued functions F (., ., .) and G(., ., .),with domains in R n × R × R n → R F (x, u, p) = λu + inf a∈A sup b∈B {p · (−f (x, a, b) − l(x, a, b))} , G(x, u, p) = λu + sup b∈B inf a∈A {p · (−f (x, a, b) − l(x, a, b))}. The a-player is the collection of m 1 evaders, labelled 1, . . . , m 1 , and the b-player the collection of m 2 pursuers, labelled m 1 + 1, . . . , m 1 + m 2 . The states of individual pursuers and evaders x 1 , . . . , x m1 and x m+1 . . . , x m1+m2 are governed by the equations . . . , m 1 , and b i ∈ B i , i = 1, . . . , m 2 .Here, f i (., .) : R n × R ri → R n , i = 1, . . . , m 1 + m 2 are given functions, and A i ⊂ R ri , 1, . . . , m 1 and B i ⊂ R ri+m 1 , 1, . . . , m 2 , are given subsets. We regard a 1 , . . . , a m1 and b 1 , . . . , b m2 as block components of a single evader control and pursuer control respectively. Take the state to be x = col{x 1 . . . , x m1+m2 }. The open loop policy spaces for evader and pursuer are A := {meas. mappings a i : [0, ∞) → R ri , i = 1, . . . , m 1 \| a i (t) ∈ A i a.e. for each i}, B := {meas. mappings b i : [0, ∞) → R ri+m 1 , i = i, . . . , m 2 \| b i (t) ∈ B i a.e. for each i} . Proposition 3. 1 1For i = 1, . . . m, let u i (.) be the upper value for (P 1 i ). Assume (a): For i = 1, . . . m, u i (.) is a continuous viscosity solution of (7) on R m+1 \T i . (b): For any i, j ∈ {1, . . . , m}, i = j, and (x 1 , . . . , x m+1 ) ∈ R m+1 \T such that u i (x 1 , . . . , x m+1 ) = u j (x 1 , . . . , x m+1 ), u i (.) and u j (.) are Lipschitz continuous on a neighborhood of (x 1 , . . . , x m+1 ). Then u(x 1 , . . . , x m+1 ) := min{u 1 (x 1 , . . . x m ), . . . , u m+1 (x 1 , . . . , x m+1 )} is a continuous viscosity solution of (7) on (R×. . . R)\T . Proposition 3. 2 2Let u i (.) be the upper value for (P 2 i ), for i = 2, . . . m + 1. Assume (a): For i = 2, . . . m + 1, u i (.) is a continuous viscosity solution of (7) on (R 2 ) m+1 \T i . (b): For any i, j ∈ {2, . . . , m+1}, i = j, and (x 1 , . . . , x m+1 ) ∈ (R 2 ) m+1 \T such that u i (x 1 , . . . , x m+1 ) = u j (x 1 , . . . , x m+1 ), u i (.) and u j (.) are Lipschitz continuous on a neighborhood of (x 1 , . . . , x m+1 ). 1 , . . . , x m+1 ) := min{u 1 (x 1 , . . . x m+1 ), . . . , u m+1 (x 1 , . . . , x m+1 )} is a continuous viscosity solution of (15) on (R 2 ) m+1 \T . F 21 and F 22 , evaluated at ((x 1 1 , x 1 2 ), . . . , Lemma 3. 3 3Letā(.) be the open loop strategyā(.) ≡ +1 for the evader, and let a(.) be any other open loop strategy. Take initial states for the evader z = (z 1 , z 2 ) Fig. 1 .Fig. 2 . 12Value function for a one-pursuer one-evader game Optimal trajectories of the agents in the first component over time. X denote the point of capture in which f 22 (((p 1 1 , p 1 2 ), . . . , ∇u 2 (z) = (−(1 − α) −1 , (1 − α) −1 , 0), and ∇u 3 (z) = ((1 − α) −1 , 0, −(1 − α) −1 ) . AcknowledgementsThis work has been supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN SADCO, "Sensitivity Analysis for Deterministic Controller Design".Appendix: Proof of Lemma 3.3Consider first the case z = z . Fix t > 0. We examine the optimal control problem ofMinimize −y1(t) subject to (ẏ1(s),ẏ2(s)) = (y2(s), −d(y2(s)) + a(s)), a.e. s ∈ [0, t] , a(s) ∈ [−1, +1], a.e. s ∈ [0, t], (y1(0), y2(0)) = (z1, z2) .(Notice that the controlled differential equation in this problem is that governing the motion of the evader.) The data for the problem satisfy standard hypotheses for the existence of a minimizer a * (.) on [0, t], with corresponding state trajectory y * (.) (see, e.g.[16,Chap. 2]). We can establish, by means of a simple contradiction argument, that the nonsmooth Maximum Principle (see[16,Thm. 6.2.3]) applies in normal form. We deduce the existence of a costate arc p(.) = (p1(.), p2(.)) such that p1(.) ≡ +1, and p2(.) satisfies the differential equation and right endpoint boundary conditionHere, ξi(.) is a Lipschitz continuous function satisfying ξ(s) ∈ co ∂L d1(y * 2 (s)) a.e. , in which ∂L d1 is the limiting subdifferential. The solution p2(.) is strictly positive on [0, t). We have shown that, for any t ≥ 0 and initial condition z, a(.) =ā(.) maximizes y1(t). This confirms (19) when z = z. We now show that (19) is true also when z 1 = z1 and z 2 > z 2 . In view of the preceding analysis, we can assume that a(.) = a(.). Write (y1(.), y2(.)) and (y 1 (.), y 2 (.)) for the solutions to the state equation, for initial states z = (z1, z2) and z = (z 1 , z 2 ) respectively. Take any timet > 0. By assumptioṅ y 2 (0) >ẏ2(0). So there are two cases to consider (a):ẏ 2 (t) >ẏ2(t) for all t ≥ 0. In this case, since y 1 (0) − y1(0) > 0, we have, as required,In this case we show as, in the previous case, that y 1 (t ) − y1(t ) > 0. We deduce from the uniquess of solutions to the differential equatioṅ y2(t) = d(y2) + 1 , on [t ,t], for fixed initial condition, that y 2 (t) = y2(t) for t ∈ [t ,t]. Hence, again, the required relation y 1 (t) − y1(t) = (y 1 (t ) − y1(t )) + t t (ẏ 2 (t) −ẏ2(t))dt > 0 . Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser. M Bardi, I Capuzzo-Dolcetta, BostonM. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston 1997. M Bardi, T E S Raghavan, T Parthasarathy, Stochastic and Differential Games: Theory and Numerical Methods. BostonM. Bardi, T.E.S. Raghavan, T. Parthasarathy, Stochastic and Differential Games: Theory and Numerical Methods, Birkhäuser, Boston, 1999. G Barles, Solutions de viscositè des equations d'Hamilton-Jacobi. Springer-VerlagG. Barles, Solutions de viscositè des equations d'Hamilton- Jacobi, Springer-Verlag, 1998. P Cannarsa, C Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser, BostonP. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Birkhäuser, Boston, 2004. Differential games of evasion with many pursuers. W Chodun, J. Math. Anal. Appl. 1422W. Chodun, Differential games of evasion with many pursuers, J. Math. Anal. Appl., 142(2) (1989) pp. 370-389. Values in differential games. R J Elliott, N J Kalton, Bull. Amer. Math. Soc. 783R.J. Elliott and N.J. Kalton, Values in differential games, Bull. Amer. Math. Soc., 78(3) (1972) pp. 427-431. A decomposition technique for pursuit evasion games with many pursuers. A Festa, R B Vinter, Proceedings of 52nd IEEE Control and Decision Conference (CDC). 52nd IEEE Control and Decision Conference (CDC)A. Festa and R.B. Vinter, A decomposition technique for pursuit evasion games with many pursuers, Proceedings of 52nd IEEE Control and Decision Conference (CDC), (2013), pp. 5797-5802. A Friedman, Differential Games. New York, USAJohn Wiley & SonsA. Friedman, Differential Games, John Wiley & Sons, New York, USA, 1971. Optimal pursuit of an evader by countably many pursuers. G I Ibragimov, Differ. Equ. 415G.I. Ibragimov, Optimal pursuit of an evader by countably many pursuers, Differ. Equ., 41(5) (2005) pp. 627-635. R Isaacs, Differential Games. New York, USAJohn Wiley & SonsR. Isaacs, Differential Games, John Wiley & Sons, New York, USA, 1965. Time optimality for the pursuit of several objects with simple motion in a differential game. R P Ivanov, Yu S Ledyaev, Trudy Mat. Inst. Steklov. 158R.P. Ivanov and Yu. S. Ledyaev, Time optimality for the pursuit of several objects with simple motion in a differential game, Trudy Mat. Inst. Steklov., 158 (1981) pp. 87-97. Game-Theoretical Control Problems. N N Krasovskii, A I Subbotin, SpringerNew YorkN.N. Krasovskii and A.I. Subbotin, Game-Theoretical Control Problems, Springer, New York, 1988. Sub-and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions. Y Ledyaev, J S Treiman, Russ. Math. Surv. 672Y. Ledyaev and J.S. Treiman, Sub-and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions, Russ. Math. Surv., 67(2) (2012) pp. 345-373. Simple pursuit by several objects. B N Pshenichnii, Cybern. Syst. Anal. 123B.N. Pshenichnii, Simple pursuit by several objects, Cybern. Syst. Anal., 12(3) (1976) pp. 484-485. Probabilistic Pursuit-Evasion Games: Theory, Implementation, and Experimental Evaluation. R Vidal, O Shakernia, J Kim, H Shim, S Sastry, IEEE T. Robotic. Autom. 185R. Vidal, O. Shakernia, J. Kim, H. Shim and S. Sastry, Probabilistic Pursuit-Evasion Games: Theory, Implementation, and Experimental Evaluation, IEEE T. Robotic. Autom., 18(5) (2002) pp. 662-669. . R Vinter, Optimal Control. R. Vinter, Optimal Control, Birkhäuser, Boston, 2000.	[]
[ "Analytic Functions of a General Matrix Variable", "Analytic Functions of a General Matrix Variable" ]	[ "Charles Schwartz [email protected] \nDepartment of Physics\nUniversity of California Berkeley\n94720California\n\nIntroduction\n\n" ]	[ "Department of Physics\nUniversity of California Berkeley\n94720California", "Introduction\n" ]	[]	Recent innovations on the differential calculus for functions of noncommuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to first order in Delta for general matrix variables X and Delta that do not commute with each other. *	null	[ "https://arxiv.org/pdf/0804.2869v2.pdf" ]	115,160,577	0804.2869	240525a43b01b89a49923be17166ba9ce7532c6e	Analytic Functions of a General Matrix Variable Charles Schwartz [email protected] Department of Physics University of California Berkeley 94720California Introduction Analytic Functions of a General Matrix Variable arXiv:0804.2869v2 [math.FA] 7 Jul 2008 July 7, 2008, added Appendix A Recent innovations on the differential calculus for functions of noncommuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to first order in Delta for general matrix variables X and Delta that do not commute with each other. * Introduction In a recent paper [1] I showed how to expand F (x + δ) = F (x) + F (1) (x) + O(δ 2 ) (1.1) when both x and δ were general quaternionic variables, thus did not commute with each other: F (1) (x) = F ′ (x)δ 1 + [F (x) − F (x * )](x − x * ) −1 δ 2 , δ = δ 1 + δ 2 ,(1.2) with specific formulas on how to construct the two components of δ. Now we shall extend that analysis to a more general situation. Consider the N x N matrices X over the complex numbers and arbitrary analytic functions F (X) with such a matrix as its variable. We seek a general construction for the first order term F (1) (X) when we expand F (X +∆) given that ∆ is small but still a general N x N matrix that does not commute with X. The first step, as before, is to represent the function F as a Fourier transform, F (X) = dpf (p)e ipX (1.3) where the integral may go along any specified contour in the complex p-plane; and then we also make use of the expansion, e (X+∆) = e X [1 + Diagonalization The first step is to assume that we can find a matrix S that will diagonalize the matrix X at any given point in the space of such matrices. A = S X S −1 , A i,j = δ i,j λ i , i, j = 1, . . . , N (2.1) and we carry out the same transformation on the matrix ∆: B = S ∆ S −1 (2.2) but, of course, the matrix B will not be diagonal. Our task is to separate the matrix B into separate parts, each of which will behave simply in the expansion of Eq. (1.5). The first step is to recognize that the diagonal part of the matrix B, call it B 0 , commutes with the diagonal matrix A and thus we have e tA B 0 e −tA = B 0 , (2.3) where we have t = −isp from (1.3), (1.4). Next we look separately at each off-diagonal element of the matrix B: that is, B (i,j) is the matrix that has only that one off-diagonal (i = j) element of the given matrix B, and all the rest are zeros. The first commutator is simply [A, B (i,j) ] = r ij B (i,j) , r ij = λ i − λ j (2.4) and then the whole series can be summed: e tA B (i,j) e −tA = e tr ij B (i,j) . (2.5) Putting this all together, we have S F (1) S −1 = dpf (p) ip e ipA 1 0 ds [B 0 + i =j e −ispr ij B (i,j) ]. (2.6) It is trivial to carry out the integrals over s; and we thus come to the final answer F (1) (X) = F ′ (X) ∆ 0 + i =j [F (X) − F (X − r ij I)] r −1 ij ∆ (i,j) (2.7) where ∆ µ ≡ S −1 B µ S, µ = 0, (i, j). (2.8) Discussion The general structure of the result, Eq.(2.7), is similar to what we found in earlier work, Eq.(1.2): the first term F ′ (X) looks like ordinary differential calculus and goes with that part of ∆ that commutes with the local coordinate X; the remaining terms are non-local, involving the function F evaluated at discrete points separated from X by specific multiples of the unit matrix I. While this final formula appears not as explicit as the previous result found for quaternionic variables (or for variables based upon the algebra of SU(2)), in any practical situation we have computer programs that can calculate the matrix operations referred to above with great efficiency. The quantities r ij , which may be real or complex numbers, can be called the "roots" following the familiar treatment of Lie algebras. They have some properties, such as r ij = −r ji and sum rules that involve traces of the original matrix X and powers of X. What happens if the eigenvalues of X are degenerate? Suppose, for example, that λ 1 = λ 2 . This means that r 12 and r 21 are zero. If we look at Eq. (2.7), we see that the terms ∆ (1,2) and ∆ (2,1) then have the coefficient F ′ (x). Thus they simply add in with ∆ 0 . In the extreme case when all the eigenvalues are identical, then the answer is F (X + ∆) = F (X) + F ′ (X) ∆ + O(∆ 2 ), which is old fashioned differential calculus for commuting variables. Appendix A A more abstract form of the result Eq. (2.7) is the following. Here the matrix C is defined implicitly, through its commutator with X, rather than explicitly; and the matrix ∆ 0 is the same as previously discussed. One may readily confirm the correctness of this formula in the case of F (X) = X n . This alternative formalism may also be applied to the case of a quaternionic variable x, which was studied in reference [1]. In that case we find ∆ 0 = δ 1 and C = 1 4r 2 [x, δ]. It is interesting that this new formalism manages to hide the non-locality, which was a prominent feature of the original analysis. sX ∆ e sX + O(∆ 2 )]. (1.4) Another well-known expansion, relevant to what we see in (1.4), is e A B e −A = B + [A, B] , [A, [A, B]]] + . . . , (1.5) involving repeated use of the commutators, [A, B] = AB − BA. F ( 1 ) 1(X) = F ′ (X) ∆ 0 + [C, F (X)], (A.1) [C, X] = ∆ − ∆ 0 , [∆ 0 , X] = 0. (A.2) . C Schwartz, arXiv:0803.3782math.FAC. Schwartz, arXiv:0803.3782 [math.FA]	[]
[ "SUMMATION FORMULA FOR GENERALIZED DISCRETE q-HERMITE II POLYNOMIALS", "SUMMATION FORMULA FOR GENERALIZED DISCRETE q-HERMITE II POLYNOMIALS" ]	[ "Sama Arjika " ]	[]	[]	In this paper, we provide a family of generalized discrete q-Hermite II polynomials denoted byh n,α (x, y\|q). An explicit relations connecting them with the q-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula is derived by using different analytical means on their generating functions.the usual generalized basic or q-hypergeometric function of degree n in the variable x (see Slater [10, Chap. 3], Srivastava and Karlsson [11, p.347, Eq. (272)] for details). For µ = 0 2010 Mathematics Subject Classification. 33C45, 33D15, 33D50.	null	[ "https://arxiv.org/pdf/1902.09994v1.pdf" ]	119,721,905	1902.09994	df7c3a196e73343f2c16db0de0757710fe582f0b	SUMMATION FORMULA FOR GENERALIZED DISCRETE q-HERMITE II POLYNOMIALS 26 Feb 2019 Sama Arjika SUMMATION FORMULA FOR GENERALIZED DISCRETE q-HERMITE II POLYNOMIALS 26 Feb 2019 In this paper, we provide a family of generalized discrete q-Hermite II polynomials denoted byh n,α (x, y\|q). An explicit relations connecting them with the q-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula is derived by using different analytical means on their generating functions.the usual generalized basic or q-hypergeometric function of degree n in the variable x (see Slater [10, Chap. 3], Srivastava and Karlsson [11, p.347, Eq. (272)] for details). For µ = 0 2010 Mathematics Subject Classification. 33C45, 33D15, 33D50. Introduction In their paper,Àlvarez-Nodarse et al [2], have introduced a q-extension of the discrete q-Hermite II polynomials as: (1 − aq k ), n = 1, 2, · · · , the q-shifted factorial, and (1.3) r Φ s q −n , a 2 , · · · , a r b 1 , b 2 , · · · , b s q; x = in (1.1), the polynomials H (0) n (x; q) correspond to the discrete q-Hermite II polynomials [1,8], i.e., H (0) n (x; q 2 ) = q n(n−1)h n (x; q). They show that the polynomials H (µ) n (x; q) satisfy the orthogonality relation (1.4) ∞ −∞ H (µ) n (x; q)H (µ) m (x; q)ω(x)dx = π q −n/2 (q 1/2 ; q 1/2 ) n (q 1/2 ; q) 1/2 δ nm on the whole real line R with respect to the positive weight function ω(x) = 1/(−x 2 ; q) ∞ . A detailed discussion of the properties of the polynomials H (µ) n (x; q) can be found in [2]. Recently, Saley Jazmat et al [7], introduced a novel extension of discrete q-Hermite II polynomials by using new q-operators. This extension is defined as: h 2n,α (x; q) = (−1) n q −n(2n−1) (q; q) 2n (q 2α+2 ; q 2 ) n L (α) n x 2 q −2α−1 ; q 2 (1.5)h 2n+1,α (x; q) = (−1) n q −n(2n+1) (q; q) 2n+1 (q 2α+2 ; q 2 ) n+1 x L (α+1) n x 2 q −2α−1 ; q 2 . For α = −1/2 in (1.5), the polynomialsh n,− 1 2 (x; q) correspond to the discrete q-Hermite II polynomials, i.e.,h n,− 1 2 (x; q) =h n (x; q). The generalized discrete q-Hermite II polynomials (1.5) satisfy the orthogonality relation (1.6) +∞ −∞h n,α (x; q)h m,α (x; q)ω α (x; q)\|x\| 2α+1 d q x = 2q −n 2 (1 − q)(−q, −q, q 2 ; q 2 ) ∞ (−q −2α−1 , −q 2α+3 , q 2α+2 ; q 2 ) ∞ (q; q) 2 n (q; q) n,α δ n,m on the whole real line R with respect to the positive weight function ω α (x) = 1/(−q −2α−1 x 2 ; q 2 ) ∞ . A detailed discussion of the properties of the polynomialsh n,α (x; q) can be found in [7]. Srivastava and Jain [12,6], investigated multilinear generating functions for q-Hermite, q-Laguerre polynomials and other special functions. Relevant connections of these multilinear generating functions with various known results for the classical or q-Hermite polynomials are also indicated. They also proved many combinatorial q-series identities by applying the theory of q-hypergeometric functions (see [6], for more details). Motivated by Saley Jazmat's work [7], our interest in this paper is to introduce new family of "generalized discrete q-Hermite II polynomials (in short gdq-H2P)h n,α (x, y\|q)" which is an extension of the generalized discrete q-Hermite II polynomialsh n,α (x; q) and investigate summation formulae. The paper is organized as follows. In Section 2, we recall notations to be used in the sequel. In Section 3, we define a gdq-H2Ph n,α (x, y\|q) and investigate several properties. In Section 4, we derive summation and inversion formulae for gdq-H2Ph n,α (x, y\|q). In Section 5, concluding remarks are given. Notations and Preliminaries For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. We refer to the general references [4,8] and [7] for the definitions and notations. Throughout this paper, we assume that 0 < q < 1, α > −1. For a complex number a, the q-shifted factorials are defined by: (2.1) (a; q) 0 = 1; (a; q) n = n−1 k=0 (1 − aq k ), n = 1, 2, · · · ; (a; q) ∞ = ∞ k=0 (1 − aq k ) and the q-number is defined by: (2.2) [n] q = 1 − q n 1 − q , n! q := n k=1 [k] q , 0! q := 1, n ∈ N. Let x and y be two real or complex numbers, the Hahn [5] q-addition ⊕ q of x and y is given by: x ⊕ q y n : = (x + y)(x + qy) . . . (x + q n−1 y) = (q; q) n n k=0 q ( k 2 ) x n−k y k (q; q) k (q; q) n−k , n ≥ 1, x ⊕ q y 0 := 1, (2.3) while the q-subtraction ⊖ q is given by (2.4) x ⊖ q y n := x ⊕ q (−y) n . The generalized q-shifted factorials [7] are defined by the recursion relations Remark that, for α = −1/2, we have (2.8) (q; q) n,−1/2 = (q; q) n , [n] q,−1/2 ! = (1 − q) n (q; q) n . We denote (2.9) (q; q) 2n,α = (q 2 ; q 2 ) n (q 2 α+2 ; q 2 ) n , and (2.10) (q; q) 2n+1,α = (q 2 ; q 2 ) n (q 2 α+2 ; q 2 ) n+1 . The two Euler's q-analogues of the exponential functions are given by [4] (2.11) E q (x) = ∞ k=0 q ( k 2 ) x k (q; q) k = (−x; q) ∞ and (2.12) e q (x) = ∞ k=0 x k (q; q) k = 1 (x; q) ∞ , \|x\| < 1. For m ≥ 1 and by means of the generalized q-shifted factorials, we define two generalized q-exponential functions as follows (2.13)Ẽ q m ,α (x) := ∞ k=0 q mk(k−1)/2 x k (q m ; q m ) k,α , and (2.14)ẽ q m ,α (x) := ∞ k=0 x k (q m ; q m ) k,α , \|x\| < 1. Remark that, for m = 1 and α = − 1 2 , we have: (2.15)Ẽ q,α (x) = E q (x),ẽ q,α (x) = e q (x). For m = 2, the following elementary result is useful in the sequel to establish the summation formulae for gdq-H2P: (2.16)ẽ q 2 ,− 1 2 (x)Ẽ q 2 ,− 1 2 (y) =ẽ q 2 ,− 1 2 (x ⊕ q 2 y), (2.17)ẽ q,− 1 2 (x)Ẽ q 2 ,− 1 2 (−y) =ẽ q (x ⊖ q,q 2 y),ẽ q 2 ,− 1 2 (x)Ẽ q 2 ,− 1 2 (−x) = 1, where (2.18) (a ⊖ q,q 2 b) n := n! q n k=0 (−1) k q k(k−1) (n − k)! q k! q 2 a n−k b k , (a ⊖ q,q 2 b) 0 := 1. Generalized discrete q-Hermite II polynomials In this section, we introduce a sequence of gdq-H2P {h n,α (x, y\|q)} ∞ n=0 . Several properties related to these polynomials are derived. Definition 3.1. For x, y ∈ R, the gdq-H2P {h n,α (x, y\|q)} ∞ n=0 are defined by: (3.1)h n,α (x, y\|q) := (q; q) n ⌊ n/2 ⌋ k=0 (−1) k q −2nk+k(2k+1) x n−2k y k (q; q) n−2k,α (q 2 ; q 2 ) k and (3.2)h n,α (x, 0\|q) := (q; q) n (q; q) n,α x n . Remark that, (1) for y = 1, we get (3.3)h n,α (x, 1\|q) =h n,α (x; q) whereh n,α (x; q) is the generalized discrete q-Hermite II polynomial [7]; (2) for α = −1/2 and y = 1, we have (3.4)h n,−1/2 (x, 1\|q) =h n (x; q). whereh n (x; q) is the discrete q-Hermite II polynomial [1,8]. (3) Indeed since (3.5) lim q→1 (q a ; q) n (1 − q) n = (a) n one readily verifies that (3.6) lim q→1h n,− 1 2 ( 1 − q 2 x, 1\|q) (1 − q 2 ) n/2 = h α+ 1 2 n (x) 2 n where h α+ 1 2 n (x) is the Rosenblums generalized Hermite polynomial [9]. Lemma 3.1. The following recursion relation for gdq-H2P {h n,α (x, y\|q)} ∞ n=0 holds true. (3.7) 1 − q n+1+θn(2α+1) 1 − q n+1h n+1,α (x, y\|q) = xh n,α (x, y\|q) − y q −2n+1 (1 − q n )h n−1,α (x, y\|q). Proof. To prove the assertion (3.7), we consider separately even and odd cases of the expression (3.8) xh n,α (x, y\|q) − y q −2n+1 (1 − q n )h n−1,α (x, y\|q). For n even, we have: xh 2n,α (x, y\|q) = (q; q) 2n (q; q) 2n,α x 2n+1 + (q; q) 2n n k=1 (−1) k q −2nk+k(2k+1) x 2n−2k+1 y k (q; q) 2n−2k,α (q 2 ; q 2 ) k . The right-hand side of the last relation can be written as (3.9) (q; q) 2n (q; q) 2n,α x 2n+1 + (q; q) 2n × n k=1 (−1) k q −2k(2n+1)+k(2k+1) x 2n+1−2k y k (q; q) 2n+1−2k,α (q 2 ; q 2 ) k q 2k (1 − q 2n+2+2α−2k ) . In the same way, − y q −4n+1 (1 − q 2n )h 2n−1,α (x, y\|q) = −y q −4n+1 (q; q) 2n (3.10) × n−1 k=0 (−1) k q −2k(2n+1)+k(2k+1) x 2n+1−2(k+1) y k (q; q) 2n+1−2(k+1),α (q 2 ; q 2 ) k . Change k to k − 1 in (3.10), one obtains (3.11) (q; q) 2n n k=1 (−1) k q −2k(2n+1)+k(2k+1) x 2n+1−2k y k (q; q) 2n+1−2k,α (q 2 ; q 2 ) k (1 − q 2k ). Then combining (3.9) and (3.11), we have (3.12) xh 2n,α (x, y\|q) − y q −4n+1 (1 − q 2n )h 2n−1,α (x, y\|q) = (q; q) 2n (q; q) 2n,α x 2n+1 + (q; q) 2n n k=1 (−1) k q −2k(2n+1)+k(2k+1) x 2n+1−2k y k (q; q) 2n+1−2k,α (q 2 ; q 2 ) k × q 2k (1 − q 2n+2+2α−2k ) + (1 − q 2k ) . After simplification, it is equal to (q; q) 2n (q; q) 2n,α x 2n+1 + (1 − q 2n+2+2α )(q; q) 2n n k=1 (−1) k q −2k(2n+1)+k(2k+1) x 2n+1−2k y k (q; q) 2n+1−2k,α (q 2 ; q 2 ) k . The last expression can be written as (3.13) 1 − q 2n+2+2α 1 − q 2n+1h 2n+1,α (x, y\|q). Summarizing the above calculations in (3.12)-(3.13), we get the assertion (3.7) for n even. In the odd case, the proof follows the same steps as the even case. Theorem 3.1. We have: (3.14) lim α→+∞h 2n,α (x, y\|q) = q −n(2n−1) (q; q) 2n (−y) n S n x 2 y −1 q −1 ; q 2 and (3.15) lim α→+∞h 2n+1,α (x, y\|q) = q −n(2n+1) (q; q) 2n+1 x (−y) n S n x 2 y −1 q −1 ; q 2 where S n (x; q) are the Stieltjes-Wigert polynomials [8]. In order to prove Theorem 3.1, we need the following Lemma. = q −n(2n+1) (q; q) 2n+1 (q 2α+2 ; q 2 ) n+1 x (−y) n L (α+1) n x 2 y −1 q −2α−1 ; q 2 . In order to prove Lemma 3.2, we need the following Proposition. Proposition 3.1. For α > −1, the sequence of gdq-H2P {h n,α (x, y\|q)} ∞ n=0 can be written in terms of basic hypergeometric functions as (3.18)h n,α (x, y\|q) = (q; q) n (q; q) n,α x n 2 Φ 1 q −n , q −n−2α 0 q 2 ; − y q 2α+3 x 2 . Proof. In fact, for n even, and by using (3.19) (q; q) 2n−2k,α = (q 2 ; q 2 ) n−k (q 2α+2 ; q 2 ) n−k , the gdq-H2Ph n,α (x, y\|q) defined in (3.1) can be rewritten as (3.20)h 2n,α (x, y\|q) = (q; q) 2n n k=0 (−1) k q −4nk+k(2k+1) x 2n−2k y k (q 2 ; q 2 ) n−k (q 2α+2 ; q 2 ) n−k (q 2 ; q 2 ) k . From the formula [8, p.9, Eq. (0.2.12)] (3.21) (a; q) n−k = (a; q) n (a −1 q 1−n ; q) k − q a k q ( k 2 )−nk , we have for a = q 2 and q 2α+2 , h 2n,α (x, y\|q) = (q; q) 2n x 2n (q; q) 2n,α n k=0 (−1) k q −4nk+k(2k+1) (q −2n , q −2n−2α ; q 2 ) k (q 2 ; q 2 ) k q 4( k 2 )−4nk−2αk y x 2 k . After simplification, the last equation reads (3.22)h 2n,α (x, y\|q) = (q; q) 2n (q; q) 2n,α x 2n n k=0 (q −2n , q −2n−2α ; q 2 ) k (q 2 ; q 2 ) k − y q 2α+3 x 2 k . In the odd case, the proof follows the same steps as the even case. Now, we are in position to prove Lemma 3.1. Proof. (of Lemma 3.1) For n even, the relation (3.18) becomes: (3.23)h 2n,α (x, y\|q) = (q; q) 2n (q; q) 2n,α x 2n 2 Φ 1 q −2n , q −2n−2α 0 q 2 ; − y q 2α+3 x 2 . By taking a −1 = q −2α−2 and z = −q 2n+1 x 2 y −1 and the formula [8, p.17, Eq. (0.6.17)] (3.24) 2 Φ 1 q −n , a −1 q 1−n 0 q; aq n+1 z = (a; q) n (qz −1 ) n 1 Φ 1 q −n a q; z we have (3.25) 2 Φ 1 q −2n , q −2n−2α 0 q 2 ; − y q 2α+3 x 2 = (q 2α+2 ; q 2 ) n − y x 2 n q −2n 2 +n 1 Φ 1 q −2n q 2+2α q 2 ; − q 2n+1 x 2 y . By using (1.2), the relation (3.25) can be written as (3.26) q −2n 2 +n (q 2 ; q 2 ) n − y x 2 n L (α) n x 2 y −1 q −2α−1 ; q 2 . The assertion (3.16) of Lemma 3.1 follows by summarizing the above calculations in (3.23)-(3.26). In the odd case, the proof follows the same steps as the even case. Proof. (of Theorem 3.2) By taking the limit α → +∞ in the assertions (3.16) and (3.17) of Lemma 3.1, respectively, we get the assertions (3.14) and (3.15) of Theorem 3.2. Connection formulae for the generalized discrete q-Hermite II polynomials {h n,α (x, y\|q)} ∞ n=0 We begin this section with the following theorem: Theorem 4.1. The sequence of gdq-H2P {h n,α (x, y\|q)} ∞ n=0 , which is defined by the relation (3.1), satisfies the connection formula (4.1)h n,α (x, ω\|q) = (q; q) n ⌊ n/2 ⌋ k=0 q −2nk+k(2k+1) (−ω ⊕ q 2 y) k (q 2 ; q 2 ) k (q; q) n−2kh n−2k,α (x, y\|q). To prove Theorem 4.1, we need the following Lemma. q ( n 2 ) t n (q; q) nh n,α (x, y\|q), \|yt\| < 1. (4.2) Proof. Let us consider the function (4.3) f q (t; x, y) := ∞ n=0 q ( n 2 ) t n (q; q) nh n,α (x, y\|q). By replacing in (4.3) gdq-H2Ph n,α (x, y\|q) by its explicit expression (3.1) we obtain f q (t; x, y) = ∞ n=0 t n   ⌊ n/2 ⌋ k=0 (−1) k q ( n 2 )−2nk+k(2k+1) x n−2k y k (q; q) n−2k,α (q 2 ; q 2 ) k   . (4.4) The right-hand side of (4.4) also reads (4.5) ∞ n=0 ⌊n/2⌋ k=0 (−1) k q ( n−2k 2 ) (yt 2 ) k (xt) n−2k (q; q) n−2k,α (q 2 ; q 2 ) k . Next, changing n − 2k by r, r = 0, 1, · · · , the last relation becomes (4.6) ∞ n=0 (−yt 2 ) n (q 2 ; q 2 ) n ∞ r=0 q ( r 2 ) (xt) r (q; q) r,α . Hence, (4.7) f q (t; x, y) =ẽ q 2 ,− 1 2 (−yt 2 )Ẽ q,α (xt). Now, we are in position to prove Theorem 4.1. Proof. (of Theorem 4.1) Replacing t by u ⊕ q t in (4.2), we find the following generating function (4.8)Ẽ q,α (u ⊕ q t)x ẽ q 2 ,− 1 2 − y(u ⊕ q t) 2 = ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, y\|q) which by using (2.17), becomes (4.9)Ẽ q,α (u ⊕ q t)x =Ẽ q 2 ,− 1 2 y(u ⊕ q t) 2 ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, y\|q). Replacing y by ω and (4.9), respectively, in (4.8), we get (4.10) ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, ω\|q) = =ẽ q 2 ,− 1 2 − ω(u ⊕ q t) 2 Ẽ q 2 ,− 1 2 y(u ⊕ q t) 2 ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, y\|q). By using (2.17), the last relation reads (4.11) ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, ω\|q) =ẽ q 2 ,− 1 2 (−ω ⊕ q 2 y)(u ⊕ q t) 2 ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, y\|q). According to (2.12), the right-hand side of (4.11) can be written as (4.12) ∞ r=0 (−ω ⊕ q 2 y) r (u ⊕ q t) 2r (q 2 ; q 2 ) r ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, y\|q) . Let us substitute n + 2r = k =⇒ r ≤ ⌊ k/2 ⌋ in (4.12), then we have: (4.13) ∞ n=0   ⌊ n/2 ⌋ k=0 (q ( n−2k 2 ) (−ω ⊕ q 2 y) k (q 2 ; q 2 ) k (q; q) n−2kh n−2k,α (x, y\|q)   (u ⊕ q t) n . Next, replacing (4.13) in (4.11), we obtain (4.14) ∞ n=0 q ( n 2 ) (u ⊕ q t) n (q; q) nh n,α (x, ω\|q) = ∞ n=0   ⌊ n/2 ⌋ k=0 (q ( n−2k 2 ) (−ω ⊕ q 2 y) k (q 2 ; q 2 ) k (q; q) n−2kh n−2k,α (x, y\|q)   (u ⊕ q t) n . Finally, on equating the coefficients of like powers of (u ⊕ q t) n /(q; q) n in (4.14), we get the desired identity. We have the following special cases of Theorem 4.1 of particular interest. (4.15)h n,α (x, ω\|q) = (q; q) n ⌊ n/2 ⌋ k=0 (−1) k q −2nk+k(2k+1) x n−2k ω k (q 2 ; q 2 ) k (q; q) n−2k,α ; (ii) ω = 0 in the assertion (4.1) of Theorem 4.1, and using (3.2), we get the inversion formula for gdq-H2P (4.16) x n = (q; q) n,α ⌊ n/2 ⌋ k=0 q −2nk+3k 2 y k (q 2 ; q 2 ) k (q; q) n−2kh n−2k,α (x, y\|q). iii) For y = 1, the summation formulae (4.1) can be expressed in terms of generalized discrete q-Hermite II polynomialsh n,α (x; q). Also, the summation formulae (4.1) can be written in terms of discrete q-Hermite II polynomialsh n (x; q) by choosing y = 1 and α = −1/2. Concluding remarks In the previous sections, we have introduced gdq-H2Ph n,α (x, y\|q) and derived several properties. Also, we have derived implicit summation formula for gdq-H2Ph n,α (x, y\|q) by using different analytical means on their generating function. This process can be extended to summation formulae for more generalized forms of q-Hermite polynomials. This study is still in progress. We note that the generating function of even and odd gdq-H2Ph n,α (x, y\|q) are given by ∞ n=0 (−t 2 ) n q n(2n−1) (q; q) 2nh 2n,α (x, y\|q) = q α(α+ 1 2 ) (q 2 ; q 2 ) ∞ (q 2α+2 ; q 2 ) ∞ x −α J (2) α (2xq −α− 1 2 ; q 2 ) (y t 2 ; q 2 ) ∞ and ∞ n=0 (−1) n q n(2n+1) t 2n+1 (q; q) 2n+1 h 2n+1,α (x, y\|q) = q α(α+1) (q 2 ; q 2 ) ∞ (q 2α+2 ; q 2 ) ∞ x −α J (2) α (2xq −α ; q 2 ) (y t 2 ; q 2 ) ∞ where J (2) ν (z; q) is the q-analogue of the Bessel function [8]. Indeed, it is well known that from (4.2), the generating function of gdq-H2Ph n,α (x, y\|q) is given by (5.1)Ẽ q,α (xt)ẽ q 2 ,− 1 2 (−yt 2 ) = ∞ n=0 q n(n−1)/2 t n (q; q) nh n,α (x, y\|q) which on separating the power in the right-hand side into their even and odd terms by using the elementary identity (−1) n q n(2n+1) t 2n+1 (q; q) 2n+1h 2n+1,α (x, y\|q) = Sin q,α (xt)ẽ q 2 ,− 1 2 (yt 2 ) where the generalized q-Cosine and q-Sine are defined as: Cos q,α (x) : = ∞ k=0 (−1) n q n(2n−1) x 2n (q; q) 2n,α , (5.6) Sin q,α (x) : = ∞ k=0 (−1) n q n(2n+1) x 2n+1 (q; q) 2n+1,α . (5.7) By using (2.9) and (2.10), respectively, the relations (5.6) and (5.7) can be expressed in terms of basic hypergeometric functions as Cos q,α (x) = 0 Φ 1 − q 2α+2 q 2 ; −qx 2 (5.8) Sin q,α (x) = x 1 − q 2α+2 0 Φ 1 − q 2α+4 q 2 ; −q 2 x 2 . (5.9) The q-analogue of the Bessel function is defined [8, p.20, Eq.(0.7.14)] by (5.10) J (2) ν (z; q) = (q ν+1 ; q) ∞ (q; q) ∞ z 2 ν 0 Φ 1 − q ν+1 q; − q ν+1 z 2 4 from which the generating functions of (5.8) and (5.9) follow. n (x; q) are the q-Laguerre polynomials given byL (α)n (x; q) : = (q α+1 ; q) n (q; q) n 1 Φ 1 q −n q α+1 q; −q n+α+1 x ; q) n+1,α = (1 − q)[n + 1 + θ n (2α + 1)] q (q; q) Lemma 3. 2 . 2For α > −1, the sequence of gdq-H2P {h n,α (x, y\|q)} ∞ n=0 can be written in terms of q-Laguerre polynomials L n (x; q) as (3.16)h 2n,α (x, y\|q) = q −n(2n−1) (q; q) 2n (q 2α+2 ; q 2 ) n (−y) n L (α) n x 2 y −1 q −2α−1 ; q 2 and (3.17)h 2n+1,α (x, y\|q) Lemma 4. 1 . 1The following generating function for gdq-H2P {h n,α (x, y\|q)} ∞ n=0 holds true. Corollary 4. 1 . 1Letting: (i) y = 0 in the assertion (4.1) of Theorem 4.1, we get the definition of gdq-H2P (3.1),i.e., 2n+1) t 2n+1 (q; q) 2n+1h 2n+1,α (x, y\|q).Now replacing t by i t in (5.3) and equating the real and imaginary parts of the resultant equation, we get the generating function of even and odd gdq-H2P h n,α (x, y\|q) n q n(2n−1) t 2n (q; q) 2nh 2n,α (x, y\|q) = Cos q,α (xt)ẽ q 2 Some orthogonal q-polynomials. W A Al-Salam, L Carlitz, Math. Nach. 30W. A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nach. 30, (1965) 47-61. A q-extension of generalized Hermite polynomials with the continuous orthogonality property on R. R Àlvarez-Nodarse, M K Atakishiyeva, N M Atakishiyev, Int. J. Pure. Appl. Math. 103R.Àlvarez-Nodarse, M. K. Atakishiyeva and N. M. Atakishiyev, A q-extension of generalized Hermite polynomials with the continuous orthogonality property on R, Int. J. Pure. Appl. Math. 10 (3) (2014) 331-342. G E Andrews, R Askey, R Roy, of Encyclopedia of Mathematics and its Applications Sciences. CambridgeCambridge University Press71Special functionsG. E. Andrews, R. Askey and R. Roy , Special functions, vol. 71 of Encyclopedia of Mathematics and its Applications Sciences, Cambridge University Press, Cambridge, (1999). G Gasper, M Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its application. Cambridge, UKCambridge Univ. Press35G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its appli- cation. vol, 35, Cambridge Univ. Press, Cambridge, UK (1990). Beiträge zur Theorie der Heineschen Reihen. W Hahn, Math. Nachr. 2W. Hahn, Beiträge zur Theorie der Heineschen Reihen, Math. Nachr. 2, (1949) 340-379. Some families of multilinear q-generating functions and combinatorial q-series identities. V K Jain, H M Srivastava, J. Math. Anal. Appl. 192V. K. Jain and H. M. Srivastava, Some families of multilinear q-generating functions and combinatorial q-series identities, J. Math. Anal. Appl. 192 (1995) 413-438. Generalized q-Hermite Polynomials and the q-Dunkl Heat Equation. M Jazmati, K Mezlini, N Bettaibi, Bull. Math. Anal. Appl. 64M. Jazmati, K. Mezlini and N. Bettaibi, Generalized q-Hermite Polynomials and the q-Dunkl Heat Equation, Bull. Math. Anal. Appl. 6 (4) (2014) 16-43. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. R Koekoek, R Swarttouw, 98-17The NetherlandsDelft ReportR. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft Report 98-17, The Netherlands (1998). M Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus In: Operator theory: Advances and Applications. BaselBirkhäuser Verlag73M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus In: Operator theory: Advances and Applications, vol. 73, Basel: Birkhäuser Verlag (1994) 369-396. Generalized Hypergeometric Functions. L J Slatter, Cambridge Univ. PressCambridge/London/New YorkL. J. Slatter, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cam- bridge/London/New York, (1966). H M Srivastava, P W Karlsson, Multiple Gaussian Hypergeometric Series. Halsted (Ellis Horwood, ChichesterH. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted (Ellis Hor- wood, Chichester); . Wiley, New YorkWiley, New York, (1985). Some multilinear generating functions for q-Hermite polynomials. H M Srivastava, V K Jain, J. Math. Anal. Appl. 144Faculty of Sciences and Technics, University of AgadezNiger E-mail address: [email protected]. M. Srivastava and V. K. Jain, Some multilinear generating functions for q-Hermite polynomials, J. Math. Anal. Appl. 144 (1989) 147-157. Faculty of Sciences and Technics, University of Agadez, Niger E-mail address: [email protected]	[]
[ "Controlador LQR y SMC Aplicado a Plataformas Pendulares", "Controlador LQR y SMC Aplicado a Plataformas Pendulares" ]	[ "Miguel F Arevalo-Castiblanco ", "C H Rodriguez-Garavito ", "A Patio-Forero ", "Jos F Salazar-Cceres " ]	[]	[]	Una plataforma pendular es una estructura robtica comnmente empleada en el diseo de controladores dada su dinmica no lineal; este trabajo presenta el modelamiento, diseo e implementacin de un controlador ptimo LQR y un controlador en modo deslizante SMC aplicado a dos plataformas comerciales, el pndulo rotatorio invertido de Quanser (RotPen) y el pndulo mvil de Lego (NxtWay). El aporte de este trabajo es presentar una metodologa de implementacin de controladores sobre plataformas pendulares, atendiendo las respectivas restricciones de hardware y software en prototipos comerciales. El artculo presenta el comportamiento de los controladores diseados sobre el modelo analtico comparado con su implementacin real. Index Terms-Control ptimo, Control en modo deslizante, Sistemas Embebidos, LQR, Modelo Pendulo Invertido Quanser, Modelo Segway NxtWay. arXiv:2002.04684v1 [eess.SY]	10.4995/riai.2017.9101	[ "https://arxiv.org/pdf/2002.04684v1.pdf" ]	117,214,535	2002.04684	81e02c2485f183f7644d9c4605254143431d79f8	Controlador LQR y SMC Aplicado a Plataformas Pendulares Miguel F Arevalo-Castiblanco C H Rodriguez-Garavito A Patio-Forero Jos F Salazar-Cceres Controlador LQR y SMC Aplicado a Plataformas Pendulares 1Index Terms-Control ptimoControl en modo deslizanteSistemas EmbebidosLQRModelo Pendulo Invertido QuanserModelo Segway NxtWay Una plataforma pendular es una estructura robtica comnmente empleada en el diseo de controladores dada su dinmica no lineal; este trabajo presenta el modelamiento, diseo e implementacin de un controlador ptimo LQR y un controlador en modo deslizante SMC aplicado a dos plataformas comerciales, el pndulo rotatorio invertido de Quanser (RotPen) y el pndulo mvil de Lego (NxtWay). El aporte de este trabajo es presentar una metodologa de implementacin de controladores sobre plataformas pendulares, atendiendo las respectivas restricciones de hardware y software en prototipos comerciales. El artculo presenta el comportamiento de los controladores diseados sobre el modelo analtico comparado con su implementacin real. Index Terms-Control ptimo, Control en modo deslizante, Sistemas Embebidos, LQR, Modelo Pendulo Invertido Quanser, Modelo Segway NxtWay. arXiv:2002.04684v1 [eess.SY] I. INTRODUCTION Es comn encontrar aplicaciones en donde se emplean estructuras pendulares con aplicacin en robots humanoides, el auge de estas aplicaciones ha incentivado a la comunidad cientfica a plantearse nuevas investigaciones en el campo del modelamiento y el control, con el fin de optimizar los procesos en donde estas estructuras se encuentran presentes [1]. Una de las estructuras robticas mviles que ha tenido mayor acogida en los ltimos aos es el pndulo invertido de dos ruedas, con plataformas comerciales como el Segway o el Hoverboard; la investigacin en el rea del control ha abordado este tipo de plataformas con mucho inters, dada su fcil construccin y su comportamiento dinmico no lineal, junto con su conocida inestabilidad. Recientemente, varios son los fabricantes que desarrollan esta clase de estructuras, con fines acadmicos o industriales, el fabricante Quanser con sus plataformas de pndulo rotatorio invertido y doble pndulo rotatorio invertido, o el propio fabricante Hoverboard con las plataformas del mismo nombre, son algunos de los ejemplos conocidos que se encuentran comercialmente [2], [3]. El objetivo de funcionamiento de estas estructuras pendulares, es lograr la estabilizacin del pndulo alrededor del punto de equilibrio con la menor energa invertida en su actuacin [4]; es comn encontrar diferentes autores que desarrollan controladores para estructuras pendulares bajo diferentes metodologas, algunos de los casos documentados sern descritos a continuacin. Los primeros controles que se implementaron en pndulos invertidos, fueron lazos de control clsico, tipo PID o realimentacin de estados, los cuales, al ser diseados sin criterios ptimos, generan ineficiencia y restringen su aplicacin a posiciones o condiciones especficas [5]. Con la ampliacin de las teoras de control modernas, el uso de controladores ptimos aumentaron la eficiencia de diferentes sistemas, incluyendo las estructuras pendulares; el controlador ptimo ms utilizado en la literatura es el controlador LQR (Linear Quadratic Regulator), este pondera heursticamente las variables de estado de un sistema [6], y usa constantes de realimentacin que minimizan al mximo el error presente en las seales controladas [7]; es comn que las variables a minimizar cuenten con un componente de ruido coloreado, que afecta el rendimiento del controlador; por esta razn, de manera alternativa se desarrolla el controlador LQG (Linear Quadratic Gaussian), en donde a partir del diseo de un Filtro de Kalman, se minimiza este ruido hasta el limite del ruido blanco, y se mejora el rendimiento del sistema [5]. Estas caractersticas han hecho que esta tcnica de control sea una de las ms utilizadas y documentadas para estructuras pendulares, sin embargo, no es la nica metodologa eficiente para ser implementada en pndulos; diferentes autores a lo largo de los ltimos aos han desarrollado nuevas tcnicas de control que permiten mejorar las aplicaciones en donde se involucren esta clase de estructuras [8], [9], [10]. Entre las tcnicas ms utilizadas de manera alternativa a los controladores LQR se encuentran tcnicas como el control en modo deslizante (Sliding Mode Control-SMC), en el cual se escoge una superficie de deslizamiento segn el comportamiento de las variables de estado, y se seleccionan las constantes de realimentacin que permitirn que las variables tomen la trayectoria definida de forma estable, a travs de los criterios de estabilidad de Lyapunov [11] [12]. Es comn encontrar tambin controladores de carcter predictivo, en donde a partir de un modelo completo del sistema pendular, es posible modelar sus perturbaciones, para que el controlador sea capaz de rechazarlas al momento que sucedan; es una estrategia de control eficiente debido a que no solo tiene en cuenta el modelamiento de la planta, sino que adems incluye una definicin analtica de las perturbaciones que pueden intervenir en el modelo; esta estrategia requiere un pleno conocimiento del modelo dinmico no lineal y linealizado del sistema y sus perturbaciones [4] [13]; tarea que en ocasiones suele ser ms complicada que el diseo de un controlador en estructuras pendulares [14]. Por ltimo, la tcnica que ha tenido ms desarrollo a lo largo de los ltimos aos, ha sido la relacionada con el control robusto (H 2 /H ∞ ) [10]; esta tcnica maneja intervalos de tolerancia para las variables de estado, los cuales permiten que el sistema tenga un comportamiento ms amplio y con inmunidad a cambios no drsticos, sin embargo la dificultad matemtica que representa este tipo de controladores, hace que su implementacin posea un grado de complejidad elevado, que dificulta su ejecucin en sistemas de procesamiento de datos. [9]. Las diferentes tcnicas desarrolladas en los ltimos aos para el control de estructuras pendulares, ha permitido una mayor eficiencia en sus realizaciones, trayendo consigo, un incremento en su uso comercial [4]; es as como es posible observar en la industria diversas aplicaciones en sectores como: aeroespacial, biomecnica o transporte; con usos como el control activo para despegue de cohetes [15], la estabilidad de prototipos bpedos caminantes [16] o el mencionado medio de transporte Segway [17]. El objetivo de la mayora de los trabajos relacionados, se enfoca en el desarrollo de las metodologas de control en un nico prototipo [9], [10], [11], [1], [8], [6], [5]; sin embargo, un anlisis comparativo del comportamiento de las tcnicas de control sobre diferentes plantas permite evaluar la factibilidad de la implementacin de controladores, teniendo en cuenta restricciones fsicas como la facilidad de medida de las variables de estado o el costo computacional relacionado con la velocidad de procesamiento del hardware donde se embebe el controlador. El enfoque de este trabajo es llevar a cabo la comparacin de las tcnicas de control LQR y SMC en dos diferentes prototipos, un pndulo rotatorio invertido de Quanser (RotPen), y una pndulo invertido mvil de Lego (NxtWay), buscando que los resultados del control sean genricos y sea posible hacer una comparacin, validando su implementacin en diferentes plantas ante una misma metodologa de control. Cada uno de los pasos de la metodologa se presenta en las siguientes secciones as: la seccin II hace nfasis en el modelo dinmico de ambas estructuras, posteriormente en la seccin III, se tomar este modelo, y a partir de su linealizacin y su representacin en espacio de estados, se realiza el clculo de cada uno de los controladores; en la seccin IV estos controladores sern implementados en los respectivos entornos, y por ltimo en la seccin V, se validar y comparar el rendimiento de los controladores bajo ndices de calidad. II. MODELO DINMICO Para el control de un sistema, es necesario tener un conocimiento pleno de su modelo dinmico; como se menciona en la seccin anterior, las plataformas a analizar son el pndulo rotatorio invertido de Quanser (RotPen) y el pndulo mvil de Lego (NxtWay), en las figuras 1 y 2 se observan las estructuras base de cada plataforma, las cuales definen las posiciones y velocidades que sern utilizadas como variables de estado en sus respectivos modelamientos. El modelo dinmico en ambos casos se obtiene a partir de un mtodo basado en energa, que parte de la definicin del operador lagrangiano [18]; es importante aclarar que las energas surgen a partir del anlisis de dos eslabones fsicos para el modelo RotPen ([q 1 , q 2 ]), como se observa en la figura 1 y para la plataforma NxtWay mostrada en la figura 2 se presenta un movimiento libre de dos grados de libertad definido por su posicin q 1 y orientacin respecto a z, q 3 que para propsitos de simplificacin se har igual a cero, y por la posicin del polo q 2 . El listado de los parmetros utilizados para el modelamiento de cada estructura se observa en las tablas I y II. El objetivo de cada modelamiento es llegar a la definicin de la ecuacin de la robtica [19]; en cada caso la entrada ser (a) Vista frontal de la plataforma (b) Vista superior de la plataforma el voltaje de los actuadores; su definicin matricial se presenta en la ecuacin (1); en las ecuaciones (2) y (3) se especifican cada uno de los trminos involucrados en esta ecuacin para el RotPen y el NxtWay respectivamente. M x (q)q + C x (q,q)q + G x (q) = V x(1) donde para el RotPen las matrices se representan como: M Q = γ(mr Lr 2 2 + mpL 2 r + Jr + mp Lp 2 2 s 2 (q2)) − 1 2 mpLpLrc (q2) γ − 1 2 mpLpLrc (q2) mp Lp 2 2 + Jp C Q = 2s (q2) c (q2) mp Lp 2 2 γq2 + γfr + KmKg 1 2 mpLpLrs (q2)q2γ −q1mp Lp 2 2 s (q2) c (q2) fp G Q = 0 − Lp 2 mpgsin (q2) Rm KmKg(2) Y para el NxtWay como: M N = 1 α 2mR 2 + M R 2 + 2J w + 2n 2 J m M LRc(q 2 ) − 2n 2 J m −(M LRc(q 2 ) − 2n 2 J m ) −(M L 2 + J q2 + 2n 2 J m ) C N = 1 α 2 (β + f w ) −2β − M LRq 2 s(q 2 ) 2β −2β G N = 1 α 0 M gLs(q 2 )(3) (a) Vista frontal de la plataforma (b) Vista superior de la plataforma Las matrices obtenidas definen el comportamiento no lineal de cada uno de los sistemas; para un correcto funcionamiento x(t) = A x x(t) + B x u(t) (4) y(t) = C x x(t) + D x u(t)(5) El vector de estados de cada plataforma esta compuesto a partir de las posiciones mostradas en las figuras 1 y 2 y sus respectivas velocidades, x(t) = [q 1 , q 2 ,q 1 ,q 2 ]; los valores de cada matriz para cada estructura se encuentran definidos en las tablas III y IV; los valores de las matrices C y D, se encuentran definidas de manera estndar. III. TÉCNICAS DE CONTROL En esta seccin se presenta el diseo de dos tcnicas de control moderno para validar la metodologa propuesta: Control ptimo a partir de un regulador cuadrtico lineal (LQR) y un Control en modo deslizante (SMC). A. Control LQR Con las matrices de estado, es posible realizar los clculos para los controladores ptimos LQR; esta tcnica de control parte del modelo dinmico de cada sistema, para obtener una matriz de realimentacin que minimice un indice de calidad energtico dado en [6], como se muestra en la ecuacin (6). En el caso del NxtWay, se emplean dos lazos de control, un lazo proporcional integrativo, PI, para la posicin del mvil q 1 y un segundo lazo de control LQR para el equilibrio del pndulo, donde interviene parte del vector de estado [q 2 ,q 1 ,q 2 ], la separacin del espacio de estados se puede efectuar gracias al desacople dinmico de q 1 en las ecuaciones de aceleracinq 1 yq 2 ; el resultado de este anlisis se evidencian en la tabla IV. Jq 2 = M L 2 3 kgm 2 Momento de inercia pitch Jq 3 = M (W 2 +D 2 ) 12 kgm 2 Momento de inercia yaw Jm = 1x10 −5 kgm 2 Momento de inercia motor fm=0.0022 N ms rad Friccin dinmica cuerpo-motor f W =0 N ms rad Friccin dinmica rueda-suelo Rm=6.69 Ω Resistencia del motor CC K b =0.468 V s rad Constante Contra electromotriz motor CC Kt=0.317 N m A Constante torque del motor η = 1 Eficiencia de transmisin α = nK t Rm Constante del motor β = ( nK t K b Rm ) + fm Constante del motor Vm=10 V Voltaje nominal del motor J = ∞ 0 x T Qx + u T Ru dt(6)0 A Q(13,24) 1 ∆ Q A Q32 1 4∆ Q mp 2 Lp 2 Lrgγ A Q33 − 1 ∆ Q (frγ + KmKg) Jp + 1 4 mpLp 2 A Q34 − 1 2∆ Q mpLpLrfpγ A Q42 1 2∆ Q Lpmpgγ mr Lr 2 2 + mpL 2 r + Jr A Q43 − 1 2∆ Q mpLpLr (frγ + KmKg) A Q44 − 1 ∆ Q fpγ mr Lr 2 2 + mpL 2 r + Jr B Q(11,21) 0 B Q31 1 ∆ Q Jp + 1 4 mpLp 2 B Q41 1 2∆ Q mpLpLr ∆ Q = γ mr Lr 2 2 + mpL 2 r + Jr mp Lp 2 2 + Jp − 1 4 m 2 p L 2 p L 2 r Q y R son matrices definidas heursticamente de acuerdo a una ponderacin realizada por el diseador para las variables de estado del sistema en ambos lazos; estas matrices buscan minimizar a partir de la ecuacin (6) la energa empleada por las variables de estado (Q) y por su entrada (R); dando como resultado una realimentacin ptima segn la ley de control (u = −Kx), en donde la ganancia, al tener un comportamiento continuo en el tiempo, se vuelve una funcin, que se obtiene a partir de la derivacin e igualacin a cero de la ecuacin (6), dando como resultado la funcin de la ecuacin (7). K(t) = −R −1 B T P (t)(7) Donde en este caso, P es la solucin de la ecuacin de Riccati [8], mostrada en la ecuacin (8). A T P + P A − P BR −1 B T P + Q = 0 (8) B. Control por modos deslizantes (SMC) para sistemas en tiempo-discreto El control por modos deslizantes (SMC) es un técnica de diseño de controladores, la cual apareció en el contexto de sistemas dinámicos de estructura variable en la década de los 70s, estos sistemas se caracterizan por la presencia de discontinuidades en sus dinámicas, lo cual hace que sus trayectorias y volumen de fase sean discontinuos en algunos puntos, o usando lenguaje matemático, sean sistemas Lipschitz de forma local. La idea tras los modos deslizantes es aprovechar estas discontinuidades para hacer que las trayectorias alcancen una variedad en el volumen de fase y se estabilice en esta zona, llevando los estados asintóticamente al origen. Las principales ventajas de esta técnica se fundamenta en: control y estabilización en presencia de incertidumbres acotadas, parámetros desconocidos del sistema, o dinámicas parasitas. El desarrollo de esta técnica ha sido basada en los trabajos e investigación de [21] con aplicaciones recientes. 0 A N (13,24) 1 A N 32 − gM L(M LR−2n 2 Jm) ∆ N 3A N 33 − 1 ∆ N 2 (β + fw) M L 2 + Jq 2 + 2n 2 Jm + β M LR − 2n 2 Jm A N 34 2β(M L 2 +Jq 2 +M LR) ∆ N A N 42 M gL(2mR 2 +M R 2 +2Jw +2n 2 Jm) ∆ N 3A N 43 1 ∆ N 2 (β + fw) (M LR − 2n 2 Jm+ β(2mR 2 + M R 2 + 2Jw+ 2n 2 Jm)) A N 44 − 2β(M LR+2mR 2 +M R 2 +2Jw ) ∆ N B N (11,12,21,22) 0 B N (31,32) α(M L 2 +Jq 2 +M LR) ∆ N B N (41,42) − α(M LR+2mR 2 +M R 2 +2Jw ) ∆ N ∆ N = (2m + M ) R 2 + 2J W + 2η 2 Jm M L 2 + Jq 2 + 2η 2 Jm − M LR − 2η 2 Jm 2 Para el diseo se trabaja con un equivalente discreto del sistema original basado en una técnica de muestreo y retención aplicando el retenedor de orden cero (ZOH), un espacio de estados discreto es establecido. De forma más simple, la aproximación es, desde el punto de vista del controlador, la forma como 'observa' el sistema dinámico. Para el diseño del controlador es importante definir dos momentos en la metodología, el primero consiste en la definición adecuada de una variedad estable, luego, se selecciona una ley de control en lazo cerrado que incluye una ganancia de realimentación y una función discontinua definida a partir de un análisis basado en el teorema de Lyapunov. Se tiene una representación en espacio de estados definida como: x 1(k+1) = A 11 x 1(k) + A 12 x 2(k) x 2(k+1) = A 21 x 1(k) + A 22 x 2(k) + u. Donde x 1(k) ∈ R n−m , x 2(k) ∈ R m ,A i,j es una matriz real de parámetros constantes. Entonces se define la variedad estable como: s k = Cx 1(k) + x 2(k) = 0 ⇒ x 2(k) = −Cx 1(k) . Donde C ∈ R m×(n−m) . Reemplazando esta ley de control en el sistema original del bloque superior, se obtiene: x 1(k+1) = (A 11 − A 12 C) x 1(k) . Haciendo una selección adecuada de los valores de la matriz C se pueden ajustar los autovalores de la matriz resultante A 11 − A 12 C dentro del círculo unitario. Una vez se ha considerado la selección de la variedad estable entonces se procede a determinar la ley de control por modos deslizantes como: u k = −ksign(s k ). Donde sign es la función signo y k es un valor a ser determinado basado en el análisis de estabilidad aplicando el segundo método de Lyapunov. Para ello sea V : R k → R una función que cumple con V (0) = 0 y V > 0 estrictamente mayor a 0. Su primera variacin a lo largo de las trayectorias esta dada por ∆V (x k ) = V (x k+1 ) − V (x k ) la cual cumple con ∆V (x k ) ≤ −T s αV (x k ) 1 2 α siendo real positivo y T s el tiempo de muestreo sea suficientemente pequeo, lo cual garantiza que el sistema alcance estabilidad en tiempo finito. Considerando la siguiente función de Lyapunov V = 1 2 s 2 k . Aplicando la primera variación ∆V = 1 2 s 2 k+1 − s 2 k . Para aplicar esta función de Lyapunov se tiene: s k = α 1 x 1(k) + α 2 x 1(k) + α 3 x 3(k) + x 4(k) = 0. En forma compacta s k = Lx k , donde L ∈ R 1×n y x k es el vector de estado. Aplicando el desplazamiento en k + 1 s k+1 = Kx + u. Donde K es un combinación de parámetros entre la variedad definida y la matriz de estados del sistema original. Juntando términos se obtiene: 1 2 x T K T Kx + 2(Kx + u k ) + u 2 k − x T k L T Lx k > 0. Considerando en este proceso que s k = 0, se puede inferir además que s k+1 = 0, y sustituyendo por la ley de control se tiene: u k = −Kx − k sign(s k ). Simplificando la ecuación y encontrando una cota para la selección de k: 1 √ 2 k 2 − 1 1 2 \|s k \| ≤ −T s α √ 2 \|s k \| k ≤ −T s α √ 2 2 + 1 1 2 α > 0 . Donde T s es el tiempo de muestreo, con esta condición se garantiza en el sistema su estabilidad en tiempo finito. IV. IMPLEMENTACIÓN El proceso de implementacin inicia a partir de las tcnicas diseadas, tomando en cuenta las restricciones hardware y software de las plataformas abordadas. Las restricciones software de cada plataforma se relacionan con los entornos de desarrollo. La plataforma Quanser utiliza el entorno de programacin grfico de Matlab, haciendo uso del Quarc Toolbox; mientras que la plataforma NxtWay utiliza el entorno de desarrollo RobotC de Mindstorms. As, una caracterstica propia en el ambiente de desarrollo Matlab, es la posibilidad de implementar algoritmos en tiempo continuo, donde el proceso de discretizacin se hace transparente en el desarrollo de aplicaciones que contienen modelos en Simulink. Caso contrario, ocurre en RobotC, donde la ley de control que se implementa debe ser discreta, dado que la programacin se realiza a bajo nivel en lenguaje C. Otra restriccin de software se encuentra en relacin al muestreo de las variables de estado, para el caso de RobotC, la nica manera de exportar datos del bloque controlador NXT, es a travs de su consola de texto, que se sincroniza con los procesos internos del bloque cada 10ms como mximo, tiempo que no siempre coincide con el tiempo de aplicacin. En Matlab por su parte, el muestro de las variables de estado, si se puede efectuar en tiempo de aplicacin, pero existe un limite de datos que se pueden capturar, dependiendo de los recursos fsicos como memoria, que se reservan para almacenar datos dentro de los indicadores grficos de Simulink. Por otro lado, en cuanto a las restricciones hardware donde se implementaron los algoritmos, se tiene que las frecuencias de procesamiento de las estructuras son diferentes, la frecuencia del bloque controlador Quanser, limitada por su tarjeta de adquisicin de datos, es el doble de la frecuencia de bloque NxtWay, por lo que la cantidad de datos capturados deben ser diferentes para realizar comparaciones en un tiempo de simulacin igual. A. Control LQR Para el controlador LQR se usan los siguientes valores en el diseo: Las matrices Q x y R x empleadas en ambas plataformas para el clculo del controlador son: Q Q =    5 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1    R Q = 1 Q N =      1 0 0 0 0 0 6x10 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4x10 2      R N = 1x10 3 0 0 1x10 3 Se observa en la ecuacin IV-A, que para el caso del NxtWay, se incluye una columna en los valores de Q y R, por el primer lazo de control utilizado, en donde se adiciona una quinta variable de estado definida por la integral de q 1 para garantizar la posicin de la plataforma dada su independencia del modelo dinmico. Los lazos de control se presentan en los diagramas de bloques de las figuras 3 y 4. La figura 3 muestra el diagrama de bloques de la plataforma RotPen, que toma la forma de un sistema de control realimentado LQR; por otro lado, la figura 4 expone el diagrama de bloques de la plataforma NxtWay; adicionalmente, se realiza una correccin de la posicin del mvil debido a los deslizamientos y asincronismos en el movimiento que puedan ocurrir cuando la energa de actuacin de la rueda virtual se deriva para cada uno de los dos actuadores fsicos, este bloque se denomina Steer. As mismo, se incluye un bloque de saturacin sobre la seal de actuacin, con un valor igual al mximo voltaje en operacin normal, para los motores de cada planta. Los valores se observan en las Tablas I y II. K N i = −0.4472(10) El comportamiento dinmico del modelo RotPen simulado e implementado se presenta en la figura 5 ante una perturbacin, que inicia despus de 60 segundos, la cual consiste en aadir un voltaje extra al actuador, proporcional al voltaje mximo permitido de cada plataforma. Para cada sistema, el punto de referencia se encuentra en 0 (Independientemente de las unidades), en la figura 5 se observa que el controlador estabiliza cada una de las variables de estado en su referencia, en un tiempo de 2 a 3 segundos aproximadamente; es decir, en el rango de 62 a 63 segundos como se muestra en las figuras. Asimismo, presenta un sobreimpulso menor con el modelo simulado que con el modelo implementado y una seal de actuacin menor a 3 voltios en escala de 0 a 10 v. De igual manera, se observa en la figura 6, la respuesta de la implementacin en el pndulo mvil NxtWay, al igual que en el RotPen, la referencia de todas las variables de estado se encuentra en 0 unidades, a diferencia de la respuesta de la planta de Quanser, las variables de estado poseen un grado de oscilacin, en especial aquellas relacionadas con las ruedas, al tener una dinmica independiente del polo. B. Control por modos deslizantes (SMC) Para la implementacin de la tcnica por modos deslizantes se consideran los siguiente parmetros de diseo para ambas plataformas. El vector L N y L Q están asociados a las variedades estable consideradas en la tcnica Estos valores deben garantizar que los autovalores se encuentre en el círculo unitario, una vez estos valores son obtenidos se considera la expresión III-B, para seleccionar el vector de realimentación K N y L Q , el cual garantice la desigualdad planteada. La respuesta de la implementacin en el pndulo mvil NxtWay, al igual que en los casos anteriores, plantea una referencia de todas las variables de estado en 0 unidades, a diferencia de la respuesta del control LQR, esta respuesta maneja una oscilacin en su seal de actuacin provocada por el uso de una ley de control discontinua o con scattering (Conmutacin de las dinmicas de control en las dos fases SMC) en el clculo de la salida del controlador, con los valores de las ecuaciones (11), (12) y (13), se obtienen los resultados de las variables de estado y seal de actuacin en la figura 7. De igual manera, la respuesta de implementacin del pndulo rotatorio RotPen, plantea un experimento con la referencia de nuevo en 0 unidades, esta respuesta se observa en la figura 8, al igual que el caso del NxtWay maneja un nivel de Scattering en sus seales, provocada por la implementacin propia del control, sin embargo al contar Quanser con una plataforma robusta para el procesamiento y discretizacin de las seales incluidas, su oscilacin no maneja valores tan altos como el NxtWay. Como datos adicionales a discutir en el proceso de implementacin, se destaca el uso de filtros derivadores en cada uno de los entornos, desarrollados a partir de la respuesta en alta frecuencia de los giroscopios y encoders de las plataformas, filtrado necesario debido al proceso de discretizacin de los algoritmos de lectura de las variables, adicional a la respuesta trmica de los elementos y al ruido electromagntico presente en el sistema; de igual manera, se destaca el manejo del tiempo de muestreo de cada algoritmo, que esta limitado por los entornos y es importante para las operaciones realizadas, como el clculo de las integrales o de las derivadas de las variables obtenidas. V. ANLISIS DE RESULTADOS A partir de la respuesta en lazo cerrado de cada uno de los sistemas controlados, se lleva a cabo una comparacin de rendimiento bajo ndices de calidad; Los ndices de validacin son: nivel de estabilizacin de la variable de estado asociada al polo, energa de la seal de control (Norma infinito de la seal), mxima velocidad de la variable asociada al polo, y consideraciones de sntesis de controladores cuando se procede al diseo. Evidenciando estos valores en las figuras 5, 6, 7 y En la tabla V se observan mtricas en las variables de relevancia del sistema para medir desempeo de los dos controladores en el mdulo NXT. Esta informacin puede ser usada para determinar un criterio de seleccin en aplicaciones futuras, por ejemplo, en cuanto al uso energtico es mejor usar un controlador LQR que uno basado en SMC, esto se puede inferir a partir de la consideracin de la mxima potencia aplicada en el servomotor y el criterio de energa usado en la sintetizacin del controlador. En la tabla VI, a diferencia del caso anterior, se tiene la comparacin de tcnicas en el sistema Quanser, mostrando un comportamiento mejorado por la suavidad de sus seales. Si se considera una seleccin basada en la robustez de la tcnica, es mejor seleccionar SMC, debido a que en la sintetizacin es posible incluir perturbaciones acotadas, lo cual reformula la desigualdad de Lyapunov, permitiendo recalcular las constantes de realimentacin y lograr estabilidad en tiempo finito. Es importante destacar que las grficas visualizadas no muestran una relacin directa entre implementacin y simulacin, debido a que en los clculos no se maneja la totalidad de las variables fsicas involucradas en el modelamiento de las plataformas; y as mismo, las variables involucradas, poseen un cierto grado de incertidumbre asociado. El uso de un controlador ptimo, permite mediante criterios de optimizacin, llevar los polos del sistema a una regin estable, sin llegar a seleccionarlos como ocurre con el control en modo deslizante, lo que permite tener un mejor proceso de diseo en el clculo de sus constantes; por otro lado, el control en modo deslizante permite incluir en sus dinmicas y metodologas de diseo, perturbaciones externas al sistema, sin afectar las constantes a obtener, a diferencia del control LQR, en donde el diseo se realiza unicamente sobre el modelo de estados de la planta, e incluir estas dinmicas, haran necesario reiniciar el proceso nuevamente, perdiendo robustez en su diseo. VI. CONCLUSIONES En este trabajo, fueron seleccionados dos estructuras pendulares disponibles para experimentacin. Inicialmente, se realiz el modelo dinmico de las plataformas usando las ecuaciones de Euler-Lagrange, cada modelo fue validado mediante el uso de herramientas computacionales; ambos sistemas tuvieron una respuesta similar en cuanto a la posicin del polo, la posicin del brazo por estar restringida fsicamente en la plataforma RotPen, describe una dinmica y respuesta diferente. A partir de los modelos y de su linealizacin, se desarrollaron controladores LQR y SMC, cada uno de estos fue probado en Simulink. Adems, se implementaron los controladores en los entornos de cada fabricante, el pndulo rotatorio RotPen se implement en la herramienta de Simulink de Matlab, mientras que el pndulo mvil NxtWay, mediante texto estructurado, se realiz en el software RobotC. En cada caso se aadi una perturbacin para obtener los resultados donde se midi la norma infinito de la seal de control, la calidad de la estabilizacin, y la norma infinito de la velocidad del polo. La perturbacin consisti en una seal sinusoidal de pulsos, con un voltaje pico equivalente a la mitad de la potencia de cada actuador a una frecuencia de 0.0167Hz. Esta estrategia permiti validar la calidad de funcionamiento del control; tomando en cuenta las restricciones de hardware y software que cada dispositivo posea; no obstante, se observa que sin importar estas restricciones, el diseo de un controlador ptimo y uno basado en modos deslizantes, son estrategias adecuadas para la estabilizacin de estructuras pendulares, evidencia que se puede observar en las grficas de funcionamiento de cada dispositivo con cada uno de los controladores empleados. A modo de comparacin, se observ un mejor rendimiento en el pndulo mvil NxtWay, por la simplicidad de su entorno de desarrollo para embeber los algoritmos de control, esto permiti una mayor velocidad en la ejecucin de una inferencia de control, lo que llev a tener una respuesta ms rpida ante perturbaciones, cuantificado en un segundo aproximadamente para la plataforma Rotpen, y en 1.5 segundos para la plataforma NxtWay; tomando en cuenta que la plataforma RotPen empleaba una frecuencia de procesamiento de 500 inferencias por segundo, mientras que la plataforma NxtWay presentaba una frecuencia de 250 inferencias por segundo. Un aspecto a tener en cuenta para la implementacin de controladores en esta clase de dispositivos, es la respuesta en alta frecuencia de los sensores, la cual afecta el clculo de las derivadas e integrales de las seales involucradas, al adicionar componentes de ruido que enmascaran la correcta lectura de las variables de estado del sistema; por esta razn se hace necesario el diseo de filtros pasa-bajos que eliminen estas componentes y permitan una lectura adecuada de las variables. TRABAJO FUTURO Como trabajo futuro, se plantea la identificacin de los sistemas controlados, con el fin de implementar una tcnica de control no lineal, que aumente la regin de operacin de los dispositivos, sin disminuir su rendimiento ni incrementar su gasto computacional; la tcnica de identificacin seleccionada es la tcnica difusa Takagi-Sugeno, la cual se ajusta a una propuesta de controlador no lineal que acta como mltiples controladores en paralelo ponderados no linealmente [22]. Esta identificacin se comparar con la versin no lineal del controlador SMC. AGRADECIMIENTOS Los autores agradecen al laboratorio de la facultad de Ingeniera en Automatizacin de La universidad de La Salle por apoyo tcnico y logstico, a la Vicerrectora de Investigacin y Transferencia Salle, y a Colciencias por financiar este proyecto a travs de las convocatorias VRIT 2432 y Jovnes Investigadores e Innovadores Colciencias 761 de 2016. Se hace un reconocimiento adicional nuevamente a Colciencias, por la beca doctoral Francisco Jos de Caldas Generacin Bicentenario de 2009 otorgada a C. H. Rodriguez-Garavito, de donde se deriv este proyecto de investigacin. Fig. 1 . 1Esquema del pndulo rotatorio invertido de Quanser, la M mostrada en ambas figuras corresponde al servomotor empleado. Fig. 2 . 2Esquema del pndulo invertido mvil NxtWay Fig. 4 . 4Diagrama de bloques control LQR Pndulo NxtWayDe la deduccin de las ecuaciones (6) y (8) se obtienen las constantes de realimentacin de control de los sistemas, estas se observan en la ecuacin (9) para Quanser y (10) para NxtWay.K Q = [−2.2361 25.4512 − 2.4613 3.6332] (9) K N = [−0.8211 − 69.4743 − 1.0739 − 9.0738] L Q = [1155.5 − 4.161 − 68.3] Fig. 5 . 5Respuesta control LQR Pndulo RotPen Fig. 6 . 6Respuesta control LQR Pndulo NxtWay 8; se observa que ambas plataformas logran la estabilizacin de sus estructuras ante una perturbacin. Para resumir algunos resultados de importancia para la aplicacin, se proponen las siguientes tablas resumen. Fig. 7 . 7Respuesta control SMC Pndulo NxtWay Fig. 8 . 8Respuesta control SMC Pndulo Quanser TABLE I PARMETROS IFSICOS ROTPEN. [3] Parmetro Und. Descripcin g=9.81 m s 2 Gravedad mp=0.127 kg Masa del pndulo Lp=0.337 m Longitud total del pndulo Jp=0.0012 kgm 2 Inercia del pndulo mr=0.257 kg Masa del brazo rotatorio Lr=0.216 m Longitud del brazo rotatorio Jr=9.98x10 −3 kgm 2 Inercia del brazo fp=0.0024 N ms rad friccin brazo -pndulo fr=0.0024 N ms rad Friccin motor -brazo Rm=2.6 Ω Resistencia del motor CC Lm=0.18 mH Inductancia del motor CC Km=0.00767 V s rad Constante Contra electromotriz motor CC Kt=0.00767 N m A Constante torque del motor ηg=0.9 Eficiencia caja reductora ηm=0.69 Eficiencia del motor Kenc=4096 conteos rev Resolucin encoder Kg=70 Relacin caja de engranajes γ = Rm K t Kg ηg ηm Constante del motor Vm=6 V Voltaje nominal del motor TABLE II PARMETROS IIFSICOS NXTWAY.[20] Parmetro Und. Descripcin g=9.81 m s 2 Gravedad m=0.03 kg Peso de la rueda R=0.02 m Radio de la rueda J W = mR 2 2 kgm 2 Inercia de la rueda M=0.6 kg Peso del cuerpo W=0.14 m Ancho del cuerpo D=0.04 m Profundidad del cuerpo H=0.27 m Altura del cuerpo L=0.12 m Distancia centro masa eje rueda TABLE III PARMETROS IIIMATRICES DE ESTADO PLATAFORMA ROTPENTrmino Definicin A Q(11,12,14,21−23,31,41) TABLE IV PARMETROS IVMATRICES DE ESTADO PLATAFORMA NXTWAYTrmino Definicin A N (11,12,14,21,22,23,31,41) TABLE V VDESEMPEO NXT Estado q 2 Nivel Estabilizacin Potencia% Potencia Mximaq 2 Velocidad Máxima Criterio Energa Mnima Robustez LQR Acotado(Suave) 5% 1.6 Si No SMC Acotado (Scattering) 27% 1.2 No Si TABLE VI DESEMPEO VIQUANSEREstado q2 Nivel Estabilizacin Potencia (V) Potencia Mximaq 2 Velocidad Maxima Criterio Energa Mnima Robustez LQR Acotado (Suave) 3.6 1.5 Si No SMC Acotado (Bajo Scattering) 10 3.6 No Si de los controladores implementados, es necesaria la linealizacin de los sistemas en un punto de operacin estable, esta linealizacin es llevada a cabo mediante series de Taylor; y su resultado toma la forma de un modelo en variable de estado, como se muestra en la ecuacin (4) y(5). 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[ "Transport Induced by Mean-Eddy Interaction: II. Analysis of Transport Processes", "Transport Induced by Mean-Eddy Interaction: II. Analysis of Transport Processes" ]	[ "Kayo Ide \nDepartment of Atmospheric and Oceanic Science\nCenter for Scientific Computation and Mathematical Modeling\nInstitute for Physical Science and Technology, & Earth System Science Interdisciplinary Center\nUniversity of Maryland\nCollege ParkUSA\n", "Stephen Wiggins \nSchool of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK\n", "Stephen Wiggins ", ") " ]	[ "Department of Atmospheric and Oceanic Science\nCenter for Scientific Computation and Mathematical Modeling\nInstitute for Physical Science and Technology, & Earth System Science Interdisciplinary Center\nUniversity of Maryland\nCollege ParkUSA", "School of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK" ]	[]	We present a framework for the analysis of transport processes resulting from the mean-eddy interaction in a flow. The framework is based on the Transport Induced by the Mean-Eddy Interaction (TIME) method presented in a companion paper[1]. The TIME method estimates the (Lagrangian) transport across stationary (Eulerian) boundaries defined by chosen streamlines of the mean flow. Our framework proceeds after first carrying out a sequence of preparatory steps that link the flow dynamics to the transport processes. This includes the construction of the so-called "instantaneous flux" as the Hovmöller diagram. Transport processes are studied by linking the signals of the instantaneous flux field to the dynamical variability of the flow. This linkage also reveals how the variability of the flow contributes to the transport. The spatio-temporal analysis of the flux diagram can be used to assess the efficiency of the variability in transport processes. We apply the method to the double-gyre ocean circulation model in the situation where the Rossby-wave mode dominates the dynamic variability. The spatio-temporal analysis shows that the inter-gyre transport is controlled by the circulating eddy vortices in the fast eastward jet region, whereas the basin-scale Rossby waves have very little impact.	10.1016/j.cnsns.2014.06.019	[ "https://arxiv.org/pdf/1107.5182v1.pdf" ]	119,247,176	1107.5182	3ce26e42aeb726c3c81029178ce87ab6c29ee8bf	Transport Induced by Mean-Eddy Interaction: II. Analysis of Transport Processes 26 Jul 2011 Kayo Ide Department of Atmospheric and Oceanic Science Center for Scientific Computation and Mathematical Modeling Institute for Physical Science and Technology, & Earth System Science Interdisciplinary Center University of Maryland College ParkUSA Stephen Wiggins School of Mathematics University of Bristol BS8 1TWBristolUK Stephen Wiggins ) Transport Induced by Mean-Eddy Interaction: II. Analysis of Transport Processes 26 Jul 2011Preprint submitted to Elsevier 27 July 2011Email addresses: [email protected] (Kayo Ide), [email protected] (Stephen Wiggins). URLs: http://www.atmos.umd.edu/ ide (Kayo Ide),Eulerian TransportLagrangian TransportMean-Eddy InteractionDynamical Systems ApproachWind-Driven Ocean Circulation We present a framework for the analysis of transport processes resulting from the mean-eddy interaction in a flow. The framework is based on the Transport Induced by the Mean-Eddy Interaction (TIME) method presented in a companion paper[1]. The TIME method estimates the (Lagrangian) transport across stationary (Eulerian) boundaries defined by chosen streamlines of the mean flow. Our framework proceeds after first carrying out a sequence of preparatory steps that link the flow dynamics to the transport processes. This includes the construction of the so-called "instantaneous flux" as the Hovmöller diagram. Transport processes are studied by linking the signals of the instantaneous flux field to the dynamical variability of the flow. This linkage also reveals how the variability of the flow contributes to the transport. The spatio-temporal analysis of the flux diagram can be used to assess the efficiency of the variability in transport processes. We apply the method to the double-gyre ocean circulation model in the situation where the Rossby-wave mode dominates the dynamic variability. The spatio-temporal analysis shows that the inter-gyre transport is controlled by the circulating eddy vortices in the fast eastward jet region, whereas the basin-scale Rossby waves have very little impact. 1 Introduction Analysis of geophysical flows often employs techniques that decompose the velocity field in a manner that will yield a desired insight [2]. A commonly used technique is the mean-eddy decomposition u(x, t) = u(x) + u (x, t) , (1a) Q(x, t) = Q(x) + Q (x, t) ,(1b) where the field is described by the unsteady eddy activity around a mean state; henceforth, {·} and {·} denote the time average (mean) and the residual (unsteadiness or eddy), respectively. Here u = (u 1 , u 2 ) T is the velocity field and Q represents any property of the flow such as temperature, chemical or biological properties. In the absence of unsteadiness, the kinematic transport occurs only along the mean streamlines on which u(x) is everywhere tangent. An effect of the unsteadiness is to stir the flow instantaneously and induce mixing across the mean streamlines over time. It is also well accepted that instantaneous pictures of the unsteady flow themselves do not indicate transport explicitly. A variety of Eulerian and Lagrangian methods have been developed to study transport observationally, analytically, and numerically. The companion paper [1] presented a new transport method that is a hybrid combination of Lagrangian and Eulerian methods. This paper develops a framework for the analysis and diagnosis of transport processes based on this new method. Our main focus is on two-dimensional geophysical flows that may be compressible. To introduce our method, we begin with a brief discussion of Lagrangian and Eulerian methods specific to our needs. The basis of any Lagrangian method involves the tracking of individual fluid particles by solving the initial value problem of d dt x = u(x, t). Starting from x 0 at time t 0 , a particle trajectory x(t; x 0 , t 0 ) at time t is given by the temporal integral of the local velocity field along itself: x(t; x 0 , t 0 ) − x 0 = t t 0 u(x(τ ; x 0 , t 0 ), τ )dτ .(2) A very rudimentary description of Lagrangian transport may be obtained from a so-called "spaghetti diagram" which is constructed by simply plotting the trajectories. Typically this results in a complex tangle of curves from which detailed a detailed assessment of Lagrangian transport may prove difficult. In recent years the mathematical theory of dynamical systems has provided a new point of view and tools for classifying, organizing, and analyzing detailed and complex trajectory information by providing a theoretical and computational framework for an understanding of the geometric properties of "flow structures". Recent reviews of the dynamical systems approach to transport are given in [3,4,5]. Nevertheless, there still remain many challenging problems to be tackled by Lagrangian methods. Quantifying Lagrangian transport is extremely elaborate in general. While techniques based on dynamical systems theory are conceptually ideal for tracking transport of fluid particles, they have not proven as useful for studying transport of Q, unless Q is uniform and passive. Moreover, Lagrangian methods are not suitable for separating and/or isolating the roles played by the mean state u(x) and the unsteadiness u (x, t) in the trajectory x(t; x 0 , t 0 ). In contrast to Lagrangian methods, transport quantities computed with Eulerian methods utilize information taken at pre-selected stationary points. At a station x E , the most basic Eulerian transport may be given by the temporal integral of the local velocity during a time interval [t 0 , t 1 ]: t 1 t 0 u(x E , t)dt = (t 0 − t 1 )u(x E ) .(3) The resulting transport is associated with u(x), but not u (x, t) by default [compare with (2) for the Lagrangian case]. For transport across a stationary Eulerian curve E = {x E (p)} where p is a parameter along E, the total transport during [t 0 , t 1 ] over a spatial segment [p A , p B ] is t 1 t 0 p B p A d dp x E (p) ∧ u(x E (p), t) dpdt = (t 1 − t 0 ) p B p A d dp x E (p) ∧ u(x E (p)) dp .(4) An advantage of the Eulerian methods is the ability to compute the transport of Q as well, by replacing u(x, t) with Q(x, t)u(x, t). Overtime, the Eulerian methods gives Q(x, t)u(x, t) = Q(x)u(x)+Q (x, t)u (x, t). Hence, the Eulerian methods account for the statistical contribution to transport at the secondorder. Our new transport method [1] has the unique ability to identify the effects of the mean-eddy interaction in a way that neither Lagrangian nor Eulerian methods have accomplished. This advantage that comes from blending of the Lagrangian and Eulerian approaches. The method uses information on a stationary (Eulerian) boundary curve C to estimate (Lagrangian) transport of both fluid particles and Q across C without requiring tracking of individual fluid particles. The method estimates the transport by integrating the instantaneous effects of the unsteady flux while taking the particle advection of the mean flow into account. We refer to our method that quantifies the Transport Induced by the Mean-Eddy interaction as "TIME." By construction, the TIME method offers a framework for a detailed analysis of the spatio-temporal structure of transport processes. The goal of this paper is to present this framework through a study of the inter-gyre transport processes in a wind driven, three-layer quasi-geostrophic ocean model ( Figure 1). Due to its relevance to the mid-latitude ocean circulation, the dynamics of wind-driven double-gyre ocean models have been actively studied from various points of view over the last few decades (for review, see [6] and references therein). [ Fig.1] Although the details of transport are highly dependent on the dynamics of the flows, there are five common preparatory steps for the analysis using the TIME method. The initial two steps of the five concern obtaining an understanding of the flow dynamics based on the mean-eddy decomposition (1). The first step is to examine the global flow structure given by the mean flow u(x), which we call the reference state ( Figure 1a for the wind-driven ocean; see Section 2.1 for the details). The second step is to understand the nature of the unsteady eddy activity in u (x, t). Unsteady eddy activity is also referred to as the variability. In geophysical flows, variability is often associated with the temporal evolution of the spatially coherent structures (Figure 1b; see Section 2.2 for the details). It should be clear that these coherent structures are defined in the instantaneous Eulerian field and different from the so-called "Lagrangian coherent structures" [7]. With understanding and insights of the mean and the variability at hand, the third step is to compute the the instantaneous flux that stirs the flow. The instantaneous flux is expressed naturally in terms of a "mean-eddy interaction". ( Figure 1c for the wind-driven ocean; see Section 2.3). The fourth step is to select the Eulerian boundary, C, of interest based on the mean flow structure. Any mean streamline with reasonable length can be a potential C. The actual choice of C should be left up to the specific geophysical interests . In Figure 1a we show our choice of C for the intergyre transport in the wind-driven ocean; which we discuss further in Section 2.4). The last step is to extract the information of the instantaneous flux on C. The signals of the variability in the mean-eddy interaction are conveniently represented by a space-time diagram (i.e., the Hovmöller diagram), which we call the flux diagram to emphasize its role in the TIME method (see Section 2.5). An important outcome of the five preparatory steps is that they link the flow dynamics to the transport processes, and vice versa, in terms of the meaneddy interaction. This link, and the spatio-temporal integration employed by the TIME method, comprise the foundation for the graphical approach to the study of the transport processes (see Section 3). In the double-gyre application, the analysis will reveal how/when/where the circulating eddy vortices in a localized area over the eastward jet are responsible for the inter-gyre transport, whereas the basin-scale Rossby waves play a very small role (see Section 4). We use the same flow field as in [1] that is nearly periodic in time and has a heteroclinic connection in the mean. It is worth noting that the TIME method does not require either of such conditions (i.e., presence of the time periodicity and the heteroclinic connection). The outline of the paper is as follows. Section 2 presents the preparatory steps using the application to the inter-gyre transport in the double-gyre winddriven ocean circulation. We extend the TIME functions defined by [1] and present a graphical approach that facilitates the analysis of transport processes in Section 3. Inter-gyre transport processes in the double-gyre ocean are analyzed in Section 4. Section 5 summarizes the results and provides a discussion. Building the links between variability and transport In this section we define the five preparatory steps mentioned in the previous section in detail and carry them out in the context of the analysis of inter-gyre transport in a wind driven double-gyre ocean circulation model. The flow field is obtained by numerical simulation of a three-layer quasi-geostrophic model with the model parameters chosen to be consistent with the mid-latitude, wind-driven ocean circulations [6,8]. As a result of a constant wind-stress curl 0.165dyn/cm 2 applied at the ocean surface, the basin-scale circulation fluctuates almost periodically around the mean state with dominant spectral peak at period T ≈ 151days after the initial 30,000-day spin-up from the rest. For our analysis time starts after this spin-up. In this study, we analyze the inter-gyre transport processes in the top layer. The instantaneous flow patterns are given by the streamfunction ψ(x, t) that is related to the velocity u(x, t) by u(x, t) = (− ∂ ∂y , ∂ ∂x )ψ(x, t). Mean flow Given u(x), the corresponding streamfunction ψ(x) is the reference state that provides a geometrical structure of the global flow. Figure 1a shows ψ(x) for the double-gyre circulation in which the axis of the eastward jet divides the ocean basin into two gyres. Because the ocean model has gone through the pitchfork bifurcation prior to the Hopf bifurcation [6] (in terms of increasing wind-stress curl), an asymmetry exists between the cyclonic subpolar gyre and the anticyclonic subtropical gyre. At the confluence of the southward and northward western boundary currents, the cyclonic subpolar gyre and anti-cyclonic subtropical gyre form an asymmetric dipole (along x J and x N indicated by the diamonds in Figure 1a). We refer to this region as the "dipole region." The eastward jet defined between the center of the subpolar vortex and that of the subtropical gyre and carries the mean net transport ψ N T ≡ \|ψ sp − ψ st \| = 24028, where ψ sp (< 0) is the minimum ψ(x) over the subpolar vortex and ψ st (> 0) is the maximum over the subtropical vortex. The asymmetry of the two gyres is measured by the transport difference, [9] ψ T D ≡ \|ψ sp \| − \|ψ tr \| = 5438. Dynamic variability of the flow in u (x, t) An understanding of the eddy activity (variability) in u (x, t) is the key for the analysis of transport processes using the TIME method. In the double-gyre application, the model ocean dynamics is almost periodic in time resulting from the Rossby-wave mode [6,8]. In the left panel of Figure 2, the evolution of the ocean dynamics is shown by two time series. The net transport of the jet N T (t) ≡ \|ψ sp (t) − ψ st (t)\|/ψ N T normalized by the mean state net transport ψ N T indicates the fluctuation of the jet strength around the mean. The transport difference T D(t) = (\|ψ sp (t)\| − \|ψ st (t)\|)/ψ T D normalized by ψ T D measures the fluctuation of the asymmetry between the subpolar and subtropical gyres around the mean. Here ψ sp (t) is the minimum ψ(x, t) over the subpolar gyre and ψ st (t) is the maximum over the subtropical gyre. The two time series show that the amplitude of the fluctuation is of the order smaller than 0.1 with respect to the mean. This fact is important since the validity of the TIME functions requires the fluctuations to be small compared to the mean. For convenience, we define the period of the k-th ocean oscillation by T [k] = [t * 38 + (k − 1)T, t * 38 + kT ) starting from t * 38 when a minimum of N T (t) occurs first time after the spin-up; the number in the subscript of t * represents time in day from here on. We also define T [ Fig.2] The eddy streamfunction field ψ (x, t) associated with u (x, t) provides an instantaneous pattern of the eddy activity. In the double-gyre application, ψ (x, t) shows two types of eddy activity (Figure 1b). One is the westward propagation of Rossby waves, whose latitudinally elongated structures are especially visible in the eastern basin. Typically there are three waves in the en-tire basin, although the one in the western basin is distorted around the dipole region. The travel time of the wave from the eastern to the western boundary is 453(=151×3)days. Each wave has the longitudinal width 333(=1000/3)km and travels with the propagation phase speed 2.2(=1000/453)km/day. As mentioned briefly in the introduction, for our study of inter-gyre transport in the wind-driven ocean, the axis of the eastward jet is chosen to be C. It is the separatrix that connects the western boundary to the eastern boundary. We will discuss this in more detail in Section 2.4. Although the westward propagation of Rossby waves is seen in almost the entire ocean basin, the dipole region has the eddy activity that is more energetic. The left panels of Figure 3 show the phases of these eddy vortices during T [7] that starts at t * 944 . At t * 944 + T /4(= t * 982 ) when T D(t) is minimum (corresponding to Figure 1b), a strongly positive eddy vortex is located near x 1 . This eddy vortex weakens as the center moves from x 1 along C as shown at t * 944 + T /2(= t * 1020 ) when N T (t) is maximum. It intensifies once again as the center reaches x 2 as shown at t * 944 + 3T /4(= t * 1058 ) when T D(t) is maximum. It moves further along C as shown at t * 944 for T later (t * 944 + T = t * 1095 ), until the center leaves C near x N as shown at t * 982 for 5T /4 later (t * 944 + 5T /4(= t * 1131 )). And it continues to make a cyclic rotation around the sub-polar vortex. One cyclic rotation takes 2T . The negative eddy vortex located in the west of this positive eddy vortex at t * 944 + T /4 follows the same cyclic motion but T /2 behind with the opposite phase of N T (t) and T D(t) with respect to the positive eddy vortex. [ Fig.3] This cyclic rotation of the eddy vortices is far from uniform because the eddy vortices tend to pulsate, i.e., when they intensify during T [k. 1] or T [k.2] (e.g., t * 944 + T /4 or t * 944 + 3T /4 in Figure 3), the centers hardly move; when they weaken at the end of T [k. 1] or T [k.2] (e.g., t * 944 or t * 944 + T /2), the centers move very quickly. As we shall see below, this complexity in the flow dynamics strongly influences the inter-gyre transport processes. We emphasize that these eddy vortices are not Lagrangian, i.e., particles don't move with them. These two types of eddy activity, westward propagation of the Rossby wave and cyclonic circulation of the eddy vortices, are synchronized. In particular they merge in the eastern part of the dipole region. At t * 944 + T /4, a negative eddy vortex connects to a negative half of the Rossby wave around x 3 as shown in Figures 1 and 3. This merger occurs every T /2 and is associated with the alternating sign of ψ (x, t). Flux variability of φ(x, t) The instantaneous flux across the mean flow φ(x, t) ≡ u(x) ∧ u (x, t) (5) is explicitly defined in terms of the "mean-eddy interaction" induced by the instantaneous spatial interaction of u(x) and u (x, t). Geometrically, φ(x, t) is the signed area of the parallelogram defined by u(x) and u (x, t) with the unit of φ(x, t) being flux per unit time over the length \|u(x)\|. The amplitude of φ(x, t) depends not only \|u(x)\| and \|u (x, t)\|, but also on the angle between u(x) and u (x, t). We refer to the coherent structures associated with φ(x, t) as the flux zones. In the double-gyre application, the two types of eddy activity in u (x, t) lead to distinct evolution of flux zones in φ(x, t). In the eastern basin where the Rossby waves dominate the variability, the direction change of u(x) in both gyres breaks the wave structure latitudinally into three flux zones with alternating signs (Figure 1c). Near the jet axis, a sequence of the Rossby waves lead to a sequence of flux zones with alternating signs; the positive flux zones correspond to northward flux, while the negative ones correspond to the southward flux. The width and the westward propagation speed of these flux zones are the same as those of the Rossby waves in ψ (x, t). In the dipole region where u(x) is stronger and non-uniform while u (x, t) consists of the cyclic rotation of the four circulating eddy vortices, the meaneddy interaction is complex ( Figure 3). Most significantly, the eddy vortices along the jet lead to the flux zones with the alternating signs that are visible particularly where \|u(x C (s))\| is large along the mean jet axis. These flux zones in φ(x, t) pulsate in sync with the eddy vortices in ψ (x, t) (compare the right panels of Figure 3 with the left panels). The shapes of the flux zones are more loosely defined than those of the eddy vortices due to the spatially nonlinear interference by u(x). As the circulating eddy vortices intensify when T D(t) is minimum (t * 944 + T /4) or maximum (t * 944 + 3T /4), three relatively well-defined flux zones are visible over [x J , x 1 ], [x 1 , x 2 ], and [x 2 , x N ] between the eddy vortex centers with the middle one stronger than the other two. In between consecutive intensifications, the flux zones weaken and quickly propagate along C as N T (t) reaches minimum (t * 944 ) or maximum (t * 944 +T /2). The propagation speed and direction of the three flux zones are the same as those of the circulating eddy vortices. As the eddy vortex leaves the mean jet axis near at x N , a weak fourth flux zone is generated over [x N , x 3 ] that propagate towards the upstream of the mean flow. Because the eddy vortices and the Rossby waves merge around x 3 over [x N , x S ], the flux zones associated with them also merge there. Boundary curve C and parameterization of the reference trajectory The TIME method uses the streamlines associated with u(x) as the Eulerian boundaries C across which the transport is estimated. Along C, the flighttime s is a natural choice of the coordinate variable because C = {x C (s)} is obtained by solving for d ds x C (s) = u(x C (s)) with a choice of initial position x C (0). Particle advection along C in the mean flow is referred to as the reference trajectory. Starting from x C (s 0 ) at t 0 , the reference trajectory is uniquely parameterized by s 0 − t 0 and can be written as (s, t) = (s 0 − t 0 + t, t) using the flight-time coordinate. Some key locations on C are shown in Figure 1. In ψ(x), x J is the location that is far enough from the hyperbolic stagnation point of C on the western boundary point in s so that u(x J ) becomes non-negligible to induce the instantaneous flux; x N and x S are the locations where the (meandering) jet axis makes sharp turns. In ψ (x, t), x 1 and x 2 are the locations where the centers of the eddy vortices pause to intensify; x N is where the eddy centers leave C; around x 3 , two types of variability, the circulating eddy vortices in the upstream and the Rossby waves in the downstream, meet on C. Accordingly, x J , x 1 , x 2 , x N and x 3 are the boundary points of the flux zones along C in φ (x, t) as the eddy vortices pause to intensify and meet the Rossby waves in φ(x, t). The flight-time coordinates of x J , x 1 , x 2 , x N , x 3 , and x S are s J = 110, s 1 = 114.5, s 2 = 118.5, s N = 129, s 3 = 150, and s S = 174.5 in the unit of days, respectively, starting with x C (0) = (2×10 −38 km, 1011.8km) located very close to the western boundary. Flux diagram The Hovmöller diagram [10,11] of the instantaneous flux in the (s, t) space: µ C (s, t) ≡ φ(x C (s), t) = u(x C (s)) ∧ u (x C (s), t)(6) is fundamental to the TIME method because it contains the stirring information locally and instantaneously extracted from φ(x, t) along C. We refer to it as the flux diagram. At a given instance t, a continuous segment of s with µ C (s, t) > 0 corresponds to a positive flux zone where the instantaneous flux goes from the right to the left across C with respect to the direction of increasing s. The direction of the flux is reversed for µ C (s, t) < 0. The reference trajectory (s, t) = (s 0 − t 0 + t, t) is a diagonal line going through (s 0 , t 0 ) with the unit slope (see the main panel of Figure 2). The nature of the signals in µ C (s, t) is dependent on both the system and the choice of C. These signals can be complex, as we shall observe in the doublegyre application ( Figure 2). Nonetheless, having systematically examined the mean u(x) (in terms of ψ(x); Section 2.1), dynamic variability u (x, t) (in terms of ψ (x, t); Section 2.2), flux variability φ(x, t) (Section 2.3), and the geographic location of C in the flow field (Section 2.4), the physical interpretation of the signals in µ C (s, t) is straightforward. Any signals can be traced back to certain flux zones in φ(x, t), and hence the mean-eddy interaction process between u(x) and u (x, t). By the construction of the flux diagram along C, the propagation speed and direction of the signals in µ C (s, t) are defined with respect to the particle advection along the reference trajectory of the mean flow. Signals with positive slopes correspond to the downstream propagation of the coherent structures in ψ (x, t). In contrast, signals with negative slopes are related to the upstream propagation of the coherent structures. If the slope is steeper than 1, then the propagation speed is slower than the particle advection along C in ψ(x). In the double-gyre application, µ C (s, t) is almost periodic with period T (Figure 2), i.e., µ C (s, t) = µ C (s, t + T ), due to the periodic dynamics in ψ (x, t). Within one period, the positive and negative phases are almost antisymmetric, i.e., µ C (s, t) ≈ −µ C (s, t + T /2). For s < s J , µ C (s, t) is very small because of near-zero u(x C (s)) and changes the sign in synchrony with T C (s, t). The slope is negative over S d because of the upstream propagation of the circulating eddies as their centers leave C around x N (Figure 3). The westward propagation of the Rossby waves is seen for s > s 3 . The slope is negative and less than 1 because the Rossby waves propagate against the mean flow with the propagation speed faster than the particle advection along C. Magnitude of µ C (s, t) rapidly decays to zero for s > 500 because of extremely small \|u(x C (s))\| and \|u (x C (s), t)\| there (not shown). The change in the propagation slope over s ∈ [200day, 240day] is mainly caused by the meander of C (Figure 1). Over S e ≡ [s S , s N ), the signals of the Rossby waves and those of the circulating eddy vortices are mixed because the two types of variability merge between x N and x S in ψ (x, t) (Sections 2.2 and 2.3). Accordingly, the (s, t) space can be divided into sub-domains based on the types of the mean-eddy interaction. Table 1 3 The TIME functions and the graphical approach to the analysis of transport processes The companion paper [1] introduced the TIME functions. The main focus in [1] was the mathematical formulation, the validity of perturbation approximations, as well as the verification of the method by comparing with the Lagrangian method based on the lobe dynamics for the case of C chosen as a heteroclinic connection of the reference state. This section refines the TIME functions for a much more detailed analysis of the transport processes due to the variability in the flow. We note that although the TIME functions as developed in [1] are based on a perturbation approach in the sense that the fluctuating part of the velocity field is "small" compared to the reference state, none of the development up to this point in the paper requires this smallness requirement-all that has been required is the decomposition of the velocity field into a (steady) reference state and a fluctuation about the reference state. The accumulation function The accumulation function along C is defined as m C (s, t; t 0 : t 1 ) = t 1 t 0 µ C (s − t + τ, τ ) dτ .(7) The left-hand side of (7) denotes the amount of fluid transport that occurs during the accumulation time interval [t 0 , t 1 ] evaluated at (s, t). The right-hand side of (7) expresses the transport in terms of the spatio-temporal integration of µ C (s−t+τ, τ ) during t 0 ≤ τ ≤ t 1 . Thus it can be thought that accumulation is advected, while it may continue to occur, with the reference trajectory going through (s, t). The sign of m C (s, t; t 0 : t 1 ) corresponds to the direction of transport across C; m C (s, t; t 0 : t 1 ) > 0 is from right to left across C; the direction of the flux is reversed for m C (s, t; t 0 : t 1 ) > 0. The basic idea of (7) is that the transport can be estimated just for the time period of interest [t 0 , t 1 ]. It can be extended to study "when", "where", and how" variability of the flow contributes to the transport. This is done by restricting the integration in (7) to a specific space-time domain D that contains the particular signals of interest (Section 2.5). Formally and practically this leads to the slight modification to the accumulation function: m C (s, t; D) = H(s − t + τ, τ ; D)µ C (s − t + τ, τ )dτ ,(8a) where Table 1 for the definition of D(t)). H(s − t + τ, τ ; D) =      1 if (s − t + τ, τ ) ∈ D, For the analysis of transport processes, there are properties of m C (s, t; D) that are useful (see [1] for technical details). One is the invariance property along the individual reference trajectory by advection of the accumulation: m C (s, t; D) = m C (s + δ, t + δ; D)(9) for any δ, but with D fixed. The invariance property implies a conservation law of the accumulation by the advection. At any (s, t) along a reference trajectory, m C (s, t; D) is independent of t as long as D is independent of t. The other useful property is the (piece-wise) independence property of D. By breaking up the domain D into L non-overlapping sub-domains D 1 , . . . , D L , the piece-wise independence property implies that the accumulation function can be written as For example, in the double-gyre application, the effect of the circulating eddy vortices and that of the Rossby wave propagation can be examined separately, while the overall transport is the sum of the two, i.e., m C (s, t; D all ) = m C (s, t; D cv ) + m C (s, t; D rw ). Transient transport can be also decomposed into the sub-domains as long as t is the same for all, e.g., m C (s, t; D all (t)) = k m C (s, t; D [k] (t)). A key feature of the TIME method is the graphical approach using the Hovmöller diagram of µ C (s, t). By definition (8), transport processes are described by the manner in which the reference trajectory passes through the signals in µ C (s, t) over the specific domain of interest D (see Figure2). Conversely, it discloses the dynamical origins of m C (s, t; D) by relating the signals in µ C (s, t) to dynamic activities in u (x, t) through φ(x, t) (Section 2). Thus, the graphical approach unveils the dynamical processes of the mean-eddy interaction that are responsible for transport. Moreover, the graphical approach can assess the efficiency of the dynamic activity u (x, t) in contributing to m C (s, t; D). The accumulation is, in general, extremely efficient if a signal in µ C (s, t) propagates with the same unit slope because it keeps accumulating the same signed instantaneous flux. In other words, transport processes are the most efficient if the (Eulerian) eddy activity in ψ (x, t) propagates at the same speed as the (Lagrangian) particle advection of the mean flow ψ(x) along C. In contrast, if the eddy activity is temporally periodic and propagates upstream of the mean flow, then the net contribution to the accumulation adds up to zero. This is because the reference trajectory cuts across the signals of µ C (s, t) whose sign alternates periodically, resulting in the cancellation of the accumulation. The displacement function The second type of TIME function quantifies the geometry associated with transport, up to the leading-order in the "size" of the unsteady part of the velocity field. To describe the geometry in the two-dimensional flow, we use an orthogonal coordinate set (l, r) for x near C, where the arc-length coordinate l = l C (s) of x C (s) is defined along C by d ds l C (s) = \|u(x C (s))\|. By taking the normal projection x C (s) of x onto C, the signed distance coordinate r = r C (l C (s)) is defined by r C (l C (s)) ≡ (u(x C (s))/\|u(x C (s))\|)∧(x−x C (s)), where \|u(x C (s))\| = 0. If r C (l C (s)) > 0, then x lies in the left side of C with respect to the direction defined by the direction of increasing s. The side of x is reversed for r C (l C (s)) < 0. The displacement distance function is defined as r C (l C (s), t; D) = a C (s, t; D) \|u(x C (s))\| , (11a) where a C (s, t; D) = H(s − t + τ, τ ; D)e C (s − t + τ : s)µ C (s − t + τ, τ )dτ : (11b) is the displacement area per unit s and H is defined in (8b). Compressibility of the flow is taken into account by e C (s − t + τ : s) = exp s s−t+τ D x u(x C (σ))dσ ,(11c) so that the flux that has occurred at (s − t + τ, τ ) may expand or compress as it advects to (s, t) with the reference particle advection. If the mean flow is incompressible as in the quasi-geostrophic model used for the double-gyre application, then m C (s, t; D) and a C (s, t; D) are the same since e C (s − t + τ : s) ≡ 1 holds for any s − t + τ and s. The technical details can be found in [1]. The advantage of r C (l, t; D) is that it gives the geometry of the so-called "pseudo-lobes" that represent the spatial coherency of the transport. Pseudolobes are the chain-like geometry of transport defined by the areas surrounded by C = {(l, r) \| r = 0} and the curve {(l, r) \| r = r C (l, t; D)} at a given time t. A positive pseudo-lobe corresponds to a coherent area defined by {(l, r) \| 0 ≤ r ≤ r C (l, t; D)}, while a negative pseudo-lobe is a coherent area defined by {(l, r) \| r C (l, t; D) ≤ r ≤ 0}. The relation of the pseudo-lobes to the Lagrangian lobes is discussed in [1]. Application to the Inter-gyre Transport in the Double-Gyre Ocean Having carried out the preparatory steps (Section 2) and defined the TIME functions (Section 3), we now use the TIME method to analyze inter-gyre transport processes in the the double-gyre circulation. We focus on the following three aspects of transport processes using the specific types of sub-domain D for m C (s, t; D) (see Table 1 for the definition of these sub-domains): using D(t), we study how the variability give rise to the development of the pseudolobes; using D cv (t) and D rw (t), we compare the impact of the circulating eddy vortices and that of the Rossby waves to the inter-gyre transport; using D all along with D(t) and D [k] , we analyze the inter-gyre transport processes of the Lagrangian lobes and examine the impact of variability during T [k] . We also use m C (s, t; D) in place for a C (l, t; D), with which we describe the geometry r C (l, t; D) associated with the transport, particular in terms of the pseudo-lobes. Figure 4 shows the transient accumulation m C (s, t; D(t)) in the Hovmöller diagram from the past up to the evaluation (present) time t as t progresses upward. Due to the nearly periodic variability in ψ (x, t), m C (s, t; D(t)) is [ Fig.4] also nearly periodic in t, i.e., m C (s, t; D(t)) = m C (s, t + T ; D(t + T )). Because of the anti-symmetry in the oscillation, m C (s, t; D(t)) is also anti-symmetric with respect to a T /2-shift in t, i.e., m C (s, t; D(t)) = −m C (s, t + T /2; D(t + T /2)). For s < s J at any t, m C (s, t; D(t)) is nearly zero because hardly any accumulation happens in the upstream direction of s J due to extremely small µ C (s, t) there. For s > s S , m C (s, t; D(t)) ≈ m C (s + δ, t + δ; D(t + δ)) along the reference trajectory (e.g., diagonal line in Fig 4) with δ > 0 indicates very little accumulation there. It suggests that the Rossby waves contribute very little to the inter-gyre transport in the downstream direction of s S . By a comparison of Figure 4 with Figure 2, accumulation processes in m C (s, t; D(t)) are closely related to the evolution of the flux zones in µ C (s, t) (Section 2.5). Below, we follow the development of a positive pseudo-lobe in Figure 4 starting from T [7.1] as t progresses. This pseudo-lobe first gains positive accumulation over S a where a positive flux zone pulsates during T [7.1] . If there is no flux in the downstream of S a (i.e., µ C (s, t) = 0 for s > s 1 ), then this positive pseudo-lobe will spread over [s J , s 1 + T /2] at the end of T [7.1] because the accumulation is simply advected along the reference trajectory; this corresponds to the invariance property (9). However, positive accumulation that occurs over S a is canceled shortly after it is advected into S b where a strong negative flux zone pulsates during T [7.1] . Hence, a small, narrow (in terms of s) positive pseudo-lobe develops mostly over S a during T [7.1] . At the end of T [7.1] when the positive flux zone over S a moves quickly to S b , the positive pseudo-lobe follows it. In Figure 5, m C (s, t; D(t)) at t = t * 944 + T /2 is shown by the dashed line and this positive pseudo-lobe is indicated by T /2. [ Fig.5] During T [7.2] , the positive pseudo-lobe continues to develop over S b where the positive flux zone pulsates with large amplitude. Because this flux zone is stronger than any flux zones, the pseudo-lobe slowly spreads into the downstream region by the advection of the accumulation. At the end of T [7.2] when the positive flux zone quickly moves to S c , the positive pseudo-lobe once again follows it. In Figure 5, m C (s, t; D(t)) is plotted at t = t * 944 + T by the solid line. The spread of the pseudo-lobe is observed over S d . During T [8.1] , the pseudo-lobe continues to develop over S c where the positive flux zone pulsates before disappearing at s N . It also continues to spread into the downstream region because the flux zones are much weaker there. Speed of the spread increases in the downstream direction of s 3 because of a weaker, positive flux zone induced by the Rossby waves over S e . At the same time, the spread of the negative pseudo-lobe that started in T [7.2] gradually pushes this positive pseudo-lobe from the upstream region. At the end of T [8.1] , the main part of the positive pseudo-lobe is located over S e (Figure 5). During T [8.2] , the positive pseudo-lobe first gains accumulation from a positive flux zone induced by the Rossby waves over S e , but loses it quickly because the negative flux zone induced by the Rossby waves moves into S e . In the meantime, the positive pseudo-lobe continues to spread into the downstream region. Once the accumulation passes s S , it simply advects along the reference trajectory. Near the end of T [8.2] , the positive pseudo-lobe reaches the final form as the entire pseudo-lobe passes s S . Our analysis of m C (s, t; D(t)) therefore reveals that the majority of inter-gyre transport occurs over a very limited segment [s J , s S ]. Accumulation processes are synchronized with the evolution of the circulating eddy vortices, and are fed mostly by the three flux zones over 3T /2 in t while they are mostly the same signed over the spatial segments S a through S c ; in the subsequent T /2 in t, the Rossby waves help form the final shape of the pseudo-lobe with the width T /2 in s. After 2T in t, the pseudo-lobe moves by the advection along the reference trajectory because there are no further accumulation in the downstream direction of S S . Thus, it takes almost 2T in t to develop a fullygrown pseudo-lobe of the width T /2 in s. While developing, the propagation speed of the pseudo-lobe is much slower than the particle advection along the reference trajectory: over 2T in t, the pseudo-lobe moves about T /2 in s starting from S a . The analysis also reveals that the Rossby waves have very little impact on the inter-gyre transport for two reasons. One reason is that the signals of µ C (s, t) in D rw are weak (see Section 2.5). The other reason comes from the fact that the upstream propagation of the signal in µ C (s, t) in D rw is nearly periodic in time (see Section 3.1). Figure 6 shows the decomposition of the transient inter-gyre transport m C (s, t; D(t)) into the part induced by the circulating eddy vortices m C (s, t; D cv (t)) and that induced by the Rossby waves m C (s, t; D rw (t)) at t = t * 36 + kT (end of T [k] ), where m C (s, t; D(t)) = m C (s, t; D cv (t)) +m C (s, t; D rw (t)) by the piece-wise independence property (10). By the definition of D rw , m C (s, t; D rw (t)) = 0 for s < s 3 means that the [ Fig.6] Rossby waves impact the inter-gyre transport only in the downstream of s 3 (dash-dot line). The amount of accumulation, m C (s, t; D rw (t)), is small except over [s 3 , s S ] where the signals of the circulating eddy vortices and that of the Rossby waves are mixed in µ C (s, t) (Section 2.5). Lagrangian lobes have proven to be extremely useful and insightful in a va-riety of transport studies and they provide precise amounts of Lagrangian transport of fluid [12,4,5,3]. In the companion paper [1], it was shown that m C (s, t; D all ) provides a good approximation to the amount of transport carried by individual lobes in the inter-gyre transport (see also [13]). Figure 7 shows m C (s, t; D all ) by the dash line at the end of T [k] . It is doubly periodic in s and t because of the periodic dynamics in ψ (x, t). Using the TIME method, we analyze the transport processes associated with the Lagrangian lobes. The transient transport m C (s, t; D(t)) in the past up to the present time t is shown in Figure 7 by the solid line. By the piece-wise independence (10), the difference m C (s, t; D all ) − m C (s, t; D(t)) corresponds to the transport that will occur in the future of t. A significant difference is observed mainly for s < s N where m C (s, t; D(t)) does not include the active accumulation over [s J , s N ] yet. A slight difference occurs over [s N , s J + T ] due to the flux zones of the Rossby waves there. For s > s J + T , m C (s, t; D all ) and m C (s, t; D(t)) are almost indistinguishable, suggesting that no further transport will occur in the future over the segment of C. Once again we confirm that the impact of Rossby waves in the downstream region is negligible for inter-gyre transport. Note that m C (s, t; D cv ) ≈ m C (s, t; D cv (t)) and m C (s, t; D rw ) ≈ m C (s, t; D rw (t)) hold over there as well (see Figure6). [ Fig.7] To examine how much transport occurs during one period T [k] of the unsteady eddy activity in ψ (x, t), Figure 7 shows m C (s, t; D Concluding remarks Building on the transport method developed in the companion paper [1], we have formulated a framework for the analysis of the dynamical processes that influence transport. The transport method, called the "Transport Induced by the Mean-Eddy interaction" (TIME), is a hybrid combination of Lagrangian and Eulerian transport approaches. Our analysis proceeds by a step by step approach. In particular, the steps are to determine the mean flow structure of u(x), determine the dynamic variability in u (x, t), construct the instantaneous stirring chart φ(x, t) = u(x) ∧ u (x, t) induced by the mean-eddy interaction, choose an Eulerian boundary C = {x C (s)}, and compute the flux diagram µ C (s, t) = φ(x C (s), t). The signals in µ C (s, t) are define relative to the reference particle advection along C, which is a diagonal line in the Hovmöller diagram. The fundamental constructions underlying the TIME method involve computing transport, either as accumulation m C (s, t; D) or displacement area a C (l, t; D) (which gives displacement distance r C (l, t; D)), which rely on the spatio-temporal integration of µ C (s, t). This provides the two-way link between the variability of the flow and the actual transport processes. It is a unique feature of the TIME method that neither the Eulerian nor Lagrangian methods alone can provide. These fundamentals also provides a platform for a novel graphical approach to the analysis of transport processes. While transport is highly system dependent, there are some common features that can hold in general that we can understand from our graphical approach for the analysis of transport processes. For example, the accumulation is most effective if the signals in µ C (s, t) propagate with the same unit slope as the reference trajectory, i.e., dynamic variability propagates with the particle advection in the mean flow. However, if the dynamic variability is temporally near periodic, and the signal has a negative slope, then the net effect would be almost zero. This may happen when the wave propagates upstream in the mean flow, like in the westward Rossby wave propagation in the double-gyre application. The role of variability in transport is analytically studied in [14] and is based on a related spatio-temporal scale analysis. We have applied our framework to the analysis of intergyre transport processes in the double-gyre ocean circulation where the Rossby-wave mode dominates the dynamic variability with a period T . The spatio-temporal analysis shows that the intergyre transport is controlled by a complex rotation of eddy vortices in the fast eastward jet near the western boundary current. The emergence of the pseudo-lobes is synchronized with the circulating eddy vortices in u (x, t). Pseudo-lobes having alternating signed area emerge every T /2 to transport water across the mean jet axis between the subpolar and subtropical gyres, while each pseudo-lobe sepends almost 2T over a very limited segment in the upstream dipole region to fully develop. During the development period, the pseudo-lobes propagate at a much slower speed than the reference particle advection. However, once fully developed, they propagate downstream of the mean jet axis at the same speed as the reference particle advection. The basin-scale Rossby wave has very little impact on the intergyre transport. Table 1 Definition of the sub-domains for the double-gyre application. List of Tables {(s, t) \| s ≥ s tr } D all entire domain D cv ∪ D rw D [k] domain for the k-th oscillation period {(s, t) \| t ∈ T [k] } D(t) transient (with respect to present time t) {(s, τ ) \| τ < t} D [k] (t) transient (with respect to present time t) dur- ing the k-th oscillation period in u (x, t) {(s, τ ) \| τ < t and τ ∈ T [k] } List of Figures 1 Wind-driven double-gyre circulation at wind-stress curl 0.165dyn/cm 2 : a) meaan streamline field ψ(x) averaged over T with contour interval 2000; b) eddy streamline field ψ (x, t) at t = t * 944 + T /4(= t * 982 ) with contour interval 500; and c) instantaneous flux field φ(x, t) at t = t * 944 + T /4(= t * 982 ) with contour interval 50. (see also Figure 3). In all panels, the dashed contours correspond to the negative values in all panels the Eulerian boundary C for the inter-gyre transport shown by a thick solid line In ψ(x), x J , x N , and x S are shown by the diamonds; In ψ (x, t) and φ(x, t), x 1 , x 2 , and The four phases of ψ (x, t) (left) and φ(x, t) (right) during T [7] at t * 944 when N T (t) is minumum t * 944 + T /4(= t * 982 ) when T D(t) is minimum, t * 944 + T /2(= t * 1020 ) when N T (t) is maximum, and t * 944 + 3T /4(= t * 1058 ) when T D(t) is maximum with time increasing upward. The contour intervals are the same as in Figure 1b for ψ (x, t) and Figure 1c for φ(x, t). The solid line with the diamonds and triangles is C: from the upstream, the diamons show x J , x N , and x S ; the triangles show x 1 , x 2 , and +3T/4) Fig. 3. The four phases of ψ (x, t) (left) and φ(x, t) (right) during T [7] at t * 944 when N T (t) is minumum t * 944 + T /4(= t * 982 ) when T D(t) is minimum, t * 944 + T /2(= t * 1020 ) when N T (t) is maximum, and t * 944 + 3T /4(= t * 1058 ) when T D(t) is maximum with time increasing upward. The contour intervals are the same as in Figure 1b for ψ (x, t) and Figure 1c for φ(x, t). The solid line with the diamonds and triangles is C: from the upstream, the diamons show x J , x N , and x S ; the triangles show x 1 , x 2 , and x 3 as shown in Figure1c Fig. 4. Transient accumulation m C (s, t; D(t)). The contour interval is 50 with the dashed contours for the negative values; the diamonds show s J , s N , and s S while the triangles show s 1 , s 2 , and s 3 from upstream to downstream. A reference trajectory is plotted for (s, t) = (s 0 − t 0 + t, t) that goes through s S at the end of T [7] with (s 0 , t 0 ) = (s S , t * 944 + T ). Figure 5), m C (s, t; D cv (t)) by the circulating eddy vortices (dash line), and m C (s, t; D rw (t)) by the circulating eddy vortices (dash-dot line) at t = t * 36 + kT (i.e., the end of T [k] ); the figure is made at t * 1095 using k = 7. Dark diamonds and dark triangles are the same as in Figure 5. , m C (s, t; D all ) for total transport (dash line) at t = t * 36 + kT (i.e., the end of T [k] ); the figure is made at t * 1095 using k = 7. Diamonds and triangles are the same as in Figure 5. Figures 24 [k.1] and T [k.2] as the first and the second half of T [k] , respectively. During T [k.1] , N T (t) increases from a minimum to a maximum while T D(t) reaches a minimum after about T /4. Conversely during T [k.2] , N T (t) decreases while T D(t) reaches a maximum after about 3T /4 from the beginning of T [k] . [k.1] for µ C (s, t) > 0 and T [k.2] for µ C (s, t) < 0. The strongest signals in µ C (s, t) are concentrated over the segment [s J , s N ] where the mean jet in ψ(x) is fast and the circulating eddy vortices in ψ (x, t) are energetic. Due to the pulsation of the circulating eddy vortices as they propagate along C (Section 2.3), the corresponding flux zones also pulsate simultaneously over the three consecutive segments, S a ≡ [s J , s 1 ), S b ≡ [s 1 , s 2 ), and S c ≡ [s 2 , s N ), with the alternating signs of µ C (s, t). The widths of S a , S b , and S c are narrow (6.5, 4, and 10.5days, respectively) with respect to the period of intensification T /2 (75.5days) during which the flux zones hardly move. Thus the slopes of the dominant signals in µ C (s, t) are steep. At the end of T [k.1] and T [k.2] , these flux zones weaken and quickly propagate downstream to the next segment along C. A positive flux zone over S a during T [k.1] connects to S b during T [k.2] , and then to S c during T [k+1.1] in a sequence; conversely a negative flux zone over S a during T [k.2] moves over to S b during T [k+1.1] , and then to S c during T [k+1.2] . Over the subsequent segment S d ≡ [s N , s 3 ) along C, weak signals of the circulating eddy vortices are observed in µ summarizes the main domains for the double-gyre application. Two types of the variability are associated with the two domains; D cv where the signals are associated with the circulating eddy vortices in the dipole region and D rw where the signals are associated with the Rossby wave in the downstream region of the eastward jet. The total domain is D all = D cv ∪ D rw . Temporally periodic variability leads to the temporal decomposition of D all based on T [k] , i.e., D all = ∪D [k] . Another useful definition D is transient D(t) that covers the (s, t) space up to the present time t.[Tab.1] a switch to turn on and off the instantaneous flux depending on whether or not (s−t+τ, τ ) is in D at time τ . For example using D(t) as D, m C (s, t; D(t)) is the transient transport by accounting for the accumulation up to the present time t (see [k] ) by the dash-dot line. Over [s J , s J + T ] of the length T in s, m C (s, t; D[k] ) and m C (s, t; D(t)) are almost indistinguishable because both include the active accumulation over [s J , s N ]. Accordingly, during T [k] , only one positive pseudo-lobe grows close to its final form over [s J +T /2, s J +T ] with width T /2 in s, but no negative pseudolobe can grow to the final form. If the period is shifted by T /2, then only one negative pseudo-lobe grows into its final form over the same [s J + T /2, s J + T ] segment. For s > s J + T , m C (s, t; D [k] ) differs from m C (s, t; D(t)) because of the accumulation in m C (s, t; D(t)) that has occurred prior to T[k] . x 3 are shown by the triangles. The flight-time coordinates of x J , x 1 , x 2 , x N , x S and x 3 are s J=110 , s 1 = 114.5, s 2 = 118.5, s N = 129, s S = 174.5 and s 3 = 150 days. 26 2 The flux diagram µ C (s, t) of the intergyre transport along the mean jet axis C as the Hovmöller diagram with t runs vertically. The contour interval is 20 (km 2 / day 2 ) with the dashed lines representing negative values. On the abscissa, the locations of s J=110 , s N = 129, and s S = 174.5 are indicated by the diamonds, while s 1 = 114.5, s 2 = 118.5, and s 3 = 150 are indicated by the triangles. In µ C (s, t), D [7] are indicated by the two horizontal lines (solid) at t = t * 944 and t = t * 944 (= t * 1095 ), while the boundary between D cv and D rw is shown by the vertical line (dashed) at s = s 3 . The two diagonal lines (solid) are examples of the reference trajectory, i.e., (s 0 − t 0 + t, t) with (s 0 , t 0 ) = (s J , 900) and (s 0 , t 0 ) = (250, 900). The left panel shows N T (t) (solid line) and T D(t) (dashed line) vs. t. The top panel shows \|u(x C (s))\| vs. s with the same abscissa as µ C (s, t). 27 3 Fig. 1 .Fig. 2 . 12Wind-driven double-gyre circulation at wind-stress curl 0.165dyn/cm 2 : a) meaan streamline field ψ(x) averaged over T with contour interval 2000; b) eddy streamline field ψ (x, t) at t = t * 944 + T /4(= t * 982 ) with contour interval 500; and c) instantaneous flux field φ(x, t) at t = t * 944 + T /4(= t * 982 ) with contour interval 50. (see alsoFigure 3). In all panels, the dashed contours correspond to the negative values in all panels the Eulerian boundary C for the inter-gyre transport shown by a thick solid line In ψ(x), x J , x N , and x S are shown by the diamonds; In ψ (x, t) and φ(x, t), x 1 , x 2 , and x 3 are shown by the triangles. The flight-time coordinates of x J , x 1 , x 2 , x N , x S and x 3 are s J=110 , s 1 = 114.5, s 2 = 118.5, s N = 129, s S = 174.5 and s 3 = 150 days. The flux diagram µ C (s, t) of the intergyre transport along the mean jet axis C as the Hovmöller diagram with t runs vertically. The contour interval is 20 (km 2 / day 2 ) with the dashed lines representing negative values. On the abscissa, the locations of s J=110 , s N = 129, and s S = 174.5 are indicated by the diamonds, while s 1 = 114.5, s 2 = 118.5, and s 3 = 150 are indicated by the triangles. In µ C (s, t), D[7] are indicated by the two horizontal lines (solid) at t = t* 944 and t = t * 944 (= t * 1095 ), while the boundary between D cv and D rw is shown by the vertical line (dashed) at s = s 3 . The two diagonal lines (solid) are examples of the reference trajectory, i.e., (s 0 − t 0 + t, t) with (s 0 , t 0 ) = (s J , 900) and (s 0 , t 0 ) = (250, 900). The left panel shows N T (t) (solid line) and T D(t) (dashed line) vs. t. The top panel shows \|u(x C (s))\| vs. s with the same abscissa as µ C (s, t). Fig. 5 . 5Pseudo-lobes of m C (s, t; D(t)) at the end of T [k.1] (t = t * 36 + (k − 1/2)T ; dash line) and at the end of T [k.2] (t * 36 + kT ; solid line) for k ≥ 1; the figure is made using k = 7. The positive pseudo-lobe that starts developing from the begining of T[7.1] at t * 944 (= t * 36 + (k − 1)T with k = 7) is indicated by: T /2 at t = t * 944 + T /2; T at t = t * 944 + T ; 3T /2 at t * 944 + 3T /2; and 2T at t * 944 + 2T . Dark diamonds show s J , s N , and s S ; dark triangles show s 1 , s 2 , and s 3 ; lighter diamonds show s J + T , s N + T , and s S + T ; and light triangles show s 1 + T , s 2 + T , and s 3 + T . Fig. 6 . 6Pseudo-lobes of m C (s, t; D(t)) for the total transient transport (solid line; same as in Fig. 7 . 7Pseudo-lobes of m C (s, t; D(t)) for the total transient transport (solid line; same as in Figures 5), m C (s, t; D [k] ) for transport induced during T [k] (dash-dot line) Definition of the sub-domains for the double-gyre application. 23 {(s, t) \| s < s tr }1 Domain Description Definition D cv domain associated with the circulating eddy vortices D rw domain asscoiated with the Rossby wave propagation .D [7] s J s N s S s 1 s 2 s 3 s (day) t (day) 100 200 300 400 900 1000 1100 1200 1300 AcknowledgmentThis research is supported by ONR Grant No. N00014-09-1-0418, (KI) and ONR Grant No. N00014-01-1-0769 (SW). Transport induced by mean-eddy interaction: I. Theory and relation to Lagrangian lobe dynamics. K Ide, S Wiggins, submitted to Physica DK. Ide, S. Wiggins, Transport induced by mean-eddy interaction: I. Theory and relation to Lagrangian lobe dynamics, submitted to Physica D. Physics of Climate. J Peixoto, A Oort, progress in oceanography. 1American Institute of PhysicsJ. Peixoto, A. Oort, Physics of Climate, progress in oceanography, vol.1, Edition, American Institute of Physics, 1992. The dynamical systems approach to Lagrangian transport in oceanic flows. S Wiggins, Ann. Rev. Fluid Mech. 37S. Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows, Ann. Rev. Fluid Mech. 37 (2005) 295-328. A tutorial on dynamical systems concepts applied to Lagrangian transport in oceanic flows defined as finite time data sets: Theoretical and computational issues. A Mancho, D Small, S Wiggins, Phys. Rep. 437A. Mancho, D. Small, S. Wiggins, A tutorial on dynamical systems concepts applied to Lagrangian transport in oceanic flows defined as finite time data sets: Theoretical and computational issues, Phys. Rep. 437 (2006) 55-124. Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach. R Samelson, S Wiggins, Springer-VerlagNew YorkR. Samelson, S. Wiggins, Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach, Springer-Verlag, New York, 2006. Nonlinear Physical Oceanography A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino. H A Dijkstra, Springer2nd EditionH. A. Dijkstra, Nonlinear Physical Oceanography A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino, 2nd Edition, Springer, 2005. Lagrangian coherent structures from approximate velocity data. G Haller, Phys. Fluids A. 14G. Haller, Lagrangian coherent structures from approximate velocity data, Phys. Fluids A 14 (2002) 1851-1861. Low-frequency variability in shallow-water models of the wind-driven ocean circulation. part ii: Timedependent solutions. E Simonnet, M Ghil, K Ide, R Temam, S Wang, J. Phys. Oceanogr. 33E. Simonnet, M. Ghil, K. Ide, R. Temam, S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. part ii: Time- dependent solutions, J. Phys. Oceanogr. 33 (2003) 729-752. Transition to aperiodic variability in a wind-driven double-gyre circulation model. K.-I Chang, M Ghil, K Ide, C.-C A Lai, J. Phys. Oceanogr. 31K.-I. Chang, M. Ghil, K. Ide, C.-C. A. Lai, Transition to aperiodic variability in a wind-driven double-gyre circulation model, J. Phys. Oceanogr. 31 (2002) 1260-1286. The trough and ridge diagram. E Hovmöller, Tellus. 1E. Hovmöller, The trough and ridge diagram, Tellus 1 (1949) 62-66. A refined Hovmöller diagram. O Martis, C Schwierz, H C Davies, Tellus. 58O. Martis, C. Schwierz, H. C. Davies, A refined Hovmöller diagram, Tellus 58A (2006) 221-226. Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time dependence, with applications to Rossby wave flow. N Malhotra, S Wiggins, J. Nonl. Sci. 8N. Malhotra, S. Wiggins, Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time dependence, with applications to Rossby wave flow, J. Nonl. Sci. 8 (1998) 401-456. Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics. C Coulliette, S Wiggins, Nonl. Proc. Geophys. 7C. Coulliette, S. Wiggins, Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics, Nonl. Proc. Geophys. 7 (2000) 59-85. Transient accumulation m C (s, t; D(t)). The contour interval is 50 with the dashed contours for the negative values; the diamonds show s J , s N , and s S while the triangles show s 1 , s 2 , and s 3 from upstream to downstream. K Ide, S Wiggins, Role of variability in transport, in preparation. A reference trajectory is plotted for (s, t) = (s 0 − t 0 + t, t) that goes through s S at the end of T [7] with (s 0 , t 0 ) = (s S , t * 944 + TK. Ide, S. Wiggins, Role of variability in transport, in preparation. Transient accumulation m C (s, t; D(t)). The contour interval is 50 with the dashed contours for the negative values; the diamonds show s J , s N , and s S while the triangles show s 1 , s 2 , and s 3 from upstream to downstream. A reference trajectory is plotted for (s, t) = (s 0 − t 0 + t, t) that goes through s S at the end of T [7] with (s 0 , t 0 ) = (s S , t * 944 + T ). Dark diamonds show s J , s N , and s S ; dark triangles show s 1 , s 2 , and s 3 ; lighter diamonds show s. + 2t . ; J + T , S N + T, S S + T , + 2T . Dark diamonds show s J , s N , and s S ; dark triangles show s 1 , s 2 , and s 3 ; lighter diamonds show s J + T , s N + T , and s S + T ; )) for the total transient transport (solid line; same as in Figure 5), m C (s, t; D cv (t)) by the circulating eddy vortices (dash line), and m C (s, t; D rw (t)) by the circulating eddy vortices (dash-dot line. Pseudo-lobes of m C (s, t; D(t. at t = t * 36 + kT (i.e., the end of T [kPseudo-lobes of m C (s, t; D(t)) for the total transient transport (solid line; same as in Figure 5), m C (s, t; D cv (t)) by the circulating eddy vortices (dash line), and m C (s, t; D rw (t)) by the circulating eddy vortices (dash-dot line) at t = t * 36 + kT (i.e., the end of T [k] ) )) for the total transient transport (solid line; same as in Figures 5), m C (s, t; D [k] ) for transport induced during T [k] (dash-dot line), m C (s, t; D all ) for total transport. Pseudo-lobes of m C (s, t; D(t. dash line. at t = t * 36 + kT (i.e., the end of T [k] )Pseudo-lobes of m C (s, t; D(t)) for the total transient transport (solid line; same as in Figures 5), m C (s, t; D [k] ) for transport induced during T [k] (dash-dot line), m C (s, t; D all ) for total transport (dash line) at t = t * 36 + kT (i.e., the end of T [k] );	[]
[ "An Iterative Method for Constructing Equilibrium Phase Models of Stellar Systems", "An Iterative Method for Constructing Equilibrium Phase Models of Stellar Systems" ]	[ "1⋆S A Rodionov \nSobolev Astronomical Institute\nSt. Petersburg State University\nUniversitetskij pr. 28198504St. Petersburg\n\nStary Peterhof\nRussia\n", "E Athanassoula \nLaboratoire d'Astrophysique de Marseille (LAM)\nUMR6110\nCNRS\nUniversité de Provence\nTechnopôle de Marseille-Etoile\n38 rue Frédéric Joliot Curie13388Marseille Cédex 20France\n", "N Ya Sotnikova \nSobolev Astronomical Institute\nSt. Petersburg State University\nUniversitetskij pr. 28198504St. Petersburg\n\nStary Peterhof\nRussia\n" ]	[ "Sobolev Astronomical Institute\nSt. Petersburg State University\nUniversitetskij pr. 28198504St. Petersburg", "Stary Peterhof\nRussia", "Laboratoire d'Astrophysique de Marseille (LAM)\nUMR6110\nCNRS\nUniversité de Provence\nTechnopôle de Marseille-Etoile\n38 rue Frédéric Joliot Curie13388Marseille Cédex 20France", "Sobolev Astronomical Institute\nSt. Petersburg State University\nUniversitetskij pr. 28198504St. Petersburg", "Stary Peterhof\nRussia" ]	[ "Mon. Not. R. Astron. Soc" ]	We present a new method for constructing equilibrium phase models for stellar systems, which we call the iterative method. It relies on constrained, or guided evolution, so that the equilibrium solution has a number of desired parameters and/or constraints. This method is very powerful, to a large extent due to its simplicity. It can be used for mass distributions with an arbitrary geometry and a large variety of kinematical constraints. We present several examples illustrating it. Applications of this method include the creation of initial conditions for N -body simulations and the modelling of galaxies from their photometric and kinematic observations.	10.1111/j.1365-2966.2008.14110.x	[ "https://arxiv.org/pdf/0808.1980v2.pdf" ]	16,127,076	0808.1980	65440fa5cb5906eac51c41606a9a08367a706a13	An Iterative Method for Constructing Equilibrium Phase Models of Stellar Systems 2008 1⋆S A Rodionov Sobolev Astronomical Institute St. Petersburg State University Universitetskij pr. 28198504St. Petersburg Stary Peterhof Russia E Athanassoula Laboratoire d'Astrophysique de Marseille (LAM) UMR6110 CNRS Université de Provence Technopôle de Marseille-Etoile 38 rue Frédéric Joliot Curie13388Marseille Cédex 20France N Ya Sotnikova Sobolev Astronomical Institute St. Petersburg State University Universitetskij pr. 28198504St. Petersburg Stary Peterhof Russia An Iterative Method for Constructing Equilibrium Phase Models of Stellar Systems Mon. Not. R. Astron. Soc 0002008(MN L A T E X style file v2.2) Accepted ???? ??? ??. Received ???? ??? ??; in original form ???? ??? ??galaxies: kinematics and dynamics -methods: N-body simulations We present a new method for constructing equilibrium phase models for stellar systems, which we call the iterative method. It relies on constrained, or guided evolution, so that the equilibrium solution has a number of desired parameters and/or constraints. This method is very powerful, to a large extent due to its simplicity. It can be used for mass distributions with an arbitrary geometry and a large variety of kinematical constraints. We present several examples illustrating it. Applications of this method include the creation of initial conditions for N -body simulations and the modelling of galaxies from their photometric and kinematic observations. INTRODUCTION In astronomy there are at least two problems where equilibrium phase models of stellar systems need to be constructed. The first one is the construction of phase models for real galaxies from observational data, i.e. the modelling of observational data. The second problem is the construction of initial conditions for N -body simulations of stellar systems. It is obvious that these two problems are tightly connected, and yet they have, so far, been solved by different methods. The Schwarzschild method (Schwarzschild 1979) and its modifications is often used for modelling of observational data (e.g Häfner et al. 2000;van den Bosch et al. 2006;Thomas 2007;van den Bosch et al. 2008;de Lorenzi et al. 2008), but has almost never been used so far to produce initial conditions for simulations. For N -body initial conditions, a wide variety of methods has been used, based on the Jeans theorem (e.g. Zang 1976;Athanassoula & Sellwood 1986;Kuijken & Dubinski 1995;Widrow & Dubinski 2005;McMillan & Dehnen 2007), or on Jeans' equations (e.g. Hernquist 1993). In the case of multi-component systems, e.g. disc galaxies with a bulge and a halo, the components are built separately and then either simply superposed, or the potential of the one is adiabatically grown in the other (e.g. Barnes 1988;McMillan & Dehnen 2007;Athanassoula 2007) before superposition. For real galaxies the phase space density is generally unknown, but we do have some information about it. For example, we know more or less accurately a distribution of ⋆ E-mail: [email protected] mass for the visible components (notwithstanding uncertainties due to the mass to light ratio) and we often have some constraints on the velocity distribution. It is also reasonable to assume that the galaxy is in an equilibrium state. So in general, the problem of constructing a model in phase space is equivalent to constructing an equilibrium phase model with a given mass distribution and, in many cases, given kinematic constraints. In the case of modelling observational data (first of the two above mentioned problems) the kinematic parameters are the observed velocities integrated along the line of sight. In the case of initial conditions for N -body simulations, a wide variety of kinematic parameters is possible, depending on the problem the simulation addresses. We have developed a new method for constructing equilibrium phase models with a given mass distribution and with given kinematic parameters, which we call the iterative method. It can be applied to systems with arbitrary geometry, so that the requested mass distribution can be arbitrary. The idea and a first implementation was presented in . In Rodionov & Orlov (2008) we improved it, and applied it to construct an N -body model of the stellar disk of our Galaxy for two realistic mass models of the Milky Way. Here we present a final version of this method, fully allowing kinematical constraints. In the previous articles we had concentrated on constructing equilibrium phase models with a given mass distribution, so that kinematic parameters were either not considered or only in terms of auxiliary parameters, such as the total angular momentum Rodionov & Orlov 2008). This, however, limited the applicability of our method, both for initial conditions and for modelling real galaxies. Indeed, initial kinematics play a crucial role in determining the evolution of N -body systems, while observational constraints more often than not include kinematics. In this paper we give equal attention to the mass distribution and kinematical constraints, so that the iterative method can now be used for a number of interesting applications. In principle, in our method, both the kinematic constraints and the mass distribution can be arbitrary. But the part of our algorithm that concerns the kinematic constraints is not universal, contrary to the part that handles the mass distribution, but is tailored to the specific constraint. Here we consider several types of constraints. Once these are understood, it is rather easy to extend the algorithm for every new type of kinematic parameter (see below). The power of the iterative method stems from its simplicity. The iterative method is based on a simple and, in a way, obvious idea, which is implemented in a simple algorithm. The purpose of this article is to fully describe this method. We first introduce the basic concept in Section 2, where we also explain the different modules of the algorithm and the way they should be applied. In section 3 we illustrate the use of the method with three examples, namely a triaxial system, a multi-component model of a disk galaxy (including live disk, bulge and halo components) and a disk constructed with given line-of-sight kinematic. We briefly conclude in section 4. THE ITERATIVE METHOD General Idea of the Iterative Method The aim of the iterative method is to construct equilibrium N -body models with a given mass distribution and with given kinematic properties, parameters, or constraints. This method relies on the fact that any non-equilibrium system will tend, more or less fast, to a stable equilibrium. We thus start by constructing any arbitrary, non-equilibrium N -body system, and let it evolve. Such an evolution changes both its mass distribution and its kinematics, so that the final system does not have the desired properties. To achieve the latter, we developed a new method which we call the iterative method and which relies on a constrained, or guided, evolution. We will describe it fully in this section. This idea is of course applicable for any arbitrary dynamical system and is even widely used in every day life. To give an example, let us consider a donkey walking by itself in a field. After some time the donkey can be anywhere in the field. Now consider another donkey which we attach to a tree by a rope. This donkey also walks in the field, but it will have to stay inside a circle of radius equal to the length of the rope. This is an example of a constrained evolution, which will necessarily lead to a final state within a circle around the tree. The crucial point now is how to achieve this constraint, i.e. what will the equivalent of the rope be in the case of galaxy models. The general scheme of our method is presented in Fig. 1 and can be applied to any arbitrary dynamical system. Let our task be to find an equilibrium state of some dynamical system obeying given constraints or having specific values of some given parameters. We start from any arbitrary state of The scheme of the iterative method for the case of an N -body system with a given mass distribution and given kinematical parameters. our dynamical system and allow the system to evolve during a short time interval. We then need to make sure that the given parameters have the required values. In order to achieve this we need to modify the system so that the given parameters have the necessary values, while making sure that the other quantities or parameters are kept unchanged, so as to retain memory of the evolution. As shown in Fig. 1, we iterate these two steps, alternating short evolutions and modifications of the system to fix the set parameters. We thus constrain the evolution in order for it to reach an equilibrium state with the desired set of constraints. We stop when we consider we are sufficiently near the desired equilibrium state of the system. Let us now consider a case, in which we wish to construct an equilibrium N -body system with a given mass distribution and with or without given kinematic constraints. The scheme is outlined schematically in Fig. 2. We initially create an N -body system with a given mass distribution but with arbitrary particle velocities (for example velocities equal to zero). We then start the iterative procedure, by letting the system go through a sequence of evolutionary steps of short duration. At the end of each one of these steps, and before the new step is started, we need to set the appropriate parameters. Let us first consider the case where we wish to have a specific mass distribution, but have no kinematic constraints. To achieve this, we construct a new N -body system, with the desired mass distribution but with velocities chosen according to the velocity distribution obtained from the evolution. In other words, we "transfer" the velocity distribution from the system obtained from the evolution to a new system, which will have the desired mass distribution and an evolved velocity distribution. The algorithm performing this "transfer" is the core of the iterative method, and will be discussed in more detail in section 2.2. If we have kinematic constraints as well, we need to modify the velocities of the particles in such a way that the constraints are fulfilled and, at the same time, as little as possible so that some memory of the evolution is kept. How this is done in practise depends on the imposed constraints and will be described, for a number of cases, in section 2.3. Procedures for further types of constraints can be easily found following similar techniques. In all cases we have a new system which has the desired mass distribution, obeys the necessary velocity constraints, while being nearer to equilibrium, since it retains partial memory of the evolution. We repeat this iterational procedure a number of times, alternating one evolution phase and one phase where the necessary parameters are set, until we come as near to the desired equilibrium state as desired. Transfer of velocity distribution The transfer problem can be formulated as follows. Any evolution step ends with a model, which we will refer to as the "old" model. This step is followed by a constraining, or fixing step, during which we create a "new" model with the desired mass distribution. We now need to transfer the velocity information from the "old" to the "new" model. There is more than one way to achieve this transfer. used an algorithm based on moments of the velocity distribution, which, however, proved to be rather complicated and cumbersome. Here we suggest a much simpler and more reliable algorithm, which is in fact an improvement of an algorithm initially used in Rodionov & Orlov (2008). The basic idea of our velocity transfer algorithm is as follows. We assign to the new-model particles the velocities of those particles from the old model that are nearest to the ones in the new model. The simplest (although, as we show below, not necessarily optimum) implementation of this idea is evident. One can prescribe to each particle in the new model the velocity of the nearest particle from the old model. Let us formulate this proposition more strictly. For each i-th particle of the new model, one finds the jth particle in the old model with the minimum value of \|r new i − r old j \|. Here, r new i is the radius vector of the i-th particle in the new model, and r old j is the radius vector of the j-th particle in the old model. Hereafter we will always imply this definition when we talk of the nearest or closest particle. One then takes as the velocity of the i-th particle in the new model the velocity of the j-th particle in the old model. This simple algorithm has, however, one essential drawback. If the numbers of particles in the old and new models are the same then only about one-half of the particles in the old model participate in the velocity transfer. The reason is that many old-model particles transfer their velocities to several particles in the new model. As a result of this, almost onehalf of the particles in the old model do not transfer their velocities at all. This means that a significant amount of information on the velocity distribution will be lost in the transfer process. The noise will therefore grow, as we verified in numerical experiments. Thus, this transfer algorithm is not optimum. This shortcoming can, nevertheless, be overcome by modifying this transfer scheme. For this, we introduce an input parameter, which we call the "number of neighbours" n nb . We also introduce, for each particle in the old model, the parameter n used , which denotes the number of times this particle has been used for velocity copying. At the beginning of the transfer procedure we set n used = 0 for each particle in the old model, since its velocity information has not been yet transferred to any of the new model particles. For any given particle in the new model we find the nearest n nb neighbours in the old model (the closeness being understood as defined above) and from these we single out the subgroup of particles that have a minimum n used . From this subgroup we find the particle that is the closest one to the position of the new-model particle, add one to its n used value and prescribe its velocity to the new-model particle we are examining. We note that if n nb = 1 this algorithm is the same as the previous one and about half of the particles will not take part in the velocity distribution transfer. If, however, we take n nb = 10, only a small fraction (a few per cent) of old-model particles will not take part in the transfer process. We adopt this improved transfer method since we showed that it gives good results in the iterative procedure. If the desired model has some symmetry, it can be useful to take it into account in the algorithm of velocity transfer by redefining the distance between two particles. For example, if we wish to build an axisymmetric system, we search for the nearest particles in the two-dimensional space R − z (where R the cylindrical radius) instead of the three-dimensional space x − y − z. We then transfer the velocity of this nearest old-model particle (in cylindrical coordinates) to the newmodel particle. It is important to adopt this new definition of the distance in order to fix not only the mass distribution, but also fix the velocity distribution and to make it fully axisymmetric. We have thus introduced three variants of the velocity transfer algorithm. We will refer to them as "transvel 3d", "transvel cyl" and "trasvel sph". (i) "transvel 3d": This is the basic algorithm, for the case when the desired system has no assumed symmetry. By using this algorithm in the iterative method we only fix the mass distribution and leave the velocity distribution unchanged. In the current work we use this algorithm when constructing triaxial models. (ii) "trasnvel cyl": This is a modification of the basic algorithm for axisymmetric systems and was described just above. We use this algorithm when both the desired mass and velocity distribution are axisymmetric. In the current work we use this algorithm for constructing all models except for the triaxial ones. (iii) "transvel sph": This is a modification of the basic algorithm for spherical systems. In this version of the algorithm we search for the nearest particles in one dimensional "r" space, where r is the spherical radius. By using this algorithm in the iterative method we fix both the mass and the velocity distribution to make the system fully spherically symmetric. This was used in Sotnikova & Rodionov (2008). Fixing the Kinematic Parameters Here we describe algorithms for fixing different kinematic parameters. The general algorithm is as follows. We slightly change the particle velocities to fix given kinematic parameters, but we keep as many other parameters as possible unchanged. Here we describe in detail only a number of algorithms, which we use in this paper. But it is easy to develop algorithms for any other kinematic parameter. It is only necessary to follow the general principle: "keep unchanged the parameters that do not need to be fixed". Fixing the radial velocity dispersion profile σR(R) We use this algorithm in order to fix in stellar disks the radial velocity dispersion profile to a given function σR(R) (for an application, see section 3.2). Let σR(R) be the given radial velocity dispersion profile which we want to fix, where R is the cylindrical radius. After each evolutionary step (see Sect.2.1) we need to change slightly the radial velocities of particles in order to fix this profile. The model is divided into n div concentric cylindrical annuli, each containing the same number of particles. For each annulus j, we calculate the target value of the radial velocity dispersion σ j R = σR(Rj) ,(1) where Rj is the mean value of the R coordinate of all particles in part j. The new radial velocity of the i-th particles in the j region is then prescribed as follows. vRi = v ′ Ri σ j R /σ j′ R ,(2) where v ′ Ri is the current value of the i-th particle radial velocity, vRi is the corrected i-th particle radial velocity and σ j′ R is the current value of radial velocity dispersion in part j. We note that in this scheme we have assumed that the mean radial velocity is equal to zero. Fixing the radial anisotropy profile This algorithm is very similar to previous one and is very useful for building spherical models with a given profile of velocity anisotropy. We will use it for building the halo model in section 3.2 and it has also been used in Sotnikova & Rodionov (2008). Let σ θ , σϕ and σr be the velocity dispersions in the θ, ϕ and r directions of spherical coordinate system and let us aim e.g. for a model with a given profile, β(r), of the σ θ /σr ratio. The model is divided in concentric spherical shells, each containing the same number of particles. For each shell j, we calculate the target value of σ θ /σr βj = β(rj) ,(3) where rj is the mean value of the r coordinate of all particles in shell j. We will attempt to obtain this ratio by changing appropriately the θ component of particle velocities (alternatively, we could have changed the r component). The new θ velocity component of the ith particle in the jth region will then be prescribed by v θi = v ′ θi βj σ j′ r σ j′ θ ,(4) where v ′ θi and v θi are, respectively, the current and the corrected values of the i-th particle θ velocity component and σ ′ r and σ ′ θ are the current values of the radial and θ velocity dispersion, respectively, in part j. Fixing the line-of-sight mean velocity or the line-of-sight velocity dispersion in the case of an edge-on disk In this section, we describe two algorithms, one for fixing the line-of-sight mean velocity and the other for fixing the line-of-sight dispersion of an axisymmetric disk. To set the notation, let us assume that the stellar disk rotates about the z-axis, the disk plane lies in the (x, y) plane and the line of sight is along the y axis (edge-on disk). We invert the sign of vy for each particle with x < 0 in order to make the half disk with x < 0 kinematically identical to the half with x > 0 and flip the x < 0 particles on the x > 0 part. We then divide the disk in slits parallel to the (y, z) plane and at different distances from the centre, i.e. at different values of x, in such a way that all slits have the same number of particles. Let us denote byv los (x) the desired profile of the line-ofsight mean velocity, i.e. the mean value of vy after integration along the line of sight. For each slit j we calculate the target value of the line-of-sight mean velocityv j los =v los (xj) (where xj is the mean value of \|x\| for particles in part j) and the current value of the line-of-sight mean velocityv j′ los (as the mean value of vy for all particles in slice j). The new y velocity component of particle i in region j should then be vyi = v ′ yi + (v j los −v j′ los ),(5) where v ′ yi is the current value of the i-th particle y velocity and vyi is the corrected i-th particle y velocity. Particles which were flipped to x > 0 part have to be flipped back and the sign of their y velocity component reversed. The particles are then azimuthally mixed to make the velocity distribution axisymmetric and the step is concluded. Of course, in this way we have tampered with v los , but this unavoidable. Nevertheless, after a number of iterations, both the axisymmetry and the desired v los will be achieved. The algorithm for fixing σ los (x) is very similar, except that we have to calculate in each slit the current value of the line-of-sight velocity dispersion σ j′ los as the dispersion of vy for all particles in slit j. Let σ los (x) be the desired profile of the line-of-sight velocity dispersion. In order to achieve this, the new y velocities should be modified as follows vyi = (v ′ yi −v j′ los ) σ j los σ j′ los +v j′ los ,(6) whereσ j los = σ los (xj) is the target value of the line-of-sight velocity dispersion. We note that in this algorithm we have to take into account that the value ofv j′ los need not necessarily be equal to zero. As previously, we still have to flip back particles which were flipped to the x > 0 part, invert the sign of their y velocity component with and mix the particles azimuthally. Fixing velocity isotropy conditions An isotropic velocity distribution depends only on the velocity module and not on the direction of the velocity. Our algorithm for fixing it is very simple. For each particle, we keep the velocity module unchanged and randomise its direction, thus ensuring that the velocity distribution is isotropic. For spherical isotropic models, the distribution function (DF) is known, at least formally, or numerically. Thus the construction of such models can be considered as a test of the iterative method, and we have verified in a number of cases that the models constructed by the iterative method are identical to the models constructed by using known equilibrium DF. The construction of spherical isotropic models with the iterative method was first described in . That work, however, used an old and rather complicated algorithm for transferring the velocity distribution. Here we use a different, superior algorithm, based on the description in Sect. 2.2. We again made sure that the models thus constructed are identical to the models obtained by using a known DF. Furthermore, we also used this algorithm for constructing models which are not fully isotropic models, but rather not-very-far from isotropic (see section 3.1 and 3.2). How many parameters should we fix? The goal of the iterative method is to construct equilibrium N -body models with given parameters (i.e. with a given mass distribution and with given kinematic constraints). There are in general three possible cases with respect to the number of constraints. In the first case the number of given parameters, or constraints are such that only one equilibrium model can exist. In this case, we can expect that the iterative method will converge to this unique equilibrium model, independent of the initial state from which the iteration is started. For example, it is known that for a spherical model with a given mass distribution only one isotropic equilibrium DF exists. If we construct a spherical model by using the iterative method and we fix velocity isotropy as a kinematical constraint (section 2.3.4), then the iteration always converges to the same model, independent of the initial state, as expected. In the second case, the number of give parameters is such that many equilibrium models can exist, i.e. this number is insufficient for determining uniquely the equilibrium model. In this case we can expect that the result of the iterative method will depend on the choice of the initial model. The iteration will converge to the equilibrium model which is "nearest" in some sense to the initial model. Alternatively, the iteration will converge to some specific, in some sense, model. For example, when constructing a triaxial model in section 3.1 we fix only the mass distribution and do not set any kinematic constraints. Of course in this case the result of the iterations will depend on the initial state (see section 3.1 for details). Another, more involved, example is the construction of a disk model with given total angular momentum. In principle, many such equilibrium models are possible, yet the iterations of Rodionov & Orlov (2008) always converged to the same model. It is unclear why this is the case, but it could be due to a specificity of the model (see Rodionov & Orlov 2008 for details). The last possibility is that for the adopted parameters, no equilibrium model exists, i.e. we have fixed too many parameters. In this case the iteration will either not converge at all, or it will converge to a system with the parameters we have fixed, which is in non-equilibrium, but close in some sense to equilibrium. Technical comments In this section we elaborate a few important technical points, useful for anybody wishing to apply the iteration method. One of the free parameters of the iterative method is the duration ti of each iteration, i.e. the time interval over which the system is evolved during each iteration. How should the numerical value of ti be chosen? It is clear that this time should not be too short, so as to allow the system to evolve sufficiently during one iteration step. On the other hand, it should not be too long either, so as not to permit the evolution of various instabilities; otherwise, these instabilities may change the system substantially. For example, when constructing a disk system it is necessary to use iteration steps considerably shorter that the growth time of the bar instability, which of course varies strongly from one model to another. For this reason, there is no strict criterion and ti should be chosen empirically. Our experiments have shown that it is usually better to try relatively big ti values, thus ensuring a much faster convergence. Moreover, in some situations the iterations for relatively small ti don't converge at all, while iterations for relatively big ti do. This was, for example, the case when we constructed a model with relatively cold stellar disk. So if iterations don't converge or they converge too slowly, it is often useful to consider bigger ti (within of course reasonable limits). Examples of appropriate ti values are given in all examples in Sect. 3. Moreover, our simulations have shown that, if we take ti within reasonable limits, the result of the iteration is the same (within the noise limits) and independent of ti, provided of course the chosen number of parameters and constraints allow a single solution. If the latter is not the case, and the result of the iteration depends on its starting model, then of course the result can depend also on ti. Another parameter of the iterative method is the parameter n nb in the algorithm of velocity transfer (see section 2.2). Its value has been chosen in a more or less "ad hoc" manner and, by trial and error, we have found that a value of n nb = 10 is often satisfactory. Our test simulations have shown that the results of the iterative method for n nb = 10 and n nb = 100 are practically the same, at least for a total number of particles as those used in our trials, i.e. of the order of a few hundred thousands to a couple of millions. The most computer costly part of our method is the computing of the evolution of the system in each iteration, since the computing cost of all other parts of the method is very small. For this reason it is recommended to use a fast N -body code and we have adopted gyrfalcON (Dehnen 2000(Dehnen , 2002. Furthermore, our test simulations have shown that computation of the evolution can be carried out with relatively low accuracy. This is mainly due to the fact that we need to calculate the evolution only during short time intervals, so that errors do not have sufficient time to accumulate. Therefore, in the iterative method we use Gyrfal-cON with relatively big values of the tolerance parameter θt and of the time-step (see section 3). Usually the total computing cost is considerably smaller, but still of the same order as that necessary to run a simulation with the constructed model. This of course will depend on whether we can start the iterations form a model reasonable close to the final one, or whether lack of any a priori knowledge leads us to start, e.g. from zero initial velocities. A 'trick' which helps reducing the computing cost is to make a few iterations initially with a small number N of particles and then gradually increase N to the required number. In the procedure of velocity transfer described in section 2.2 the number of particles in the old and the new system can be different. So in the next iteration step we can get a system with a larger number of particles. EXAMPLES OF MODELS In this section we consider three examples of models constructed by our method, namely a triaxial model of a spheroid, a multi-component model of a disk galaxy and a model with given line-of-sight kinematics. Triaxial model Our first example is that of a triaxial model. As mass distribution we use a Plummer sphere flattened in two dimensions. ρ(x, y, z) = 3M pl 4 π a b c a 2 pl x 2 a 2 + y 2 b 2 + z 2 c 2 + a 2 pl −5/2 ,(7) where M pl and a pl are the total mass and the scale-length of the model and a, b, c are rescaling parameters. In the present specific example we discuss a model with the following parameters: M pl = 1, a pl = 3π/16, a = 1, b = 0.8 and c = 0.7. Scaling our model to an elliptical galaxy with a = 3 kpc and M pl = 10 11 M⊙, gives a time unit tu ≈ 17 Myr. Our target is to construct an equilibrium N -body system with this given mass distribution. We didn't impose any well-defined restriction on the kinematics of the system, but aimed instead for a velocity distribution not very far from isotropic. Our initial model was cold with velocities equal to zero. Our target model was reached in 50 iterations. For the first 10 we fixed a condition of velocity isotropy (section 2.3.4), while for the last 40 we didn't fix any kinematic parameters. If we performed all 50 iterations without fixing any kinematic parameters, we would also obtain an equilibrium model but with a higher degree of anisotropy. We chose N = 500 000 and an iteration time ti = 10. The integration step and the softening length were taken dt = 1/2 7 and ǫ = 0.01, respectively. The tolerance parameter for gyr-falcON was set to θt = 0.9 (see section 2.5) and we used the "transvel 3d" modification of the algorithm of velocity transfer (see section 2.2). In such a case it is crucial that the final model be sufficiently close to equilibrium so that the axial ratios do not tend to unity after some time evolution, as it happens for many other techniques used in calculating triaxial equilibria. To test this we evolved our model for 50 time units, i.e. roughly 50 crossing times for the scale-length of the system a pl . The integration step and softening length were taken dt = 1/2 8 and ǫ = 0.005, respectively -in agreement with the recommendations of Rodionov & Sotnikova (2005) (see also Athanassoula et al. 2000) -and the tolerance parameter for gyrfalcON was set to θt = 0.6. Fig. 3 shows the evolution of the ellipticity of the model for three different projections. This was calculated as the ratio of the medians of the absolute values of corresponding particle coordinates. As can be seen, the shape of the model is practically unchanged during the evolution. We also made sure that the model also conserved all its other properties, thus demonstrating that it is indeed very close to equilibrium. Multi-Component model of a disk galaxy As a second example, we constructed a model of a disk galaxy consisting of three components: a stellar disk with a given profile of radial velocity dispersion, a non-spherical bulge and a halo with a given anisotropy profile. To start, we need to define the mass distribution in each of the components. The disk model is an exponential disk with density: ρ d (R, z) = M d 4πR 2 d z0 exp − R R d sech 2 z z0 ,(8) where M d is the total disk mass, R d is the disk scale length, z d is its scale height and R is the cylindrical radius. The halo model is a truncated NFW halo (Navarro et al. 1996) ρ h (r) = C h exp(−r 2 /r 2 th ) (r/r h )(1 + r/r h ) 2 , where r h is the halo scale length, C h is a parameter defining the mass of the halo and r th is the truncation radius of the halo. For the bulge we used a truncated and flattened Hernquist sphere (Hernquist 1990) with density ρ b (R, z) = M ′ b r b 2πq exp(−d 2 /r 2 tb ) d(d + r b ) 3 ,(10) where d = R 2 + z 2 q 2 ,(11) r b is the bulge scale length, M ′ b is the total bulge mass before truncation, r tb is the truncation radius of the bulge and q is the flattening parameter. For the parameter values we chose for the disk: M d = 1, R d = 1, z0 = 0.2, for the halo: r h = 4, C h = 0.01, r th = 14 and for the bulge: r b = 0.2, M ′ b = 0.2, r tb = 2, q = 0.7. For these parameters the total mass of the bulge and of the halo are M b ≈ 0.15 and M h ≈ 4.98, respectively. We use units such that the constant of gravity is G = 1. Scaling our model to a disk galaxy with R d = 3.5 kpc and M d = 5 · 10 10 M⊙, gives a time unit tu ≈ 13.8 Myr and a velocity unit vu ≈ 247.9 km/s. The rotation curve for our model is shown in Figure 4, which also displays the contribution of the disk, halo and bulge components separately. To make the exercise more realistic, we still need to choose kinematical constraints for each of the components, although, as we have already mentioned, these are not obligatory for our method. We created the disk with the following profile σR(R) = 0.3 · exp (−R/3) + 0.2 · exp (−R/0.5) ,(12) where σR is the radial velocity dispersion. From the mass model of the galaxy and from profile (12) we can calculate the radial profile of the Toomre parameter Q (Toomre 1964), shown in figure 5. Our main target here is to demonstrate that our method can construct an equilibrium model of the disk with any realistic profile of σR(R). The choice of profile in eq. (12) is more or less arbitrary, but demonstrates that our method can work with more elaborate profiles than a single exponential function. When constructing the bulge, we did not impose any specific kinematic constraints. Instead, we aimed for a model not-very-far from isotropic, as in the case of the triaxial model of Sect. 3.1. For the halo we adopted a velocity anisotropy profile, so as to test a different kind of constraint. More specifically we chose σ θ (r) σr(r) = 0.2 r 0.9 2 + 1 + 0.8 ,(13) where σ θ and σr are the velocity dispersion in the θ and the r direction in a spherical coordinate system. Note that we constructed the phase model of the halo in the presence of the non-spherical potential of the disk and bulge, i.e. we don't have spherical symmetry and in the halo equilibrium model σ θ should not be equal to σϕ (the velocity dispersion in ϕ direction). So when we constructed the halo model we fixed only the fraction σ θ /σr, but did not fix the fraction σϕ/σr. This is different from the case of the isolated spherical NFW halo, constructed in Sotnikova & Rodionov (2008). And again we want to underline that kinematical constraints are not obligatory. We could also construct the halo, or the bulge, without any kinematical constraints, or with another type of kinematical constraints. For example we can construct a rotating halo with given total angular momentum or a rotating halo with a given profile of the mean azimuthal velocity. Once the mass models and the required kinematical parameters for each of three components are defined, we can apply the iterative method for constructing an equilibrium N -body model for the whole system. In this specific example we chose N d = 200000, N b = 30324, N h = 995978 for the number of particles in disk, bulge and halo, respectively. With these numbers, the mass of particles in all components is the same. We constructed each of these components separately in the rigid potential of all other components. In order to take into account the external potential, we need to make only one small evident modification of the iterative method. Namely, when we need calculate the evolution of the system during the iteration time (see fig. 2), we simply need to do it in the presence of the external potential. This can be done either by introducing an analytical external potential to the gyrfalcON program, or we can add it as a rigid N -body system. In current work we use the latter. For example, in order to add a rigid halo we simply add in the system rigid particles according to the mass distribution of the halo. Let us first describe the disk construction. Our initial model was a cold disk where all particles move on circular orbits. Indeed, the circular velocity can be easily calculated, since the mass distribution in the model is known. Had we, instead, started off with zero disk velocities, we would have again obtained an equilibrium model, but with counter rotating subsystems. This will happen because we fix only the profile of σR(R) and do not fix any parameters defining the direction of rotation. It is therefore better to start off the iterations with a rotating disk, unless of course a disk with counter-rotating components is specifically sought. Note that the result of the iteration is independent of the initial iterative guess for the disk rotation. For example it will be the same if initially all the disk particles have tangential velocities equal to half of the circular velocity, or twice that. We made 50 iterations, each with ti = 20. The integration step and softening length were taken dt = 1/2 4 and ǫ = 0.04, respectively and the tolerance parameter for gyr-falcON was set θt = 0.9. In order to fix the σR(R) profile we used the algorithm described in section 2.3.1. The number of layers in this algorithm was n div = 200 (see section 2.3.1). We used the "transvel cyl" modification of the algorithm of velocity transfer (see section 2.2). This algorithm also was used for constructing the bulge and halo components. We call this disk model DISK.SVR. For constructing the bulge we also made 50 iterations. The other parameters for this construction were ti = 10, dt = 1/2 6 , ǫ = 0.02 and θt = 0.9. Our initial model was a cold model with velocities equal to zero. During the first 10 iterations we fixed a condition of velocity isotropy (section 2.3.4), while we did not set any kinematical constraints during the last 40 iterations. To construct the halo we used again 50 iterations and the remaining parameters were taken as follows : ti = 50, dt = 1/2 4 , ǫ = 0.04 and θt = 0.9. For fixing the velocity anisotropy radial profile (13) we used the algorithm described in section 2.3.2. The number of layers in this algorithm was n div = 500. Once all three components of our model were constructed, we simply stacked them in order to obtain the complete system. In order to check whether this was indeed near equilibrium, as it should, we simply evolved with a full N -body simulation, using again gyrfalcON, now with an integration step and softening length of dt = 1/2 7 and ǫ = 0.02 (parameters were chosen according to recommendations of Rodionov & Sotnikova (2005)). The tolerance parameter for gyrfalcON was set θt = 0.6. The evolution of the total system over 160 time units is illustrated separately for the disk, bulge and halo components in figures 6, 7 and 8, respectively. These show that all three components of the constructed model conserve their structural and dynamical properties very well, demonstrating that the constructed model is indeed close to equilibrium as it should. An interesting question arises in connection with the equilibrium of our model: how well do the moments of the velocity distribution in the constructed disk satisfy the equilibrium Jeans equations (see Binney & Tremaine 1987)?            v 2 ϕ = v 2 c + σ 2 R − σ 2 ϕ + R ρ d ∂(ρ d σ 2 R ) ∂R , σ 2 ϕ = σ 2 R R 2vϕ ∂vϕ ∂R + vϕ R , ∂(ρ d σ 2 z ) ∂z = −ρ d ∂Φtot ∂z .(14) Here Φtot is the potential generated by all the components of our model (disk, halo and bulge),vϕ, σR, σϕ and σz are four moments of the velocity distribution in the disk (mean azimuthal velocity and velocity dispersions in the R, ϕ and z directions, respectively). Fig. 9 comparises the radial profiles ofvϕ, σϕ, and σz calculated from the constructed disk and from the Jeans equations (14). It can be seen that the model follows the Jeans equations very well. Note that the moments of velocity distribution both in the Jeans equations and in the constructed disk depend on z. In fig. 9, all moments calculated by means of Jeans equations were calculated for z = 0. We thus took only particles with \|z\| < 0.05 to calculate them from simulations. We also checked that our disk follows the Jeans equations very well in the rest part of the space (not shown here). We want to underline that, according the Jeans equations, the σz(R, z) in our disk is unambiguously defined by the chosen mass model of the galaxy (see third equation of system (14)), as already discussed by . Models with given line-of-sight kinematics In this section we demonstrate the capability of the iterative method to construct models with given line-of-sight kinematics. Let us first calculate the edge-on line-of-sight mean velocityv los (x) and the edge-on line-of-sight velocity dispersion σ los (x) of the disk model constructed in the previous section (model DISK.SVR in section 3.2). These profiles are presented on figure 10. We construct two disk models. The first one, called DISK.MVLOS, with a givenv los (x) and the second one, called DISK.SVLOS. with a given σ los (x). Since we use the disk galaxy mass model described in the previous section, the process is similar to reconstructing DISK.SVR by using line-of-sight kinematic profiles obtained from "observation". . Initial evolutionary stages for the disk of the constructed disk galaxy model. The evolution of the model was calculated with live disk, halo, and bulge components (see also fig. 7 and 8). From left to right, the upper snapshots show the disc views face-on for times 0, 40, 80, 120 and 160 and the grey scales are logarithmically spaced. The middle and bottom panels show the dependence of various disc quantities on the cylindrical radius R at the same times. Here n is the number of particles in concentric cylindrical layers; z 1/2 is the median of the value \|z\|, i.e a measure of the disc thickness (see andvϕ, σ R , σϕ and σz are four moments of the velocity distribution. At the beginning of the evolution (t = 0) the disk has, by construction, the radial dispersion profile given by eq. (12). Our initial model for the iterative method was a "cold" disk where all particles move on circular orbits (see previous section where we constructed DISK.SVR). We made 100 iterations, each with ti = 20. Note that this is twice the number of iterations used for DISK.SVR, because the converge in the case of line-of-sight kinematics is slower. The remaining parameters were taken dt = 1/2 4 , ǫ = 0.04 and θt = 0.9. In order to fix the profile ofv los (x) (for DISK.MVLOS) and the profile of σ los (x) (for DISK.SVLOS) we used the algorithms described in section 2.3.3. The number of layers in these algorithms was n div = 200. Let us check the equilibrium of the DISK.MVLOS and the DISK.SVLOS disks. In both cases we use the halo and bulge constructed in the previous section, because the mass model is the same. For the self-consistent evolution we used the same parameters as in previous section. The evolution of the disks in these models is illustrated in figures 11 and 12, respectively. These show that the constructed disks are as close to equilibrium as they should. It is interesting to compare DISK.SVR and the two disks constructed from it by using line-of-sight kinematics. The three radial profiles of σR and σϕ are visibly different, as can be seen in figure 13. The velocity dispersion in the disk plane is visibly bigger for DISK.MVLOS than for DISK.SVR, especially near the disk periphery. This is also the case for DISK.SVLOS, but to a lesser extent. In general, it is clear that both models DISK.MVLOS and DISK.SVLOS are different from DISK.SVR. This was not expected and must be due to the fact that more than one equilibrium solution exists for the adopted constraints. This must be kept in mind when we apply our method for constructing phase models of real galaxies and we will discuss it more extensively in a forthcoming paper. Figure 7. Initial evolutionary stages for the bulge of the constructed disk galaxy model. The evolution of the model was calculated with live disk, halo, and bulge (see also fig. 6 and 8). The upper snapshots show the bulge viewed edge-on for two moments of time (0, 160); the grey intensities correspond to the logarithms of particle numbers in the pixels. The bottom panels show the dependence of two parameters of the bulge on the spherical radius r for various moments of time. Here n is the number of particles in concentric spherical layers; σr is the velocity dispersion in the r direction (in spherical coordinate system). We demonstrate these parameters in order to show that the model is close to the equilibrium. For astrophysical applications it should be taken into account that the bulge in our model is not spherical. CONCLUSIONS We presented a new method for constructing equilibrium phase models for stellar systems -the iterative method. The aim of this method is to construct equilibrium N -body models with given parameters, or constraints. More specifically, these are a given mass distribution and, if desired, given kinematic properties, parameters, or constraints. Our method is straightforward both conceptually and in its implementation. We believe that it is this simplicity that makes this method so powerful. It simply relies on a constrained, or guided evolution. We let the system reach equilibrium via a dynamical evolution in a number of successive steps. In between two such steps we make sure that the parameters are set to their desired value and/or that the constraints are fulfilled. This means that the evolution is guided towards an equilibrium with the desired parameters and/or constraints. Setting a mass distribution is of course obligatory, but kinematical constraints are not. If we wish to include them, we have the choice of a large number of possibilities, such as setting the radial profile(s) of one, or more moments of the velocity distribution. In this article we described only a few types of kinematic constraints: the profile of radial velocity dispersion, the profile of velocity anisotropy, a condition of velocity isotropy and line-of-sight kinematics. Procedures for further types of kinematic constraints can be easily found following similar techniques. Furthermore, our implementation of the iterative method can be directly applied to systems with arbitrary geometry, i.e. the given mass distribution can be arbitrary and need not have any symmetries. Thus our method can be used in many different applications. We used our iterative method to construct several models. The first one is a triaxial system. The second one is Figure 8. Initial evolutionary stages for the halo of the constructed disk galaxy model. The evolution of the model was calculated with live disk, halo, and bulge components (see also fig. 6 and 7). We show the dependence of various halo parameters on the spherical radius r for various moments of time. Here n is the number of particles in concentric spherical layers (upper left panel); σ θ /σr is the ratio of velocity dispersion in the θ and r directions (upper right panel); σr is velocity dispersion in the radial direction (bottom left panel); σϕ is the velocity dispersion in the ϕ direction (bottom right panel). At the initial moment of time the halo has the profile of σ θ /σr given by (13). a multi-component model of a disk galaxy consisting of a stellar disk with a given radial velocity dispersion profile, a non-spherical bulge and a halo with a given anisotropy profile. We also constructed two disk models with given line-ofsight kinematic. Using self-consistent N -body simulations, we made sure that the models we constructed are indeed very close to equilibrium (see figs. 3, 6, 7, 8, 12 and 11). The iterative method has a number of further applications. It can of course be used for constructing equilibrium initial conditions for N -body modelling of stellar systems. For instance, the iterative method allows one to investigate bar formation in galaxies with a halo having different kinematics. Also the iterative method can be applied for constructing phase models of real galaxies. For example, we can model observational data by constructing phase models with given line-of-sight kinematics, as shown in section 3.3. This paves the way for studies of e.g. the distribution of dark matter in ellipticals, or obtaining phase space models of observed disk galaxies. A further interesting application is the study of the properties of several equilibrium models for a given mass distribution, as for example triaxial systems. The software necessary for the implementation of this method should be thought of in a very modular way, e.g. with different units for the various kinematical constraints, and is very straightforward to write. Nevertheless, we will make our own software publicly available as soon as this paper is accepted, at the address http://www.astro.spbu.ru/staff/seger/soft/. This package will contain also step-by-step examples for constructing models by using the iterative method, including the models described in this article. Our software uses the N -body code gyrfalcON (Dehnen 2000(Dehnen , 2002 and the NEMO package (http://astro.udm.edu/nemo; Teuben 1995). Figure 1 . 1General scheme of the iterative method in the case of an arbitrary dynamical system. Figure 2 . 2Figure 2. The scheme of the iterative method for the case of an N -body system with a given mass distribution and given kinematical parameters. Figure 3 . 3Evolution of the ellipticity for the three projections of the triaxial model constructed with our iterative method. Note that they do not evolve, i.e. that the model is in equilibrium. Figure 4 . 4Rotation curve for the disk galaxy model. The solid line is the total rotation curve. We also show the contributions from the disk, halo and bulge components. Figure 5 . 5Radial profile of the Toomre parameter Q for the disk in the model described in Sect. 3.2. Figure 10 . 10The edge-on line-of-sight mean velocityv los (x) and and velocity dispersion σ los (x) for model DISK.SVR. These parameters are defined in section 2.3.3. Figure 6 6Figure 6. Initial evolutionary stages for the disk of the constructed disk galaxy model. The evolution of the model was calculated with live disk, halo, and bulge components (see also fig. 7 and 8). From left to right, the upper snapshots show the disc views face-on for times 0, 40, 80, 120 and 160 and the grey scales are logarithmically spaced. The middle and bottom panels show the dependence of various disc quantities on the cylindrical radius R at the same times. Here n is the number of particles in concentric cylindrical layers; z 1/2 is the median of the value \|z\|, i.e a measure of the disc thickness (see Sotnikova & Rodionov 2006) andvϕ, σ R , σϕ and σz are four moments of the velocity distribution. At the beginning of the evolution (t = 0) the disk has, by construction, the radial dispersion profile given by eq. (12). Figure 9 . 9Comparison of profiles of the velocity distribution moments calculated from the Jeans equations and from the disk of the constructed disk galaxy model (DISK.SVR). All moments for the disc were calculated inside the region \|z\| < 0.05. Left panel: the solid line corresponds to the valuevϕ for the model, and the dashed line corresponds to the same value calculated from the Jeans equation (the first equation of the system (14)) for z = 0, and where the values σ R and σϕ were taken from the model. Middle panel: the solid line corresponds to the value σϕ for the model, and the dashed line corresponds to the same value calculated from the Jeans equation (the second equation of the system (14)), where the valuesvϕ and σ R were taken from the model. Right panel: the solid line corresponds to the value σz for the model, and the dashed line corresponds to the same value calculated from the Jeans equation (the third equation of the system(14)). Figure 11 . 11Initial evolutionary stages for the model DISK.MVLOS. The evolution of the model was calculated with live disk, halo, and bulge components (we used the halo and bulge constructed in section 3.2). The same values are shown as in middle and bottom panels ofFig. 6. Foundation for Basic Research (grants 06-02-16459 and 08-02-00361a) and by a grant from the President of the Russian Federation for support of Leading Scientific Schools (grant NSh-8542.2006.02). Figure 13 . 13The dependence of four moments of the velocity distribution,namelyvϕ, σ R , σϕ and σz as a function of the cylindrical radius R for DISK.SVR, DISK.MVLOS and DISK.SVLOS. ACKNOWLEDGEMENTSWe thank the referee, J. Dubinski, for helpful comments.This work was partially supported by grants ANR-06-BLAN-0172 and ANR-XX-BLAN-XXXX, by the Russian . E Athanassoula, MNRAS. 3771569Athanassoula E., 2007, MNRAS, 377, 1569 . E Athanassoula, E Fady, J C Lambert, A Bosma, MNRAS. 314475Athanassoula E., Fady, E., Lambert J. C., Bosma A., 2000, MNRAS, 314, 475 . E Athanassoula, J A Sellwood, MNRAS. 213232Athanassoula E., Sellwood J. A., 1986, MNRAS, 213, 232 . J Barnes, ApJ. 331699Barnes J., 1988, ApJ, 331, 699 An Iterative Method for Constructing Equilibrium Phase Models of Stellar Systems 13. 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T A Zang, PhD thesisZang T. A., 1976, Unpub. PhD thesis, MIT	[]
[ "A Thousand Problems in Cosmology: Interaction in the Dark Sector", "A Thousand Problems in Cosmology: Interaction in the Dark Sector" ]	[ "Yu L Bolotin *[email protected] \nA.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine\n", "V A Cherkaskiy \nA.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine\n", "O A Lemets \nA.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine\n", "I V Tanatarov \nA.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine\n\nV.N. Karazin\nKharkov National University\nSvobody Sq. 461077KharkovUkraine\n", "D A Yerokhin \nA.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine\n" ]	[ "A.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine", "A.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine", "A.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine", "A.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine", "V.N. Karazin\nKharkov National University\nSvobody Sq. 461077KharkovUkraine", "A.I.Akhiezer Institute for Theoretical Physics\nNational Science Center \"Kharkov Institute of Physics and Technology\"\nAkademicheskaya Str. 161108KharkovUkraine" ]	[]	The evolution of any broadly applied model is accompanied by multiple generalizations that aim to resolve conceptual difficulties and to explain the ever-growing pool of observational data. In the case of Standard Cosmological Model one of the most promising directions of generalization is replacement of the cosmological constant with a more complicated, dynamic, form of dark energy and incorporation of interaction between the dark components-dark energy (DE) and dark matter (DM). Typically, DE models are based on scalar fields minimally coupled to gravity, and do not implement explicit coupling of the field to the background DM. However, there is no fundamental reason for this assumption in the absence of an underlying symmetry which would suppress the coupling. Given that we do not know the true nature of either DE or DM, we cannot exclude the possibility that there is some kind of coupling between them. Whereas interactions between DE and normal matter particles are heavily constrained by observations (e.g. in the solar system and gravitational experiments on Earth), this is not the case for DM particles. In other words, it is possible for the dark components to interact with each other while not being coupled to standard model particles. Therefore, the possibility of DE-DM interaction should be investigated with utmost gravity.	null	[ "https://arxiv.org/pdf/1312.6556v1.pdf" ]	17,010,268	1312.6556	37913f4f2f2b4c9f4262841ee30410fbe5b03b48	A Thousand Problems in Cosmology: Interaction in the Dark Sector 18 Dec 2013 Yu L Bolotin [email protected] A.I.Akhiezer Institute for Theoretical Physics National Science Center "Kharkov Institute of Physics and Technology" Akademicheskaya Str. 161108KharkovUkraine V A Cherkaskiy A.I.Akhiezer Institute for Theoretical Physics National Science Center "Kharkov Institute of Physics and Technology" Akademicheskaya Str. 161108KharkovUkraine O A Lemets A.I.Akhiezer Institute for Theoretical Physics National Science Center "Kharkov Institute of Physics and Technology" Akademicheskaya Str. 161108KharkovUkraine I V Tanatarov A.I.Akhiezer Institute for Theoretical Physics National Science Center "Kharkov Institute of Physics and Technology" Akademicheskaya Str. 161108KharkovUkraine V.N. Karazin Kharkov National University Svobody Sq. 461077KharkovUkraine D A Yerokhin A.I.Akhiezer Institute for Theoretical Physics National Science Center "Kharkov Institute of Physics and Technology" Akademicheskaya Str. 161108KharkovUkraine A Thousand Problems in Cosmology: Interaction in the Dark Sector 18 Dec 2013arXiv:1312.6556v1 [physics.ed-ph] The evolution of any broadly applied model is accompanied by multiple generalizations that aim to resolve conceptual difficulties and to explain the ever-growing pool of observational data. In the case of Standard Cosmological Model one of the most promising directions of generalization is replacement of the cosmological constant with a more complicated, dynamic, form of dark energy and incorporation of interaction between the dark components-dark energy (DE) and dark matter (DM). Typically, DE models are based on scalar fields minimally coupled to gravity, and do not implement explicit coupling of the field to the background DM. However, there is no fundamental reason for this assumption in the absence of an underlying symmetry which would suppress the coupling. Given that we do not know the true nature of either DE or DM, we cannot exclude the possibility that there is some kind of coupling between them. Whereas interactions between DE and normal matter particles are heavily constrained by observations (e.g. in the solar system and gravitational experiments on Earth), this is not the case for DM particles. In other words, it is possible for the dark components to interact with each other while not being coupled to standard model particles. Therefore, the possibility of DE-DM interaction should be investigated with utmost gravity. 6. Find the effective state parameters w (de)ef f and w (dm)ef f for the case of the warm dark matter (w dm = 0) and analyze the features of dynamics in this case. 7. Show that the quintessence coupled to DM with certain sign of the coupling constant behaves like a phantom uncoupled model, but without negative kinetic energy. 8. In order to compare dynamics of a model with observational results it is useful to analyze all dynamic variables as functions of redshift rather than time. Obtain the corresponding transformation for the system of interacting dark components. 9. Show that energy exchange between dark components leads to time-dependent effective potential energy term in the first Friedman equation. [4] 10. Show that the system of interacting components can be treated as the uncoupled one due to introduction of the partial effective pressure of the dark components Π de ≡ Q 3H , Π dm ≡ − Q 3H . 11. Assume that the mass m dm of dark matter particles depends on a scalar field ϕ. Construct the model of interacting dark energy and dark matter in this case. 12. Assume that the mass m dm of DM particles depends exponentially on the DE scalar field m = m e −λϕ . Find the interaction term Q in this case. 13. Find the equation of motion for the scalar field interacting with dark matter if its particles' mass depends on the scalar field. 14. Make the transformation from the variables (ρ de , ρ dm ) to r = ρ dm ρ de , ρ = ρ dm + ρ de for the system of interacting dark components. 15. Generalize the result of previous problem to the case of warm dark matter. 16. Calculate the derivatives dr/dt and dr/dH for the case of flat universe with the interaction Q. 17. It was shown in the previous problem thaṫ r = r ρ dm ρ dm −ρ de ρ de = 3Hr w de + 1 + r ρ dm Q 3H = (1 + r) 3Hw de r 1 + r + Γ , Γ ≡ Q ρ de . Exclude the interaction Q and reformulate the equation in terms of ρ de , H and its derivatives. 18. Generalize the result, obtained in the previous problem, for the case of non-flat Universe [5]. 19. Show that critical points in the system of equations obtained in problem 14 exist only for the case of dark energy of the phantom type. 20. Show that the result of previous problem holds also for warm dark matter. 21. Show that existence of critical points in the system of equations obtained in problem 14 requires a transfer from dark energy to dark matter. 22. Show that the result of previous problem holds also for warm dark matter. 23. Assume that the ratio of the interacting dark components equals r ≡ ρ dm ρ de ∝ a −ξ , ξ ≥ 0. Analyze how the interaction Q depends on ξ. 24. Show that the choice r ≡ ρ dm ρ de ∝ a −ξ , (ξ ≥ 0) guarantees existence of an early matter-dominated epoch. 25. Find the interaction Q for the Universe with interacting dark energy and dark matter, assuming that ratio of their densities takes the form r ≡ ρ dm ρ de = f (a), where f (a) is an arbitrary differentiable function of the scale factor. Let Q =ḟ (t) f (t ρ dm . Show that the sign of the deceleration parameter is defined by the ratiȯ f f H . 27. Show that in the model, considered in the previous problem, the transition from the accelerated expansion to the decelerated one can occur only due to time dependence of the interaction. SIMPLE LINEAR MODELS 28. Find the scale factor dependence for the dark matter density assuming that the interaction between the dark matter and the dark energy equals Q = δ(a)Hρ dm . 29. Obtain the equation for the evolution of the DE energy density for Q = δ(a)Hρ dm . 30. Find ρ dm and ρ de in the case Q = δHρ dm , δ = const, w de = const. 31. As was shown above, interaction between dark matter and dark energy leads to non-conservation of matter, or equivalently, to scale dependence for the mass of particles that constitute the dark matter. Show that, within the framework of the model of previous problem (Q = δHρ dm , δ = const, w de = const) the relative change of particles mass per Hubble time equals to the interaction constant. 32. Find ρ dm and ρ de in the case Q = δHρ de , δ = const, w de = const. 33. Find ρ dm and ρ de in the case Q = δ(a)Hρ de , δ(a) = β 0 a ξ , w de = const [6]. 34. Let's look at a more general linear model for the expansion of a Universe that contains two interacting fluids with the equations of state p 1 = (γ 1 − 1)ρ 1 , p 2 = (γ 2 − 1)ρ 2 , and energy exchangeρ 1 + 3Hγ 1 ρ 1 = −βHρ 1 + αHρ 2 , ρ 2 + 3Hγ 2 ρ 2 = βHρ 1 − αHρ 2 . Here α and β are constants describing the energy exchanges between the two fluids. Obtain the equation for H(t) and find its solutions [7,8] H →H = −H, ρ →ρ = ρ, p →p = −2ρ − p, γ ≡ ρ + p ρ →γ ≡ρ +p ρ = −γ. Generalize the duality transformation to the case of interacting components [9]. COSMOLOGICAL MODELS WITH A CHANGE OF THE DIRECTION OF ENERGY TRANSFER Let us consider one more type of interaction Q, whose sign (i.e., the direction of energy transfer) changes when the mode of decelerated expansion is replaced by the mode of accelerated expansion, and vice versa. The simplest interaction of this type is the one proportional to the deceleration parameter. An example of such interaction is Q = q(αρ + βHρ), where α and β are dimensionless constants, and ρ can be any of densities ρ de , ρ dm or ρ tot . In the following problems for simplicity we restrict ourselves to the decaying Λ model, for whicḣ ρ de =ρ Λ = −Q and p de = −ρ de . (after [10]) 37. Construct general procedure to the Hubble parameter and the deceleration parameter for the case Q = q(αρ dm + βHρ dm ). 38. Find deceleration parameter for the case α = 0 in the model considered in the previous problem. 39. Consider the model Q = q(αρ Λ + 3βHρ Λ ). Obtain the Hubble parameter H(a) and deceleration parameter for the case α = 0. NON-LINEAR INTERACTION IN THE DARK SEC-TOR The interaction studied so far are linear in the sense that the interaction term in the individual energy balance equations is proportional either to dark matter density or to dark energy density or to a linear combination of both densities. Also from a physical point of view an interaction proportional to the product of dark components seems preferred: an interaction between two components should depend on the product of the abundances of the individual components, as, e.g., in chemical reactions. Moreover, such type of interaction looks more preferable then the linear one when compared with the observations. Below we investigate the dynamics for a simple two-component model with a number of non-linear interactions. In problems 40-43 let a spatially flat FLRW Universe contain perfect fluids with densities ρ 1 and ρ 2 . Consider a nonlinear interaction of the form Q = γρ 1 ρ 2 [11] 40. Consider a model in which both fluids are dust. Find r(t) ≡ ρ 1 (t)/ρ 2 (t). 41. Consider a Universe with more than two CDM components interacting with each other. What is the asymptotic behavior of the individual densities of the components in the limit t → ∞? 42. Consider a Universe containing a cold dark matter and a dark energy, in which the dark energy behaves like a cosmological constant. Show that in such model dark energy is a perpetual component of the Universe. 43. Consider a two-component Universe with the interaction Q = γρ 1 ρ 2 . Let one component is CDM (ρ 1 = ρ dm , w 1 = 0), and the second is the dark energy with arbitrary state equation (ρ 2 = ρ de , w 2 = γ de − 1). (The case considered in the previous problem corresponds to γ de = 0.) Find the relation between the dark energy and dark matter densities. 44. Let interaction term Q be a non-linear function of the energy densities of the components and/or the total energy density. Motivated by the structure ρ dm = r 1 + r ρ, ρ de = r 1 + r ρ, ρ ≡ ρ dm + ρ de , r ≡ ρ dm ρ de consider ansatz Q = 3Hγρ m r n (1 + r) s . where γ is a positive coupling constant. Show that for s = −m interaction term is proportional to a power of products of the densities of the components. For (m, n, s) = (1, 1, −1) and (m, n, s) = (1, 0, −1) reproduce the linear case [12]. where K and n are the polytropic constant and polytropic index, respectively. Find dependence of DE density on the scale factor under assumption that the interaction between the dark components is Q = 3αHρ de [13]. 49. Show that under certain conditions the interacting polytropic dark energy with Q = 3αHρ de behaves as the phantom energy. 50. Find deceleration parameter for the system considered in the problem 48 PHASE SPACE STRUCTURE OF MODELS WITH INTERACTION The evolution of a Universe filled with interacting components can be effectively analyzed in terms of dynamical systems theory. Let us consider the following coupled differential equations for two variablesẋ = f (x, y, t), y = g(x, y, t). (1) We will be interested in the so-called autonomous systems, for which the functions f and g do not contain explicit time-dependent terms. A point (x c , y c ) is said to be a fixed (a.k.a. critical) point of the autonomous system if f (x c , y c ) = g(x c , y c ) = 0. A critical point (x c , y c ) is called an attractor when it satisfies the condition (x(t), y(t)) → (x c , y c ) for t → ∞. Let's look at the behavior of the dynamical system (1) near the critical point. For this purpose, let us consider small perturbations around the critical point x = x c + δx, y = y c + δy. Substituting it into (1) leads to the first-order differential equations: d dN δx δy =M δx δy . Taking into account the specifics of the problem that we are solving, we made the change d dt → d dN , where N = ln a. The matrixM is given bŷ M =   ∂f ∂x ∂f ∂y ∂g ∂x ∂g ∂y   The general solution for the linear perturbations reads δx = C 1 e λ 1 N + C 2 e λ 2 N , δy = C 3 e λ 1 N + C 4 e λ 2 N , The stability around the fixed points depends on the nature of the eigenvalues. Let us treat the interacting dark components as a dynamical system described by the equations ρ ′ de + 3(1 + w de )ρ de = −Q ρ ′ dm + 3(1 + w dm )ρ dm = Q Here, the prime denotes the derivative with respect to N = ln a. Note that although the interaction can significantly change the cosmological evolution, the system is still autonomous. We consider the following specific interaction forms, which were already analyzed above: Q 1 = 3γ dm ρ dm , Q 1 = 3γ de ρ de , Q 1 = 3γ tot ρ tot 51. Find effective EoS parameters w (dm)ef f and w (de)ef f for the interactions Q 1 , Q 2 and Q 3 . Find the critical points of equation for ratio r = ρ dm /ρ de if Q = 3αH(ρ dm + ρ de ), where the phenomenological parameter α is a dimensionless, positive constant, w dm = 0, w de = const. 53. Show, that the remarkable property of the model, considered in the previous problem, is that for the interaction parameter α, consistent with the current observations α < 2.3 × 10 −3 the ratio r tends to a stationary but unstable value at early times, r + s , and to a stationary and stable value, r − s (an attractor), at late times. Consequently, as the Universe expands, r(a) smoothly evolves from r + s to the attractor solution r − s . 54. Transform the system of equations ρ ′ de + 3(1 + w de )ρ de = −Q, ρ ′ dm + 3(1 + w dm )ρ dm = Q, into the one for the fractional density energies [14]. 55. Analyze the critical points of the autonomous system, obtained in the previous problem 56. Construct the stability matrix for the dynamical system considered in the problem 54 and determine its eigenvalues. 57. Using result of the previous problem, determine eigenvalues of the stability matrix for the following cases: i) Ω dm = 1, Ω de = 0, f j = 0; ii) Ω dm = 0, Ω de = 1, f j = 0; iii) f j = 0. 58. Obtain position and type of the critical points obtained in the previous problem for the case of cosmological constant interacting with dark matter as Q = 3γ dm ρ dm . 59. Construct the stability matrix for the following dynamical system ρ ′ = − 1 + w de 1 + r ρ, r ′ = r w de − (1 + r) 2 rρ Π , and determine its eigenvalues [12]. PECULIARITY OF DYNAMICS OF SCALAR FIELD COUPLED TO DARK MATTER Interacting quintessence model Given that the quintessence field and the dark matter have unknown physical natures, there seem to be no a priori reasons to exclude a coupling between the two components. Let us consider a two-component system (scalar field ϕ + dark matter) with the energy density and pressure ρ = ρ ϕ + ρ dm , p = p ϕ + p dm (we do not exclude the possibility of warm DM (p dm = 0).) If some interaction exists between the scalar field and DM, thenρ dm + 3H(ρ dm + p dm ) = Q ρ ϕ + 3H(ρ ϕ + p ϕ ) = −Q. Using the effective pressures Π ϕ and Π dm , Q = −3HΠ dm = 3HΠ ϕ one can transit to the systemρ dm + 3H(ρ dm + p dm + Π dm ) = 0, ρ ϕ + 3H(ρ ϕ + p ϕ + Π ϕ ) = 0. 60. Obtain the modified Klein-Gordon equation for the scalar field interacting with the dark matter. 61. Consider a quintessence scalar field ϕ which couples to the dark matter via, e.g., a Yukawalike interaction f (ϕ/M P l )ψψ, where f is an arbitrary function of ϕ and ψ is a dark matter Dirac spinor. Obtain the modified Klein-Gordon equation for the scalar field interacting with the dark matter in such way [15]. 62. The problems 62-66 are inspired by [16]. Show that the Friedman equation with interacting scalar field and dark matter allow existence of stationary solution for the ratio r ≡ ρ dm /ρ ϕ . 63. Find the form of interaction Q which provides the stationary relation r for interacting cold dark matter and quintessence in spatially flat Universe. 64. For the interaction Q which provides the stationary relation r for interacting cold dark matter and quintessence in spatially flat Universe (see the previous problem), find the dependence of ρ dm and ρ ϕ on the scale factor. 65. Show that in the case of interaction Q obtained in the problem 63, the scalar field ϕ evolves logarithmically with time. 66. Reconstruct the potential V (ϕ), which realizes the solution r = const, obtained in the problem 63. 67. Let the DM particle's mass M depend exponentially on the DE scalar field as M = M * e −λϕ , where λ is positive constant and the scalar field potential is V (ϕ) = V * e ηϕ . Obtain the modified Klein-Gordon equation for this case [17]. 68. Let the DM particle's mass M depend exponentially on the DE scalar field as M = M * e −α , and the scalar field potential is V (ϕ) = V * e β , where α, β > 0. Obtain the modified Klein-Gordon equation for this case. Interacting Phantom Let the Universe contain only noninteracting cold dark matter (w dm = 0) and a phantom field (w de < −1). The densities of these components evolve separately: ρ dm ∝ a −3 and ρ de ∝ a −3(1+w de ) . If matter domination ends at t m , then at the moment of time t BR = w de 1 + w de t m the scale factor, as well as a series of other cosmological characteristics of the Universe become infinite. This catastrophe has earned the name "Big Rip". One of the way to avoid the unwanted big rip singularity is to allow for a suitable interaction between the phantom energy and the background dark matter. 69. Show that through a special choice of interaction, one can mitigate the rise of the phantom component and make it so that components decrease with time if there is a transfer of energy from the phantom field to the dark matter. Consider case of Q = δ(a)Hρ dm and w de = const. 70. Calculate the deceleration parameter for the model considered in the previous problem. 71. Let the interaction Q of phantom field ϕ with DM provide constant relation r = ρ dm /ρ ϕ . Assuming that w ϕ = const, find ρ ϕ (a), ρ ϕ (ϕ) and a(ϕ) for the case of cold dark matter (CDM) [18]. 72. Construct the scalar field potential, which realizes the given relation r for the model considered in the previous problem. Tachyonic Interacting Scalar Field Let us consider a flat Friedmann Universe filled with a spatially homogeneous tachyon field T evolving according to the Lagrangian L = −V (T ) 1 − g 00Ṫ 2 . The energy density and the pressure of this field are, respectively ρ T = V (T ) 1 −Ṫ 2 and p T = −V (T ) 1 −Ṫ 2 . The equation of motion for the tachyon is T 1 −Ṫ 2 + 3HṪ + 1 V (T ) dV dT . (Problems 73-77 are after [19].) 73. Find interaction of tachyon field with cold dark matter (CDM), which results in r ≡ ρ dm /ρ T = const. 74. Show that the stationary solutionṙ = 0 exists only when the energy of the tachyon field is transferred to the dark matter. R µν − f rac12Rg µν = 8πG T µν + Λ 8πG g µν . According to the Bianchi identities, (i) vacuum decay is possible only from a previous existence of some sort of non-vanishing matter and/or radiation, and (ii) the presence of a time-varying cosmological term results in a coupling between T µν and Λ. We will assume (unless stated otherwise) coupling only between vacuum and CDM particles, so that u µ , T (CDM )µν ;ν = −u µ Λg µν 8πG ;ν = −u µ (ρ Λ g µν ) ;ν where T (CDM ) µν = ρ dm u µ u ν is the energy-momentum tensor of the CDM matter and ρ Λ is the vacuum energy density. It immediately follows thaṫ ρ dm + 3Hρ dm = −ρ Λ . Note that although the vacuum is decaying, w Λ = −1 is still constant, the physical equation of state (EoS) of the vacuum w Λ ≡= p Λ /ρ Λ is still equal to constant −1, which follows from the definition of the cosmological constant. (see [20,21] ) 78. Since vacuum energy is constantly decaying into CDM, CDM will dilute in a smaller rate compared with the standard relation ρ dm ∝ a −3 . Thus we assume that ρ dm = ρ dm0 a −3+ε , where ε is a small positive small constant. Find the dependence ρ Λ (a) in this model. 79. Solve the previous problem for the case when vacuum energy is constantly decaying into radiation. 80. Show that existence of a radiation dominated stage is always guaranteed in scenarios, considered in the previous problem. 81. Find how the new temperature law scales with redshift in the case of vacuum energy decaying into radiation. Since the energy density of the cold dark matter is ρ dm = nm, there are two possibilities for storage of the energy received from the vacuum decay process: (i) the equation describing concentration, n, has a source term while the proper mass of CDM particles remains constant; (ii) the mass m of the CDM particles is itself a time-dependent quantity, while the total number of CDM particles, N = na 3 , remains constant. Let us consider both the possibilities. Vacuum decay into CDM particles 82. Find dependence of total particle number on the scale factor in the model considered in problem 78. 85. Show that the model considered in the previous problem correctly reproduces the scale factor evolution both in the radiation-dominated and non-relativistic matter (dust) dominated cases. 86. Find the dependencies ρ γ (a) and Λ(a) both in the radiation-dominated and non-relativistic matter dominated cases in the model considered in problem 84. 96. Find statefinder parameters for interacting dark energy and cold dark matter. Show that for the Λ(t) models T dS dt = −ρ Λ a 3 . Consider a two-component 97. Show that the statefinder parameter r is generally necessary to characterize any variation in the overall equation of state of the cosmic medium. 98. Find relation between the statefinder parameters in the flat Universe. 99. Express the statefinder parameters in terms of effective state parameter w (de)ef f , for whicḣ ρ de + 3H(1 + w (de)ef f )ρ de = 0. 100. Find the statefinder parameters for Q = 3δHρ dm , assuming that w de = const. 101. Find statefinder parameters for the case ρ dm /ρ de = a −ξ , where ξ is a constant parameter in the range [0, 3] and w de = const. 102. Show that in the case ρ dm /ρ de = a −ξ the current value of the statefinder parameter s = s 0 can be used to measure the deviation of cosmological models from the SCM. 103. Find how the statefinder parameters enter the expression for the luminosity distance. INTERACTING HOLOGRAPHIC DARK ENERGY The traditional point of view assumed that dominating part of degrees of freedom in our World are attributed to physical fields. However it became clear soon that such concept complicates the construction of Quantum Gravity: it is necessary to introduce small distance cutoffs for all integrals in the theory in order to make it sensible. As a consequence, our World should be described on a three-dimensional discrete lattice with the period of the order of Planck length. Lately some physicists share an even more radical point of view: instead of the three-dimensional lattice, complete description of Nature requires only a two-dimensional one, situated on the space boundary of our World. This approach is based on the so-called "holographic principle". The name is related to the optical hologram, which is essentially a two-dimensional record of a three-dimensional object. The holographic principle consists of two main statements: 1. All information contained in some region of space can be "recorded" (represented) on the boundary of that region. 2. The theory, formulated on the boundaries of the considered region of space, must have no more than one degree of freedom per Planck area: N ≤ A A pl , A pl = G c 3 .(3) Thus, the key piece in the holographic principle is the assumption that all the information about the Universe can be encoded on some two-dimensional surface -the holographic screen. Such approach leads to a new interpretation of cosmological acceleration and to an absolutely unusual understanding of Gravity. The Gravity is understood as an entropy force, caused by variation of information connected to positions of material bodies. More precisely, the quantity of information related to matter and its position is measured in terms of entropy. Relation between the entropy and the information states that the information change is exactly the negative entropy change ∆I = −∆S. Entropy change due to matter displacement leads to the so-called entropy force, which, as will be proven below, has the form of gravity. Its origin therefore lies in the universal tendency of any macroscopic theory to maximize the entropy. The dynamics can be constructed in terms of entropy variation and it does not depend on the details of microscopic theory. In particular, there is no fundamental field associated with the entropy force. The entropy forces are typical for macroscopic systems like colloids and biophysical systems. Big colloid molecules, placed in thermal environment of smaller particles, feel the entropy forces. Osmose is another phenomenon governed by the entropy forces. Probably the best known example of the entropy force is the elasticity of a polymer molecule. A single polymer molecule can be modeled as a composition of many monomers of fixed length. Each monomer can freely rotate around the fixation point and choose any spacial direction. Each of such configurations has the same energy. When the polymer molecule is placed into a thermal bath, it prefers to form a ring as the entropically most preferable configuration: there are many more such configurations when the polymer molecule is short, than those when it is stretched. The statistical tendency to transit into the maximum entropy state transforms into the macroscopic force, in the considered case-into the elastic force. Let us consider a small piece of holographic screen and a particle of mass m approaching it. According to the holographic principle, the particle affects the amount of the information (and therefore of the entropy) stored on the screen. It is natural to assume that entropy variation near the screen is linear on the displacement ∆x: ∆S = 2πk B mc ∆x.(4) The factor 2π is introduced for convenience, which the reader will appreciate solving the problems of this section. In order to understand why this quantity should be proportional to mass, let us imagine that the particle has split into two or more particles of smaller mass. Each of those particles produces its own entropy change when displaced by ∆x. As entropy and mass are both additive, then it is natural that the former is proportional to the latter. According to the first law of thermodynamics, the entropy force related to information variation satisfies the equation F ∆x = T ∆S.(5) If we know the entropy gradient, which can be found from (4), and the screen temperature, we can calculate the entropy force. An observer moving with acceleration a, feels the temperature (the Unruh temperature) k B T U = 1 2π c a.(6) Let us assume that the total energy of the system equals E. Let us make a simple assumption that the energy is uniformly distributed over all N bits of information on the holographic screen. The temperature is then defined as the average energy per bit: E = 1 2 Nk B T.(7) Equations (4)-(7) allow one to describe the holographic dynamics, and as a particular case-the dynamics of the Universe, and all that without the notion of Gravity. 104. For the interacting holographic dark energy Q = 3αHρ L , with the Hubble radius as the IR cutoff, find the depending on the time for the scale factor, the Hubble parameter and the deceleration parameter. 105. Show that for the choice ρ hde ∝ H 2 (ρ hde = βH 2 , β = const)an interaction is the only way to have an equation of state different from that of the dust. 106. Calculate the derivative dρ de d ln a for the holographic dark energy model, where IR cut-off L is chosen to be equal to the future event horizon [24]. 107. Find the effective state parameter value w ef f , such that ρ ′ de + 3(1 + w ef f )ρ de = 0 for the holographic dark energy model, considered in the previous problem, with the interaction of the form Q = 3αHρ de 108. Analyze how fate of the Universe depends on the parameter c in the holographic dark energy model, where IR cut-off L is chosen to be equal to the future event horizon. 109. In the case of interacting holographic Ricci dark energy with interaction is given by Q = γHρ R ,(8) where γ is a dimensionless parameter, find the dependence of the density of dark energy and dark matter on the scale factor. 110. Find the exact solutions for linear interactions between Ricci DE and DM, if the energy density of Ricci DE is given by ρ x = 2Ḣ + 3αH 2 /∆, where ∆ = α − β and α, β are constants. 111. Find the equation of motion for the relative density TRANSIENT ACCELERATION Unlike fundamental theories, physical models only reflect the current state of our understanding of a process or phenomenon for the description of which they were developed. The efficiency of a model is to a significant extent determined by its flexibility, i.e., its ability to update when new information appears. Precisely for this reason, the evolution of any broadly applied model is accompanied by numerous generalizations aimed at resolving conceptual problems, as well as a description of the ever increasing number of observations. In the case of the SCM, these generalizations can be divided into two main classes. The first is composed of generalizations that replace the cosmological constant with more complicated dynamic forms of DE, for which the possibility of their interaction with DM must be taken into account. Generalizations pertaining to the second class are of a more radical character. The ultimate goal of these generalizations (explicit or latent) consists in the complete renunciation of dark components by means of modifying Einstein's equations. The generalizations of both the first and second classes can be demonstrated by means of a phenomenon that has been termed "transient acceleration". A characteristic feature of the dependency of the deceleration parameter q on the redshift z in the SCM is that it monotonically tends to its limit value q(z) = 1 as z → 1. Physically, this means that when DE became the dominant component (at z ∼ 1), the Universe in the SCM was doomed to experience eternal accelerating expansion. In what follows, we consider several cosmological models that involve dynamic forms of DE that lead to transition acceleration, and we also discuss what the observational data says about the modern rate of expansion of the Universe. Barrow [26] was among the first to indicate that transient acceleration is possible in principle. He showed that within quite sound scenarios that explain the current accelerated expansion of the Universe, the possibility was not excluded of a return to the era of domination of nonrelativistic matter and, consequently, to decelerating expansion. Therefore, the transition to accelerating expansion does not necessarily mean eternal accelerating expansion. Moreover, in Barrows article, it was shown to be neither the only possible nor the most probable course of events. Consider a simple model of transient acceleration with decaying cosmological constanṫ ρ m + 3ȧ a ρ m = −ρ Λ ,(9) where ρ m and ρ Λ energy density DE and cosmological constant Λ. At the early stages of the expansion of the Universe, when ρ Λ is quite small, such a decay does not influence cosmological evolution in any way. At later stages, as the DE contribution increases, its decay has an ever increasing effect on the standard dependence of the DM energy density ρ m ∝ a −3 on the scale factor a. We consider the deviation to be described by a function of the scale factor -ǫ(a). ρ m = ρ m,0 a −3+ǫ(a) ,(10) where a 0 = 1 in the present epoch. Other fields of matter (radiation, baryons) evolve independently and are conserved. Hence, the DE density has the form ρ Λ = ρ m0 1 a ǫ(ã) +ãǫ ′ ln(ã) a 4−ǫ(ã) dã + X ,(11) where the prime denotes the derivative with respect to the scale factor, and X is the integration constant. If radiation is neglected, the first Friedmann equation takes the form H = H 0 Ω b,0 a −3 + Ω m0 ϕ(a) + Ω X,0 1/2 ,(12) Using the assumption that the function ǫ(a) has the following simple form ǫ(a) = ǫ 0 a ξ = ǫ 0 (1 + z) −ξ ,(13) where ǫ 0 and ξ can take both positive and negative values, find function ϕ(a) and relative energy density Ω b (a), Ω m (a) and Ω Λ (a). 113. Using the results of the previous problem, find the deceleration parameter for this model is V (φ) = ρ φ 0 [1 − λ 6 (1 + α √ σφ) 2 )] exp [−λ √ σ(φ + α √ σ 2 φ 2 )],(14) where ρ φ 0 is a constant energy density, σ = 8πG/λ, and α and λ are two dimensionless, positive parameters of the model, that the deceleration parameter is non-monotonically dependent on the scale factor. Plot the deceleration parameter as a function of the scale factor. 115. Consider a simple parameterization: Q = 3β(a)Hρ de(15) with a simple power-law ansatz for β(a), namely: β(a) = β 0 a ξ .(16) Substituting this interaction form into conservation equations for DM and DE: ρ dm + 3Hρ dm = Q,(17) ρ de + 3H(ρ de + p de ) = −Q, we get ρ de = ρ de0 a −3(1+w 0 ) · exp 3β 0 (1 − a ξ ) ξ ,(19) where the integration constant ρ de0 is value of the dark energy at present, and the dark energy EoS parameter w ≡ p de /ρ de is a constant-w 0 . Substituting Eq. (19) into Eq. (18), we get the dark matter energy density, ρ dm = f (a)ρ dm0 ,(20) where f (a) ≡ 1 a 3 1 − Ω de0 Ω dm0 3β 0 a −3w 0 e 3β 0 ξ ξ · a ξ E3w 0 ξ 3β 0 a ξ ξ − a 3w 0 E3w 0 ξ 3β 0 ξ ,(21) where ρ dm0 is dark matter density at present day, and E n (z) = ∞ 1 t −n e −xt dt the usual exponential integral function. Note however that Eq. (20) is an analytical expression, while in the corresponding expressions were left as integrals and were calculated numerically. Obviously, in the case of non-interaction (that is, for β 0 = 0), Eq. (20) recovers the standard result ρ dm = ρ dm0 /a 3 . For the special case ξ = 0 find dimensionless Hubble parameter E 2 (z) ≡ H 2 H 2 0 , the evolution of the density parameters Ω b (a), Ω dm (a) and Ω de (a) and q(a). For what values of β 0 the cosmic acceleration is transient? 116. Consider the flat FLRW cosmology with two coupled homogeneous scalar fields Φ and Ψ: ρ b = −3Hγ b ρ b (22) Φ = −3HΦ − ∂ Φ V (23) Ψ = −3HΨ − ∂ Ψ V (24) H = −4πG(γ m ρ m + γ r ρ r + γ Q ρ Q ) ,(25)H 2 = 8πG 3 (ρ m + ρ r + ρ Q ) − k a 2 .(26) Here a dot denotes a derivative with respect to the cosmic time t, the subscript b refers to the dominant background quantity, either dust (m) or radiation (r) while Q refers to the Dark Energy sector, here the two quintessence scalar fields. The quintessence fields with potential V have the following energy density and pressure: ρ Q = 1 2Φ 2 + 1 2Ψ 2 + V (Φ, Ψ)(27)p Q = 1 2Φ 2 + 1 2Ψ 2 − V (Φ, Ψ)(28) with p Q = (γ Q − 1)ρ Q . It is convenient to define the following new variables : X Φ = 8πG 3H 2Φ √ 2 , X Ψ = 8πG 3H 2Ψ √ 2 , X V = 8πG 3H 2 √ V .(29) Find expressions for the Φ ′ , Ψ ′ , X ′ Φ , X ′ Ψ , X ′ V where a prime denotes a derivative with respect to the quantity N, the number of e-folds with respect to the present time, N ≡ ln a a 0 ,(30) and we have also H =Ṅ. 117. Using result from the previous problem find the relative energy density for matter, radiation and quintessence, Ω m , Ω r and Ω Q , the deceleration parameter q, the Hubble-parameter-free luminosity distance D L and the age of the Universe t 0 . 45 . 45Find analytical solution of non-linear interaction model covered by the ansatz of previous problem for (m, n, s) = (1, 1, −2), Q = 3Hγρ de ρ dm /ρ 46. Find analytical solution of non-linear interaction model for (m, n, s) = (1, 2, −2), Q = 3Hγρ 2 dm /ρ. 47. Find analytical solution of non-linear interaction model for (m, n, s) = (1, 0, −2), Q = 3Hγρ 2 de /ρ. 48. Consider a flat Universe filled by CDM and DE with a polytropic equation of state 83 . 83Find time dependence of CDM particle mass in the case when there is no creation of CDM particles in the model considered in problem 78. 84. Consider a model where the cosmological constant Λ depends on time as Λ = σH. Let a flat Universe be filled by the time-dependent cosmological constant and a component with the state equation p γ = (γ − 1)ρ γ . Find solutions of Friedman equations for this system [22] . Universe filled by matter with the state equation p = wρ and cosmological constant and rewrite the second Friedman equation in the following form 89. Consider a two-component Universe filled by matter with the state equation p = wρ and cosmological constant with quadratic time dependence Λ(τ ) = Aτ 2 and find the time dependence for of the scale factor. 90. Consider a flat two-component Universe filled by matter with the state equation p = wρ and cosmological constant with quadratic time dependence Λ(τ ) = Aτ ℓ . Obtain the differential equation for Hubble parameter in this model and classify it. 91. Find solution of the equation obtained in the previous problem in the case ℓ = 1. Analyze the obtained solution. 92. Solve the equation obtained in the problem 90 for ℓ = 2. Consider the following cases a) λ 0 > −1/(3γτ 0 ) 2 , b) λ 0 = −1/(3γτ 0 ) 2 , c) λ 0 < −1/(3γτ 0 ) 2 (see the previous problem). Analyze the obtained solution. 93. Consider a flat two-component Universe filled by matter with the state equation p = wρ and cosmological constant with the following scale factor dependence Λ = B a −m . (2) Find dependence of energy density of matter on the scale factor in this model. 94. Find dependence of deceleration parameter on the scale factor for the model of previous problem. 9 STATEFINDER PARAMETERS FOR INTERACTING DARK ENERGY AND COLD DARK MATTER.95. Show that in flat Universe both the Hubble parameter and deceleration parameter do not depend on whether or not dark components are interacting. Become convinced the second derivativeḦ does depend on the interaction between the components[23]. q(a). Draw the graph deceleration parameter as a function of log(a) for various values of ǫ 0 and ξ: ξ = 1.0 and ǫ 0 = 0.1, ξ = −1.0 and ǫ 0 = 0.1, ξ = 0.8 and ǫ 0 = 0.5, ξ = −0.5 and ǫ 0 = −0.1.. 114. Consider the possibility of an accelerating transient regime within the interacting scalar field model using potential of the form 36. The Hubble parameter is present in the first Friedmann equation quadratically. This gives rise to a useful symmetry within a class of FLRW models. Because of this quadratic dependence, Friedmann's equation remains invariant under a transformation H → −H for the spatially flat case. This means it describes both expanding and contracting solutions. The transformation H → −H can be seen as a consequence of the change a → 1/a of the scale factor of the FLRW metric. If, instead of just the first Friedmann equation, we want to make the whole system of Universe-describing equations invariant relative to this transformation, we must expand the set of values that undergo symmetry transformations. Then, when we refer to a duality transformation, we have in mind the following set of transformations). 35. Show that the energy balance equations (modified conservation equations) for Q ∝ H do not depend on H. 75 . 75Find the modified Klein-Gordon equation for arbitrary interaction Q of tachyon scalar field with dark matter. 76. Find the modified Klein-Gordon equation for the interaction Q obtained in the problem 73 and obtain its solutions for the caseφ = const.77. Show that sufficiently small values of tachyon field provide the accelerated expansion of Universe. 8 REALIZATION OF INTERACTION IN THE DARK SECTOR 8.1 Vacuum Decay into Cold Dark Matter Let us consider the Einstein field equations M Hoffman, arXiv:astro-ph/0307350Cosmological constraints on a dark matter-dark energy interaction. M. Hoffman, Cosmological constraints on a dark matter-dark energy interaction, arXiv:astro-ph/0307350 Josue De-Santiago, David Wands, Yuting Wang, arXiv:astro-ph/1209.0563Inhomogeneous and interacting vacuum energy. Josue De-Santiago, David Wands, Yuting Wang, Inhomogeneous and interacting vacuum en- ergy, arXiv:astro-ph/1209.0563 Reconstruction of the interaction term between dark matter and dark energy using SNe Ia, BAO, CMB, H(z) and X-ray gas mass fraction. 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[ "Permutation-like Matrix Groups with a Maximal Cycle of Prime Square Length", "Permutation-like Matrix Groups with a Maximal Cycle of Prime Square Length" ]	[ "Guodong Deng \nSchool of Mathematics and Statistics Central\nChina Normal University\n430079WuhanChina\n", "Yun Fan \nSchool of Mathematics and Statistics Central\nChina Normal University\n430079WuhanChina\n" ]	[ "School of Mathematics and Statistics Central\nChina Normal University\n430079WuhanChina", "School of Mathematics and Statistics Central\nChina Normal University\n430079WuhanChina" ]	[]	A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime.	10.1016/j.laa.2014.02.048	[ "https://arxiv.org/pdf/1311.6583v1.pdf" ]	119,302,527	1311.6583	d21d17a934d3f5980f990a9c38704732d3f86252	Permutation-like Matrix Groups with a Maximal Cycle of Prime Square Length 26 Nov 2013 Guodong Deng School of Mathematics and Statistics Central China Normal University 430079WuhanChina Yun Fan School of Mathematics and Statistics Central China Normal University 430079WuhanChina Permutation-like Matrix Groups with a Maximal Cycle of Prime Square Length 26 Nov 2013arXiv:1311.6583v1 [math.GR]matrix groupmatrix similaritypermutation-like grouppermutation matrix group Mathematics Subject Classification 2010: 15A1815A3020H20 A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime. Introduction A multiplicative group consisting of complex invertible matrices of size n × n is said to be a matrix group of dimension n. A matrix group G is said to be permutation-like if any matrix of G is similar to a permutation matrix, see [2,3]. If there exists an invertible matrix Q such that Q −1 AQ is a permutation matrix for all A ∈ G, then we say that G is similar to a permutation matrix group, or G is a permutation matrix group for short. A matrix is called a maximal cycle if it is similar to a permutation matrix corresponding to a cycle permutation with cycle length equal to the dimension. G. Cigler in [3] showed that a permutationlike matrix group is not a permutation matrix group in general, and suggested a conjecture as follows. Conjecture. A permutation-like matrix group containing a maximal cycle is similar to a permutation matrix group. G. Cigler in [3] proved it affirmatively in two cases: the dimension ≤ 5, or the dimension is a prime integer and the cyclic subgroup generated by the maximal cycle is normal. In this paper we extend the result of [3] to the case where the length of the maximal cycle is a square of a prime. Theorem 1.1. Let G be a permutation-like matrix group of dimension p 2 where p is a prime. If G contains a maximal cycle C such that the subgroup C generated by C is normal in G, then G is a permutation matrix group. In Section 2 we state some preliminaries as a preparation. The theorem will be proved in Section 3. Preparation The complex field is denoted by C. For a positive integer n, by Z * n we denote the multiplicative group consisting of units of the residue ring Z n of the integer ring Z modulo n. A diagonal blocked matrix    B 1 . . . B k    is denoted by B 1 ⊕ · · · ⊕ B k for short. The identity matrix of dimension n is denoted by I n×n . All complex invertible matrices of dimension n consist the so-called general linear group, denoted by GL n (C). We denote the characteristic polynomial of a complex matrix A by char A (x). Lemma 2.1. The following two are equivalent to each other: (i) A is similar to a permutation matrix; (ii) A is diagonalizable and char A (x) = i (x ℓi − 1). If it is the case, then each factor x ℓi − 1 of char A (x) corresponds to exactly one ℓ i -cycle of the cycle decomposition of the permutation of the permutation matrix. Proof. It is clear. We'll apply the lemma to the case where A p 2 = 1 with p being a prime, at that case it is easy to check the condition (ii) of the lemma, because there are only three divisors 1, p, p 2 of p 2 which form a chain with respect to the division relation, and x p 2 − 1 = Φ 1 (x)Φ p (x)Φ p 2 (x) where Φ p i (x) denotes the p i 'th cyclotomic polynomial. Let C ∈ GL n (C) be a maximal cycle of dimension n, and λ be a primitive n'th root of unity. Then there is a basis of the vector space C n : e 0 , e 1 , · · · , e n−1 , (2.1) such that C i e j = λ ij e j for all i, j = 0, 1, · · · , n−1; with respect to this basis, C is a diagonal matrix C = 1⊕λ⊕· · ·⊕λ n−1 ; see [3, §4]. And, such basis is unique up to non-zero scales, since the 1-dimensional subspace Ce j , for j = 0, · · · , n − 1, is just the eigen-subspace of the eigenvalue λ j , for j = 0, · · · , n − 1 respectively, of the matrix C; or, in representation theoretic notations, Ce j for j = 0, · · · , n − 1 are just all irreducible modules of the cyclic group C , see [1, §15 Example 1]. Taking any non-zero complexes c 0 , · · · , c n−1 and setting f = n−1 j=0 c j e j , we obtain another basis of C n : f, Cf, · · · , C n−1 f, (2.2) with which C is a cycle permutation matrix, see [3,Lemma 4.1]. Let B ∈ GL n (C) with ord(B) = ℓ, where ord(B) denotes the order of B. Assume that B normalizes the group C . Since the automorphism group of the cyclic group C is isomorphic to Z * n , there is an r ∈ Z * n such that B −1 C i B = C ri , ∀ i ∈ Z n ; (2.3) thus the action by conjugation of B on C is determined by the action of µ r on Z n , where µ r (a) = ra for all a ∈ Z n . Further, C i Be j = B · B −1 C i Be j = B · C ri e j = B · λ rij e j = λ rij Be j ; taking i = 1, we see that Be j is an eigenvector of the eigenvalue λ rj of C, i.e. BCe j = Ce rj , j = 0, 1, · · · , n − 1. (2.4) Thus, B permutes the eigen-subspaces Ce 0 , Ce 1 , · · · , Ce n−1 of C in the same way as µ r permutes Z n . Let Γ 1 , Γ 2 , · · · , Γ m be orbits of the action of the group B on the set of 1-dimensional subspaces {Ce 1 , Ce 2 , · · · , Ce n }. Assume that the length of Γ k is n k for k = 1, · · · , m. Since \| B \| = ℓ where \| B \| denotes the order of the group B , we have that n k \|ℓ for k = 1, · · · , m. Lemma 2.2. Let notation be as above. (1) For each k, take any one Ce j k ∈ Γ k and any non-zero e ′ k ∈ Ce j k , set V k = n k −1 h=0 B h Ce ′ k and E k = {e ′ k , Be ′ k , · · · , B n k −1 e ′ k }; then B n k e ′ k = ω k e ′ k where ω k is an (ℓ/n k )'th root of unity, V k is a B-invariant subspace of C n , and E k is a basis of V k , with which the matrix of B restricted to V k is B\| V k =       0 · · · 0 ω k 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       n k ×n k . (2.5) (2) C n = V 1 ⊕ · · · ⊕ V m , the union E = E 1 ∪ · · · ∪ E m is a basis of C n and, with respect to the basis E, the matrix of B is B = B\| V1 ⊕ · · · ⊕ B\| Vm . (2.6) Proof. (1). Since the length of Γ k is n k , Γ k = Ce ′ k , BCe ′ k , · · · , B n k −1 Ce ′ k and B n k Ce ′ k = Ce ′ k , hence there is an ω k ∈ C such that B n k e ′ k = ω k e ′ k . Then it is clear that V k is B-invariant, E k is a basis of V k and Eqn (2.5) is the matrix of B\| V k . Since B ℓ = I n×n , we have (B\| V k ) ℓ = I n k ×n k ; but by Eqn (2.5), (B\| V k ) n k = ω k I n k ×n k ; so ω k is an (ℓ/n k )'th root of unity. (2). Applying (1) to all orbits Γ 1 , · · · , Γ m , by Eqn (2.1) one can check the conclusions in (2) easily. Proposition 2.3. Let notation be as in Lemma 2.2. Assume that the following condition is satisfied: (SC) For any e j and B i , if B i Ce j = Ce j then B i e j = e j . Then the matrix group C, B generated by C and B is a permutation matrix group. Proof. We keep the notations in Lemma 2.2 and its proof. We have seen that B n k Ce ′ k = Ce ′ k ; by the condotion (SC) we have B n k e ′ k = e ′ k , i.e. ω k = 1 and B\| V k =       0 · · · 0 1 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       n k ×n k ; (2.7) hence B n k −1 h=0 B h e ′ k = n k −1 h=0 BB h e ′ k = n k −1 h=0 B h e ′ k . Now we set f = m k=1 n k −1 h=0 B h e ′ k ; then Bf = B m k=1 n k −1 h=0 B h e ′ k = m k=1 n k −1 h=0 BB h e ′ k = m k=1 n k −1 h=0 B h e ′ k = f. By Eqn (2.2) the set of the vectors: f, Cf, · · · , C n−1 f, is a basis of C n ; and with respect to this basis C is a cycle permutation matrix. Further, by Eqn (2.3) we have BC i f = BC i B −1 Bf = C r −1 i f ; that is, with respect to the basis f, Cf, · · · , C n−1 f , the B is also a permutation matrix. In conclusion, the matrix group C, B generated by C and B is a permutation matrix group. We'll quote a result of [3] repeatedly, so state it as a lemma: Lemma 2.4. ([3, Proposition 4.2]) If C, B is an abelian permutation-like matrix group where C is a maximal cycle, then B ∈ C . We state some group-theoretic information as a remark for later quotations. We say that an action of a group G on a set X is free if the stabilizer of any X ∈ X in G is trivial; at that case, X is partitioned in to G-orbits such that each orbit is a regular G-set (i.e. equivalent to the set G on which the group G acts by left translation). Remark 2.5. Let p be an odd prime. (1) Let C = C be a cyclic group of order p 2 . It is easy to see that C p = C p where C p = {X p \| X ∈ C}; and, mapping X ∈ C to X p ∈ C p is a surjective homomorphism from C onto C p with kernel C p . (2) Let G be a finite group containing a normal cyclic subgroup C = C of order p 2 such that C is self-centralized (i.e. the centralizer C G (C) = C). Then G/C is isomorphic to a subgroup of the automorphism group of C, hence to a subgroup of the multiplicative group Z * p 2 which is a cyclic group of order p(p−1); so there are an r ∈ Z * p 2 and a B ∈ G such that G = B, C and C B = C r where C B = B −1 CB denotes the conjugate of C by B; and the order ord(r) = \|G/C\|. (5) If B p = 1 and r = p + 1 as in (4), then it is easy to check that: (i) B centralizes the subgroup C p = C p of C, and acts freely by conjugation on the difference set C − C p ; in particular, C − C p is partitioned into p − 1 conjugacy classes by B , the length of every class is p. (ii) For any X ∈ C the product p−1 j=0 (X) B j ∈ C p ; the mapping X to p−1 j=0 (X) B j is a surjective homomorphism from C onto C p ; hence the homomorphism induces a bijection from the set of the conjugacy classes in C − C p on to the set C p − {1}. Proof of Theorem 1.1 If p = 2 then p 2 = 4 and the conclusion of Theorem 1.1 has been checked in [3]. In the following, we always assume that p is an odd prime and G is a permutation-like matrix group of dimension p 2 which contains a normal cyclic subgroup C generated by a maximal cycle C; and prove that G is a permutation matrix group. If G is abelian, by Lemma 2.4, G = C which is a permutation matrix group. So we further assume that G is non-abelian. Let λ be a primitive p 2 'th root of unity, By Eqn (2.1) there is a basis e 0 , e 1 , · · · , e p 2 −1 of C p 2 such that C i e j = λ ij e j , i, j = 0, 1, · · · , p 2 − 1. Let C = C and q = \|G/C\|. By Lemma 2.4, C is self-centralized in G; by Remark 2.5(2), there are a B ∈ G, an r ∈ Z * p 2 and integers s, t such that • q = p δ s where δ = 0 or 1, p − 1 = st; • G = B · C and the quotient B B ∩ C ∼ = r ≤ Z * p 2 ; • B −1 CB = C r , i.e. the action by conjugation of B B ∩ C on C is equivalent to the action of µ r on the residual set Z p 2 , where µ r (a) = ra for all a ∈ Z p 2 . The residual set Z p 2 is a disjoint union of two µ r -stable subsets: Z p 2 = Γ 0 ∪ Z * p 2 , where Γ 0 = Z p 2 − Z * p 2 = {0, p, 2p, · · · , (p − 1)p}; (3.1) in fact, Γ 0 corresponds to the subgroup C p of C. We prove the theorem in three cases. There are t orbits of µ r on Γ 0 − {0}; taking representatives v 1 , · · · , v t from the t orbits, we can write the orbits of µ r on Γ 0 as follows: Γ 00 = {0}, Γ 01 = {v 1 , rv 1 , · · · , r s−1 v 1 }, · · · , Γ 0t = {v t , rv t , · · · , r s−1 v t }. There are m orbits of µ r on Z * p 2 where m = p(p−1) s = pt; taking representatives w 1 , · · · , w m from these m orbits, we have the orbits of µ r on Z * p 2 as follows: Γ 1 = {w 1 , rw 1 , · · · , r s−1 w 1 }, · · · , Γ m = {w m , rw m , · · · , r s−1 w m }. Accordingly, we apply Lemma 2.2 and its notation to get the basis of C p 2 : E = E 00 ∪ E 01 ∪ · · · ∪ E 0t ∪ E 1 ∪ · · · ∪ E m , and write matrices with respect to this basis. So C = 1 ⊕ s−1 ⊕ i=0 λ r i v1 ⊕ · · · ⊕ s−1 ⊕ i=0 λ r i vt ⊕ s−1 ⊕ i=0 λ r i w1 ⊕ · · · ⊕ s−1 ⊕ i=0 λ r i wm , and there is an s'th root ε 0 of unity such that B = ε 0 ⊕ t+m P ⊕ · · · ⊕ P , where P is the cycle matrix of dimension s (see Lemma 2.2 and Eqn (2.7)): P =       0 · · · 0 1 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       s×s . Then the characteristic polynomial of B is char B (x) = (x − ε 0 )(x s − 1) t+m ; since B is similar to a permutation matrix, by Lemma 2.1, we have ε 0 = 1. So the condition (SC) of Proposition 2.3 is satisfied: (SC) For any e j and B i , if B i Ce j = Ce j then B i e j = e j ; hence G is a permutation matrix group. Case 2. s = 1, i.e. q = p. By Remark 2.5(4), we can assume that B p = 1 and the conjugation of B on C is equivalent to the action of µ p+1 on Z p 2 , where µ p+1 a = (p + 1)a for a ∈ Z p 2 ; further, µ p+1 centralizes the subset Γ 0 in Eqn (3.1), and µ p+1 partitions Z * p 2 into p−1 µ p+1 -orbits, see Remark 2.5(5.i); take representatives u 1 , · · · , u p−1 from each µ p+1 -orbit, the p − 1 orbits can be written as: Γ 1 = {u 1 , (p + 1)u 1 , · · · , (p + 1) p−1 u 1 }, Γ 2 = {u 2 , (p + 1)u 2 , · · · , (p + 1) (p−1) u 2 }, · · · Γ p−1 = {u p−1 , (p + 1)u p−1 , · · · , (p + 1) (p−1) u p−1 }. Accordingly, we apply Lemma 2.2 and its notation to get the basis of C p 2 : E = E 0 ∪ E 1 ∪ · · · ∪ E p−1 , where E k is corresponding to Γ k for k = 1, · · · , p − 1 as above, while E 0 is corresponding to Γ 0 ; and we write matrices with respect to this basis. So C = D 0 ⊕ D 1 ⊕ · · · ⊕ D p−1 , where D 0 = 1 ⊕ λ p ⊕ · · · ⊕ λ (p−1)p ,(3. 2) D i = λ ui ⊕ λ (p+1)ui ⊕ · · · ⊕ λ (p+1) p−1 ui , i = 1, · · · , p − 1; and B = B 0 ⊕ p−1 P ⊕ · · · ⊕ P , where B 0 = ε 0 ⊕ ε 1 ⊕ · · · ⊕ ε p−1 (3.3) with ε 0 , ε 1 , · · · , ε p−1 being p'th roots of unity and P is the cycle matrix of dimension p (see Lemma 2.2 and Eqn (2.7)): P =       0 · · · 0 1 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       p×p . Any element of G has the form C k B h , 0 ≤ k ≤ p 2 − 1, 0 ≤ h ≤ p − 1; and C k B h = D k 0 B h 0 ⊕ D k 1 P h ⊕ · · · ⊕ D k p−1 P h . Obviously, D k 0 B h 0 = ε h 0 ⊕ λ pk ε h 1 ⊕ · · · ⊕ λ p(p−1)k ε h p−1 . (3.4) It is easy to calculate the characteristic polynomials: char D k i P h (x) = x p − λ j∈Γ i jk = x p − λ uipmk , i = 1, · · · , p − 1; where pm = 1 + (p + 1) + · · · + (p + 1) p−1 = (p+1) p −1 p , hence m is an integer coprime to p. If k ≡ 0 (mod p), by Remark 2.5(5.ii), λ uipmk for i = 1, · · · , p − 1 are just all primitive p'th root of unity. Thus the characteristic polynomial of the matrix C k B h is char C k B h (x) = char D k 0 B h 0 (x) · (x p − 1) p−1 , k ≡ 0 (mod p); char D k 0 B h 0 (x) · Φ p 2 (x), k ≡ 0 (mod p); where Φ p 2 (x) denotes the p 2 'th cyclotomic polynomial. Since C k B h is similar to a permutation matrix, by Lemma 2.1, for any k, h we obtain that char D k 0 B h 0 (x) = x p − 1 or (x − 1) p , k ≡ 0 (mod p); x p − 1, k ≡ 0 (mod p). (3.5) By Eqn (3.2), we can view D 0 as a maximal cycle of dimension p; by Eqn (3.3), the matrix B 0 of dimension p commutes with D 0 ; by Lemma 2.1, from Eqn (3.5) we see that the abelian matrix group D 0 , B 0 of dimension p is a permutation-like matrix group of dimension p; so, by Lemma 2.4, we have an integer 0 ≤ ℓ ≤ p − 1 such that (ε 0 , ε 1 , · · · , ε p−1 ) = (1, λ pℓ , · · · , λ p(p−1)ℓ ). Then, from Eqn (3.4) it is easy to calculate the characteristic polynomial char D k 0 B h 0 (x) = (x − 1)(x − λ p(k+ℓh) ) · · · (x − λ p(p−1)(k+ℓh) ), so char D k 0 B h 0 (x) = (x − 1) p , k + ℓh ≡ 0 (mod p); x p − 1, k + ℓh ≡ 0 (mod p). (3.6) Suppose that 0 < ℓ ≤ p − 1; taking k ≡ 0 (mod p), we have an h such that k + ℓh ≡ 0 (mod p), then, by Eqns (3.5) and (3.6) we have x p − 1 = char D k 0 B h 0 (x) = (x − 1) p , which is impossible. Thus ℓ = 0, i.e. (ε 0 , ε 1 , · · · , ε p−1 ) = (1, 1, · · · , 1). Summarizing the above, we obtain that and write matrices with respect to this basis. Then C = 1 ⊕ s−1 ⊕ i=0 λ r i v1 ⊕ · · · ⊕ s−1 ⊕ i=0 λ r i vt ⊕ q−1 ⊕ i=0 λ r i w1 ⊕ · · · ⊕ q−1 ⊕ i=0 λ r i wt ; because p\|v j for j = 1, · · · , t (see Eqn (3.1)), λ r i vj p = 1 for 0 ≤ i ≤ s − 1 and 1 ≤ j ≤ t, so C ap = I p×p ⊕ q−1 ⊕ i=0 λ r i w1ap ⊕ · · · ⊕ q−1 ⊕ i=0 λ r i wtap . And, there are complexes ε 0 and ε j , ω j for j = 1, · · · , t such that B = ε 0 ⊕ P 1 ⊕ · · · ⊕ P t ⊕ Q 1 ⊕ · · · ⊕ Q t , where P j , Q j are as described in Eqn (2.5) (see Lemma 2.2) : P j =       0 · · · 0 ε j 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       s×s . Q j =       0 · · · 0 ω j 1 . . . . . . 0 . . . . . . . . . . . . 0 · · · 1 0       q×q ; hence B ps = ε ps 0 ⊕ ε p 1 I s×s ⊕ · · · ⊕ ε p t I s×s ⊕ ω 1 I q×q ⊕ · · · ⊕ ω t I q×q . Since B ps = C ap see Eqn (3.7) , the collection of q ω 1 , · · · , ω 1 , · · · , q ω t , · · · , ω t (3.8) is coincide with the collection of λ w1ap , λ rw1ap , · · · , λ r q−1 w1ap , · · · , λ wtap , λ rwtap , · · · , λ r q−1 wtap . (3.9) Note that w 1 , rw 1 , · · · , r q−1 w 1 , · · · , w t , rw t , · · · , r q−1 w t = Z * p 2 . If 0 < a < p (cf. Eqn (3.7)), then the collection (3.9) is just all primitive p'th roots of unity with multiplicity p for each one, see Remark 2.5(1); on the other hand, the number of the elements appeared in the collection (3.8) is at most t; but ts = p − 1 and s > 1, so t is less than the number of primitive p'th roots; that is a contradiction to the coincidence of the collections (3.8) and (3.9). In conclusion, a = 0 and B ps = 1. Since p and s are coprime, we have B = B p × B s , \| B p \| = s, \| B s \| = p. We have considered C, B p and C, B s in Case 1 and Case 2 respectively, and have concluded that in both cases the condition (SC) in Proposition 2.3 is satisfied. Assume that B i Ce j = Ce j ; taking integers k, h such that ph + sk = 1, we have B i = B phi B ski , B phi ∈ B p , B phi Ce j = Ce j and B ski ∈ B s , B ski Ce j = Ce j ; by the conclusions in Case 1 and in Case 2, B phi e j = e j and B ski e j = e j ; hence B i e j = e j . Thus, by Proposition 2.3, G is a permutation matrix group. The proof of Theorem 1.1 is completed. ( 3 ) 3If ord(r) (p − 1), then ord(B) = \|G/C\| and the action by conjugation of B on C is equivalent to the action of r by multiplication on Z p 2 , the latter is denoted by µ r , i.e. µ r (a) = ra for all a ∈ Z p 2 ; it is easy to check that the group B acts freely by conjugation on the difference set C − {1} this is a specific case of[4, Corollary 4.35] .(4) If ord(r) = \|G/C\| = p, then we can choose B such that B p = 1 and r = ap + 1 with 0 < a < p; further, replacing B by a suitable power of B, we can get r = p + 1; see [1, §8 Proposition 10] for details. Case 1 . 1δ = 0, i.e. q = s \| (p − 1). By Remark 2.5(3), \| B \| = s and B ∼ = r ≤ Z * p 2 . The group µ r fixes 0 ∈ Z p 2 , and acts freely on both Z * p 2 and Γ 0 − {0}. AcknowledgementsThe research of the authors is supported by NSFC with grant numbers 11171194 and 11271005.Similar to Case 1, the group B satisfies the condition (SC) of Proposition 2.3:Thus G is a permutation matrix group.Case 3. q = ps and s > 1. First we show that B ps = C ap with 0 ≤ a < p, hence ord(B) = ps, a = 0; p 2 s, 0 < a < p. For: otherwise B ps ∈ C − C p , then C = B ps , hence B centralizes C , which contradicts to that G is non-abelian. It is easy to see that the group µ r of order q = ps acts freely on Z * p 2 , hence partitions Z * p 2 in to t orbits of length q; taking representatives w 1 , · · · , w t from these t orbits, we have the orbits of µ r on Z * p 2 as follows:On the other hand, since the group µ r s of order p centralizes Γ 0 , the group µ r fixes 0 and partitions Γ 0 − {0} in to t orbits of length s (cf. Case 2); taking representatives v 1 , · · · , v t from these t orbits, we have the orbits of µ r on Γ 0 as follows:According to the orbits Γ 00 , Γ 01 , · · · , Γ 0t , Γ 1 , · · · , Γ t , we apply Lemma 2.2 and its notation to get the basis of C p 2 : E = E 00 ∪ E 01 ∪ · · · ∪ E 0t ∪ E 1 ∪ · · · ∪ E t , J L Alperin, R B Bell, Groups and Representations. New YorkSpringer-Verlag162J. L. Alperin, R. B. Bell, Groups and Representations, GTM 162, Springer- Verlag, New York, 1997. Groups of matrices with prescribed spectrum, Doctoral dissertation. G Cigler, G. Cigler, Groups of matrices with prescribed spectrum, Doctoral disser- tation, 2005, http://matknjiz.si/doktotati/2005/10921-83.pdf Permutation-like matrix groups. G Cigler, Linear Algebra and its Applications. 422G. Cigler, Permutation-like matrix groups, Linear Algebra and its Appli- cations 422(2007) 486-505. I M Isaacs, Graduate Studies in Mathematics. Providence, Rhode IslandAmer. Math. Soc92Finite Group TheoryI. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics Vol. 92, Amer. Math. Soc., Providence, Rhode Island, 2008.	[]

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